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Multidimensional Nonlinear Integral Inequalities
Chapter Five Multidimensional Nonlinear Integral Inequalities 5.1 Introduction In recent years considerable interest has been shown in developing various aspects of the theory of partial differential and integral equations both for their own sake and for their applications in science and technology. One of the most useful tools in the development of the qualitative theory of partial differential and integral equations is integral inequalities involving functions of many independent variables, which provide explicit bounds on the unknown functions. In the past few years, many new and useful integral inequalities involving multivalued functions and their partial derivatives have been found. This chapter gives some basic nonlinear multidimensional integral inequalities recently discovered in the literature. These inequalities can be used as ready and powerful tools in the study of various problems in the theory of certain partial differential, integral and integro-differential equations. Some applications are discussed to illustrate how these inequalities may be used to study the qualitative behaviour of solutions of certain partial differential and integro-differential equations, and miscellaneous inequalities which can be used in certain applications are also given.
459
460
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
5.2 Generalizations of Wendroff's Inequality The fundamental role played by Wendroff' s inequality and its generalizations and variants in the development of the theory of partial differential and integral equations is well known. In this section we present some basic nonlinear generalizations of Wendroff's inequality established by Bondge and Pachpatte (1979b, 1980a) and some new variants, which can be used as tools in the study of certain partial differential and integral equations. Bondge and Pachpatte (1979b) proved the following useful nonlinear generalization of Wendroff's inequality. Theorem 5.2.1 Let u(x, y) and p(x, y) be nonnegative continuous functions defined for x, y ~ R+. Let g(u) be a continuously differentiable function defined for u > O, g(u) > 0 for u > 0 and g' (u) > 0 for u > O. If x
u(x, y) < a(x) + b(y) +
y
ff 0
(5.2.1)
t)g(u(s, t)) ds dt,
0
for x, y ~ R+, where a(x) > O, b(y) > O, a'(x) > O, b'(y) > 0 are continuous functions defined for x, y ~ R+, then for 0 < x < Xl, 0 < y < Yl, u(x, y)
<
-
f2(a(0) + b(y)) +
~"2- 1
x
y
+ffp(s,t)dsdt 0
0
]
g(a(s) + b(0))
ds
0
(5.2.2)
,
where
ds
f2(r) --
g(s)
r > 0, ro > 0,
(5.2.3)
ro
g2-1 is the inverse function of f2 and Xl, yl are chosen so that x
f2(a(0) + b(y)) +
x
y
i g(a(s)a'
0
for all x, y lying in the subintervals 0 < x <
0 Xl,
0 ~ y < Yl of R+. Vl
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
461
Proof: We note that since g'(u) > 0 on R+, the function g(u) is monotonically increasing on (0, 0r Define a function z(x, y) by the fight-hand side of (5.2.1), then z(x, O) = a(x) 4- b(O), z(O, y) -- a(O) 4- b(y), and Zxy(X,
y ) - p(x, y)g(u(x, y)).
(5.2.4)
Using u(x, y) < z(x, y) in (5.2.4) and the fact that z(x, y) > 0, we observe that
Y) < p(x, y). g(z(x, y)) Zxy( x,
(5.2.5)
From (5.2.5) and by using the facts that Zx(X, y ) > 0, z(x, y) > O, g'(z(x, y)) > 0 for x, y 6 R+, we observe that Zxy( X, Y)
< p(x, y) + g(z(x, y)) -
Zy(X, y ) >
0,
Zx(x, y)g' (z(x, y))Zy (x, y) [g(z(x, y))]2
i.e.
( Zx(X, y) ) < p(x, y). Oy k,g(-~, y))_ -
(5.2.6)
Keeping x fixed in (5.2.6), we set y - t; then, integrating with respect to t from 0 to y and using the fact that z(x, O) - a(x) + b(O), we have y
Zx(X, y) < -t- f p(x, t) dt. a'(x) g(z(x, y)) - g(a(x) 4- b(O))
(5.2.7)
0
From (5.2.3) and (5.2.7) we observe that
a Ox
m f2 (z(x, y ) )
--
Zx(X, y) g(z(x, y)) y
a'(x) / < 4- p(x, t) dt. - g(a(x) 4- b(0))
(5.2.8)
0
Keeping y fixed in (5.2.8), set x = s; then, integrating with respect to s from 0 to x and using the fact that z(O, y) - a(O) + b(y) we have x
f2(z(x, y)) < ~(a(0) 4- b(y)) 4- f g(a(s)a'(s) 4- b(0)) ds 0 x y
+ffp(s,t)dsdr 0 0
(5.2.9)
462
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Now substituting the bound on z(x, y) from (5.2.9) in u(x, y) < z(x, y), we obtain the desired bound in (5.2.2). The subintervals for x and y are obvious. II R e m a r k 5.2.1 From the proof of Theorem 5.2.1, it is easy to observe that, in addition to (5.2.2), we can conclude that
b'(t) u(x, y) < f2 -1 I f2(a(x) + b(0)) + / g(a(O) + b(t)) dt
xy
]
0
0 0
where the expression in the square bracket on the fight-hand side of (5.2.10) belongs to the domain of ~2-1 . We also note that the above conclusion applies to the following Theorems 5.2.3 and 5.2.4 and also some of the results given in our subsequent discussion. A Bondge and Pachpatte (1980a) gave the following generalization of Wendroff' s inequality. Theorem 5.2.2 Let u(x, y), a(x, y), b(x, y) and c(x, y) be nonnegative
continuous functions defined for x, y ~_R+. Let g(u), h(u) be continuously differentiable functions defined for u > O, g(u) > O, h(u) > 0 for u > 0 and g'(u) > 0, h'(u) > Ofor u > O, and let g(u) be subadditive and submultiplicative for u > O. If u(x,y)
(5.2.11)
0
for x, y ~ R+, then for 0 < x < x2, 0 < y <_ Y2, r"
u(x, y) < a(x, y) + b(x, y)h ( G -1 IG(A(x, y))
rrxY + - - - - c(s, 0 0
\
L
t))ds dt
])
,
(5.2.12)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
463
where x
y
ff fds
A(x, y ) -
0
t))dsdt,
(5.2.13)
0
F
G(r)-
g(h(s))
ro
,
r>0,
r0>0,
(5.2.14)
G -1 is the inverse function of G and x2, y2 are chosen so that x
y
G(A(x, y)) + f f c(s, t)g(b(s, t)) ds dt ~ Dom(G -1), 0
0
for all x, y lying in the subintervals 0 <_x <_x2, 0 <_ y <_ y2 of R+. P r o o f : From the hypotheses on g and h, we note that the functions g and h are monotonically increasing on (0, cx~). Define a function z(x, y) by x
Z(X, y) =
y
f f c(s 0
ds dt.
(5.2.15)
0
From (5.2.15) and using the fact that u(x, y) < a(x, y) + b(x, y)h(z(x, y)) from (5.2.11) and the hypotheses on g we have x
z(x, y) < A(x, y) +
y
f f c(s. 0
t))g(h(z(s, t)))as dt,
(5.2.16)
0
for x, y 6 R+, where A(x, y) is defined by (5.2.13). Now fix c~,/~ 6 R+ such that 0 < x _< oe, 0 < y
y
z(x, y) < A(c~, ~) + f f c(s, t)g,b(s, t))g(h(z(s, t)))ds dt, 0
(5.2.17)
0
for 0 < x < c~, 0 < y
y) -- c(x, y)g(b(x, y))g(h(z(x, y))) < c(x, y)g(b(x, y))g(h(v(x, y))).
(5.2.18)
464
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Now first assume that A(ct,/~) > 0; then from (5.2.18) we observe that 1;xy(X, Y)
< c(x, y)g(b(x, y)). g(h(v(x, y))) -
(5.2.19)
As in the proof of Theorem 5.2.1, from (5.2.19) we observe that
O < -c-(-x(g(h(v(x, 'Vy )xg( X( b' Y( x)y))) ')y ) ) o y-
(5.2.20)
By keeping x fixed in (5.2.20), setting y = t, and then integrating with respect to t from 0 to ~ we have
vx,x,o,
#
Vx(X, ~) < g(h(v(x, ~))) - g(h(v(x, 0)))
+f
c(x, t)g(b(x, t)) dt.
(5.2.21)
o
From (5.2.14) and (5.2.21) we observe that /,b
0
G(v(x, [3))< -ff-G(v(x,0)) + I c(x,t)g(b(x,t))dt. Ox Ox J
(5.2.22)
o
Now keeping y fixed in (5.2.22), setting x = s, and then integrating with respect to s from 0 to c~ we get
(5.2.23)
G(v(ct, f l ) ) < _ G ( A ( ~ , # ) ) + f f c ( s , t ) g ( b ( s , t ) ) d s d t , 0 o
Since z(ct, fl) < v(c~, fl) and ct, fl 6 R+ are arbitrary from (5.2.23) we have
[
z(x, y) < G -1 G(A(x, y)) +
:: o o
c(s, t)g(b(s, t))dsdt
]
,
(5.2.24)
for 0 _< x < X1, 0 ~ y _< Yl. The desired bound in (5.2.12) follows by using (5.2.24) in u(x, y) <_a(x, y) + b(x, y)h(z(x, y)). If A(ct,/~) in (5.2.17) is nonnegative, we carry out the above procedure with A(c~,/~) + E instead of A(ct,/~), where E > 0 is an arbitrary small constant, and subsequently pass to the limit as E - , 0 to obtain (5.2.12).
I
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES 465 The following theorem provides another useful generalization of Wendroff' s inequality. Theorem 5.2.3 Let u(x, y), c(x, y) and p(x, y) be nonnegative continuous functions defined for x, y ~ R+. Let g(u), g' (u), a(x), a'(x), b(y) and b'(y) be as in Theorem 5.2.1 and g(u) be submultiplicative on R+. If x
y
u(x,y)<_a(x)+b(y)+ffc(s,t)u(s,t)dsdt 0 x
0
y
+ffp(s,t)g(u(s,t))dsdt, 0
(5.2.25)
0
for x, y ~ R+, then for 0 < x < x3, 0 < y < Y3, u(x, y) < q(x, y) [~2- 1
j
f2(a(O) + b(y)) +
xy
+
f f p(s t)g(q(s, t))ds dt
]]
a'(s) g(a(s) + b(O)) ds
0
,
(5.2.26)
q'xy) exp(joc'st)
(5.2.27)
0
0
where
ds dt) ,
and f2, g2-1 are as defined in Theorem 5.2.1 and x3, Y3 are chosen so that x
a'(s) g(a(s) + b(0)) ds
f2(a(0) + b(y)) + 0 x
+
y
ff 0
t)g(q(s, t))ds dt ~ Dom(f2 -1),
0
for all x, y lying in the subintervals 0 < x < x3, 0 <_ y < y3 of R+.
D
Proof: We note that, since a'(x) > O, b'(y) > O, g'(u) > 0 for x, y, u R+, the functions a(x), b(y), g(u) are monotonically increasing on (0, c~).
466
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
Define a function m(x, y) by x
m(x, y) -- a(x) + b(y) +
y
f f p(s, t)g(u(s, t)) ds dt,
(5.2.28)
0 0
then m(x, O) - a(x) + b(O), m(O, y) - a(O) + b(y) and (5.2.25) can be restated as x
y
u(x,y)<_m(x,y)+ffc(s,t)u(s,t)dsdt.
(5.2.29)
0 0
Since m(x, y) is positive and nondecreasing in each variable x, y ~ R+, by applying Theorem 4.2.2 to (5.2.29) we have
(5.2.30)
u(x, y) < m(x, y)q(x, y),
where q(x, y) is defined by (5.2.27). From (5.2.28) and (5.2.30) we have
mxy(X, y ) -
p(x, y)g(u(x, y))
< p(x, y)g(m(x, y)q(x, y)) < p(x, y)g(m(x, y))g(q(x, y)), i.e.
mxy(X,
y)
< p(x, y)g(q(x, y)). g(m(x, y)) -
(5.2.31)
Now by following the same arguments as in the proof of Theorem 5.2.1 below the inequality (5.2.5) we get
m(x, y) < S2-~
~(a(O) + b(y)) +
g(a(s) + b(O)) 0
xy +ff.(s,
t)g(q(s, t ) ) d s dt
1
.
(5.2.32)
0 0
Using (5.2.32) in (5.2.30) we get the required inequality in (5.2.26). II A slight variant of Theorem 5.2.3 is given in the following theorem.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
467
Theorem 5.2.4 Let u(x, y), c(x, y), p(x, y), g(u), g'(u), a(x), a'(x), b(y) and b'(y) be as in Theorem 5.2.3. If u(x, y) < a(x) + b(y) +
j
iJ
c(s, y)u(s, y)ds +
0
p(s, t)g(u(s, t)) ds dt,
0 0
(5.2.33) for x, y ~ R+, then for 0 < x < x4, 0 < y < Y4,
[[
u(x, y) < f (x, y) S2-1
j
f2(a(O) + b(y)) +
g(a(s) + b(O))
as
0
xy p(s, t)g(F(s, t))dsdt
+ fl
1] ,
(5.2.34)
0 0
where
y, exp( c s y,ds) for x, y ~ R+, f2, so that
~'2- 1 , a r e
as defined in Theorem 5.2.1 and
(5.2.35)
X4,
24 are chosen
a'(s) ds g(a(s) + b(0))
f2(a(0) + b(y)) + 0 x y
+fJ'p(s,t)g(F(s,t))dxdt6Dom(f2-1), 0 0
for all x, y lying in the subintervals 0 < x <
X4,
0 < y < Y4 of R+. D
The proof of this theorem can be completed by following the proof of Theorem 5.2.3, with suitable changes. The details are omitted here. R e m a r k 5.2.2 We note that, if the inequality (5.2.33) in Theorem 5.2.4 is replaced by y
x
y
u(x,y)<_a(x)+b(y)+lc(x,t)u(x,t)dt+ffp(s,t)g(u(s,t))dsdt, 0
0 0
468
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
then the bound obtained in (5.2.34) is replaced by
[[
u(x, y) < Fo(x, y) ~2-1
f2(a(0) + b(y)) +
]]
xy
ja s,
g(a(s) + b(0)) ds
0
0 0
where
o'x y' exp(/oc'x"dt) for x, y 6 R+, and the expression in the inner square bracket on the righthand side belongs to the domain of f2 -1. A
5.3 Wendroff-type Inequalities During the past twenty years or so many authors have developed extensions and variants of Wendroff's inequality and exhibited applications in partial differential and integral equations. In this section we shall deal with the Wendroff-type inequalities investigated by Pachpatte (1995e) and Bondge and Pachpatte (1979b,c) which can be used in the study of certain partial differential and integral equations. Pachpatte (1995e) established the Wendroff-type inequalities in the following theorem. Theorem 5.3.1 Let u(x, y), a(x, y) and b(x, y) be nonnegative continuous functions defined on R2+ and L" R3+ -+ R+ be a continuous function which satisfies the condition 0 < L(x, y, v ) - L(x, y, w) < M(x, y, w ) ( v - w),
(L)
for x, y ~ R+ and v > w > O, where M" R 3 --+ R+ is a continuous function.
(i) if x
y
u~ y, <_a~x,y, + b~x y, / f L~s , u~s ,,, ~ d, 0 0
(5.3.1)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES 469
for x, y ~ R+, then u(x, y) < a(x, y) + b(x, y)A(x, y)exp ( / / M ( s ,
t, a(s, t))b(s, t)dsdt) , (5.3.2)
for x, y ~ R+, where x
y
A(x,y)=ffL(s,t,a(s,t))dsdt, 0
(5.3.3)
0
for x, y ~ R+. (ii) Let F(u) be a continuous, strictly increasing, convex and submultiplicative function for u > 0, limu_+~ F(u) -- oe, F -1 denotes the inverse function of F, u(x, y), ~(x, y) be continuous and positive functions defined on R 2 and or(x, y) + ~(x, y) = 1. If u(x, y) < a(x, y) -t- b(x, y)F-l ( / /
L(s, t, F(u(s, t))) ds dt) ,
(5.3.4)
for x, y ~ R+, then u(x, y) < a(x, y) + b(x, y)F -1
L(s, t, ~(s, t)
x F(a(s, t)ot-1 (s, t))) ds dt) /
•
exp ( / / M ( s ,
t, or(s, t)F(a(s, t)c~-1 (s, t)))
xfl(s,t)F(b(s,t)fl-l(s,t))dsdt)], for x, y ~ R+.
(5.3.5)
470
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
(iii) Let g(u), g'(u), h(u), h'(u) be as in Theorem 5.2.2. If (5.3.6)
u(x, y) < a(x, y) + b(x, y)h (fo fo L(s, t, g(u(s, t)))dsdt ) , for x, y E R+, then for O < x < xl, O < y < yl, /
u(x, y) < a(x, y) + b(x, y)h
(G -1 \
[G(B(x, I"
Y))
I..
xy
])
+//M(s,t,g(a(s,t)))g(b(s,t))dsdt
,
(5.3.7)
0 0
where
x y
B(x,y)-/fL(s,t,g(a(s,t)))dsdt,
(5.3.8)
0 0
G, G -1 be as defined in Theorem 5.2.2 and xl, yl be chosen so that x y
G(B(x, y)) +
ff
M(s, t, g(a(s, t)))g(b(s, t)) ds dt ~ Dom(G -1),
0 0
for all x, y lying in the subintervals 0 < x < Xl, 0 < y < Yl of R+. IS]
Proof:
(i) Define a function z(x, y) by x
y
z(x,y)-f/L(s,t,u(s,t))dsdt. 0
(5.3.9)
0
From (5.3.9) and using the fact that u(x, y) <_ a(x, y) + b(x, y)z(x, y) and the conditions on the function L we observe that x
y
z(x,y)
= A(x, y ) +
,, a(s, t) + b(s, ,)z(s, t)) 0 0
- L(s, t, a(s,
t))] ds dt
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES x
471
y
< A(x, Y) + / / M ( s ,
t, a(s, t))b(s, t)z(s, t)dsdt, (5.3.10)
t i l l
0
0
where A(x, y) is defined by (5.3.3). Clearly A(x, y) is nonnegative and nondecreasing in each variable x, y e R+. Now an application of Theorem 4.2.2 yields
z(x, y)
(5.3.11)
< a(x, y) + b(x, y)z(x, y) we get the required
u(x, y) < or(x, y)a(x, y)ot-1 (x, y)+ ~(x, y)b(x, y)fl-1 (x, y) x F -1
(/0/
)
L(s, t, F(u(s, t)))ds dt .
0
(5.3.12)
Since F is convex, submultiplicative and monotonic, from (5.3.12) we see that
F(u(x, y)) < or(x, y)F(a(x, y)ot-1 (x, y)) + 13(x, y)F(b(x, y)fl-l(x, y)) x
y
xJ'fL(x,t,F(u(s,t)))dsdt. 0
(5.3.13)
0
The estimate given in (5.3.5) follows by first applying the inequality established in (i) to (5.3.13) and then applying F -1 to both sides of the resulting inequality. (iii) Define a function z(x, y) by x P
y
P
Z(X, y)= I l L ( s , t, g(u(s, t))) ds d,. t.i
qt/
0
0
(5.3.14)
From (5.3.14) and using the fact that u(x, y) < a(x, y)+ b(x, y)h(z(x, y)) and, as noted in the proof of Theorem 5.2.2, that the functions g and h are monotonically increasing on (0, c~), and using the conditions on the function
472
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
L we observe that x
y
z(x, y) < / f
L(s, t, g(a(s, t)+ b(s, t)h(z(s, t))))ds dt
0 0 x y
ff
t, g(a(s, t))+ g(b(s, t))g(h(z(s, t))))ds dt
0 0 x y
t, g(a(s, t)) + g(b(s, t))g(h(z(s, t))))
= B(x, y) + 0 0
- L(s, t, g(a(s, t)))] ds dt x y
< B(x, y) +
ff
g(a(s, t)))g(b(s, t))
0 0
x g(h(z(s, t))) ds dt,
(5.3.15)
where B(x, y) is defined by (5.3.8). The rest of the proof follows by the same arguments as in the proof of Theorem 5.2.2 below the inequality (5.2.16), with suitable changes. Further details are omitted here. II The next two theorems proved by Bondge and Pachpatte (1979b) can be used more effectively in certain situations. Theorem 5.3.2 Let u(x, y), Ux(X, y), uy(x, y)and U~y(X, y)be nonnegative
continuous functions defined for x, y ~ R+, u(x, 0) = u(O, y) = O, and p(x, y) > 1 be a continuous function defined for x, y ~. R+. Let g(u), g'(u), a(x), a'(x), b(y), b'(y) be as in Theorem 5.2.1. If Uxy(X, y) < a(x) + b(y) + M [u(x, y) W f /o p(s, t)g(Ust(S, t)) ds dt] (5.3.16)
for x, y ~ R+, where M > 0 is a constant, then for 0 < x < x2, 0 < y < Y2, Uxy(X, y) < H -1
(a(0) + b(y)) +
a(s) + b(O) + g(a(s) + b(0)) 0
xy
0
0
]
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
where
473
r
H(r) --
f
r > 0, r0 > 0,
s + g(s)
(5.3.18)
ro
H -1 is the inverse function of H, and x2, y2 are chosen so that x
H(a(O) + b(y)) +
f
a'(s) ds a(s) + b(O) + g(a(s) + b(O))
0 x
y
+Mffp(s,t)dsdt
E Dom(H-1),
0 0
for all x, y lying in the subintervals 0 <_x <_x2, 0 <_ y < y2 of R+. D Proof: We note that since g'(u) >_ 0 on R+, the function g is monotonically increasing on (0, oo). Define a function z(x, y) by the right-hand side of (5.3.16), then z(x, O) = a(x) + b(O), z(O, y) = a(O) + b(y), Uxy(X, y) <_ z(x, y) and
Zxy(X, y) = M[uxy(X, y) + p(x, y)g(Uxy(X, Y))I < M p(x, y)[z(x, y)4-g(z(x, Y))I, i.e.
Zxy( X,
Y)
< M p(x, y). z(x, y) + g(z(x, y)) -
(5.3.19)
Now by following the same arguments as in the proof of Theorem 5.2.1 below the inequality (5.2.5) with suitable modifications we obtain the estimate for z(x, y) such that
z(x, y) < H -1 -
[o
(a(0) + b(y)) +
xy
]
r a(s) + b(O) +a g(a(s) s, + b(O)) ds 0
0 0
Using (5.3.20) in uxy(x, y) < z(x, y) we obtain the required bound in (5.3.17). II
474
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Theorem 5.3.3 Letu(x, y), Uxy(X, y), p(x, y), g(u), g'(u), a(x), a'(x), b(y) and b' (y) be as in Theorem 5.3.2. If x y
lgxy(X, y) < a(x) + b(y) +
f f p(s,t)g(u(s, t) + Ust(S, t)) ds dt,
(5.3.21)
0 0
for x, y e R+, then for 0 < x < x3, 0 < y < Y3,
Uxy(X, y) < a(x) + b(y) +
xy(
f f p(st)g
n -1
(a(0) + b(t))
0 0 s
+f
a'(sl) dSl a(sl) + b(O) + g(a(sl) + b(0))
+ j fo p(sl, tl)dsl dtl]
(5.3.22)
dsdt,
where H, H -1 are as defined in Theorem 5.3.2 and x3, y3 are chosen so that x
a'(sl) a(sl) + b(O) + g(a(sl) + b(0))
H(a(O) + b(y)) +
ds1
0 x y
+/fp(sl,
tl)dsldtl 6 Dom(H-1),
0 0
for all x, y lying in the subintervals 0 < x < x3, 0 < y < Y3 of R+. Proof: Since g'(u)>_ 0 on R+, the function g(u) is monotonically increasing on (0, oo). Define a function z(x, y) by the right-hand side of (5.3.21), then z(x, O) - a(x) + b(O), z(O, y) - a(O) + b(y),
Uxy(X, y) < z(x, y),
(5.3.23)
and Zxy(X, y ) -
p(x, y)g(u(x, y ) + Uxy(X, y)).
(5.3.24)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES From (5.3.23) and using
475
u(x, 0) - u(0, y) - 0, it is easy to observe that x y
(5.3.25)
u(x,y)<_ffz(s,t)dsdt. 0 0
Using (5.3.23) and (5.3.25) in (5.3.24) we get
Zxy(X,
( j/
y) < p(x, y)g z(x, y)+
z(s, t)ds dt
0 0
Define a function
)
.
(5.3.26)
v(x, y) by x
y
v(x,y)-z(x,y)+ffz(s,t)dsdt. oo From (5.3.27) and using the facts that Zxy(X, y)< p(x, y)g(v(x, (5.3.26) and z(x, y) < v(x, y) from (5.3.27) we observe that Vxy(X,
y)
= Zxy(X,
(5.3.27)
y)) from
y) + z(x, y)
< p(x, y)[v(x, y)+ g(v(x, y))], i.e.
Vxy(X, Y) < p(x, y). v(x, y) + g(v(x, y)) -
(5.3.28)
Now by following the same arguments as in the proof of Theorem 5.2.1 below the inequality (5.2.5) with suitable modifications we obtain the estimate for v(x, y) such that
v(x, y) < H -~
IH
(a(0) +
b(y)) +
xy
] tl)dS,
/
a'(sl) a(sl) + b(O) + g(a(sl) + b(0))
dsl
0
(5.3.29)
0 0
Using (5.3.29) in (5.3.26), and first keeping x fixed, setting y = t and integrating from 0 to y, then keeping y fixed, setting x - s in the resulting
476
MULTIDIMENSIONALNONLINEARINTEGRALINEQUALITIES
inequality and integrating from 0 to x we obtain
xy
Z(X, y) < a(x) + b(y) +
f f p(,,t)g 0
( H
-1
H(a(0) + b(t))
0
s
a'(sl) dsl a(sl) 4- b(0) -t- g(a(sl) + b(0))
+
])
st 0
(5.3.30)
0
Using (5.3.30) in (5.3.23) we get the desired inequality in (5.3.22). II
The following two theorems established by Bondge and Pachpatte (1979c) can be used in certain applications. Theorem 5.3.4 Let q~(x, y), a(x, y), b(x, y) be nonnegative continuous functions defined for x, y ~ R+, and u(x, y) be a positive continuous function defined for x, y ~ R+. Let F(u), t~(x, y), fl(x, y) be as in Theorem 5.3.1 part (ii). If
u(s, t) > q~(x, y) - a(s, t)F-l ( f f b(sl, tl)F(q~(Sl, tl))dsl d t l ) , \x
y
(5.3.31) for O < x < s < oo, O< y < t < oo, then
U(S, t) > Or(S,t)F -1 (o~-1 (s, t)F(qb(x, y)) exp (-fl(s, t)F(a(s, t)f1-1 (s, t))
x//b Sltl, sldtm x
(5.3.32)
y
for O < x < s < oo, 0 < y < t < c~. [3
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
477
Proof: Rewrite (5.3.31) as
~(x, y) < or(s, t)u(s, t)ot -1 (s, t) + ~(s, t)a(s, t)/~ -1 (s, t) xf-1
( f f b(sl, tl)f(qb(Sl, t l ) ) d s l d t l ) . \x
(5.3.33)
y
Since F is convex, submultiplicative and monotonic, from (5.3.33) we have
or(s, t)F(u(s, t)ot -1 (s, t)) > F(~(x, y)) - ~(s, t)F(a(s, t)~ -1 (s, t)) s
t
•
tl)F(dp(Sl, tl))dsldtl. x
(5.3.34)
y
Now an application of Theorem 4.9.4 part (i) with n = 2 yields the desired bound in (5.3.32).
II Theorem 5.3.5 Let u(x, y), a(x, y) and b(x, y) be as in Theorem 5.3.4. Let g(u) and g'(u) be as in Theorem 5.2.1. If s
t
u(s,t)>_ u ( x , y ) - a ( s , t ) f f b ( c r , x
o)g(u(a,o))dado,
(5.3.35)
y
for O < x < s < c~, O < y < t < c~, then for O < x < s < sl, O <_ y <_ t <_ tl, u(s, t) > f2 -1
[
f2(u(x, y)) - a(s, t)
JJ x
where f2,
~"2-1 are
]
b(cr, O)dcr do ,
(5.3.36)
y
as defined in Theorem 5.2.1 and s
t
.>
f2(u(x, y)) - a(s, t ) f f x
forO < x ~ s ~ s1,
O ~
d. ~ Dom(f2 -1 ),
y
y < t < tl. D
47'8
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
P r o o f : Since g'(u)> 0 on R+, the function g(u) is monotonically increasing on (0, c~). Rewrite (5.3.35) as
u(x, y ) < u ( s , t ) + a ( s , t ) f f b ( t r , x
rl)g(u(tr, 0)) dcr d0.
(5.3.37)
y
For fixed s and t in R+ we define for 0 < x < s, 0 < y < t, s
z(x, y) -- u(s, t) + a(s, t)
t
ff x
b(tr, o)g(u(tr, 0))dcr do,
(5.3.38)
y
then z(x, t) - z(s, y) - u(s, t), u(x, y) < z(x, y) and
Zxy(X, y ) = a(s, t)b(x, y)g(u(x, y)) < a(s, t)b(x, y)g(z(x, y)), i.e.
y) < a(s, t)b(x, y). g(z(x, y)) Zxy(X,
(5.3.39)
From (5.3.39) we observe that
Zxy(X, y) g' (Z(X, y))Zy (X, Y)Zx(X, y) < a(s, t)b(x, y) + g(z(x, y)) [g(z(x, y))]2 9
(5.3.40)
Here we note that Zx(X, y) and Zy(X, y) are nonpositive, which implies that Zx(X, y)zy(X, y) is nonnegative and hence (5.3.40) is true. Now (5.3.40) is equivalent to
0 (Zx(X'Y)) Oy g(z(x, y))
(5.3.41)
By keeping x fixed in (5.3.41), setting y - r/and integrating from y to t we have
Zx(X, t) g(z(x, t))
t
Zx(X, Y) a(s, t) / b(x, r/)do. g(z(x, y)) i 1
i
(5.3.42)
e,j
Y
From (5.2.3) and (5.3.42) we observe that t
-~x S2(z(x, t)) - -~x ~ ~2(z(x' y)) < a(s, t) f b(x, 0) dr/. Y
(5.3.43)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
479
Now keeping y fixed in (5.3.43), setting x - cr and integrating from x to s we have
f2(Z(X, y)) <_f2(u(s, t)) + a(s, t) f f b(cr, ~) dcrdo, x y which implies
f2(u(s,t))>~(u(x, y))-a(s,t)ffb(tr,
0) dcr d0.
(5.3.44)
x y The desired bound in (5.3.36) follows from (5.3.44). The subintervals of s and t in R+ are obvious.
B
5.4 Generalizations of Pachpatte's Inequalities Pachpatte (1974d, 1975h) investigated some nonlinear Bihari-type integral inequalities which are applicable in certain general situations. In this section we present some basic two independent variable generalizations of the certain inequalities in Pachpatte (1974d, 1975h) obtained by Bondge and Pachpatte (1979b, 1980a) which can be used as tools in the study of certain partial integro-differential and integral equations. Bondge and Pachpatte (1979b) established the following generalization of the inequality given by Pachpatte (1974d). Theorem 5.4.1 Let u(x, y) and p(x, y) be nonnegative continuousfunc-
tions definedfor x, y ~ R+. Let g(u), g' (u), a(x), a'(x), b(y) and b' (y) be as in Theorem 5.2.1. If XY
u(x,y)
(
u(s,t)
0 0
- I -t fl f)op) d( ssll,d ttl)g(u(sl, l)o
dsdt,
(5.4.1)
480
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
for x, y ~ R+, then for 0 <_x <_Xl, 0 5 Y <_ Yl,
xy
u(x, y) < a(x) + b(y) +
f f p~s,t ) . ,
(a(0) + b ( t ) )
0 0 s
f
a'(sl)
a(s1) -+- b(0) + g(a(sl) + b(0))
q-
st
dsl
]
+ J j,,,:s, ,,, :,s, ~,1 ,:,s,t
(5.4.2)
0 0
where H(r) -- f
ds s + g(s)'
(5.4.3)
r > 0, r0 > 0,
ro
H -1 is the inverse function of H, and Xl, Yl are chosen so that x
dsl H(a(O) + b(y)) + f a(sl) + b(0) a'(sl) + g(a(sl) + b(0)) 0
x y
+ffp(sl,
tl)dSldtl 6 Dom(H-1),
0 0
for all x, y lying in the subintervals 0 < x < Xl, 0 <__y < Yl of R+. D Proof: Since g'(u)> O, the function g(u) is monotonically increasing on (0, oo). Define a function z(x, y) by the fight-hand side of (5.4.1), then z(x, O) - a(x) + b(O), z(O, y) - a(O) + b(y), u(x, y) < z(x, y) and
Zxy(X,
y) = p(x, y)
(
u(x, y) +
JJ
p(sl, tl)g(u(s1, tl))ds1 dtl
0 0
< p(x, y)
( iJ Z(X, y) +
0 0
p(s1, tl)g(z(s1, tl))dsldtl
)
)
.
(5.4.4)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES 481 If we put x
v(x, y) -- Z(X, y) +
y
f f p(sl, tl)g(Z(Sl, tl)) dsl dtl, 0 0
then v(x, O) - a(x) + b(O), v(O, y) = a(O) + b(y), z(x, y) < v(x, y) and y)
Vxy(X,
Zxy(X, y) < p(x, y)v(x, y),
y) + p(x, y)g(z(x, y))
-- Zxy(X,
< p(x, y)(v(x, y)-t-g(v(x, y))), i.e.
Vxy(X' Y) < p(x, y). v(x, y) + g(v(x, y)) -
(5.4.5)
Now by following the same arguments as in the proof of Theorem 5.2.1 below (5.2.5) with suitable modifications and in view of (5.4.3) we obtain the estimate for v(x, y) such that
v(x, y) < H -1
IH
(a(0) + b(y)) +
xy
]
j
a'(sl) a(s1) + b(0) + g(a(sl) + b(0)) dsl
0
+// Sltl, ldta
(5.4.6)
0 0
Using (5.4.6) in (5.4.4), and first keeping x fixed, setting y = t and integrating from 0 to y, then in the resulting inequality keeping y fixed, setting x - s and integrating from 0 to x we obtain xy
z(x, y) < a(x) + b(y) +
f f p(s,t ) n -1
(a(0) + b(t))
0 0 s
-t-
f
a'(sl) ds1 a(sl) + b(O) -I- g(a(sl) + b(O))
0 s
t
+ffp(sl, 0 0
tl)dsldtl
]
dsdt.
(5.4.7)
482
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
Now using (5.4.7) in u(x, y) < z(x, y) we obtain the desired bound in (5.4.2). The subintervals for x and y are obvious. II
Bondge and Pachpatte (1980a) established the following three theorems which deal with the two independent variable generalizations of certain inequalities given by Pachpatte (1974d, 1975h). Theorem 5.4.2 Let u(x, y), a(x, y) and b(x, y) be nonnegative continuous functions defined for x, y ~ R+. Let g(u), g'(u) be as in Theorem 5.2.1 and in addition g(u) be subadditive on R+. If u(x, y) < a(x, y) + - - - - b(s, t)
u(s, t ) +
00
b(sl, tl)
00
• g(u(sl, tl))dsl dtl] dsdt,
(5.4.8)
xy{[ /
for x, y ~ R+, then for 0 < x < x2, 0 < y < Y2,
00 + f fo p(sl, tl)dsl dtl] ) where
xy
(
00 s1 tl
t2)g(a(s2, t2))ds2dt2
+ 0
(5.4.9)
I
dsl dtl,
(5.4.10)
0
H, H -1 are as defined in Theorem 5.4.1 and x2, y2 are chosen so that
xy
H(A(x,y))+ffp(sl, 0
t l ) d s l d t l 6 Dom(H-1),
0
for all x, y lying in the subintervals 0 < x < x2, 0 <_ y < y2 of R+.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES Theorem
483
5.4.3 Let u(x, y), a(x, y), b(x, y) and c(x, y) be nonnegative
continuous functions defined for x, y ~ R+. Let g(u) and g'(u) be as in Theorem 5.4.2. If
xy
IJJ
f f ~s o .~s.,~+
u(x, y) < a(x, y) +
s
O0
C~Sl.tl)U(S1, tl)dSl O0
+ flybys1, ,,~ + c~,, tl))g(u(sl, tl))dsl dtl 0
] dsdt, (5.4.11)
0
for x, y ~ R+, then for 0 < x <
X3,
0 < y < Y3,
.y st 0
where
dtl
0
0
{
H-1
H(B(x, y))
]}
(5.4.12)
0
xy [
s/j
B(x,y)=//,.s.. ,..
a(s,, tl) +
O0
b(s., t.)c(s.,t.)ds, dt. O0
s1 tl
+ ff~s. 0
,.~ + c~s. t2))g(a(s2, t2))ds2 dt2 ds1 dtl,
J
0
(5.4.13)
H, H -1 are as defined in Theorem 5.4.1 and x3, Y3 are chosen so that x
H(B(x, y))9-
y
ff~b~Sl, tl) -k-c(s1, tl))dsl dtl E Dom(H-1), 0
0
for all x, y lying in the subintervals 0 < x < x3, 0 < y < Y3 of R+. [2 Theorem
5.4.4 Let u(x, y), a(x, y), b(x, y), c(x, y) and k(x, y) be nonneg-
ative continuous functions defined for x, y ~ R+. Let g(u), g' (u) be as defined
484
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
in Theorem 5.2.2. If
xy
u(x, y) < a(x, y) + b(x, y)
f f c<~,t)g
( u(s, t) + b(s, t)
0 0
s,
) (5.4.14)
0 0
for x, y ~ R+, then for 0 <_x < x4, 0 <_ y <_ Y4, u(x, y) < a(x, y) + b(x, y)
(x, y) +
r
0 0
{[
Jr x. b slta Sldta]})
X
where
~-2-1
f2(L(x, y)) +
xy L(x,y)=ffc(sa, oo
c(s, t)g
(C(S1,
b(s, t)
tl) + k(sl, tl))
0 0
( tl)g
(
(5.4.15)
jtj a(sl, t l ) W b ( s l , tl)
k(s2, t2) oo
• g(a(s2, t2))ds2 dr2) ds1 dtl,
(5.4.16)
~, ~'2-1 are as defined in Theorem 5.2.1 and x4, y4 are chosen so that x y
~,~x, y~ + ff~c~sl, ta~ + ~sa, , 1 ~ ~ S l , t,~ ~s, ~,,~ ~om~ 1~, 0 0
for all x, y lying in the subintervals 0 < x < x4, 0 < y < y4 of R+.
r-q The details of the proofs of Theorems 5.4.2-5.4.4 follow by arguments similar to those in the proofs of Theorems 5.2.2 and 5.4.1 in view of the results given in Pachpatte (1974d, 1975h). Here we leave the details to the reader.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
485
In concluding, we note that the inequalities established in Sections 5.2-5.4 can be extended very easily to n (n > 3) independent variables. The precise formulations of these results are very close to those of the results given in Sections 5.2-5.4 with suitable modifications. It is left to the reader to fill in the details where needed.
5.5 Pachpatte's Inequalities I In view of wider applications, Wendroff' s inequality given in Beckenbach and Bellman (1961) has been generalized and extended in various directions. The present section is devoted to the Wendroff-like inequalities investigated by Pachpatte (1988c, 1993, 1996d) in order to apply them in the study of certain higher order partial differential equations. In what follows we shall use the notations and definitions as given in Section 4.5 without further mention. Pachpatte (1988c) established the Wendroff-like inequalities in the following two theorems. Theorem 5.5.1 Let u(x, y) and h(x, y) be nonnegative continuous functions defined for x, y ~ R+. Let a(x), b(y), p(x), q(y) be positive and twice continuously differentiable functions defined for x, y ~ R+, a' (x), b' (y), p' (x), q' (y) are nonnegative for x, y ~ R+ and define c(x, y ) - a(x) + b(y) + yp(x) + xq(y), for x, y ~ R+. Let g be a continuously differentiable function defined on R+ and g(u) > 0 on (0, cx~), g'(u) > 0 on R+ and
u(x, y) < c(x, y ) + a [ x , y,h(sl, tl)g(u(sl, tl))],
(5.5.1)
holds for x, y ~ R+. (i) If a"(x), p"(x) are nonnegative for x >_ O, then for 0 < x < Xl, O
[
X
+
cxO )
~(c(O, y)) + x k,g(c(O, y))
S
giC-~li-~
dslds
0 0 "1
+ A[x, y, h(sl, tl)]] ,
]
(5.5.2)
486
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
where
j( ds f2(r) --
g(s) '
r > 0, ro > 0,
(5.5.3)
ro ~'2-1 is the inverse of g2 and xl, Yl are chosen so that
(Cx(O,y) ) //a"(Sl)+yp"(Sl) x
f2(c(O, y)) + x
g(c(O, y))
s
+
dsl ds
g(C-~l~-~)
0 0
+ A[x, y, h(sl, tl)] E Dom(f1-1), for all x, y lying in the subintervals 0 <_x <_Xl, 0 ~ y <_ Yl of R+. (ii) If b"(y), q"(y) are nonnegative for y >_ O, then for 0 < x < x2, O
u(x, y) < ~2-1
+ffb"(tl)Wxq"(tl)dtldt+A[y,x,h(sl, g(c(O, tl ))
tl)] ]
(5.5.4)
o o
where f2, ~"2-1 are as defined in (i) and X2, 22 are chosen so that
(Cy(X,O))i/'"<,,,+xq",,,, y
a(c(x, 0)) + y k,g-(c--~,, 0))
t
g(c(O, tl))
+
dt~ dt.
0 0
+ A[y, x, h(sl, tl)] E Dom(~ -1), for all x, y lying in the subintervals 0 < x < X2, 0 ~ y < 22 of R+. [2
Theorem 5.5.2 Let u(x, y), h(x, y), a(x), b(y), p(x), q(y), a'(x), b'(y), if(x), q'(y), c(x, y), g(u) and g'(u) be as in Theorem 5.5.1 and u(x, y) <_ c(x, y ) + A[x, y, h(s1, tl)[U(S1, tl) + A[Sl, tl, h(s2, t2)g(u(s2, t2))]]], holds for x, y ~ R+.
(5.5.5)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
487
(i) If a" (x) and p" (x) are nonnegative for x ~ R+, then for 0 < x < x3, O< y < y3,
u(x,y)<_c(O,y)+xcx(O,y)+ff[a"(sl)+yp"(Sl)]dslds 0
0
+ A[x, y, h(sl, tl)Q1 (S1, tl)],
(5.5.6)
in which
Q1 (x, y) - H -1 x
[.
( cx.o.y. ) c(O, y) + g(c(O, y))
$2
//
"1-
(c(O, y)) + x
00
a"(s3)+yp"(s3) ds3 ds2 + A[x, y, h(s3, t3)]] , (5.5.7) C($3, O) -Jr-g(C(S3, 0))
where ds
H(r) --
r > 0, ro > 0,
s + g(s)
(5.5.8)
ro
H -1 is the inverse function of H and x3, Y3 are chosen so that x
H (c(O, y)) + x (
$2
Cx(O,y) ) . f+f a " ( s 3 ) + y p " ( s 3 ) ds3 ds2 c(O, y) -Jr-g(c(O, y)) c(s3, O) + g(c(s3, 0)) 0
0
+ A[x, y, h(s3, t3)] E D o m ( H -1), for all x, y lying in the subintervals 0 < x < x3, 0 < y < Y3 of R+. (ii) If b"(y), q"(y) are nonnegative for y ~ R+ then for 0 < x < x4, O< y < y4, y
u(x, y) < c(x, O) + yCy(X, O) -~-
t
f f tb"(tl)+ xq" (tl)] dtl dt 0
0
+ A[y, x, h(sl, tl)Q2(Sl, tl)],
(5.5.9)
488
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
in which
a2(x, y) - n -1
+ff
( c(x, O)cy,xo, ) + g(c(x, 0))
(c(x, 0)) + y
y t2
0 0
b"(t3)+xq"(t3) C(0, t3) + g(c(O, t3))
dt3 dt2
(5.5.10)
+ A[y, x, h(s3, t3)]] ,
where H, H -1 are as defined in (i) and x4, y4 are chosen so that y t2
(Cy(X,O) )+ffff'(t3)+xq"(t3) H(c(x, 0)) + y c(x, O) + g(c(x, g(c(O, 0))
c(0, t3) +
t3))
dt3 dt2
0 0
+ A[y, x, h(s3, t3)] E Dom(H-1), for all x, y lying in the subintervals 0 < x < x4, 0 < y < y4 of R+. D P r o o f of T h e o r e m 5.5.1" Since g'(u) >_ O, the function g(u) is monotonically increasing on (0, c~). Define a function z(x, y) by
z(x, y) = c(x, y) + A[x, y, h(sl, t l ) g ( u ( s 1 , t l ) ) ] .
(5.5.11)
From (5.5.11) it is easy to observe that z(0, y) -- c(0, y) -- a(0) + b(y) + yp(O),
(5.5.12)
z(x, O) -- c(x, O) = a(x) + b(O) + xq(O),
(5.5.13)
zx(O, y) -- cx(O, y) -- a'(O) + yp'(O) + q(y),
(5.5.14)
Zxx(X, O) -- Cxx(X, O) = a"(x),
(5.5.15)
Zxxy(X, O) -- Cxxy(X, O) "- p"(x),
(5.5.16)
and Zxxyy(X, y) -- h(x, y)g(u(x, y)).
(5.5.17)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
489
Using u(x, y) < z(x, y) in (5.5.17) we have
Zxxyy(X, y) < h(x, y)g(z(x, y)). From (5.5.18)and using the facts that z(x, y) > 0, g'(u(x, y)) > 0 for x, y ~ R+, we observe that
(5.5.18)
Zy(X, y) > O, zxxy(X, y) > O,
Zxxyy(X, y) < h(x, y)4- Zxxy(X, y)g'(z(x, y))Zy(X, y) g(z(x, y)) [g(z(x, y))l 2 i.e.
O___(Zx~y(x,y) ) ay \ g(z(x, y))
(5.5.19)
By keeping x fixed in (5.5.19), we set y = tl and then integrating with respect to tl from 0 to y and using (5.5.13) and (5.5.16) we have y
Zxxy(X, y) < p"(x) / g(z(x, y)) - g(c(x, 0)) -t- h(x, tl)dtl. 0
(5.5.20)
Again as above, from (5.5.20) and using the facts that z(x, y) > O, Zy(X, y) > O, Zxx(X, y) > 0 for x, y ~ R+, we observe that y
8 (Zxx(x,y)) Oy \ g - ~ , y))
< p"(x) 4- / h(x, tl ) dtl. - g(c(x, 0))
(5.5.e~)
0
By keeping x fixed in (5.5.21), setting y - t and then integrating with respect to t from 0 to y and using (5.5.13) and (5.5.15) we have y
g(z(x, y)) -
g(c(x, 0))
t
4-
h(x, tl) dtl dt.
(5.5.22)
0 0 As above, from (5.5.22) and using the facts that z(x, y) > O, Zx(X, y) > 0 for x, y ~ R+, we observe that
i9 ( z x ( x , y ) -~x \ g ( - ~ , y))
y
< a" (x) + yp" (x) g(c(x, 0))
t
(5.5.23) 0
0
490
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Now keeping y fixed in (5.5.23), setting x - sl and then integrating with respect to Sl from 0 to x and using (5.5.12) and (5.5.14) we have
Zx(X,y) < cx(O, y) + / a"(Sl) + yp"(Sl) g(z(x, y)) - g(c(O, y)) g(C~-liO-)) dsl 0 x
y
t
+fffh(sl,
tl)dtldtdsl.
(5.5.24)
000 From (5.5.3) and (5.5.24) we have x
f2(z(x, y)) = Zx(X, y) < cx(O, y) + f a"(Sl)q-- yp"(sl) -~x g(z(x, y)) - g(c(O, y)) gi-C-~ll-0-)) dsl x y t
+J'ffh(sl,
tl)dtldtds1.
(5.5.25)
0 0 0
Now keeping y fixed in (5.5.25), setting x - - s and then integrating with respect to s from 0 to x and using (5.5.12) we obtain
a (z(x, y)) <_ a (c(O, y)) + x g-@-~iy))
-~-
f f a " ( S l )g(c~-~l + y p "i (~-~i Sl)
dslds
0 0
+ A[x, y, h(sl, tl)].
(5.5.26)
The desired bound in (5.5.2) now follows by substituting the bound on z(x, from (5.5.26) in u(x, y) < z(x, y). The subdomains for x, y are obvious. Rewriting (5.5.11) in the form
y)
z(x, y) - c(x, y) + A[y,x, h(sl, tl)g(u(sl, tl))], since A[x, y, h(sl, tl)g(u(sl, tl))] -- A[y,x, h(sl, tl)g(u(sl, tl))] and by following the same arguments as in the proof of the inequality (5.5.2) given above with suitable modifications we obtain the required inequality in (5.5.4).
II
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
491
The proof of Theorem 5.5.2 follows by an argument similar to that in the proof of Theorem 5.5.1 and by closely looking at the proof of Theorem 4.5.2, with suitable modifications. The details are omitted here. We next establish the following inequality which can be used in the study of certain partial differential and integral equations. Theorem 5.5.3 Let u(x, y), f (x, y) and p(x, y) be nonnegative continuous functions defined for x, y ~ R+. Let g(u), g' (u) be as in Theorem 5.5.1 and g(u) be submultiplicative on R+. If xc c"
u(x, y) <_ c + / / tl
f ( s l , y)u(sl, y)dsl ds +A[x, y, p(sl,
tl)g(u(s1,
tl))],
tl
0 0
(5.5.27)
for x, y ~ R+, where c > 0 is a constant, then for 0 < x < x5, 0 < y < Y5, u(x, y) < Q(x, y){f2-1[f2(c)+ A[x, y, p(sl, tl)g(Q(Sl, tl))]]},
(5.5.28)
where
y, exp(jj ,sl y,dsls)
(5.5.29)
\0 0 ~, ~ - 1 are as defined in Theorem 5.5.1 and xs, Y5 are chosen so that
f2(c) + A[x, y, p(sl, tl)g(Q(sl, tl))] E Dom(f2 -1), for all x, y lying in the subintervals 0 < x < x5, 0 < y < Y5 of R+. D
Proof: Since g'(u)> 0 on R+, the function g(u) is monotonically increasing on (0, ec). We assume that c > 0 and that the standard limiting argument can be used to treat the remaining case. Define a function m(x, y) by
m(x, y) -- c + A[x, y, p(sl, tl)g(u(sl, tl))].
(5.5.30)
Then (5.5.27) can be restated as X
S
u(x, y) < m(x, y) + f / f (sl, y)u(sl, y) dsl ds. 0 0
(5.5.3~)
492
MULTIDIMENSIONALNONLINEARINTEGRALINEQUALITIES
Clearly m(x, y) is positive and nondecreasing in both the variables x, y 6 R+. From (5.5.31) we observe that
u(x, y) < 1 + f~ f m(x, y) --
y________~)dsl U(S1, y) ds. f (sl, Y)m(sl,
(5.5.32)
0 0
Define a function z(x, y) by the fight-hand side of (5.5.32); then
z(O, y) and
--
1,
u(x, y) m(x, y)
< z(x, y)
u(x, y) Zxx(X, y ) - f (x, y ) ~ < f (x, y)z(x, y) m(x, y) -
i.e.
Zxx(X, y) < f(x, y). z(x, y)
(5.5.33)
Now by following the proof of Theorem 5.5.1 we obtain
z(x, y) < exp
f (sl, y)dsl ds
- Q(x, y).
(5.5.34)
Using (5.5.34) in (5.5.32) we have
u(x, y) < Q(x, y)m(x, y).
(5.5.35)
From (5.5.30) and (5.5.35) we have mxxyy(X,
y ) - p(x, y)g(u(x, y)) < p(x, y)g(Q(x, y)m(x, y)) < p(x, y)g(Q(x, y))g(m(x, y)).
(5.5.36)
Now by following the proof of Theorem 5.5.1 we obtain
m(x, y)
<
~-1 [~"2(c)-~-A[x,y, p(sl, tl)g(Q(Sl, tl))]].
(5.5.37)
By using (5.5.37) in (5.5.35) we get the required inequality in (5.5.28). 1
The inequalities given in the following theorem have been recently established by Pachpatte (1993, 1996d) and are motivated by the study of certain higher order partial differential equations.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
493
Theorem 5.5.4 Let u(x, y), a(x, y) and b(x, y) be nonnegative continuous functions defined for x, y ~ R+ and h" R3+ --+ R+ be a continuous function which satisfies the condition 0 < h(x, y, V1) -- h(x, y, 1;2) ~ k(x, y, V2)(V1 -- V2), for x, y ~ R+ and Vl > (i) If
V2
(H)
>_ 0 where k" R3+ --+ R+ is a continuous function.
u(x, y) < a(x, y) + b(x, y)B[x, y, h(s, t, u(s, t))],
(5.5.38)
for x, y ~ R+ then u(x, y) < a(x, y) + b(x, y)p(x, y)exp(B[x, y, k(s, t, a(s, t))b(s, t)]), (5.5.39) for x, y ~ R+, where p(x, y) -- B[x, y, h(s, t, a(s, t))],
(5.5.40)
for x, y E R+. (ii) Let F(u) be a continuous, strictly increasing, convex, submultiplicative function for u > O, l i m u ~ F(u) -- o0, F -1 denote the inverse function of F, and ~(x, y), fl(x, y) be continuous and positive functions for x, y ~ R+ and or(x, y) + fl(x, y) -- 1. If u(x, y) <_ a(x, y) + b(x, y)F -1 (B[x, y, h(s, t, F(u(s, t)))]),
(5.5.41)
for x, y ~ R+, then u(x, y) < a(x, y) + b(x, y)F -l(B[x, y, h(s, t, t~(s, t)F(a(s, t)ct -1 (s, t)))]
x exp(B[x, y, k(s, t, ~(s, t)F(a(s, t)c~-1 (s, t))) x fl(s, t)F(b(s, t)fl -l(s, t))])),
(5.5.42)
for x, y ~ R+. (iii) Let g(u) be a continuously differentiable function defined for u >_ O, g(u) > 0 for u > 0 and g'(u) > 0 for u > 0 and g(u) is subadditive and submultiplicative for u > O. If u(x, y) < a(x, y) + b(x, y)B[x, y, h(s, t, g(u(s, t)))],
(5.5.43)
494
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for x, y ~ R+, then for 0 < x < x0, 0 < y < Yo, u(x, y) < a(x, y ) + b(x, y)f2-1[f2(q(x, y)) d- B[x, y, k(s, t, g(a(s, t)))g(b(s, t))]],
(5.5.44)
where q(x, y) = B[x, y, h(s, t, g(a(s, t)))],
(5.5.45)
g2, f2 -1 are as defined in Theorem 5.5.1 and xo, Yo are chosen so that f2(q(x, y)) + B[x, y, k(s, t, g(a(s, t)))g(b(s, t))] 6 Dom(f2 -1), for all x, y lying in the subintervals 0 < x < xo, 0 < y < Yo of R+. 0 Proof: (i) Define a function z(x, y) by
z(x, y) - B[x, y, h(s, t, u(s, t))].
(5.5.46)
From (5.5.46) and using u(x, y) < a(x, y) + b(x, y)z(x, y) and the condition (H), we observe that
z(x, y) < B[x, y, h(s, t, a(s, t) + b(s, t)z(s, t))] -- p(x, y) + B[x, y, {h(s, t, a(s, t) + b(s, t)u(s, t)) - h ( s , t, a ( s ,
t))}]
< p(x, y) + B[x, y, k(s, t, a(s, t))b(s, t)z(s, t)],
(5.5.47)
where p(x, y) is defined by (5.5.40). Clearly p(x, y) is nonnegative for x, y R+. It is sufficient to assume that p(x, y) is positive, since the standard limiting argument can be used to treat the remaining case. Now since p(x, y) is positive and monotonic nondecreasing in x, y 6 R+, from (5.5.47) we observe that
z(x, y) < l + B [x, y, k(s, t, a(s, t))b(s, t) z(s, t) ] p(x, y) p(s, t) "
(5.5.48)
Define a function v(x, y) by the fight-hand side of (5.5.48), then it is easy to observe that
D~D~v(x, y) -- k(x, y, a(x, y))b(x, y) z(x, y_____~) p(x, y) <_ k(x, y, a(x, y))b(x, y)v(x, y).
(5.5.49)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
495
The rest of the proof follows by the same arguments as in the proof of Theorem 4.5.4 with suitable changes, and hence further details are omitted. (ii) The proof follows by the same arguments as in the proof of Theorem 5.3.1 part (ii), applying the inequality given in part (i) above. The details are omitted here. (iii) Define a function z(x, y) by
z(x, y) = B[x, y, h(s, t, g(u(s, t)))].
(5.5.50)
From (5.5.50) and using u(x, y) < a(x, y) + b(x, y)z(x, y) and following the same arguments as in the proof of inequality (5.5.47) in part (i) with suitable changes we have
z(x, y) < q(x, y) + B[x, y, k(s, t, g(a(s, t)))g(b(s, t))g(z(s, t))],
(5.5.51)
where q(x, y) is defined by (5.5.45). The rest of the proof can be completed by following the proofs of Theorems 5.2.2 and 4.5.4 with suitable changes. We leave the details to the reader. II
5.6 Pachpatte's Inequalities II In the past few years, the inequality given by Snow (1971) has attracted much attention and a number of its generalizations and their applications have appeared in the literature. In this section we present some basic nonlinear integral inequalities in two independent variables investigated by Pachpatte (1980c,d) which can be used in the analysis of various problems in the theory of partial integro-differential and integral equations. Pachpatte (1980c) gave the following useful inequalities in two independent variables. Theorem 5.6.1 Let u(x, y), a(x, y), b(x, y), c(x, y), p(x, y) and q(x, y) be nonnegative continuous functions defined on a domain D. Let Po(xo, Yo) and P(x, y) be two points in D such that (x - xo)(y - Yo) > 0 and R be the rectangular region whose opposite corners are the points Po and P. Let G(r) be a continuous, strictly increasing, convex and submultiplicative function for r > 0, l i m r ~ G(r) -- c~ for all (x, y) in D, ~(x, y), fl(x, y) be positive continuous functions defined on a domain D, and or(x, y) + ~(x, y) = 1. Let
496
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
v(s, t; x, y) and w(s, t; x, y) be the solutions of the characteristic initial value problems L[v]
[p(s, t) -+- fl(s, t)G(b(s, t)f1-1 (s, t))(c(s, t)
= Vst -
+ q(s, t))]v = 0,
(5.6.1)
v(s, y) = v(x, t) = 1, and M[w]
-- Wst -
[fl(s, t)G(b(s, t)/3-1 (s, t))c(s, t) - p(s, t)]w "- O,
w(s, y) = w(x, t) = 1,
(5.6.2)
respectively and let D + be a connected subdomain of D which contains P and on which v > 0 and w > 0 (Figure 4.1). If R C D + and u(x, y) satisfies u(x, y) < a(x, y) + b(x, y)G -1 [xf / c(s, t)a(u(s, t)) ds dt
+
ff
,,
dsdt
,(5.6.3)
X [C(Sl, tl) + q(sl, tl)]V(Sl, tl; s, t)dsl dtl } dsdt] ,
(5.6.4)
q(Sl, t l ) G ( U ( S l , tl))ds1 dtl \xoYo
xoYo
then u(x, y) also satisfies u(x, y) <_a(x, y) + b(x,
y)G-1 [~~ w(s, t; x, y)
x { ol(s, t)G(a(s, t)c~-1 (s, t))c(s, t) s t "~-P(S,t) f/Ol(Sl,
tl)a(a(sl,
tl)ol-l(sl,
tl))
xo yo
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
497
Proof: Rewrite (5.6.3) as u(x, y) < a(x, y)a(x, y)ct -1 (x, y ) + fl(x, y)b(x, y)fl-1 (x, y) x G -1
c(s, t)G(u(s, t)) ds dt +
JJ
p(s, t)
xo Yo
x
q(sl, tl)G(U(Sl, tl))dsldtl
) ] dsdt
.
(5.6.5)
Since G is convex, submultiplicative and monotonic, from (5.6.5) we have G(u(x, y)) <_ ix(x, y)G(a(x, y)ct -l(x, y)) + fl(x, y)G(b(x, y)fl-l(x, y)) x
[jj
c(s, t)G(u(s, t))ds dt +
i.xo Yo
x
r
p(s, t)
xo Yo
q(sl, tl)G(u(sl,
tl))ds1 dtl
) ] dsdt
.
(5.6.6)
The estimate given in (5.6.4) follows by first applying Theorem 4.8.2 to (5.6.6) with a(x, y ) = t~(x, y)G(a(x, y)a-l(x, y)), b(x, y ) = 13(x, y)G(b(x, y)13-1 (x, y)) and u(x, y) = G(u(x, y)) and then applying G -1 to both sides of the resulting inequality. II Theorem 5.6.2 Let u(x, y), a(x, y), b(x, y), c(x, y), p(x, y), q(x, y), Po(xo, Yo) and P(x, y) be as in Theorem 5.6.1. Let H(r) be a positive, continuous, strictly increasing, subadditive and submultiplicative function for r > O, limr~c~, H (r) -- c~, H -1 is the inverse function of H. Let v(s, t; x, y) and w(s, t; x, y) be the solutions of the characteristic initial value problems L[v]
=
Vst -
[p(s,
t ) --[-
H(b(s, t))(c(s, t) + q(s, t))]v = 0,
v(s, y) -- v(x, t) = 1,
(5.6.7)
and M[w] -- Wst - [H(b(s, t))c(s, t) - p(s, t)]w -- O, w(s, y) - w(x, t) -- 1,
(5.6.8)
498
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
respectively and let D + be a connected subdomain of D which contains P and on which v > 0 and w > 0 (Figure 4.1). If R C D + and u(x, y) satisfies u(x, y) < a(x, y) + b(x, y)H-l [x~~ C(S, t)H(u(s, t))dsdt
xy (jj
+ ff.(s.,~ x0 Y0
)]
q(Sl, tl)n(u(s1, tl))ds1 dtl \xo
dsdt ,(5.6.9)
Y0
then u(x, y) also satisfies u(x, y) < H-1 [H(a(x, y)) q-H(b(x, y)) [xfo ~w(s,t; x, y )
•
{
H(a(s, t))c(s, t) + p(s, t)
JJ
H(a(sl, tl))
xo Yo
• [C(Sl, tl)+q(sl,tl)]V(Sl, tl;S,t)dsldtl}dsdt]] . (5.6.10) D Proof: Since H is subadditive, submultiplicative and monotonic, from (5.6.9) we have
H(u(x, y)) < H(a(x, y)) + H(b(x, y)) [xfo ~ c(s, t)H(u(s, t))dsdt
+ f f ~s.,~
q(sl, tl)H(U(Sl, tl))dsl dtl
dsdt .
xo Yo
(5.6.11) The desired bound in (5.6.10) follows by first applying Theorem 4.8.2 to (5.6.11) with a(x, y) - H(a(x, y)), b(x, y) - H(b(x, y)) and u(x, y) H(u(x, y)) and then applying H -1 to both sides of the resulting inequality. II
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
499
Pachpatte (1980c) also gave the following two inequalities which can be used in some applications. Theorem 5.6.3 Let u(x, y), a(x, y), b(x, y), c(x, y), p(x, y), q(x, y) and
Po(xo, Yo), P(x, y) be as in Theorem 5.6.1. Let G(r), t~(x, y), fl(x, y) be as in Theorem 5.6.1. Let v(s, t; x, y) and w(s, t; x, y) be the solutions of the characteristic initial value problems L[v] -- v~t - fl(s, t)G(b(s, t)~ -1 (s, t))[c(s, t) + p(s, t) + q(s, t)]v = O,
(5.6.12)
v(s, y) - v(x, t) - 1, and M[w]
--
Wst
-
~(s,
t)G(b(s, t)~ -1 (s, t))c(s, t)w - O, (5.6.13)
w(s, y) -- w(x, t) -- 1, respectively and let D + be a connected subdomain of D which contains P and on which v > 0 and w > 0 (Figure 4.1). If R C D + and u(x, y) satisfies u(x, y) < a(x, y) + b(x, y)G-l [xfo ~ c(s, t)G(u(s, t)) ds dt
+ // xo
P(S, t) (G(u(s, t)) + ~(s, t)G(b(s, t)~-l (s, t)) YO
//q sl l o U Sl l sld l)dsd] xo Yo
then u(x, y) also satisfies u(x, y)
x { ~(s, t)G(a(s, t)ot-l(s, t))[c(s, t) + p(s, t)] + ~(s, t)G(b(s, t)~ -1 (s, t))p(s, t)
6
500
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
s t
•
ff Ct(Sl,
tl)G(a(s1,
tl)~-l(sl, tl))
xo yo
• [C(Sl, tl) + p(sl, tl) + q(Sl, tl)] • V(S1,
tl;S, t)dsl dtl } ds dt] .
(5.6.15)
U] Theorem 5.6.4 Let u(x, y), a(x, y), b(x, y), c(x, y), p(x, y), q(x, y),
Po(xo, Yo) and P(x, y) be as in Theorem 5.6.1. Let H and H -1 be the same functions as defined in Theorem 5.6.2. Let v(s, t; x, y) and w(s, t; x, y) be the solutions of the characteristic initial value problems L[v] = Vst - H(b(s, t))[c(s, t) + p(s, t) + q(s, t)]v = O,
(5.6.16)
v(s, y) = v(x, t) = 1, and M[w] = Wst - H(b(s, t))c(s, t)w -- O,
(5.6.17)
w(s, y) -- w(x, t) = 1, respectively and let D + be a connected subdomain of D which contains P and on which v > 0 and w > 0 (Figure 4.1). If R C D + and u(x, y) satisfies
u(x, y) < a(x, y) + b(x, y ) H - l [ f / c(s, t)H (u(s, t)) ds dt
xy
(
t.xo YO
st)]
xo yo
xo Yo
(5.6.18)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
501
then u(x, y) also satisfies u(x, y) < H-l [H(a(x, Y)) + H(b(x, Y)) [xfo f w(s, t; x, Y) I"
• ~ H(a(s, t))[c(s, t) + p(s, t)] + n(b(s, t))p(s, t)
[
s
•
ff
t
H(a(sl, tl))[C(Sl, tl) -I- p(sl, tl) + q(sl, tl)]
xo Yo
x v sl tl S t Ol tll S t]]
(5.6.19) [3
The proofs of Theorems 5.6.3 and 5.6.4 can be completed by following similar arguments to those in the proofs of Theorems 5.6.1 and 5.6.2 and applying Theorem 4.8.3. The details are omitted here. R e m a r k 5.6.1 Note that in the special case when p(x, y) = q(x, y) = O, Theorems 5.6.1-5.6.4 reduce to the further generalizations of the integral inequality established by Snow (1971); see also the results given in Bondge and Pachpatte (1980b). In the special case when c(x, y) = 0, Theorems 5.6.1-5.6.4 reduce to the new inequalities which can be used in certain applications. A Pachpatte (1980d) established the following inequalities, which can be used in the analysis of a class of nonlinear non-self-adjoint hyperbolic partial integro-differential and integral equations. Theorem 5.6.5 Let u(x, y), a(x, y), b(x, y), c(x, y), p(x, y), q(x, y),
r(x, y), h(x, y) and g(x, y) be nonnegative continuous functions defined on a domain D. Let Po(xo, Yo) and P(x, y) be two points in D such that (x - xo)(y - Yo) > 0 and R is the rectangular region whose opposite corners are the points Po and P. Let G, G -1, a, ~ be as in Theorem 5.6.1. Let V (s, t; x, y) be the solution of the characteristic initial value problem M[V] = 0,
(5.6.20)
where M is the adjoint operator of the operator L defined by L[~] = ~st + al~s + a2~rt q-a3~,
(5.6.21)
502
MULTIDIMENSIONALNONLINEARINTEGRALINEQUALITIES
in which a l - - ~ G ( b ~ - l ) c q , a2--3G(b~-l)cp and a 3 = - [ g + 3G(b3-1)c(r + h)]. Let W(s, t;x, y) be the solution of the characteristic initial value problem N[W] - O, (5.6.22) where N is the adjoint operator of the operator T defined by
T[q~] -
~st --I- bl~)s qt_ b2~) t +
b3~,
(5.6.23)
in which bl - -~G(b~ -1)cq, b2 - -~G(b~ -1)cp, and b3 - -~G(b~ -1) x c ( r - h). Let D + be a connected subdomain of D which contains P and on which V > 0 and W > 0 (Figure 4.5). If R C D + and u(x, y) satisfies
[r
u(x, y) < a(x, y) + b(x, y)G -1 p(x, y)
c(s, y)G(u(s, y))ds
xo
y
+ q(x, y) / c(x, t)G(u(x, t))dt yo x y
+r(x,
y)ffc(s,t)G(u(s,t))dsdt xo Yo
xy
xo Yo
x G(u(sl, tl))dsl then u(x, y) also satisfies
(jj xo Yo
dtl)dsdt] ,
[j
u(x, y) < a(x, y) + b(x, y)a -1 p(x, y)
(5.6.24)
c(s, y)G(u(s, y))ds
xo
y
+ q(x, y) / c(x, t)G(u(x, t))dt + r(x, y)Qo(x, y) yo xy
+ ~x, y ~ / / ~ s , xo Yo
]
~o~s, ~ dsd~ ,
(5.6.25)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
503
where Qo(x, y) is defined by the right member of Q(x, y) -
xy
f f w (s,,; ~, y)c(s, t)
{ a(s, t) + b(s, t)h(s, t)
xo Yo
(ffV(Sl,tl;s,t)a(sl,tl)C(Sl,tl)ds~dtl)}dsdt, xo Yo
(5.6.26) by replacing a(x, y) by ct(x, y)G(a(x, y)ot-l(x, y)) and b(x, y) by fl(x, y) • G(b(x, y)fl-l(x, y)). Further, if q(x, y)--O, then u(x, y) <_a(x, y) + b(x, y)G -1 Jr(x, y)Qo(x, y) x
y
+h(x, y)fJ'g(s,t)Qo(s,t)dxdt xo Yo
+ p(x, y ) / c ( s , y)fo(s,
y)exp
c(sl, y)
xo
x fl(Sl, y)G(b(sl, y)fl-l(Sl, y))p(sl,
y)dsl) ds] , (5.6.27)
where f o(x, y) is defined by the right-hand side of
f (x, y)----a(x, y)+ b(x, y) [r(x, y)Q(x, y)
xy
]
(5.6.28)
xo Yo
replacing a(x, y) by ol(x, y)G(a(x, y)ct-l(x, y)), b(x, y) by fl(x, y)G(b(x, y) • fl-1 (x, y)) and Q(x, y) by Qo(x, y). Again, if p(x, y) - O, then
504
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
u(x, y) < a(x, y) + b(x, y)G -1 Jr(x, y)Qo(x, y) x
y
y
+h(x, y)//g(s,t)Qo(s,t)dsdt+q(x, y)/c(x,t)fo(x,t) xo Yo
Yo
x e x p ( / c ( x , tl)fl(X, tl)G(b(x, tl)fl-l(x, tl))
(5.6.29) where Qo(x, y) and f o(x, y) are as defined above. I-7 Proof: Rewrite (5.6.24) as
u(x, y) < or(x, y)a(x, y)ot-1 (x, y) + ~(x, y)b(x, y)~-I (x, y)
[j
xG -1 p(x, y) c(s, y)G(u(s, y))ds xo
y
+ q(x, y) / c(x, t)G(u(x, t))dt yo X
y
+ r(x, y ) f f c,s xo yo
xy
+h(x,y,ffg(s,O xO YO
t))ds dt
(jj C(Sl, tl) xo Yo
• G(u(sl, tl))ds1 dtl) dsdt I .
(5.6.30)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
505
Since G is convex, submultiplicative and monotonic we have
G(u(x, y)) < or(x, y)G(a(x, y)ot-1 (x, y)) at- ~(x, y)G(b(x, y),6 -1 (x, y)) x
[j
p(x, y)
c(s, y)G(u(s, y))ds xo
Y
+ q(x, y) / c(x, t)G(u(x, t))dt yo x
y
+r(x,y)ffc(s,t)G(u(s,t))dsdt
xy (jr
xo Yo
c s ,ta x0 Y0
XXo Y0
x G(u(sl, tl))dsl dtl) dsdt] .
(5.6.31)
The estimate in (5.6.25) follows by first applying Theorem 4.8.8 with a(x, y) - o~(x, y)a(a(x, y)c~ -1 (x, y)), b(x, y) -- fl(x, y)G(b(x, y)~-I (x, y)), and u(x, y) - G(u(x, y)) and then applying G -1 to both sides of the resulting inequality. The rest of the proof when q(x, y) = 0 and p(x, y) = 0 follows by a similar argument to that in the last part of the proof of Theorem 4.8.8, in view of the proof of the first part of this theorem, with suitable modifications. The details are omitted here. l
Theorem 5.6.6 Let u(x, y), a(x, y), b(x, y), c(x, y), p(x, y), q(x, y),
r(x, y), h(x, y), g(x, y), Po(xo, yo) and P(x, y) be as in Theorem 5.6.5. Let H, H -1 be as in Theorem 5.6.2. Let V(s, t; x, y) be the solution of the characteristic initial value problem (5.6.20) in which M is the adjoint operator of the operator L defined by (5.6.21) with a l - - -H(b)cq, a2 = - H ( b ) c p and a3 = - [ g + H(b)c(r + h)]. Let W(s, t; x, y) be the solution of the characteristic initial value problem (5.6.22) in which N is the adjoint operator of the operator T defined by (5.6.23) with bl = - H (b)cq, b2 = - H (b)cp, and
506
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
b3 - - H (b)c(r - h). Let D + be a connected subdomain of D which contains P and on which V > 0 and W > 0 (Figure 4.5). IfR C D + and u(x, y) satisfies
[j
u(x, y) < a(x, y) + b(x, y)H -1 p(x, y)
c(s, y)H(u(s, y))ds xo
Y
+ q(x, y) / c(x, t)H(u(x, t))dt Yo x
y
x
y
+r(x,y)f./'c(x,t)H(u(s,t))dsdt+h(x,y)ffg(s,t) xo Yo
xo Yo
then u(x, y) also satisfies u(x, y) < H -1 [n(a(x, y)) + H(b(x, y)) [p(x, y ) f c(s, y)H(u(s, y))ds xo Y
+ q(x, y) / c(x, t)H(u(x, t))dt + r(x, y)Ql(X, y) Yo
xy -t-h(x, y)//g(s,t)Ql(s,t)dsdt
1] ,
(5.6.33)
xo Yo
where Q1(x, y) is defined by the right-hand side of (5.6.26) by replacing a(x, y) by H(a(x, y)) and b(x, y) by H(b(x, y)). Further, if q(x, y) - O, then u(x,y)
[/c(s,y)fl(s,Y) xo
xexp(fs C(Sl'y)H(b(sl'Y))p(sl'y)dsl)ds]] ' (5"6"34)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
507
where f l(X, y) is defined by the right-hand side of (5.6.28) by replacing a(x, y) by H(a(x, y)), b(x, y) by H(b(x, y)), and Q(x, y) by Ql(X, y). Again, if p(x, y)= O, then
[
u(x, y) < H -1 f l(x, y)+ H(b(x, y))q(x, y)
[/
c(x, t)f l(x, t)
x e x p ( / c ( x , tl)H(b(x, tl))q(x, tl)dtl)dt]] , (5.6.35) where Q1 (x, y) and f l(x, y) are as defined above. D Proof: Since H is subadditive, submultiplicative and monotonic, from (5.6.32) we have
[ /
H(u(x, y)) < II(a(x, y)) + H(b(x, y)) p(x, y)
c(s, y)H(u(s, y))ds
xo Y P
+ q(x, y) / c(x, t)H (u(x, t)) dt t,i
Yo x
y
+r(x, y)f/c(s,t)H(u(s,t))dsdt
xy
xo Yo
xo Yo
(/r \ xo Yo
X H(U(Sl, tl))dsl dtl) dsdt] .
(5.6.36)
The desired bound in (5.6.33) follows by first applying Theorem 4.8.8 to (5.6.36) with a(x, y)= H(a(x, y)), b(x, y) = H(b(x, y)) and u(x, y)= H(u(x, y)) and then applying/4 -1 to both sides of the resulting inequality. Further, by setting q(x, y ) = 0 and p(x, y ) = 0 in (5.6.32) and applying Theorem 4.8.8 we obtain the desired bounds in (5.6.34) and (5.6.35).
I
508
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES R e m a r k 5.6.2 We note that the functions V(s, t; x, y) and W(s, t; x, y)
involved in Theorems 5.6.5 and 5.6.6 are the well-known Riemann functions relative to the point P(x, y). The existence and continuity of the Riemann function is well known and may be demonstrated by the method of successive approximations (Counant and Hilbert, 1962). A
5.7 Inequalities in Many Independent Variables The integral inequalities involving functions of many independent variables which provide explicit bounds on unknown functions play a fundamental role in the development of the theory of partial differential equations. The last few years have witnessed a great deal of research concerning such inequalities and their applications in the theory of partial differential equations. This section deals with some basic inequalities established in Pelczar (1963), Headley (1974), Beesack (1975) and Pachpatte (1981c) which provide a very useful and important device in the study of many qualitative properties of the solutions of various types of partial differential, integral and integrodifferential equations. Pelczar (1963) initiated the study of some inequalities for a broad class of operators. In order to present the main results in Pelczar (1963) we need the following definitions found there. We call a set P partly ordered if for some pairs of elements x, y ~ P a relation x < y is defined in such a way that: (a) for each x ~ P, x < x, (b) i f x < y and y < x, then x = y and (c) i f x < y and y < z, then x < z. Let P be a partly ordered set and Q c P; we call z the upper bound of
Q in P if z ~ P and if x ~ Q, then x < ~. We call ~ the supremum of the set Q (abbreviated sup Q) if ~ is an upper bound of Q in P and if x is an upper bound of Q in P, then x < ~. Each partially ordered set can have at most one supremum. The set P will be said to satisfy the condition (II) if the difference x - y ~ P is defined for each x, y ~ P in such a way that (d) if x < y, then for each x ~ P is x - z < y - z, (e) there exists an element 0 ~ P, such that for each x ~ P is x - 0 - x and (f) x - y if and only if x - y = 0. The set P will be said to satisfy the condition (II*) if, for each x, y ~ P, there exists in P, z = sup{x, y}.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
509
The main result established by Pelczar (1963) is embodied in the following theorem. T h e o r e m 5.7.1 A s s u m e that: (al)
The set P is not empty, partly ordered and fulfils the conditions (II) and
(a2)
the functions W (x) and L(x) are defined in the set P, a n d are such that
(ii*), W ( P ) C P and L ( P ) C P,
(a3)
if x <_ L(x), then x < O,
(a4)
if x < y, then W (x) < W (y),
(as)
if 0 < W (x) - W (y), then W (x) - W (y) < L ( x - y),
(a6)
w
(a7)
v
is a solution o f the equation w = w (w),
(5.7.1)
v 5_ W ( v ) .
(5.7.2)
~ P is such that
Then we have
v < w.
(5.7.3) [2
Proof: Let z be the supremum of the set {w, v}. Then w
and
v
From (5.7.4), condition (c) and assumption w=W(w)
and
(5.7.4)
(a4) it follows that
v
(5.7.5)
From (5.7.5) and the definition of z as the sup {w, v} it follows that z < W (z). From condition (d) we have z - w _< W(z) - w - W(z) - W(w). Then, from (5.7.4) and the assumption (as) it follows that z - w < L(z - w).
In view of the assumption
(5.7.6)
(a3), the inequality (5.7.6) implies that z - w - 0.
Hence z - w, which means (5.7.3) holds. II
510
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
R e m a r k 5.7.1 It is easy to see that we can assume that the function L(x) is defined only for 0 < x (x 6 P). It is also easy to see that under the assumptions of Theorem 5.7.1, equation (5.7.1) can have at most one solution in P. A As an application, consider the following equation u(x) = f ( x ) + f
F(x, y, u(y))dy,
(5.7.7)
t /
E
where x - (Xl . . . . . Xn), y = (Yl . . . . . Yn) and E is an n-dimensional set, and the inequality
v(x) < f (x) + / " F(x, y, v(y)) dy.
(5.7.8)
t.i
E
By making use of Theorem 5.7.1, Pelczar (1963) proved the following important result. Theorem 5.7.2 Assume that (bl) (b2) (b3) (b4)
F (x, y, z) is defined and continuous in E x E x R, R -- (-oo, oo ), f (x) is defined and continuous in E, if z < 2, then F(x, y, z) < F(x, y, z), IF(x, y, z) - F(x, y, z)l < l(x, y, Iz - zl), where the function l(x, y, z) is defined in E x E x R and such that if w(x) < f l (x, y, w(y)) dy, E
then
w(x) <_o, (bs) u(x) is a solution of equation (5.7. 7) in E, (b6) v(x) is a continuous function defined in E and fulfils the inequality (5.7.8). Then in the set E we have v(x) < u(x).
(5.7.9)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
511
Proof: It is easy to see that the set P of all continuous functions w(x) defined for x 6 E fulfils the assumptions of Theorem 5.7.1 concerning the set P. Now, for u 6 P, we define W(u) as the function which at the point u has the value at the point z defined by the formula
z(x) -- f (x) + f F (x, y, u(y)) dy. E I I
By L(u) we denote the function defined in P, which at the point u 6 P has the value at the point w defined by the formula
w(x) -- f l (x, Y, u(y)) dy. E tI
It is easy to see that the functions W(u) and L(u) fulfil all assumptions of Theorem 5.7.1. Hence for each solution w of the equation w-
W(w),
and each function, which fulfils the inequality
v <_ W(v), we have v < w. This means that for each solution of the equation (5.7.7) and for each function v(x) which fulfils the inequality (5.7.8), we have
v(x) <_ w(x), and the proof is complete. II R e m a r k 5.7.2 In particular if we put l (x, y, z) = kz and assume that the function F(x, y, z) is bounded and that k < / z ( E ) , where/z(E) is the measure of the set E, then all the assumptions of Theorem 5.7.1 are satisfied. A In order to establish the next theorem, given by Headly (1974), we require the following result, given in Beesack (1975, p. 88). T h e o r e m 5.7.3 Let G be an open set in R N, and for x, y ~ G let
G(x, y) -- {z ~ R N " zj - )Uxj + ( 1 - )U)yj,
O < )~j < 1,1 < j < N},
~12
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
denote the rectangular parallelepiped with one diagonal joining the points x, y. Let the points x ~ y ~ G be such that Go = G(x ~ y) C G, and let the functions a(x), k(x, t, z) be real-valued and continuous on Go and on Gr x R respectively, where Gr -- {(x, t ) ' x ~ Go, t ~ G(x ~ x)}. Suppose also that k is nondecreasing in z for each (x, t) ~ GT and that Ik(x, t, z)l ~ h(t)g(Izl),
(5.7.10)
for (x,t,z) ~ Gr x R, where h ~ L(Go) and g is continuous and nondecreasing on R+ with f l ds/g(s) - c~. Then the integral equation u(x)
= a(x) +
f
k(x, t, u(t))dt,
(5.7.11)
q.r
G(x~
has a solution which is continuous on Go. Moreover, if {En} is a strictly decreasing sequence with lim En = 0, and if Un is a continuous solution on Go of the integral equation Un(X) - a(x) + En -3t- l
k(x, t, Un(t))dt,
(5.7.12)
t.i
G(x~
then U(x) = lim Un(x) exists uniformly on Go, and U(x) is the maximal solution of (5.7.11). D In case x ~ _< y (that is, x ~ < Yi for 1 < i _< N), so that x ~ Go - G(x ~ y) implies x ~ x < y and t ~ G(x~ implies x~ t _< x, this result is a special case of results in Walter (1970), namely Theorem II, p. 131 (with n - 1, H i ( x ) - G(x~ together with Remark VI(/3), pp. 136-7, with n -- 1, and no /~s) and Theorem VII, p. 139. Cases where x ~ 2~ Y can be reduced to this case by appropriate change of independent variables. For example, if y~ < x~o for certain subscripts ct while x~ _< y~ for the remaining subscripts 13, then the change of variables ~ = - x ~ x ~ - x~, y---~= - y ~ , y ~ - y~, ~ - - t ~ , ~ - t~, and ~ - - x ~ , ~ - x~ reduces the integral equations (5.7.11), (5.7.12) to equations over Go - G(x-~ Y) having x-~ < y, to which Walter's results do apply.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
513
As noted in Remark VI(/3) of Walter (1970, p. 136), if hypothesis (5.7.10) is deleted, then equation (5.7.11) may have only a local continuous solution, that is a solution defined in G(x ~ yO) for some yO ~ G(xO, y). In this case, if G(x ~ yO) is also a common domain of existence of solutions of (5.7.12), then the final conclusion of the theorem still holds, but only on G(x ~ yO). Headley (1974, Theorem 1) considered the case of Theorem 5.7.3 having
k = k(t,z) continuous on Go x R and nondecreasing in z. Hypothesis (5.7.10) was overlooked in Headley (1974) and in the case k = k(t, z) of the following theorem given by Headley (1974, Theorem 2). T h e o r e m 5.7.4 Let G, Go = G(x ~ y) and the functions a, k be as in
Theorem 5.7.3, and let the function b be continuous on Go and satisfy the inequality. f v(x) < a(x) + ]
k(x, t, v(t)) dt,
x 6 Go.
(5.7.13)
tl
G(x~ Then v(x) < U(x),
x ~ Go,
(5.7.14)
where U is the maximal solution of (5.7.11) on Go. rq Proof: We may assume that x ~ ~ yi for 1 _< i _< N, since otherwise U(x) - a(x) and (5.7.13) is v(x) <_ a(x). For n _> 1, let Un be a continuous solution of the integral equation (5.7.12) on Go, where En -- n -1 for example. Then
v(x) < Un(X) for x 6 Go,
n >_ 1.
(5.7.15)
For, if this were false for some n and some x ~ Go, then because
1 v(x ~ < a(x ~ < a(x ~ + - - u. (x~ n
it would follow from the continuity of v and
Un
that for some point z ~ Go
with z ~ x ~ we would have v(x) < Un (x) for x ~ G(x ~ z ) -
Un(z). But then v(Z) < a(z) +
f [ J
G(x~
k(z, t, v(t)) dt
{z} but v(z) -
514
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
'
/
< a(z) + - + n
k(z,
t, U n ( t ) )
dt
- - Un ( z ) .
G(x~
It follows that (5.7.15) must hold. Taking limits as n ~ oe, (5.7.14) now follows from Theorem 5.7.3.
II R e m a r k 5.7.3 If hypothesis (5.7.10) is deleted, then (5.7.14) will hold on any subset G(x ~ yO) C Go such that equations (5.7.11) and (5.7.12) have continuous solutions on the common domain G(x ~ yO), if such y 0 6 Go exists. This was the case considered in Headley (1974), and for N = 2 also by Rasmussen (1976). A In the further discussion, some useful integral inequalities in n independent variables given by Pachpatte (1981 c) are presented, which are motivated by a well-known integral inequality due to Wa2eski (1969). We use the same notation as given in Section 4.9 without further mention. Pachpatte (1981c) gave the following general version of Wa2eski's inequality given in Wa2eski (1969). Theorem 5.7.5 Let u(x) and a(x) be nonnegative continuous functions defined on ft. Let k(x, y, z) and W (x, z) be nonnegative continuous functions defined on f22 • R and g2 x R, respectively, and nondecreasing in the last variables, and k(x, y, z) be uniformly Lipschitz in the last variable. If
u(x) < a(x) + W
(j) x,
k(x, y, u(y))dy
,
(5.7.16)
xo
then u(x) <_ a(x) + W(x, r(x)),
(5.7.17)
for x ~ f2, where r(x) is the solution of the equation x
r(x) -- / k(x, y, a(y) + W (y, r(y))) dy,
(5.7.18)
xo
existing on f2. 5
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
515
Proof: Define a function z(x) by x
z(x) - / k(x, y, u(y))dy,
(5.7.19)
,J
xo
then (5.7.16) can be restated as
u(x) < a(x) + W(x, z(x)).
(5.7.20)
Using the monotonicity assumption on k and (5.7.20) in (5.7.19) we obtain x
z(x) < / k(x, y, a(y) + W(y, z(y)))dy.
(5.7.21)
x0
Now a suitable application of Theorem 5.7.2 or Theorem 5.7.4 to (5.7.18) and (5.7.21) yields z(x) < r(x), (5.7.22) where r(x) is the solution of equation (5.7.18). Now, using (5.7.22) in (5.7.20) we obtain the desired bound in (5.7.17). II The following inequality established by Pachpatte (1981c) combines the features of two inequalities, namely, the n independent variable generalization of Wendroff's inequality (Bondge and Pachpatte, 1979a) and the integral inequality given by Headley (1974, Theorem 2), and can be used more effectively in the theory of certain integral and integro-differential equations involving n independent variables. Theorem 5.7.6 Let u(x), f (x), g(x), q(x), c(x) be nonnegative continuous
functions defined on f2, with f (x) > 0 and nondecreasing in x and q(x) > 1. Let k(x, y, z) and W(x, z) be nonnegative continuous functions defined on ~"22 • R and f2 x R, respectively; k(x, y, z) is nondecreasing in x and z and is uniformly Lipschitz in z and W (x, z) is nondecreasing in both x and z. If u(x) < f (x) + q(x)
[/
g(y)u(y) dy +
xo
+W(x,
fk(x,y,u(y))dy), x0
/
xo
g(y)q(y)
(/) ] c(s)u(s) ds
dy
xo
(5.7.23)
516
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for x E f2, then u(x) < Eo(x)[f (x) + W(x, r(x))],
(5.7.24)
for x ~ f2, where
EJ
Eo(x) -- q(x) 1 +
(J
g(y)q(y) exp
q(s)[g(s)+ c(s)] ds
xo
xo
)] dy
,
(5.7.25)
and r(x) is the solution of the equation x
(5.7.26)
r(x) = f k(x, y, Eo(y)[f (y) + W(y, r(y))])dy, xo
existing on f2. [3
Proof: Define a function
re(x) by
m ( x ) = f ( x ) + W ( x , fk(x,y,u(y))dy),
(5.7.27)
x0
then (5.7.23) can be restated as
u(x) < m(x) + q(x)
[j
g(y)u(y) dy +
x0
Since
m(x) is positive,
J (J) ] g(y)q(y)
c(s)u(s) ds dy .
xo
nondecreasing and
xo
q(x)
(5.7.28) > 1, we observe from (5.7.28)
that U(X)
m(x) -< q(x) I 1 f+
xo
c(s) u(s) ds dy . g(Y)m(y) dy + g(y)q(y) m(s) ) ] u(y) j (f x0
xo
(5.7.29) Define a function
z(x)
such that
g(Y)rnr dy +
z(x) = 1 + xo
z(x) -
1 on Xj = X jo,
u(s)
g(y)q(y) xo
l <_j < n , _
c(s)m(O xo
dy,
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES then
517
x
D 1 . . . DnZ(X) -- g(x) U(~'mlx)+ g(x)q(x) f c(s) m(s)U(S)ds, xo
which in view of (5.7.29) implies
D1. . . DnZ(X) <_ g(x)q(x)
[ j Z(X) +
c(s)q(s)z(s) ds
xO
]
.
(5.7.30)
If we put x
v(x) -- z(x) + / c(s)q(s)z(s) ds, (5.7.31)
x0
v(x) = z(x) on xj -- xj0 . . l <.j < .n then (5.7.32)
D1. . . Dnv(X) = D1. . . DnZ(X) -'[-c(x)q(x)z(x). Using the facts that D1 . . . D n z ( X ) < g(x)q(x)v(x) z(x) < v(x) from (5.7.31) in (5.7.32) we obtain
from (5.7.30) and
D1. .. Dnv(X) < q(x)[g(x) + c(x)]v(x).
(5.7.33)
From (5.7.33) and using the facts that Dnv(x) >_ 0, D 1 . . . Dn-lV(X) >__0 and v(x) > 0, we observe that
D1. . .Dnv(x) (Dn v ( x ) )(D1 9 9 9 D n - 1 v ( x ) ) < q(x)[g(x) + c(x)] + v(x) v2(x) i.e.
Dn ( DI ""v--~) Dn-l V(X))
< q(x)[g(x) + c(xl].
(5.7.34)
By keeping X1 . . . . . Xn-1 fixed in (5.7.34), setting x , , - sn and then integrating with respect to Sn from x n0 to xn we have x
D1. . .Dn-lV(X)
v(x)
< / q ( x l , . . . , Xn-1, Sn)[g(Xl . . . . , Xn-1, Sn ) xo "+" r
.....
X n - 1 , Sn )]
dSn.
(5.7.35)
518
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Again as above, from (5.7.35) we observe that x
Dn-1
(ol'''on-2V(X))
v--~)
<_
/
q(xl
....
, X n - 1 , Sn
)[g(Xl . . . . , X n - 1 , Sn )
xo X n - 1 , Sn)] dsn.
"1-" C ( X 1 , . . . ,
By keeping Xl . . . . , X n - 2 and X n fixed in the above inequality, setting X n - 1 - sn-1 and then integrating with respect to Sn-1 from Xn_ 1 ~ to Xn-1 we have Xn- 1 Xn 9
q(Xl . . . .
<
v(x)
o
Xn- 1
, X n - 2 , S n - 1 , Sn)
gO
x [g(x1 . . . . .
X n - 2 , S n - 1 , Sn) X n - 2 , S n - 1 , Sn)] dsn d s n - 1 .
+ C(X1,...,
Continuing in this way we have x2
Xn
DlV(X)
v(x)
< f " " f q(xl , s2, . . . , Sn )[g(xl , s2, . . . , Sn )
x~
xO
-]- C(X1, $2 . . . . .
Sn)] d s n . . ,
ds2.
Now keeping x2 . . . . , Xn fixed in the above inequality, we set Xl - - s l and then integrating with respect to Sl from x ~ to Xl we have
v(x)
Substituting this bound on v(x) in (5.7.30), setting Xn - - Yn and then integrating both sides with respect to Yn from x n0 to Xn, then setting Xn-1
Yn-1 and integrating with respect to Yn-1 from X nO_ 1 t o Xn-1, and continuing in this way, finally setting Xl - Yl and then integrating with respect to yl from x ~ to Xl, we obtain
X
( j )
z(x) < 1 + / g ( y ) q ( y ) e x p x0
q(s)[g(s) + c(s)] ds x0
dy.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
~19
Substituting this bound on z(x) in (5.7.29) we have
u(x) < Eo(x)m(x),
(5.7.36)
where Eo(x) is defined by (5.7.25). From (5.7.27) and (5.7.36) we have
u(x) < Eo(x) [ f (x) + W (x, f k(x, y, u(y)) dy) ] .
(5.7.37)
xo
Now a suitable application of Theorem 5.7.5 yields the desired bound in (5.7.24).
II R e m a r k 5.7.4 We note that in the special case when c ( x ) - 0, the inequality established in Theorem 5.7.6 reduces to another interesting inequality which can be used in some applications. A Another interesting and useful integral inequality given by Pachpatte (1981c) in n independent variables involving two nonlinear functions on the fight-hand side of the inequality is embodied in the following theorem. Theorem 5.7.7 Let u(x), f (x), g(x), q(x), k(x, y, z), W (x, z) be as in
Theorem 5.7.6 and furthermore f (x) > 1. Let H" R+ --+ R+ be a continuously differentiable function with H(u) > 0 for u > O, H'(u) > 0 for u > 0 and satisfies (1/v)H (u) < H (u/v) for v > 1, u > 0 and H (u) is submultiplicative for u > O. If
X
)
u(x) < f (x) + q(x) ;ag(y)H(u(y))dy + x0
x,
k(x, y, u(y))dy
,
xo
(5.7.38)
for x ~ f2, then for x ~ f21 C f2, u(x) < El(x)[f (x)-b W(x, r(x))], where
(5.7.39)
El(x)-q(x)G-l[G(1)Wfg(y)H(q(y))dy], (5.7.40)
xo
520
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
in which
fas v
G(v) =
H(s)'
v > 0, v0 > 0,
(5.7.41)
v0
G -1 is the inverse of G and x
G(1) + / g ( y ) H ( q ( y ) ) d y ~ Dom(G-1), xo Or
for all x ~ ~, and r(x) is a solution of the equation x D
r(x) = / k ( x , y, E~ (y)[f (y) + W(y, r(y))]) dy,
(5.7.42)
t.I
x0
existing on ~2. D Proof: We note that since H'(u) >_0 on R+, the function H is monotonically increasing on (0, ~ ) . Define a function m(x) as in the proof of Theorem 5.7.6 given by (5.7.27); then (5.7.38) can be restated as x D
u(x) < re(x) + q(x) / g(y)H(u(y))dy.
(5.7.43)
L I
x0
Since re(x)> 1 and nondecreasing and q(x)> 1, then from (5.7.43) we observe that u(x)
Define a function z(x) by x
z(x)=l+fg(y)H(u(Y>) xo
z(x) -- 1 on
Xj
=
0
X j,
then D1. . . Dnz(X) -- g(x)H
l <_j_ < n _,_
u(x) ) m(x) '
(5.7.44)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
521
which in view of (5.7.44) and the submultiplicative character of H implies (5.7.45)
D1. . . DnZ(X) < g(x)H(q(x))H(z(x)).
From (5.7.45) and using the facts that DnH(Z(X)) > O, D1...Dn-lZ(X) >_ 0 and H ( z ( x ) ) > 0, we observe that D1. . . Dn Z(X)
n (z(x) )
[Dn H (Z(X) )] [ D 1
< g(x)H(q(x)) +
Dn-1Z(X)]
9 9 9
[H (z(x) )] 2
i.e.
Dn ( D I ' ' ' D n - l z ( X ) ) it-(-~ff)-)
<_ g ( x ) H ( q ( x ) ) .
Now by following an argument similar to that in the proof of Theorem 5.7.6, with suitable modifications, we obtain X2
< H(z(x)) -
Xn
... gO
g ( x l , Y2 . . . . , y n ) H ( q ( x l , Y2 . . . . .
yn))dyn..,
dy2.
gO
(5.7.46) From (5.7.41) and (5.7.46), we observe that X2
Xn
D 1 G ( u ( x ) ) < . f . . . f g(xl , y2 . . . . . Yn ) gO gO x H ( q ( x l , Y2 . . . . .
Yn)) d y n . . ,
dy2.
By keeping x2 . . . . , Xn fixed in the above inequality, setting Xl -- yl and then integrating with respect to Yl from x ~ to Xl we have
u(x) < G -~
[ r G(1)+
xo
g(y)H(q(y))dy
]
.
The rest of the proof is immediate by analogy with the last argument in the proof of Theorem 5.7.6. The subdomain ~"~1 of ~ is obvious. II Pachpatte (1981c) gave the following inequality, which can be used in more general situations.
522
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
5.7.8 Let u(x), f (x), g(x), k(x, y, z) and W(x, z) be as in Theorem 5.7. 7. Let H ' R + --+ R+ be a continuously differentiable function with H (u) > 0 for u > O, H'(u) > 0 for u > 0 and satisfying (1/v)H (u) < H (u/v), for v > 1, u > O. If Theorem
j(j (/)
u(x) < f (x) +
g(y)
u(y) +
xo
+ W
x,
g(s)H(u(s))ds
)
dy
x0
k(x, y, u(y))dy
,
(5.7.47)
xo
for x ~ ~2, then for x ~ ~'22 C ~'2, u(x) < Ez(x)[f (x) + W(x, r(x))],
(5.7.48)
where x
[ / ] y
E2(x) -- 1 + f g(y)F -1
g(s)ds
F(1) +
x0
dy,
(5.7.49)
x0
in which o"
F(cr)
-
/
ds , s + H (s)
cr > 0, cro > 0,
(5.7.50)
o-0
F-1 is the inverse function of F and x
F(1) + / g ( s ) d s
Dom(F-1),
t] x0
for all x ~ f2e, and r(x) is a solution of the equation x
F(X)
/ k(x, y, E e ( y ) [ f (y) -F W(y, r(y))])dy,
(5.7.51)
tJ Xo
existing on f2. D The details of the proof of this theorem follow by an argument similar to that in the proof of Theorem 5.7.7, together with the proof of Theorem 5.4.1, and the details are omitted here.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
523
Pachpatte (1981c) has established the following generalization of the integral inequality given by Young (1973). Theorem 5.7.9 Let u(x), a(x), b(x), c(x) and g(x) be nonnegative continuous functions defined on f2. Let f (x), k(x, y, z) and W(x, z) be as in Theorem 5.7.6. Let v(y; x) and e(y; x) be the solutions of the characteristic initial value problems (-- l)nvyl...y n (y; x) - [a(y)b(y) + a(y)g(y) + c(y)lv(y; x) - 0 in f2,
(5.7.52) v(y; x ) -
1 on Y i - xi,
1 <_ i <_ n,
and (-- 1)neyl...y n (y; x) - [a(y)b(y) - c(y)]e(y; x) -- 0 in f2,
e(y, x ) -
1 on Yi - xi,
(5.7.53)
1 < i < n,
respectively, and let D + be a connected subdomain of f2 containing x such
]
that v > O, e > 0 for all y ~ D +. If D C D + and
X (j)
u(x)
xo
g(s)u(s) ds
dy
x0
(5.7.54)
xo
for x ~ S2, then
(5.7.55)
u(x) < E3(x)[f (x) + W(x, r(x))], where
E3 (x) -- 1 + a(x)
IJ{
e(y; x)
b(y) + c(y)
xo
xv(s;y)ds)}dy]
(J
[b(s) + g(s)]
xo
,
(5.7.56)
524
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
and r(x) is a solution of the equation x
r(x) - i k(x, y, E3(y)[f (y) + W(y, r(y))])dy,
(5.7.57)
xo
existing on f2. D Proof: Define a function m(x) as in the proof of Theorem 5.7.6 given by (5.7.27); then (5.7.54) can be restated as
u(x) < m(x) + a(x)
[:
]
i (:)
b(y)u(y) dy +
xo
c(y)
xo
g(s)u(s) ds d y .
xo
(5.7.58) Since m(x) is positive nondecreasing, we observe from (5.7.58) that
u(x) re(x) -< 1 + a(x)
[i
u(y)) dy + b(Y)m(y
xo
:(i c(y)
xo
)]
g(s) u(s) m(s) ds dy . xo
(5.7.59) Define a function z(x) such that
z(x) -- f b ( y ) u(y)...,,, ~ dy + i c(y) ( : x0
z(x) = 0 on Xi
x0
g ( s )u(s) ~ ds dy,
x0
l
- - X O,
then we obtain
D1. . .Dnz(X) = b(X)m(x) + c(x)
(:U"') g(s)~
ds ,
xo
which in view of (5.7.59) implies
Da...Dnz(x)
(5.7.60) Adding c(x)z(x) to both sides of the above inequality we have
D1. . . Dnz(X) + c(x)z(x) < b(x)[1 + a(x)z(x)] + C(X) [Z(X) + : g(s)[ 1 + a(s)z(s)] ds] . x0
(5.7.61)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
525
If we put x
q/(x) -- z(x) + f
g(s)[ 1 -I- a(s)z(s)] ds, (5.7.62)
xo
~p(x) -- z(x) -- 0 on
Xi
-"
X O,
1 < i < n,
then we obtain D1.
Using
. . Dn ~(x)
-- D1.
. . DnZ(X)
D1...Dnz(x)
-k- g ( x ) [ 1
+
(5.7.63)
a(x)z(x)].
from
+a(x)z(x)]+c(x)q/(x)
(5.7.61)
and
z(x) < q/(x) from (5.7.62) in (5.7.63) we have
L[ap] -- D 1 . . . Dn q/(x) - [a(x)b(x) + a ( x ) g ( x ) + c(x)]q/(x) < [b(x) + g(x)]. Furthermore,
all
pure
Xl,...,Xi-l,Xi+l,...,Xn
mixed
(5.7.64) derivatives
of
~
with
respect
to
up to order n - 1 vanish on xi = x ~ 1 < i < n.
If w is a function which is n times continuously differentiable in D, then n
w L ~ - ~pMw -- Z
( - 1)k-lDk[(DoD1 999Dk-1 w ) ( D k + l . . . DnDn+l ~)],
k=l
(5.7.65)
where M w = ( - 1 ) n D 1 . . . D n w ( X ) - [a(x)b(x) + a ( x ) g ( x ) + c(x)]w(x),
with Do -- Dn+l --- I the identity. By integrating (5.7.65) over D, using s as the variable of integration, and noting that ap vanishes together with all its mixed derivatives up to order n - 1 on Sk = x ~ 1 < k < n, we then obtain n
S
z<-,,'-' i
D
k=l
Sk__Xk
(5.7.66) where d s ' - d s 1 . . , dsk-1 dSk+l . . . dsn. Now let w be chosen as the function v satisfying (5.7.52). Since v -
1
on Sk = xk, 1 <_ k <_ n, it follows that D1 . . . D k - l V(S; x) -- 0 on Sk -- Xk for
526
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
2 < k < n. Thus (5.7.66) becomes
f v(s; x)L~(s) ds - / v(s;x)D2... Dn~(y) dy' - ~(x). O
(5.7.67)
Sl = X l
By the continuity of v and the fact that v - 1 on s -- x, there is a domain D + containing x on which v > 0. Now multiplying (5.7.64) throughout by v and using (5.7.62) and (5.7.67), we obtain
g(s)]v(s; x)ds.
7t(x) < f [ b ( s ) + x0
Now substituting this bound on 7t(x) in (5.7.61) we obtain
Lz -- D1... Dnz(X)- [a(x)b(x)- c(x)]z(x)
<_b(x)+c(x)(f[b(s)+g(s)]v(s;x)ds). X { (j )} x0
Again, by following the same argument as above we obtain the estimate for z(x) such that
z(x) < / e(y; x) b(y) + c(y) xo
[b(s) +
g(s)]v(s; y)ds
dy.
xo
Now substituting this bound on
z(x) in
(5.7.59) we obtain
u(x) < E3(x)m(x), where E3 (x) is defined by (5.7.56). From the definition of
(5.7.68)
m(x) and
(5.7.68)
we have
u(x) < E3(x) [f (x) + W (x, f k(x, y, u(y))dy) ] . xo
Now a suitable application of Theorem 5.7.5 yields the desired bound in
(5.7.55). II
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
527
In concluding this section we note that there is no essential difficulty in establishing the further generalizations of the integral inequalities established by Pachpatte (1979a, 1980b,c,d) and many other theorems given in earlier sections in the set-up of Theorem 5.7.9. Since the details of these results are very close to those given in the proof of Theorem 5.7.9 with suitable modifications, they are left to the reader to fill in where needed.
5.8 Pachpatte's Inequalities III Pachpatte (1996c, 1997, in press d, unpublished manuscript) investigated some useful integral inequalities in two independent variables which claim their origin in the inequalities given in Theorems 3.4.1 and 3.4.3 respectively by Ou-Iang (1957) and Dafermos (1979). In this section we shall give some basic inequalities investigated in Pachpatte (1996c, 1997, in press d, unpublished manuscript) which can play a vital role in the study of certain new classes of partial differential and integral equations. Here we shall use the notation and definitions as given in Section 4.5 without further mention. The following inequality established by Pachpatte (in press d) deals with the two independent variable generalization of the inequality given by OuIang in Theorem 4.3.1. Theorem 5.8.1 Let u(x, y) and p(x, y) be nonnegative continuous functions defined for x, y c R+. If x
U2(X, y ) < c
2
y
+2ff
p(s, t)u(s, t) ds dt,
(5.8.1)
0 0
for x, y ~ R+, where c > 0 is a constant, then x y
u xy c+fJ
p(s, t) dt ds,
(5.8.2)
0 0
for x, y ~ R+. D
Proof: It is sufficient to assume that c is positive, since the standard limiting argument can be used to treat the remaining case. Let c > 0 and define a function z(x, y) by the fight-hand side of (5.8.1), then
528
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
z(x, 0) -- z(0, y)
=
r
U(X, y) <_ ~/Z(X, y) and D2Dlz(x, y) < 2p(x, y). ~/z(x, y)
(5.8.3)
From (5.8.3) and the facts that ~/z(x, y) > 0, Dlz(x, y) > 0, D2z(x, y) > 0 for x, y 6 R+, we observe that
D2Dlz(x, y) [Dlz(x, y)][D2~/z(x, y)] < 2p(x, y)+ ~/Z(X, y) [~/Z(X, y)]2 i.e.
Dlz(x, y)) < 2p(x, y). ~/z(x, y) -
D2
(5.8.4)
By keeping x fixed in (5.8.4), setting y - t and integrating with respect to t from 0 to y, and using the fact that DlZ(X, 0) = 0, we have y
Dlz(x, y) < 2 [ p(x, t)dt. ~/z(x, y) - J
(5.8.5)
0
Now keeping y fixed in (5.8.5) and setting x = s and integrating with respect to s from 0 to x we have x
V/z(x, y)
y
< c + / / p(s, t ) dt ds.
(5.8.6)
0 0
Using (5.8.6) in u(x, y) < ~/z(x, y), we get the required inequality in (5.8.2). II Another useful integral inequality involving functions of two independent variables established by Pachpatte (in press d) is given in the following theorem. Theorem 5.8.2 Let u(x, y), p(x, y) and c be as in Theorem 5.8.1. Let
g(u) be a continuously differentiable function defined on R+ and g(u) > 0 for u > O, g'(u) >_0 for u >_O. If x
y
a~
a~
u2(x, y) <_c 2 + 2 / / p ( s , 0
0
t)u(s, t)g(u(s, t))dtds,
(5.8.7)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
529
for x, y ~ R+, then for 0 <_ x <_ Xl, 0 ~ y <_ Yl, u(x, y)
[ jj
< ~2 - 1
~'~(c)
+
p(s, t)dt ds
0 0
]
,
(5.8.8)
where ds g(s) '
~2(r) -
(5.8.9)
r > 0, r0 > 0,
ro ~'~-1
is the inverse of f2 and
Yl are chosen so that
Xl,
x
y
f2(c)+ffp(s,t)dtdseDom(S2-1), 0 0
for all x, y lying in the subintervals 0 < x < Xl, 0 <_~y < Yl of R+. D
Proof: Since g'(u) _> 0 on R+, the function g is monotonically increasing on (0, c~). As in the proof of Theorem 5.8.1, we assume that c > 0 and define a function z(x, y) by the right-hand side of (5.8.7); then z(x, O ) z(O, y) - c 2, u(x, y) < x/z(x, y) and it is easy to observe that
D2Dlz(x, y) ~/z(x, y)
< 2p(x, y)g(v/z(x, y)).
(5.8.10)
Now by following the same arguments as in the proof of Theorem 5.8.1 below (5.8.3) up to (5.8.6), from (5.8.10) we get x
V/Z(X, y ) < c +
y
S l "s ,.,.+z
(5.8.11)
0 0
Now a suitable application of Theorem 5.2.1 to (5.8.11) yields
V/z(x, y)
< ~'2- 1
[ ij ~'2(c)Jr-
0 0
p(s, t)dt ds
]
.
(5.8.12)
Using (5.8.12) in u(x, y) < ~/z(x, y), we get the required inequality in (5.8.8). The subintervals for x and y are obvious. II
530
MULTIDIMENSIONALNONLINEARINTEGRALINEQUALITIES
The following theorem deals with fairly general inequalities established by Pachpatte (1997) which can be used in some applications. Theorem 5.8.3 Let u(x, y), f (x, y), g(x, y) and h(x, y) be nonnegative continuous functions defined for x, y E R+ and c be a nonnegative constant. (al) If
U2(x,y)
x,S1,1,d,ldSl)h s ,./, ,.] d,as for x, y ~ R+, then
u(x, y) <_ p(x, y) [1 +
xy[, (jj // ,~s,~,exp
0
0
\0
(5.8.13)
,,,~Sl ,1,
0
Sl] d,o]
(5.8.14)
for x, y ~ R+, where x
y
p(x,y)-c+ffh(s,t)dtds, 00 for x, y ~ R+. (a2) If
xy[
u2(x, y) < c2-t- 2 f f 0
f (s, t)u(s, t)
0
+ h(s, t)u(s, t)] dt ds,
(jj \0
(5.8.15)
g(sl, tl)U(Sl, tl)dtl dsl
)
0
(5.8.16)
MULTIDIMENSIONAL
NONLINEAR INTEGRAL INEQUALITIES
531
for x, y ~ R+, then u(x,y)
f ( s , t ) ( S f f g(sl, tl)dtldsl)dtds), 0
\0
0
(5.8.17)
for x, y ~ R+, where p(x, y) is defined by (5.8.15). D Proof: (al) We first assume that c > 0 and define a function z(x, y) by the fight-hand side of (5.8.13). Then z(x, O)- z(O, y ) - c2, u(x, y)< ~/z(x, y) and D2Dlz(x,y)<2v/z(x,Y)[f(x,y)(v/z(x,Y)
+
) ]
xy tl)dZ(S,,,l)dtldSl -t-h(x, y) ffg(sl,
.
(5.8.18)
0 0
From (5.8.18) and using the facts that D2z(x, y) > 0 for x, y 6 R+, we observe that
OlZ,X y,) _
~/z(x, y)> O, Dlz(x, y)>0,
y,( z,x y,
D2 \ ~ ~ , y)
x y
Jr f f g ( s l , tl)V/Z(Sl, tl)dtl 0 0
ds1
) ]
-+-h(x, y) . (5.8.19)
Now by following the same arguments as in the proof of Theorem 5.8.1 below (5.8.3) up to (5.8.6), from (5.8.19) we get
y,+jj o ,s st
)
+ff~,s,,,,VZ,Sl,l,d,,ds, 0 0
d, ds
(5.8.20)
532
MULTIDIMENSIONAL N O N L I N E A R I N T E G R A L INEQUALITIES
where p(x, y) is defined by (5.8.15). Since p(x, y) is positive and monotonic nondecreasing in x, y 6 R+, by applying Theorem 4.4.2, we get
[ jj (JJ tljdtlSl)dto]
V/Z(X, y) < p(x, y) 1 +
f (s, t) exp
0 0
[f(s1, tl)
\0
0
(5.8.21)
Now by using (5.8.21) in u(x, y) < ~/z(x, y) we get the required inequality in (5.8.14). If c > 0 we carry out the above procedure with c -t- e instead of c, where E > 0 is an arbitrary small constant, and subsequently pass to the limit as E ~ 0 to obtain (5.8.14). (a2) The proof follows by the same arguments as those given in the proof of (al) above with suitable modifications, and hence the details are omitted here. II A generalization of Theorem 5.8.1 in the other direction given by Pachpatte (unpublished manuscript) is contained in the following theorem. Theorem 5.8.4 Let u(x, y), f (x, y) and g(x, y) be nonnegative contin-
uous functions defined for x, y E R+ and c be a nonnegative constant. Let L" R3+ --+ R+ be a continuous function which satisfies the condition 0 < L(x, y, v) - L(x, y, w) < k(x, y, w)(v - w),
(L)
for x, y ~ R+ and v > w > O, where k" R3+ --+ R+ is a continuous function. If x
U2(X, y )
< c2
y
+ 2 f f [f (s, t)u(s, t)L(s, t, u(s, t)) 0
0
+ g(s, t)u(s, t)] dt ds
(5.8.22)
for x, y ~ R+, then u(x, y) < p(x, y) + q(x, y) exp ( fo fo f (s, t)k(s, t, p(s, t)) dt ds ) , (5.8.23)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
a33
for x, y ~ R+, where x
y
(5.8.24)
p(x,y)-c+ffg(s,t)dtds, 0 x
0
y
q(x, y) -- f / f (s, t)L(s, t, p(s, t)) dt ds, o o
(5.8.25)
for x, y ~ R+. V]
Proof: It is sufficient to assume that c is positive, since the standard argument can be used to treat the remaining case. Let c > 0 and define a function z(x, y) by the fight-hand side of (5.8.22), then z(x, 0) = z(0, y) = c 2, u(x, y) < ~/z(x, y) and
D2Dlz(x, y) < [ f (x, y)L(x, y, v/z(x, y))+ g(x, y)]. ~/z(x, y)
(5.8.26)
Now by following the same arguments as in the proof of Theorem 5.8.1 below (5.8.3) up to (5.8.6) we get x
y
~/z~x, y~ ~_~ , y~ + / / i ~ s . , ~ s . , .
,/z~s. ,~,,,s
~ ~~
, . 1 1 1
0
Define
0
v(x, y) by x
y
y)-//f(s,
v(x,
t)L(s, t, V/Z(S, t))dt ds.
(5.8.28,
0 0
From (5.8.28) and using the fact that hypothesis (L) we observe that x
v(x, y) <
~/z(x, y) < p(x, y)+ v(x, y) and the
y
f f e(s,,~(s ,, p(s,,~ +v(s, t))dtds 0
0 x
y
= q(x, y) + / f f (s, t){L(s, t, p(s, t) 0
0
+ v(s, t)) - L(s, t, p(s, t))} dt ds
534
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES x
< q(x, y ) +
ff
y
f(s, t)k(s, t, p(s, t))v(s, t)dtds.
(5.8.29)
0 0
Clearly q(x, y) is nonnegative and monotonic nondecreasing for x, y 6 R+. Now an application of Theorem 4.2.2 to (5.8.29) yields
v(x, y) < q(x, y)exp (fo ! f(s,t)k(s,t, p(s,t))dtds ) .
(5.8.30)
The desired inequality in (5.8.23) now follows by using (5.8.30) in (5.8.27) and the fact that u(x, y) < ~/z(x, y).
II Pachpatte (1996c) has established the inequalities in the following theorem which can be used more effectively in certain applications. Theorem 5.8.5 Let u(x, y), f (x, y) and h(x, y) be nonnegative continuous functions defined for x, y ~ R+ and c be a nonnegative constant. (bl) If U2 (X,
y)
< c 2 -+- 2B[x,
y, f (s, t)u 2(s, t) + h(s, t)u(s, t)].
(5.8.31)
for x, y ~ R+, then u(x, y) < p(x, y)exp(B[x, y, f (s, t)]),
(5.8.32)
for x, y ~ R+, where p(x, y) -- c + B[x, y, h(s, t)],
(5.8.33)
for x, y ~ R+. (b2) Let g(u) be a continuously differentiable function defined on R+ and g(u) > O f or u > O, g'(u) > 0 for u > O. If uZ(x, y) < c 2 + 2B[x, y, f (s, t)u(s, t)g(u(s, t)) + h(s, t)u(s, t)], for x, y ~ R+, then for 0 < x <
X1,
(5.8.34)
0 ~ y < Yl,
u(x, y) < f2-1[f2(p(x, y ) ) + B[x, y, f (s, t)]],
(5.8.35)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
where p(x, y) is defined by (5.8.33) and f2, 5.8.2 and Xl, Yl are chosen so that
~'2-1
535
are as defined in Theorem
f2(p(x, y)) + B[x, y, f (s, t)] ~ Dom(f2 -1), for all x, y lying in the subintervals 0 < x < Xl, 0 <_ y < Yl of R+. (b3) Let L and k be as in Theorem 5.8.4. If uZ(x, y) < c 2 + 2B[x, y, f (s, t)u(s, t)L(s, t, u(s, t)) (5.8.36)
+ h(s, t)u(s, t)], for x, y ~ R+, then u(x, y) < p(x, y ) + q(x, y)exp(B[x, y, f (s, t)k(s, t, p(s, t))]),
(5.8.37)
for x, y ~ R+, where p(x, y) is defined by (5.8.33) and q(x, y) -- B[x, y, f (s, t)L(s, t, p(s, t))],
(5.8.38)
for x, y ~ R+. U]
Proof: (bl) Assume that c > 0 and define a function z(x, y) by the fighthand side of (5.8.31); then u(x, y) <_ ~/z(x, y) and Dm 2 D n1z(x, y) _< 2[f(x, y)v/z(x, y ) + h(x, y)]. ~/z(x, y)
(5.8.39)
using the facts that z(x, y ) > O, D2z(x, y ) > 0 , D~-lD'~z(x, y) > O, for x, y 6 R+, we observe that (see Theorem 4.5.4) From
(5.8.39)
and
m--1 n
( D z _ DlZ(X, Y) D2 \ ~/z(x, y)
) <_ 2[f(x, y)v/z(x, y ) + h(x, y)].
(5.8.40)
Now by following the same arguments as in the proof of Theorem 4.5.4, below the inequality (4.5.48) up to (4.5.53), with suitable modifications, we get X Sn-2
S1
y
tm-1
tl
OlZ(X, y)< 2 / f . . . J r f ...f[f(s,t)v/Z(S,t) 4 z ( x , y) 0 0 00 0 0 + h(s, t)] dt d t l . . , dtm-1 ds dsl 999 dsn-2, (5.8.41)
536
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
Now keeping y fixed in (5.8.41), setting x - - S n - 1 and then integrating with respect to s,,-1 from 0 to x and using the fact that ~/z(0, y) - c, we have
v/z(x, y) <_ p(x, y)+ B[x, y, f (s, t)v/z(s, t)],
(5.8.42)
where p(x, y) is defined by (5.8.33). Clearly p(x, y) is positive and monotonic nondecreasing in x, y ~ R+. By applying Theorem 4.5.4 part (i) we get
V/z(x, y) <_ p(x, y)exp(B[x, y, f (s, t)]).
(5.8.43)
Now using (5.8.43) in u(x, y) <_ ~/z(x, y), we get the required inequality in (5.8.32). If c _ 0, we carry out the above procedure with c + E instead of c, where E > 0 is an arbitrary small constant, and subsequently pass to the limit as E --+ 0 to obtain (5.8.32). (ba) Assume that c > 0 and define a function z(x, y) by the right-hand side of (5.8.34), then u(x, y) <_ ~/z(x, y) and
D~D'~z(x, y) _< 2[f(x, y)g(v/z(x, y))+ h(x, y)]. ~/z(x, y)
(5.8.44)
Now by following the same steps as in the proof of part (bl) up to (5.8.42) we get
v/z(x, y) <_ p(x, y)+ B[x, y, f (s, t)g(v/z(s, t))],
(5.8.45)
where p(x, y) is defined by (5.8.33). The rest of the proof can be completed by following the ideas used in the proof of Theorem 5.2.2 and Theorem 4.5.4 with suitable changes. Here we omit the further details. (b3) Assume that c > 0 and define a function z(x, y) by the fight-hand side of (5.8.36), then u(x, y) <_ ~/z(x, y) and
D'~D~z(x, y) _< 2[f(x, y)L(x, y, V/Z(X, y)) + h(x, y)]. ~/z(x, y)
(5.8.46)
Now by following the same steps as in the proof of part (bl) up to (5.8.42) we get
v/z(x, y) <_ p(x, y) + B[x, y, f (s, t)L(s, t, V/Z(S, t))],
(5.8.47)
where p(x, y) is defined by (5.8.33). Define a function v(x, y) by
v(x, y) = B[x, y, f (s, t)L(s, t, V/z(s, t))].
(5.8.48)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
537
From (5.8.48) and using the fact that ~/z(x, y) < p(x, y ) + v(x, y), and the hypothesis (L) we have
v(x, y) < B[x, y, f (s, t)L(s, t, p(s, t) + v(s, t))] = q(x, y) + B[x, y, f (s, t){L(s, t, p(s, t) + v(s, t)) - L ( s , t, p ( s ,
t))}]
< q(x, y) + B[x, y, f (s, t)k(s, t, p(s, t))v(s, t)],
(5.8.49)
where q(x, y) is defined by (5.8.38). Clearly q(x, y) is nonnegative and monotonic nondecreasing in each variable x, y 6 R+. Now an application of Theorem 4.5.4 part (i) yields
v(x, y) <_ q(x, y)exp(B[x, y, f (s, t)k(s, t, p(s, t))]).
(5.8.50)
Using (5.8.50) in (5.8.47) and then u(x, y) < ~/z(x, y), we get the desired inequality in (5.8.37). The proof of the case when c is nonnegative can be completed as mentioned in the proof of part (bl).
II
5.9 Pachpatte's Inequalities IV Inspired by the study of certain new classes of differential and integral equations, Pachpatte (1994a,b) established some useful integral inequalities in one and more than one independent variables. It seems that the inequalities given in Pachpatte (1994a,b) will quite likely be a useful source for future work. In this section, we offer some basic inequalities given in Pachpatte (1994a,b) which claims their origin to be in the inequality given by Ou-Iang (1957). This discussion uses the notation and definitions as given in Section 4.5 and also the following notation and definitions without futher mention. Let R~_ denote the product R+ • ... • R+ (n times). A point (xl, Xn) in R~_ is denoted by x and we denote d x ~ . . . d x l by dx. We define the operators lj recursively by I o u ( x ) = u(x), I j u ( x ) = bj(x)Dju(x), j 1, 2 . . . . . n with bn (x) - 1, where u(x) and bj(x) are some functions defined for x 6 R~_ and Dj = O/Oxj. For x 6 R~_ and some functions b j ( x ) > O, j = 1, 2 . . . . . n - 1, n > 2 an integer, and k(x) defined for x 6 R~_, we set . . . ,
M[x, b, k(y)] = M[xl, . . . , X n , bl, . . . , bn-1, k(y)]
538
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES Xl
Xn-1 ,.,
0
bl (Yl, X2,
. . . , Xn ) 0
bn-
1 (Yl,
99 9
Y n - 1, Xn )
Xn
x f k(y) dy. 0
Pachpatte (1994a) investigated the following useful inequality. Theorem 5.9.1 Let F(x, y) and g(x, y) be nonnegative continuous func-
tions defined for x, y ~ R+ and let p > 1 be a constant. If FP(x, y) < c + B[x, y, g(s, t)F(s, t)],
(5.9.1)
for x, y ~ R+, where c > 0 is a constant, then F(x, y)
< [c ( p - 1 ) / p - ~ - ( ( p -
1)/p)B[x, y, g(s,
t)]] 1 / ( p - l ) ,
(5.9.2)
for x, y ~ R+. D Proof: We first assume that c > 0 and define a function
z(x, y) by the
fight-hand side of (5.9.1), then F(x, y) < (z(x, y) and
D~D~z(x, y) < g(x, y)(//z(x, y).
(5.9.3)
From (5.9.3) and the facts that z(x, y) is positive and D2(P4'z(x, y)) and D~-lD~z(x, y) are nonnegative for x, y 6 R+, we observe that
D'~D~z(x, y) [D2(~z(x, y))][D~-ID~z(x, y)] <_g(x, y)+ (z(x, y) [(z(x, y)]2 i.e.
(D~-ID'~z(x, y) ) D2 ~k --P~~xl~ < g(x, y).
(5.9.4)
By keeping x fixed in (5.9.4), we set y - t and then, by integrating with respect to t from 0 to y and using the fact that D~-lD~z(x, O ) - O , m 2, 3 . . . . . we have
D2m - 1 D ~ z ( x , Y) I "Y < [ g(x, t)dr. ~z(x, y) - J
(5.9.5)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
539
Again as above, from (5.9.5) and the facts that z(x, y) is positive and D z ( ( z ( x , y)) and D m-2 e D nlZ(X, y) are nonnegative for x, y ~ R+, we observe that
(
D'~-2D'~z(x' Y) P~Z(X, y)
D2
)J <
g(x, t) dt.
(5.9.6)
-
o
By keeping x fixed in (5.9.6), setting y - tl, then integrating with respect to tl from 0 to y and using the fact that D m 2 - 2 D nlz(x, O) -- O, m -- 3, 4 . . . . , we have m-2
De
n
Y tl
DlZ(X,y)
0
Continuing in this way we obtain tl
Y tm-1
D~z(x,y) <
S S ...S ,(x,,)dtd,,d,m_,.
(z(x, y) -
0
0
(5.9.7)
0
From (5.9.7) and the facts that z(x, y) is positive and Dl((Z(X, y)) and DT-lz(x, y) are nonnegative for x, y ~ R+, we observe that y tm-1
tl
D~z(x, y ) < S . f "" S g(x' t)dt dtl . .. dtm-1 ~z(x, y) 0
0
+
0
[DI( P~/Z(X,y))][DT-lz(x, y)] [ pv/z(x, y ) ] 2
i.e.
D1
D1 -lz(x' y) ~Z(X, y)
< -
i 0
0
. ..
g(x, t) dt dtl .. 9 dtm-1.
(5.9.8)
0
By keeping y fixed in (5.9.8), setting x - s, then integrating with respect to s from 0 to x and using the fact that DT-lz(0, y) - 0 , for n -- 2, 3 , . . . , we have
D~-lz(x, y)
x
y tm-1
0 0
0
tl
(z(x, y) 0
541)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Continuing in this way, we obtain x Sn-2
Sl y tm-I
tl
DlZ(X'Y)
0 0
0
0
(5.9.9) Now by keeping y fixed in (5.9.9), setting x - Sn-1, then integrating with respect to Sn-1 from 0 to X and using the fact that z(0, y) - c, we have
(~/Z(X, y))p-1 _ ( p ~ ) p - 1 < ((p _ 1)/p)B[x, y, g(s, t)].
(5.9.10)
From (5.9.10) and using the fact F(x, y) <_ ( z ( x , y), we observe that F ( x , y) ~ [c (P-1)/p q- ( ( p -
1)/p)B[x, y, g(s, t)]] 1/(p-l).
This is the required inequality in (5.9.2). If c >_ 0, we carry out the above procedure with c + e instead of c, where e > 0 is an arbitrary small constant, and subsequently pass to the limit E --+ 0 to obtain (5.9.2). II Another useful inequality established by Pachpatte (1994a) is given in the following theorem. Theorem 5.9.2 Let u(x, y), v(x, y), hi(x, y) for i - 1, 2, 3, 4 be nonnegative continuous functions defined for x, y ~ R+ and let p > 1 be a constant. If c l, c2 and lz are nonnegative constants such that
uP(x, y) < Cl -Jr B[x, y, hi (s, t)u(s, t)] -b B[x, y, hz(s, t)~(s, t)], (5.9.11)
Vp (X, y) < C2 -[- B[x, y, h3(s, t)u(s, t)] + B[x, y, h4 (s, t)v(s, t)], (5.9.12)
for x, y ~ R+, where -~(x, y) - exp(-p/~(x + y))u(x, y) and ~(x, y) = exp(p/x(x + y))v(x, y ) f o r x, y ~ R+, then u(x, y) <_ exp(/z(x + y))[(2p-l(cl + r
(p-1)/p -[" 2 P - I ( ( p - 1)/p)
x B[x, y, h(s, t)]] 1~(p-l), V(X, y) <___[(2P-I(c1 -'[-r
(p-1)/p A t - 2 p - I ( ( p -
• B[x, y, h(s, t)]] 1~(p-l),
(5.9.13) 1)/p) (5.9.14)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
541
for x, y ~ R+, where h(x, y) - max{[hl (x, y) -+- h3(x, y)], [h2(x, y) -+- h4(x, y)]},
(5.9.15)
for x, y ~ R+.
Proof: We first multiply (5.9.11) by exp(-p/z(x + y)) and observe that exp(-p/z(x + y))uP(x, y) < Cl -t- B[x, y, hi (s, t)-u(s, t)] + B[x, y, h2(s, t)v(s, t)].
(5.9.16)
Define F(x, y) -- exp(-/x(x + y))u(x, y) + v(x, y).
(5.9.17)
By taking the pth power on both sides of (5.9.17) and using the elementary inequality f (al -k- a2) r ~ 2 r-1 (a~ + a2), (where al, a2 > 0 are reals and r > 1) and (5.9.16), (5.9.12) we observe that FP(x, y) < 2P-l[exp(--plz(x + y))uP(x, Y) + vP(x, y)]
< 2P-1 [Cl -t--C2 -k- B[x, y, [hi (s, t) -at- h3(s, t)]u(s, t)] + B[x, y, [hz(s, t) + h4(s, t)]v(s, t)]].
(5.9.18)
Now using the fact that exp(-p/z(x + y)) < exp(-/z(x + y)) and (5.9.15) in (5.9.18) we observe that FP(x, y) < 2P-I(cl -+- c2) -+-B[x, y, 2p-lh(s, t)F(s, t)].
(5.9.19)
The bounds in (5.9.13) and (5.9.14) follow from an application of Theorem 5.9.1 to (5.9.19) and splitting. II
The inequalities in the following two theorems involving functions of n independent variables given by Pachpatte (1994a) can be used in the study of certain new classes of partial differential and integral equations. Theorem 5.9.3 Let F(x) > O, k(x) > 0 and bi(x) > 0 for i = 1, 2 . . . . . n - 1, be continuous functions defined for x ~ R~_ and let p > 1 be a constant. If FP(x) < c + M[x, b, k(y)F(y)], (5.9.20)
542
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
f o r x ~ R~_, where c > 0 is a constant, then F(x) < [r
_~_ ( ( p _
1)/p)M[x, b, k(y)]] 1~(p-l)
(5.9.21)
for x ~ R~_. [B Theorem 5.9.4 Let u(x) > O, v(x) > O, bi(x) > O, i - 1, 2 . . . . . n - 1, and hj(x) > O, i = 1, 2, 3, 4, be continuous functions defined for x ~ R~_ and let p > 1 be a constant. If c l, c2 and I~ are nonnegative constants such that
uP(x) < Cl + M[x, b, hi (y)u(y)] + M[x, b, h2(y)~(y)],
(5.9.22)
Vp (X) <_ C2 -I- M[x, b, h3 (y)~(y)] + M[x, b, h4 (y)v(y)],
(5.9.23)
for x ~ R~_, where -~(x) - exp(-plzEin__lXi)U(X) for x ~ R~_ and ~(x) exp(plzEn=lxi)v(x) for x E R~_, then u(x) < exp
~
xi
[(2p-l(cl +
c2)) (p-1)/p
i=1 -t- 2p-1 ((p -- 1)/p)M[x, b, h(y)]] 1~(p-l). v(x) <__[(2p-l(cl
(5.9.24)
-1-c2)) (p-1)/p
-I--2 p-1 ((p -- 1)/p)M[x, b, h(y)]] 1~(p-l),
(5.9.25)
for x ~ R~_, where h(x) = max{[hl(x) + h3(x)],
[h2(x) -'t- h4(x)]},
(5.9.26)
for x ~ R~_.
IS] The proofs of Theorems 5.9.3 and 5.9.4 proceed much as the proofs of the theorems given above and follow by closely looking at the proofs of the main results given in Sections 5.7 and 3.10. The details are left to the reader. The following theorem, which is a slight variant of the results established by Pachpatte (1994b) involving functions of n independent variables, can be proved by using similar arguments to those mentioned above. Theorem 5.9.5 Let u(x), f (x) and h(x) be nonnegative continuous functions defined for x ~ R~_ and c be a nonnegative constant. Let bj(x) > O, j - 1, 2 , . . . , n - 1, n > 2 an integer, be continuous functions defined for x~R~_.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
543
(i) If U2 (X) < C2 -Jr-2M[x, b, f (y)u 2(y) + h(y)u(y)],
(5.9.27)
f o r x E R~_, then u(x) < p(x) exp(M[x, b, f (y)]),
(5.9.28)
p(x) -- c + M[x, b, h(y)],
(5.9.29)
for x ~ R~_, where
f o r x E R~. (ii) Let g(u), g'(u) be as in Theorem 5.8.2. If U2(X) < C2 -[-- 2M[x, b, f (y)u(y)g(u(y)) + h(y)u(y)],
(5.9.30)
f o r x E R~_, then f o r 0 <_ Xi ~ X~, i "- 1, 2 . . . . . n, u(x) <__~-l[~(p(x)) --1-M[x, b, f(y)]],
(5.9.31)
where p(x) is defined by (5.9.29), f2, f2 -1 are as defined in Theorem 5.8.2 and x ~ R~_ be chosen so that f2(p(x)) + M[x, b, f (y)] 6 Dom(f2 -1), for all X i lying in the subintervals 0 < Xi <__X~, i -- 1, 2 . . . . . n of R+. (iii) Let L" R~_ x R+ --+ R+ be a continuous function satisfying the condition 0 < L(x, v) - L(x, w) < k(x, w)(v - w), (5.9.32) for x ~ R~_ and v > w > O, where k" R~_ x R+ --+ R+ is a continuous function. If U2(x) <_ C2 + 2M[x, b, f (y)u(y)L(y, u(y)) + h(y)u(y)],
(5.9.33)
for x E R~_, then u(x) < p(x) + q(x) exp(M[x, b, f (y)k(y, p(y))]),
(5.9.34)
f o r x E R~_, where p(x) is defined in (5.9.29) and q(x) - M[x, b, f (y)L(y, p(y))],
(5.9.35)
f o r x 6 R~_. [2
544
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
5.10 Pachpatte's Inequalities V Pachpatte (1991, 1992, 1994d) investigated a number of new inequalities involving functions of two independent variables which claim their origins in the inequalities used by Haraux (1981) and Engler (1989). This section presents some basic inequalities given in Pachpatte (1991, 1992, 1994d) which can be used as tools in the study of certain new classes of partial differential and integral equations. We shall use the same notation and definitions as given in Section 4.5 without further mention. Pachpatte (1994d) investigated the inequalities given in the following two theorems. Theorem 5.10.1 Let u" E --+ R1, p" E --+ R+ be continuous functions and c > 1 be a constant, where E -- [0, ct] x [0,/3], ot > 0,/5 > 0, R1 = [1, c~) and R+ -- [0, c~). If x
y
u(x,y)<_c+ffp(s,t)u(s,t)logu(s,t)dtds, 0
(5.10.1)
0
for (x, y) ~ E, then u(x, y) <_ c Q(x'y),
(5.10.2)
Q(x, y) = exp ( fo fo p(s, t) dt ds ) ,
(5.10.3)
for (x, y) ~ E, where
for (x, y) ~ E. D Proof: Define a function v(x, y) by the fight-hand side of (5.10.1), then u(x, y) <_v(x, y) and
D2D1v(x, y ) - p(x, y)u(x, y)log u(x, y) <_ p(x, y)v(x, y)log v(x, y).
(5.10.4)
From (5.10.4) and the facts that v(x, y) > 0, DlV(X, y) > 0, D2v(x, y) > 0 for (x, y) 6 E, we observe that
D2D1v(x, y) D1v(x, y)D2v(x, y) < p(x, y)log v(x, y)+ v(x, y) Iv(x, y)]2 '
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
545
i.e.
y) ) < p(x, y)log v(x, y). v(x, y) -
D1 v(x,
D2
(5.10.5)
By keeping x fixed in (5.10.5), setting y = t, integrating from 0 to y and using the fact that DlV(X, 0) -- 0, we get y
D1 v(x, y) < / p(x, t) log v(x, t) dt, v(x, y) -
(5.10.6)
0
Keeping y fixed in (5.10.6) and setting x using the fact that v(0, y) - c, we obtain x
log v(x, y) < log c 4-
s, integrating from 0 to x and
y
ff
p(s, t)log v(s, t)dt ds.
(5.10.7)
0 0
Now a suitable application of the inequality given in Theorem 4.2.1 (see Remark 4.2.1) yields
logv(x, y) < (logc)exp ( / f o = log c Q~x'Y).
(5.10.8)
From (5.10.8) we observe that
v(x, y)
< c Q(x'y).
(5.10.9)
Now using (5.10.9) in u(x, y ) < v(x, y) we get the required inequality in (5.10.2).
II Theorem 5.10.2 Let u" E -~ R1, p" E ~ R+ be continuous functions and c >_ 1 be a constant, where E and R1 are as defined in Theorem 5.10.1. Let g(u) be a continuously differentiable function defined on R+ and g(u) > 0 on (0, c~) and g'(u) > 0 on R+. If x
y
u(x, y) < c + f / p ( s , 0 0
t)u(s, t)g(log u(s, t)) dt ds,
(5.10.10)
546
MULTIDIMENSIONAL
NONLINEAR
for (x, y) ~ E, then for (x, y) u(x, y) < exp
INTEGRAL
~ E1 C E,
( [ JJ ~-1
INEQUALITIES
S2(logc) +
p(s, t) dt ds
0 0
])
,
(5.10.11)
where f2(r)- f
ds
g(s) '
(5.10.12)
r > O, ro > 0 ,
ro
~'2 - 1
is the inverse of f2 and (x, y) x
~ E1 C
E is chosen so that
y
~2(logc)+ffp(s,t)dtds~Dom(S2-1), 0
for (x, y)
0
6 E1 C E.
D Proof: Since g'(u)>_ 0 on R+, the function g(u) is monotonically increasing on (0, c~). Define a function v(x, y) by the fight-hand side of (5.10.10). By following the arguments as in the proof of Theorem 5.10.1 up to the inequality (5.10.7) we obtain x
log v(x, y) < log c +
y
f f p(s, 0
(5.10.13)
t)g(log v(s, t))dt ds.
0
Now a suitable application of Theorem 5.2.1 yields
log v(x, y) < ~_l lf2(log c) + ~ ~o p(s, t) dt ds]
(5.10.14)
From (5.10.14) we observe that
v(x, y) < exp
( [ jj ~-I
~(log c) +
0 0
p(s, t) dt ds
l)
.
(5.10.15)
The desired inequality in (5.10.11) now follows by using (5.10.15) in u(x, y) < v(x, y). The subdomain E1 of (x, y) in E is obvious. II
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
547
Another interesting and useful inequality is embodied in the following theorem. Theorem 5.10.3 Let u" E --+ R1, p, q" E --+ R+ be continuous functions and c > 1 be a constant, where E and R1 are as defined in Theorem 5.10.1. Let g(u), g'(u) be as defined in Theorem 5.10.2 and furthermore we assume that g(u) is submultiplicative on R+. If
u(x, y) <_ c + ;;,-~- - -Y U(S, t) [ p(s, t)log u(s, t) 0 0 "1
+ q(s, t)g(log u(s, t))] dt ds,
J
(5.10.16)
for (x, y) ~ E, then for (x, y) ~ E2 C E,
[ { [ J/
u(x, y) < exp Q(x, y)
f2 -1
f2(log c ) +
q(s, t)g(Q(s, t))dt ds
0 0
]}] ,
(5.10.17) where Q(x, y) is defined by (5.10.3), ~2, g2-1 are as defined in Theorem 5.10.2 and (x, y) ~ E2 C E is chosen so that x
y
g2(log c) + / / q ( s , 0
t)g(Q(s, t)) dt ds ~ Dom(g2 -1),
0
for (x, y) E E2 C E. D Proof: Since g'(u)> 0 on R+, the function g(u) is monotonically increasing on (0, cxz). Define a function v(x, y) by the fight-hand side of (5.10.16). Then by following the arguments as in the proof of Theorem 5.10.1 up to the inequality (5.10.7) we have x y
log v(x, y) < log c +
ff
p(s, t)log v(s, t)dt ds
0 0 x
y
+f/q(s,t)g(logv(s,t))dtds. 0 0
(5.10.18)
548
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
Define a function m(x, y) by x
m(x, y ) = log c +
y
ff 0
(5.10.19)
q(s, t)g(log v(s, t))dt ds.
0
Then (5.10.16) can be restated as x
log v(x, y) < m(x, y) +
y
ff 0
(5.10.20)
p(s, t)log v(s, t)dt ds.
0
Since m(x, y) is positive and nondecreasing in both the variables x and y, by applying Theorem 4.2.2 to (5.10.20) we have log v(x, y) < Q(x, y)m(x, y),
(5.10.21)
where Q(x, y) is defined by (5.10.3). From (5.10.19) and (5.10.21) we observe that x
m(x, y) < log c +
y
ff 0
0 x
< log c +
ff 0
q(s, t)g(Q(s, t)m(s, t)) dt ds
y
q(s, t)g(Q(s, t))g(m(s, t))dt ds. (5.10.22)
0
Now a suitable application of Theorem 5.2.1 to (5.10.22) yields
[
m(x, y) < f2-1 f2(log c) +
JJ
]
q(s, t)g(Q(s, t)) dt ds .
0
(5.10.23)
0
Using (5.10.23) in (5.10.21) we get log V(X, y) < Q(x, y) { ~--~-1 [f2(1ogC) xy
0
0
]}
(5.10.24)
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES 549 From (5.10.24) we get
I ]}]
V(X, y) < exp [Q(x, y) { f2-1 f2 (log c) x
0
y
(5.10.25)
0
Using (5.10.25) in u(x, y) < v(x, y) we get the desired inequality in (5.10.17). The subdomain E2 of (x, y) in E is obvious.
II The inequalities in the following theorems are also established by Pachpatte (1994d) and can be used in some applications. Theorem 5.10.4 Let u" E --+ R1, p, q" E --+ R+ be continuous functions and c > 1 be a constant, where E and R1 are as defined in Theorem 5.10.1.
if
xy
[
j) q(Sl, tl)
logu(s,,, + O0
O0
x logu(sl, tl)dtl dsl] dtds,
(5.10.26)
for (x, y) ~ E, then u(x, y)
(5.10.27)
< c Q~
for (x, y) ~ E, where
Qo(x, y)
--
1 +
tl) -+-
----
0 0
\0
0
tl)] dtl ds1
at ds,
(5.10.28)
for (x, y) ~ E. D
551} MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES Theorem 5.11).5 Let u(x, y), p(x, y), c, g(u) and g'(u) be as in Theorem 5.10.2. If
xy
u(x, y) < c +
ff
[
jj
p(s, t)u(s, t) log u(s, t ) +
O0
p(sl, tl) O0
(5.10.29)
x g(logu(sl, t l ) ) d t l ds1] dtds,
for (x, y) ~ E, then for (x, y) ~ E3 C E,
] ]
st 0
(5.10.30)
0
where
ds
G(r)=
s + g(s)
,
r>0, ro>0,
(5.10.31)
ro
G -1 is the inverse function of G and (x, y) ~ E3 C E is chosen so that x
y
G(logc)+/fp(sl,
tl)dtadSl ~ Dom(G-1),
0 0
for (x, y) ~ g 3 C g .
D The proofs of Theorems 5.10.4 and 5.10.5 follow by the same arguments as in the proofs of Theorems 5.10.1 and 5.10.2, using Theorems 4.4.1 and 5.4.1 respectively, with suitable modifications. The details are omitted. The following theorems established by Pachpatte (1991) deals with the two independent variable generalizations of the inequalities given in Theorems 3.9.4 and 3.9.6. Theorem 5.10.6 Let u" E --> R1, p" E -~ (0, c~), q" E --> R+ be continuous functions and c >_ 1 be a constant, where E and R1 are as defined in
MULTIDIMENSIONAL NONLINEARINTEGRALINEQUALITIES 551 Theorem 5.10.1. If x y
u(x,y)<_c+ffp(s,t)u(s,t) 0 0
(
f
f q(sl, tl)log u(sl, tl)dtl dsl) dtds (5.10.32)
00
for (x, y) ~ E, then u(x, y) <
(5.10.33)
c Ql(x,y),
for (x, y) ~ E, where Ql(X,y)=exp (~o ~o p(s,t) ( ~ / oq ( s l , tl)dtldsl) dtds)
(5.10.34)
for (x, y) ~ E. 7q Theorem 5.10.7 Let u(x, y), p(x, y), q(x, y) and c be as in Theorem 5.10.6. Let g(u), g'(u) be as in Theorem 5.10.2. If x y U(X,y)<_c+ffp(s,t)u(s,t) 0 0
)
q(sl, tl)g(logu(sl, tl))dtl dsl dtds,
(5.1o.35)
for (x, y) E E, thenfor (x, y) ~ E4 C E,
+ f fxpy ( s , t ) 0 0
( ~ 0 ~0
q(sl, tl)dtldSl ) d t d]] s , (5.10.36)
552
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
where g2, f2-1 are as defined in Theorem 5.10.2 and chosen so that
y)~ E4 C E is
(x,
f2(logc) + ~ j o p(s, t) \o ( j ~ o q(sl, tl)dtl dsl) dtds E for (x, y) ~ E4 C E. 7q Proofs of Theorems 5.10.6 and 5.10.7: The details of the proof of Theorem 5.10.7 only are given; the proof of Theorem 5.10.6 can be completed similarly. Since g'(u) > 0 on R+, the function g(u) is monotonically increasing on (0, c~). Define a function z(x, y) by the fight-hand side of (5.10.35). Then
u(x, y) < z(x, y) and D2Dlz(x, y) <_p(x, y)z(x, y) X
q(Sl,
tl)g(logZ(Sl, tl))dtldSl
.
(5.10.37)
> 0,
Dlz(x, y) >_0,
0
From
(5.10.37) and using the D2z(x, y) >__0, we observe that
D2(Dlz(x'-Y))\z(x,y)
facts
that
z(x, y)
-
) "
(5.10.38) By keeping x fixed in (5.10.38), setting y - t, integrating from 0 to y and using the fact that Dlz(x, 0) - 0 we get
y (jj
Dlz(x, Y) < z(x, y) - f_ p(x, t)
q(sl, tl)g(logZ(Sl, tl))dtl
ds1
)
dt.
tJ
0
\0
0
(5.10.39) Now keeping y fixed in (5.10.39), setting x - s, integrating from 0 to x and using the fact that z(0, y) = c, we obtain
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES x
553
y
logz(x,y)<_logc+ffp(s,t) 0 0
x (~o~q(sl, l ) ) d t l dtl)g(logz(sl, s l ) d t d S . ot Define a function
(5.10.40)
v(x, y) by the right-hand side of (5.10.40).
Then it is easy
to observe that
1 D 2y)D l D2D1 ~p(x,
\
V(X,y)) -- q(x, y)g(log z(x, Y)) < q(x, y)g(v(x, y)).
From
(5.10.41)
and
using
the
facts
that
Dl((1/p(x, y))DzDlV(X,y)) > O, we observe D2
(5.10.41)
g'(u)>O, D2v(x,y)>O,
that
Dl((1/p(x, y))D2DlV(X,y))) < q(x, y). g(v(x, y)) -
(5.10.42)
By keeping x fixed in (5.10.42), setting y = tl, integrating from 0 to y and using the fact that Dl((1/p(x,
O))D2DlV(X,0))
- - 0 , we obtain
Dl((1/p(x, y))D2DlV(X,y)) < / q(x, tl)dtl. g(v(x, y))
(5.10.43)
As above from (5.10.43) we observe that y
DI( (1/p(x'y))D2Dlv(x'y))g(v(x, y)) _
(5.10.44)
0
By keeping y fixed in (5.10.44), setting x - sl, integrating from 0 to x and using the fact that (l/p(0,
y))D2Dlv(O,y) -- O, we have
D2DlV(X,y) <_p(x, y) ( ~ f q(sl, tl)dtldsl) . g(v(x, y))
(5.10.45)
554
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Again as above from (5.10.45) we observe that
D2 (DlV(X'Y))
0
q(sl, tl)dtldSl
.
(5.10.46)
Keeping x fixed in (5.10.46), setting y = t, integrating from 0 to y and using the fact that D1 v(x, 0) = 0, we have
DlV(X,Y)
<
g(v(x, y)) -
Y (iJ q(sl,
i_ p(x, t) 0
\0
tl) dtl
aS')
dt.
(5.10.47)
0
From (5.10.12) and (5.10.47) we observe that 0
D1v(x,Y)
Ox
g(v(x, y)) -
- - f2 ( v ( x , y ) ) =
<
Y (Ji
[_ p(x, t) 0
\0
q(sl, tl ) dtl
) dsl
dt.
0
(5.10.48) Now keeping y fixed in (5.10.48), setting x = s, integrating from 0 to x and using the fact that v(0, y) = log c, we have
f2(v(x, y)) < f2(logc) + ~ i o p(s, t) ( i ~ qo( s l , tl)dtl dSl) (5.10.49) Using the bound on
v(x, y)
from (5.10.49) in (5.10.40) we have
I-
log z(x, y) < f2 -1 /S2(log c) i_
rrxy (ijqs"'ld'lds')
+jjp(s,t) 0 0
]
dtds .
\0
(5.10.50)
0
From (5.10.50) we get
Z(X, y) < exp
[~"2-1 [Q(log c)
+iiP(S't) (jf q(sl'tl)dtlds1)dtds]] 9 0 0
\0
0
(5.10.51)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
555
The desired inequality in (5.10.36) now follows by using (5.10.51) in u(x, y) < z(x, y). The subdomain E4 of (x, y) in E is obvious. II Pachpatte (1992) established the inequalities given in the following theorems, which can be used in certain applications. Theorem 5.10.8 Let u" E --+ R1, h" E ~ R+ be continuous functions and c > 1 be a constant, where E and R1 are as defined in Theorem 5.10.1.
If u(x, y) < c + B[x, y, h(s, t)u(s, t) log u(s, t)],
(5.10.52)
for (x, y) 9 E, then u(x, y)
<
cexp(B[x,y,h(s,t)]),
(5.~0.53)
for (x, y) 9 E. D
Theorem 5.10.9 Let u" E --+ R1, h" E --+ R+ be continuous functions and c > 1 be a constant, where E and R1 are as defined in Theorem 5.10.1. Let g(u), g'(u) be as in Theorem 5.10.2. If u(x, y) < c + B[x, y, h(s, t)u(s, t)g(log u(s, t))].
(5.10.54)
for (x, y) 9 E, then for (x, y) 9 Eo C E, u(x, y) <_ exp(f2-1[f2(log c) -4- B[x, y, h(s, t)]]),
(5.~0.55)
where f2, f2 -1 are as defined in Theorem 5.10.2 and (x, y) 9 Eo C E is chosen so that
f2(log c) + B[x, y, h(s, t)] 9 Dom(f2 -1), for (x, y) 9 Eo C E.
89 The proofs of Theorems 5.10.8 and 5.10.9 can be completed by following the ideas used in the proof of Theorem 5.10.7 and Theorem 5.8.5 and looking closely at the proof of Theorem 4.5.4, with suitable changes. The details are omitted here. In concluding this section, we note that the inequalities established in Theorems 5.10.1-5.10.9 can be extended very easily to n > 3 independent
556
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
variables. The precise formulations of these results are very close to those of the results mentioned above, with suitable modifications. It is left to the reader to fill in the details where needed. We note that the inequalities given in Chapters 4 and 5 are recently established and still admit various generalizations and extensions in different directions. For various other inequalities related to the inequalities given in Chapters 4 and 5 and their applications, the interested reader is referred to Apartsin and Men (1979), Beesack (1985), Cheung (1993), Conlan and Daiz (1963), Corduneanu (1982, 1983, 1987), DeFranco (1976), Fink (1981), Gutowski (1978), Westphal (1949), Yang (1984, 1985) and Young (1982).
5.11 Applications In this section we present applications of some of the inequalities given in earlier sections to study the qualitative behaviour of the solutions of certain partial differential and integro-differential equations. Most of the inequalities given here are recently investigated and can be used as tools in the study of certain new classes of partial differential, integral and integro-differential equations.
5.11.1 Hyperbolic Partial Differential Equations In this section we first present an application of Theorem 5.2.4 to obtain bounds on the solutions of a nonlinear hyperbolic partial differential equation of the form
Uxy(X, y) = (b(x, y)u(x, y))y -1- f (x, y, u(x, y)),
(5.11.1)
with the boundary conditions
u(x, O) = a(x),
u(O, y) = r(y),
a(O) = r(O),
(5.11.2)
where a, z" R+ ~ R, b" R 2 ~ R, f " R2+ • R --+ R are continuous functions and Ib(x, Y)I _< c(x, y),
(5.11.3)
If(x, y, u)l _< p(x, y)g(lul),
(5.11.4)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
557
where c(x, y), p(x, y), g(r) are as defined in Theorem 5.2.4. It is easy to observe that the problem (5.11.1)-(5.11.2) is equivalent to the integral equation x
u(x, y) = a(x, y)+ /b(s, y)u(s, y)ds 0 x
y
+ f f f (s, t, u(s, t)) ds dt,
(5.11.5)
0 0
where
u(x, y) is
a solution of (5.11.1) with (5.11.2), x
a(x, y) = a(x) 4- z(y) - a(O) - / b(s, O)a(s)ds. 0
We assume that
la(x, Y)I _< k,
(5.11.6)
where k _> 0 is a constant. Using (5.11.3), (5.11.4) and (5.11.6) in (5.11.5) we have x
x
y
lu(x,y)l <_k+fc(s,y)lu(s,y)lds+ffp(s,t)g(lu(s,t)l)dsdt. 0
0 0
(5.11.7) Now a suitable application of Theorem 5.2.4 to (5.11.7) yields
lu(x, y)l < F(x, y) [~2_l [f2(k) + ~ jo p(s, t)g(F(s, t)) dsdt] ] (5.11.8) where F(x, y), g2, f2 -1 are as defined in Theorem 5.2.4. If the fight-hand side in (5.11.8) is bounded then we obtain the boundedness of the solutions of (5.11.1)-(5.11.2). . We next apply Theorem 5.5.3 to obtain the bound on the solution of a nonlinear fourth-order partial differential equation of the form
Uxxyy(X,y) 4- (a(x, y)u(x, y))yy = F(x, y, u(x, y)),
(5.11.9)
558
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
with the given boundary conditions u(x, o) = Oo(x),
bty(X, O) = ~1 (X),
u(O, y) - Oo(y),
ux(O, y) -- ~r1(y),
(5.11.10)
where a" R 2 --+ R, F" R 2 x R--+ R are continuous functions and q~o(x), ~O(y),~l(X),~rl(y) are real-valued twice continuously differentiable functions for x, y 6 R+ and ~0(0) -- ~/ro(O), ~1 (0) -- ~r6(O), ~6(0) -- ~r 1 (0), ~tl (0) -- ~rtl (0). (5.11.11) It is easy to observe that the problem (5.11.9)-(5.11.11) is equivalent to the integral equation X
S
u(x,y)-k(x,y)-f/a(sl,
y)u(sl, y)dslds
0 0
(5.11.12)
-t- A[x, y, F ( s 1 , tl, u(s1, tl)],
where u(x, y) is a solution of (5.11.9) with (5.11.10)-(5.11.11) and the notation A[x, y, F] is defined as in Theorem 5.5.3,
k(x, y) - ~o(x) + Oo(Y) -t- x~/rl(y) + Y~I (x) --
~b0(0 ) -- ~b6(0)X
- y~l(O) - xydYl(O)-k- f j a(sl, O)q~O(Sl)dsl ds. (5.11.13) 0 0
We assume that
Ik(x, y)l < c,
(5.11.14)
la(x, Y)I _< f (x, y),
(5.11.15)
IF(x, y, u)l < p(x, y)g(lul),
(5.11.16)
where c, f (x, y), p(x, y), g(r) are as defined in Theorem 5.5.3. Using (5.11.14)-(5.11.16) in (5.11.12) we have x
s
lu(x, y)l < c + / f f (sl, y)lU(Sl, y)l dsl dS 0 0
+ A[x, y, p(sl, tl)g(lU(Sl, tl)l)].
(5.11.17)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
559
Now an application of Theorem 5.5.3 yields lu(x, y)l _< Q(x,
y){f2-1[~2(c)+A[x,
y,
p(sl, tl)g(O(Sl, tl))]]}, (5.11.18)
where Q(x, y), f2, f2 -1 are as defined in Theorem 5.5.3. If the fight-hand side of (5.11.18) is bounded, then we obtain the bound on the solutions of (5.11.9)-(5.11.11). For similar applications, see Pachpatte (1979a, 1980d, 1988c)
5.11.2 Hyperbolic Partial Integro-differential Equations This section presents an application of Theorem 5.4.2 to obtain bounds on the solutions of a nonlinear hyperbolic partial integro-differential equation of the form Uxy(X, y) -- f
0
, y, u(x, y),
//
k(x, y, s, t, u(s, t)) ds d t )
(5.11.19)
0 0
with the given boundary conditions u(x, O) -- or(x),
u(O, y) -- r(y),
u(0, 0) -- 0,
(5.11.20)
where or, r ' R + --+ R, k" R 4 x R --+ R, f " R2+ x R x R --+ R are continuous functions. The problem (5.11.19)-(5.11.20) is equivalent to the integral equation
xy(
s,
st
0 0
) dsdt,
(5.11.21)
0 0
where u(x, y) is a solution of (5.11.19) with (5.11.20). We assume that
I~(x) + r(y)l ~ a(x, y),
(5.11.22)
Ik(x, y, s, t, u)l < b(s, t)g(lul),
(5.11.23)
If(x, y, u, v)l < b(x, y)[lu[ + Ivl],
(5.11.24)
560
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
where a(x, y), b(x, y), g(r) are as defined in Theorem 5.4.2. Using (5.11.22)-(5.11.24) in (5.11.21) we have
xy
(
,u~xy~,~_a~xy~+//~s,~ ,~st~,
st
0 0
)
0 0
Now an application of Theorem 5.4.2 to (5.11.25) yields
xy
{
,u~xy~ ~_a~xy~+a~xy~+//~s,~ ~1 0 0
+//b~Sltl,dsld,1 0 0
n(A(s,t))
]}
~sd,
(5.11.26)
where A(x, y), H, H -1 are as defined in Theorem 5.4.2. The fight-hand side of (5.11.26) gives the bound on the solution u(x, y) of (5.11.19)-(5.11.20). The inequalities established in Theorems 5.5.1 and 5.5.2 can be used to study the problems of boundedness of the solutions of nonlinear fourth-order partial differential and integro-differential equations of the forms Uxxyy(X,
y)- f (x, y, u(x, y)),
(5.11.27)
and Uxxyy(X,
y)- f (x, y, u(x, y), X $2
Y
t2
fff/k(x,y,
\
s3, t3, u(s3, t3))dt3dt2ds3ds2)
(5.11.28)
0 0 0 0
respectively with the given boundary conditions (5.11.10)-(5.11.11), and under some suitable conditions on the functions involved in (5.11.27) and (5.11.28). For more details, the reader is referred to Pachpatte (1979c, 1988c).
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
561
5.11.3 Higher Order Hyperbolic Partial Differential Equations In this section we present an application of Theorem 5.10.8 given by Pachpatte (1992), to obtain the bound on the solution of the following higher order hyperbolic partial differential equation of the form
D~D~u(x, y ) = h(x, y)u(x, y)log lu(x, Y)l,
(5.11.29)
with the given boundary conditions
OJu(x, o) OyJ
=otj(x),
0
= fli(Y),
O
.
.
.
.
.
(51130)
oiu(o, y) Oxi
(5.11.31)
where h ~ C[R2+, R], aj ~ c(n)[R+, R], fli ~ c(m)[R+, R] and
cxj(i) ( 0 ) - - r ' iB (j) .
. 0 < . j < .m - . 1
0 .< i <. n - 1
.
(5 11.32)
It is easy to observe that the problem (5.11.29)-(5.11.32) is equivalent to the integral equation
u(x, y) = q(x, y) + B[x, y, h(s, t)u(s, t) log lu(s, t)l],
(5.11.33)
where u(x, y) is a solution of (5.11.29) with (5.11.30)-(5.11.32), the notation B[x, y, f] is defined as in Theorem 5.10.8 and
xi_ 1 q(x, y ) -
i=l
-~
m
---------~fli-l(y) ( i - 1) + E
"
n
j=l
Xi_ 1 ~ ( i - 1)v
i--1
yj-1 ( j _ 1)vaJ -l(x)
"
"
y j - 1 0 t J -/)(0)" ( j - 1)v
"=
(5 11.34)
"
From (5.11.33) we observe that
lu(x, y)l < Iq(x, y)l + B[x, y, Ih(s, t)llu(s, t)ll log lu(s, t)ll].
(5.11.35)
From (5.11.35) we observe that 1 + lu(x, Y)I ___ 1 + Iq(x, Y)I + B[x, y, Ih(s, t)l(1 + lu(s, t)l)
• log(1 + lu(s, t)l)].
(5.11.36)
562
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
We assume that
1 + Iq(x, Y)I ~ c,
(5.11.37)
where c > 1 is a constant. Using (5.11.37) in (5.11.36) we have
1 + lu(x, Y)I <~ c + n[x, y, Ih(s, t)l(1 + lu(s, t)l) x log(1 + lu(s, t)l)].
(5.11.38)
Now an application of Theorem 5.10.8 to (5.11.38) yields 1 + lu(x, y)l ~
cexp(B[x'y'lh(s't)l]).
(5.11.39)
From (5.11.39) we observe that lu(x, Y)I ~
[cexp(B[x'y'lh(s't)l])- 1],
(5.11.40)
for (x, y ) 6 E, where E is defined as in Theorem 5.10.8. The inequality (5.11.40) gives the bound on the solution u(x, y) of (5.11.29)-(5.11.32) in terms of the known functions.
5.11.4 Multivariate Hyperbolic Partial Integro-differential Equations In this section we present applications of Theorem 5.7.6 to the boundedness and behavioural relationships of the solutions of some nonlinear hyperbolic partial integro-differential equations given by Pachpatte (1981c). As a first application, we obtain a bound on the solution of a nonlinear hyperbolic partial integro-differential equation of the form
D1. . .Dnu(X) -- A
(X / , u(x),
B(x, y, u ( y ) ) d y
)
+ F(x, u(x)),
(5.11.41)
xo
with the conditions prescribed on xi - x~ 1 < i < n, where F" f2 x R ~ R, B: f22 x R ~ R, A: f2 x R 2 ~ R are continuous functions. We assume that IB(x, y, u)l _< c(y)lul,
(5.11.42)
iA(x, u, v)[ _< g(x)(lul + Iv]),
(5.11.43)
IF(x, u)l <_ k(x, [ul),
(5.11.44)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
563
where c(y), g(x), k(x, y, 4)) - k(y, ~) are as defined in Theorem 5.7.6. Let the boundary conditions be such that the given equation (5.11.41) is equivalent to the integral equation
X (/ yu(y),
-- h(x) + [_ A
hi(X)
B(y,
Su(s)) ds
) / dy +
F(y, u(y)) dy,
d
x0
x0
x0
(5.11.45) where h(x) depends on the given boundary conditions. We assume that Ih(x)l ~ f(x),
(5.11.46)
were f(x) is defined as in Theorem 5.7.6. Using (5.11.42)-(5.11.44) and (5.11.46) in (5.11.45) we have
x
lu(x)l <_f (x) + [_g(y)lu(y)l dy
+ [~x(j --
c(s)lu(s)lds dy
t /
x0
x0
x0
+ / k(y, lu(y)l) dy.
(5.11.47)
J
x0
Now a suitable application of Theorem 5.7.6 with q(x) -- 1, w(x, ~) - 4) and k(x, y, ~) - k(y, ~) yields (5.11.48)
lu(x)l _< E; (x)[f (x) + r(x)], where
E~(x) -- 1 + [--x g(y) exp (
j [g(s) +) c(s)] ds
dy,
(5.11.49)
Or
x0
x0
and r(x) is a solution of the equation X
Y(X)
/ k(y, E; (y)[f (y) + r(y)]) dy. Or
x0
If the right-hand side in (5.11.48) is bounded then we obtain the boundedness of the solution u(x) of (5.11.41). Our second application deals with the behavioural relationship between the solutions of (5.11.41) with the conditions prescribed on x i - x~
564
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
1 < i < n, and the nonlinear hyperbolic partial integro-differential equation of the form
(X j
D1...Dnv(X) --Ao
, v(x),
Bo(x, y, v ( y ) ) d y
)
,
(5.11.50)
xo
with the conditions prescribed on xi - x~ 1 < i < n, where B" ~22 x R -+ R, Ao" ~2 x R 2 ~ R are continuous functions. We assume that IB(x, y, u) - Bo(x, y, v)l < c(y)lu - vl,
(5.11.51)
IA(x, u,-a) - Ao(x, v, v)l _< g(x)[lu - vl + lu - vl]
(5.11.52)
IF(x, u)l ~ k(x, lul),
(5.11.53)
where c(y), g(x) and k(x, ~) are as explained above. Equations (5.11.41) and (5.11.50) are equivalent to the integral equation (5.11.45) and
X ( / /
v(x) -- h(x) +
Ao
y, v(y),
x0
)
Bo(y, s, v(s))ds
dy,
(5.11.54)
x0
where h(x) depends on the given boundary conditions. From (5.11.45) and (5.11.54) we have u - v -- h(x) - h(x) +
X{( / / A
y, u(y),
xo
- Ao
( j y, v(y),
)
B(y, s, u(s))ds
xo
Bo(y, s, v(s)) ds
xo
)} J dy +
F(y, u(y)) dy.
x0
(5.11.55)
Using (5.11.51)-(5.11.53) and lul - Ivl ~ In - vl and assuming that Ih(x) h(x)l < f ( x ) , and that the solution v(x) of (5.11.50) is bounded by a constant M > 0 in (5.11.55), where f ( x ) is as defined in Theorem 5.7.6, we have lu - vl < f (x) +
/ ( / g(y)
x0
lu - vl +
c(s)lu(s) - v(s)l ds
)
dy
xo
x
+ /k(y, xo
M + lu(y) - v ( y ) l ) d y .
(5.11.56)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
565
Now a suitable application of Theorem 5.7.6 yields lu - vl < E; (x)[f (x) + r(x)],
(5.11.57)
where E*(x) is defined by (5.11.49) and r(x) is a solution of the integral equation x ari
r(x) - / k ( y , M
+ E~(y)[f (y) + r(y)]) dy.
(5.11.58)
.J
xo
If the right-hand side of (5.11.57) is bounded then we obtain the relative boundedness of the solution u(x) of (5.11.41) and the solution v(x) of (5.11.50). If f ( x ) in (5.11.57) is small enough and, say, less than ~, where E > 0 is arbitrary, if equation (5.11.58) admits only an identically zero solution, and if E~(x) in (5.11.57) is bounded and E--+ 0, then we obtain l u ( x ) - v(x)l ~ 0, which gives the equivalence between the solutions of (5.11.41) and (5.11.50).
5.12 Miscellaneous Inequalities 5.12.1 Dragomir and Ionescu (1987, 1989) Let u, a, b, be nonnegative continuous functions defined on R2+ and L" R3+ -+ R+ be a continuous function which satisfies the condition
0 < L(x, y, v) - L(x, y, w) < M(x, y, w)(v - w),
(L)
for x, y 6 R+ and v > w > 0, where M is a nonnegative continuous function defined on R3+. (i) If x y
U(X, y ) < a ( x ,
y)+b(x,
y)ffL(s,t,u(s,t))dtds, 0 0
for x, y 6 R+, then
ux,
ax, +
[ex
0)
-1],
(*)
566
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for x, y 6 R+, where P(x, y) is given by
P(x, y) - [{L(x, y, a(x, y))}2 at- {M(x, y, a(x, y))}2b2(x, y)]l/2, for x, y 6 R+. (ii) If u(x, y) satisfies the inequality (.) above, then x y
u(x, y) < a(x, y) + b(x, y)Q(x, y ) / / L ( s ,
t, a(s, t))dt ds,
0 0
for x, y 6 R+, where x y
Q(x,y)-l+ffM(s,t,a(s,t))b(s,t) 0 0
x exp
(
ffM(sl,
tl, a(sl, tl))b(sl,
tl)dtl d s l ) d t d s ,
0 0
for x, y 6 R+.
5.12.2 Pachpatte (in press k) Let u(x, y), p(x, y), q(x, y) and r(x, y) be nonnegative continuous functions defined for x, y 6 R+. Let k(x, y, s, t) and its partial derivatives el (x, y, s, t) = (O/Ox)k(x, y, s, t), k2(x, y, s, t) = (O/Oy)k(x, y, s, t), kl2(X, y, s, t) = (O2/OxOy)k(x, y, s, t) be nonnegative continuous functions for 0 < s < x < c~, 0 _< t < y < oe and L" R 3 ~ R+ be a continuous function satisfying the condition
0 < L(x, y, v) - L(x, y, w) < M(x, y, w)(v - w), for x, y 6 R+, v > w > O, where M'R3+ --+ R+ is continuous.
(i) If x y
u(x,y)<_
p(x,y)+q(x,y)ffk(x,y,s,t)[r(s,t)u(s,t) 0 0
+ L(s, t, u(s, t))] dt ds,
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES 567 for x, y 6 R+, then
u(x, y) < p(x, y) + q(x, y) ( f f A(Sl, tl)dtl dSl) \0
\0
0
0
for x, y 6 R+, where
A(x, y) -- k(x, y, x, y)[r(x, y)p(x, y) + L(x, y, p(x, y))] x
-Jr- f
kl (x,
y, s, y)[r(s, y)p(s, y)+ L(s, y, p(s, y))] ds
0 y
-k- f k2(x, y, x, t)[r(x, t)p(x, t) + L(x, t, p(x, t))] dt 0
x y +ffkl2(x,y,s,t)[r(s,t)p(s,t)+L(s,t,p(s,t))]dtds, o o
B(x, y) --k(x, y, x, y)[r(x, y)4-M(x, y, p(x, y))]q(x, y) x
+ / k l ( x , y, s, y)[r(s, y)+ M(s, y, p(s, y))]q(s, y)ds o y
+ / k2(x, y, x, t)[r(x, t) 4- M(x, t, p(x, t))]q(x, t) dt 0 x y
+
ff
kl2(X, y, s, t)[r(s, t) + M(s, t, p(s, t))]q(s, t)dt ds,
0 0
for x, y 6 R+. (ii) Let F, F -1, oe,/~ be as in Theorem 5.3.1, part (ii). If
u(x,y)
0
+ L(s, t, F(u(s, t)))] dt ds) ,
568
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for x, y 6 R+, then
u(x,y)
x exp
[(fo fo -A(sl,tl)dtldSl )
(OJOJ)] B(Sl, tl) dtl ds1
,
for x, y 6 R+, where A(x, y) and B(x, y) are defined by the fight-hand sides of A(x, y) and B(x, y) defined above by replacing p(x, y) by c~(x, y)F(p(x, y)c~-l(x, y)) and q(x, y) by fl(x, y)F(q(x, y)fl-l(x, y)) for
x, y6R+.
5.12.3 Pachpatte (1979b) Let u(x, y), b(x, y) be nonnegative continuous functions defined for x, y R+ and a(x, y) > 0 for x, y 6 R+ be a continuous function. Let g(u) be a continuously differentiable function defined for u >_O, g(u) > 0 for u > 0 and g'(u) > 0 for u >_ O. (i) If ax(X, y), ay(X, y), axy(X, y) exist and are nonnegative continuous functions defined for x, y 6 R+ and
axy(X, y) < q(x, y)g(a(x, y)), for x, y 6 R+, where q(x, y) is a nonnegative continuous function defined for x, y 6 R+, and suppose further that
x y
b(s, t)g(u(s, t)) dt ds,
u(x,y)<_a(x,y)+ff o o for x, y 6 R+; then for 0 < x _
Xl,
[
0 ~ y < Yl,
u(x, y) < f2-1 f2(a(O, y)) +
J
as(s,O)
g(a(s, 0))
0
x y
+
/f [q(s, t) + b(s, 0 0
t)] dt dsJ
ds
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
569
where f2, ~ - 1 are as defined in Theorem 5.2.1 and X l, Yl are chosen so that x
x y
as(s,O) ~(a(O, y)) + f g(a(s, 0)) d s + f f t q ~ s , t ) + b ~ s , t ~ l a t d s ~ D o m ~ _ l ) 0
0 0
for all x, y lying in the subintervals 0 _< x _< Xl, 0 < y _< yl of R+. (ii) If
ax(X, y), ay(X, y), axy(X, y)
exist and are nonnegative continuous
functions defined for x, y 6 R+ and
axy(X, y) < q(x, y)[a(x, y) + g(a(x, q(x, y)
for x, y ~ R+, where
y))],
is nonnegative continuous function defined for
x, y ~ R+ and suppose further that
xy
(
u(x,y)<_a(x,y)+ffb(s,t)
u(s,t)
s,
0 0
+ffb(sl, o o
) tl)g(u(sl, t l ) ) d t l d S l
dtds,
for x, y ~ R+, then for 0 _< x < x2, 0 < y < y2, x y
u(x, y) < a(x, O) + a(O, y) - a(O, O) +
f f q(s, t)[a(s, t) + g(a(s,
t))] dt as
0 0 xy
+ ff
b(s, t)H -1
0 0
+
asl(Sl'O) a(sl, O) + g(a(sl, 0))
(a(O, y)) +
dsl
0
s, s,, fftq
t l ) + b(sl, tl)]
dtl dsl
1
a t ds,
o o
where H, H - 1 are as defined in Theorem 5.4.1 and x2, Y2 are chosen so that x
H(a(O, y)) +
f
asl (S1, 0) a(sl, O) + g(a(sl, 0))
dsl
0 x y
+
fftq
tl)-!- b(sl, tl)] dtl ds1 c
Dom(H-1),
0 0
for all x, y lying in the subintervals 0 _< x _< x2, 0 _< y < Y2 of R+.
570
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
5.12.4 P a c h p a t t e (unpublished m a n u s c r i p t ) Let u(x, t) and p(x, t) be nonnegative continuous functions defined for x I1 -- [0, 1], t E IT -- [0, T]. Let r h(t), f ( t ) be positive and continuous functions defined for x ~ I1, t ~ IT and define c(x, t) - r + xh(t) + f (t) for x ~ I1, t ~_ IT and let g(u), g'(u) be as in Theorem 5.5.1 and x
s
t
u(x,t)
tl)g(u(sl, tl))dtldSldS,
(i) If r is twice continuously differentiable function such that r > 0, r > 0 for x ~ 11, and f(t), h(t) be continuously differentiable functions such that f ' ( t ) ___0, h'(t) > 0 for t ~ IT, then for 0 < x < Xl,
O
[
( Cx(O, t) ~"2(C(0,t)) -!- X \g(c(O, t))
U(X, t) ~ ~"2-1 x
s
)]J d-
~bt'(Sl)
g(C(S1, 0))
dslds
0 0
t
1
-Jrfffp(sl, tl)dtldsldS,J ooo where ~, f2 -1 are as defined in Theorem 5.5.1 and Xl, tl are chosen so that x
ff2(c(O, t)) + x \g(c(O, t))
+
s
if g(C(S1,0)) r
o o x
s
t
+//fp(sl, ooo
tl)dtldslds~Dom(f2-1),
for all x, t lying in the subintervals 0 < x < xx of I1 and 0 < t < tl of It. (ii) If r f(t), h(t) be continuously differentiable functions such that r >_ O, f ' ( t ) >_ O, h'(t) >_ 0 for x ~ I1, t E IT, then for 0 _< x <_ x2, O_
[
u(x, t) < ~,~-1 ~(c(x, 0)) -+-
-
-~fffp(sl, o o o
J o
[ f ' ( t l ) + xht(tl)] dtl
g(c(O, tl))
]
tl) dSldSdtl ,
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES where g2,
~"2- 1 a r e
as defined in Theorem 5.5.1 and
X2,
571
t2 are chosen so that
t
~(c(x, 0)) -+- /
[ f ' ( t l ) -i- xh'(tl)]
g(c(O, tl ))
0 t
x
dtl
s
+fffp(sl, tl,dsldsdtl E Dom(f2-1), 0
0
0
for all x, t lying in the subintervals 0 _< x _< X2 of 11 and 0 _< t _< t2 of I t .
5.12.5 Pachpatte (unpublished manuscript) Let u(x, t) and p(x, t) be nonnegative continuous functions defined for x 11 -- [0, 1], t ~ Ir -- [0, T]. Let ~b(x), h(t), f ( t ) be positive and continuous functions defined for x ~ 11, t ~ Ir and define c(x, t) = ~(x) + xh(t) + f (t) for x ~ I1, t ~ It. Let g(u), g'(u) be as in Theorem 5.5.1 and
x st( sfs/tj U(x.t)~C(X.t)-J-//f.(SI. tl) U(SI.tl)~p(S3. t3) 0
0
0
0
0
0
x g(u(s3, t3))dr3 ds3 ds2) dtl ds1 ds,
holds for x ~ I1, t E I t . (i) If ~b(x) is a twice continuously differentiable function such that ~b'(x) >_ 0, q~"(x) >_ 0 for x ~ I1 and f (t), h(t) are continuously differentiable functions such that f'(t) >__0, h'(t) >_ 0 for t ~ I t , then for 0 < x <_ x3, 0
xs u(x,t)
s
000
t
0
572
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
in which
(
Q1 (x, t) - H -1 x
+
cx,o,t, )
(c(O, t)) 4- x c(O, t) + g(c(O, t)) $2
C($3, 0) + g ( c ( s 3 , 0 ) )
0 0 x
s
]
t
+fffp(s3,
t3)dt3ds3ds ,
000
H, H -1 are as defined in Theorem 5.5.2 and x3, t3 are chosen so that
H (c(O, t)) 4- x
(
c(O, t) 4- g(c(O, t)) x
)
s2
+
C($3, 0) + g ( c ( s 3 , 0 ) )
0 0 x
s
ds3 ds2
t
+ffJ'p(s3, ooo
t3)dt3ds3ds~Dom(H-1),
for all x, t lying in the subintervals 0 < x < x3 of I1 and 0 < t _< t3 of IT. (ii) If 4~(x), f(t), h(t) are continuously differentiable functions such that tp'(x) >_ O, f'(t) >_O, h'(t) >_0 for x e I1, t e I t , then for 0 _< x _< X4,
O
f tf' (tl) + xh' (tl )] dtl
u(x, t) < c(x, O) +
0 t
x
s
+ f f f p(sl, tl)Q2(Sl, tl)dsl dsdtl, 0
0 0
in which H Q2(X, t) - - H - 1
(C(X, 0)) +
txs 0
0 0
f 0
[f'(t2) +xh'(t2)] C(0, t2) + g(c(0, t2)) dt2
]
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
~'/3
H, H -1 are as defined in Theorem 5.5.2 and X4, t4 are chosen so that t
H(c(x, 0)) +
f
[f'(t2) nt-xh'(t2)]
c(O, t2) + g(c(O, t2))
dt2
0 t
x
s
+ f f f p(s3, t3)ds3dsdt3 6 Dom(H-1), 0
0
0
for all x, t lying in the subintervals 0 < x < x4 of 11 and 0 < t < t4 of
IT.
5.12.6 Rasmussen (1976) Let P0(xo, Y0) and Pl(Xl, Yl) be points in the domain D such that (xl - x 0 ) (Yl -- Y0) > 0 and such that the closed rectangle R1 with opposite vertices P0 and P1 is contained in D. Let ~(x, y), g(x, y) and k(x, y, u) be functions, with the former two continuous on D and with the latter continuous on D x R nondecreasing in u, and satisfying the Lipschitz condition Ik(x, y, u) k(x, y, v)l < L l u - vl. If for all (x, y) in R1, x
y
c p ( x , y ) < _ g ( x , y ) + f f k ( t , s , ep(s,t))dsdt, xo yo then cp(x, y) < U(x, y) on R1, where U(x, y) is a maximal solution of the equation x P
y
P
u(x, y) -- g(x, y) + / / k ( t , ,.J
s, u(s, t))ds dt.
t,r
xo Yo
5.12.7 Pachpatte (unpublished manuscript) Let u(x, y), b(x, y), c(x, y) and p(x, y) be nonnegative continuous functions defined for x, y E R+. Let a(x, y) > 0 for x, y ~ R+ be a continuous and nondecreasing function in x. (i) Let g, g', h, h' be as in Theorem 5.2.2. If u(x,
y) < a(x, y)+ [_ 0
+h
( j
y) u(s, y)+
p(sl, y)u(sl, y)dsl
0
c(s, t)g(u(s, t)) dt ds , 0
)
ds
574
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for x, y 6 R+, then for 0 < x <
Xl,
0 _< y < Yl,
u(x,Y)<~(x,y)[a(x,y)+h(G-1 [G(fo fo c(s,t)g(a(s,t)ap(s,t))dtds) x
])]
y
0 0
where
X ~(x, y) -- 1 + f b(s,
y)exp
[b(sl,y)+ p(sl, y)] dsl
) ds,
0
G, G -1 are as defined in Theorem 5.2.2 and xl, yl are chosen so that
x
y
+ffc(s,t)g(Tt(s,t))dtds~Dom(G -1) oo for all x, y lying in the subintervals 0 < x < xl, 0 < y < Yl of R+. (ii) Let
k(x, y, u) be a nonnegative continuous
X ( j u(x, y) < a(x, y)+ f b(s, y) u(s, y)+
function defined on R 2 •
) p(sl, y)u(sl, y)dsl ds
R, which is nondecreasing in z and is uniformly Lipschitz in z. If
0 x
0
y
+ffk(s,t,u(s,t))dtds, 0 0
for x, y 6 R+, then
u(x, y) < 7t(x, y)[a(x, y)+ r(x, y)], for x, y 6 R+, where ap(x, y) is as defined above and the integral equation
r(x, y) is a solution of
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES x
575
y
r(x,y)-ffk(s,t,O(s,t)[a(s,t)+r(s,t)])dtds, oo existing for x, y 6 R+.
5.12.8 Pachpatte (unpublished manuscript) u(x, y),b(x, y) and c(x, y) be nonnegative continuous functions defined for x, y 6 R+. Let a(x, y) > 1 for x, y 6 R+ be a continuous and
Let
nondecreasing function in x. (i) Let g, g', h, h' be as in Theorem 5.2.2 and let H, H ' be as in Theorem 5.7.8. If
u(x, y) < a(x, y)+
j0 ( j0 b(s, y)
u(s, y)+
b(sl, y)H(u(sl, y ) ) d y
)
ds
for x, y 6 R+, then for 0 < x < xl, 0 < y < Yl,
U(x,Y)<--~I(X,Y)[a(x,Y)nt-h(a-1 [a (fo fo c(s,t)
) xy x g(a(s, t)Ol (s, t)) dt ds
+
o;c,s
where G, G -1 are as defined in Theorem 5.2.2,
[ j0
x
aPl(X, y)
-- 1 + f b(s, y)F -1 F ( 1 ) +
0
]
b(sl, y)dsl ds,
in which F, F - 1 a r e as defined in Theorem 5.7.8 and Xl, Yl are chosen so that x
F(1)
+ f b(sl, y)dsl ~ D o m ( F - 1 ) , 0
576
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
and
G x y
+
//
c(s, t)g(~l (s, t)) dt ds 6 Dom(G -1 ),
0 0
for all x, y lying in the subintervals 0 < x < Xl, 0 _< y _< Yl of R+. (ii) Let
k(x, y, u) be a nonnegative continuous function defined on R 2 x
X(j
R which is nondecreasing in u and is uniformly Lipschitz in z. If
u(x, y) < a(x, y) + / b(s, y)
u(s, y) +
0
b(sl, y)H(u(sl, y)) dsl
ds
0
x y
+ffk(s,t,u(s,t))dtds, 0
0
for x, y 6 R+, then for 0 < x < x2, 0 < y < y2,
u(x, y) < ~1 (x, y)[a(x, y)+ r(x, y)], where lPl (X, y) is as defined above and
r(x, y) is a solution of the integral
equation x
y
k(s, t, 7tl (s, t)[a(s, t) + r(s, t)]) dt ds, 0
0
existing for x, y 6 R+.
5.12.9 Pachpatte (1981b) Let
u(x, y), b(x, y), c(x, y), p(x, y) and q(x, y) be nonnegative continuous
functions defined on a domain D. Let g, g' be as in Theorem 5.2.3 and k > 0 is a constant. Let
Po(xo, yo) and P(x, y) be two points in D such that
( x - x o ) ( y - Yo) > 0 and let R C D be a rectangular region whose oppov(s, t; x, y) be the solution of the
site comers are the points P0 and P. Let
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
577
characteristic initial value problem
L[v]
--
Vst
b(s, t)[c(s, t) + p(s, t)]v -- O,
--
v(s, y) -- v(x, t) -- 1,
and let D + be a connected subdomain of D which contains P and on which v > 0 (Figure 4.1). If R C D + and
u(x, y) < k + b(x, y)
(Ji
+ ff
c(s, t)u(s, t)ds dt
xo Yo
xy
(jj
C(S, t)b(s, t)
o o x
u(x, y) satisfies
p(sl, tl)U(s1, tl)dSl dtl
)) as at
\o o
y
+ffq(s,t)g(u(s,t))dsdt, xo yo
for (x, y) 6 R, then for (x, y) 6 R1 C R,
[
u(x, y) < Q(x, y)S2 -1 ~2(k) +
ii
]
q(s, t)g(Q(s, t))ds dt ,
xo Yo
where
y,
y (x /{C,Sl [C($2, t2) -k- p(S2,
X
,,c,s ,, t2)]V(S2, t2; Sl, tl)ds2dt2
ds1 dtl
yo
for (x, y) 6 R, f2, ~ - I are as defined in Theorem 5.2.1 and xl, yl are chosen so that x
y
f2(k) + j" f q(s, t)g(Q(s, t)) ds dt ~ Dom(f2 -1), xo Yo
for all (x, y) 6 R1 C R.
,
578
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
5.12.10 Bondge and Pachpatte (1979a) Let
u(x) and p(x)
be nonnegative continuous functions defined on f2, where
! ai(xi) > 0, ai(xi) >_0 for 1 < i < n are continuous functions defined for xi > x ~ Let H" R+ ~ R+ be continuously differentiable function with H(u) > 0 for u > 0 and H'(u) > 0 for u > 0.
f2 is defined as in Section 4.9. Let
(i) If x
U(X)< ~-~ai(xi)-+i----1
f p(s)n(u(s))ds,
x0
for x ~ f2, then for x ~ < x < x*,
iO(aixi,+alxo,)i=
u(x) < G -1 x1
al(Sl)
J
+ .o H(y~in=3ai(xi)+a2(xO)+al(Sl))
ds1
X ]
+ f p(s) ds , xo
where
G(r)- f
ds H(s)'
r > 0, r0 > 0,
ro
and G -1 is the inverse of G, and x* be chosen so that
) xj
G
ai(xi) + a l ( x 0) i=2
+
,
al(s1) x0
n(~-~in=3ai(xi) + a2(x 0) -{- al (s1))
x
+ f p(s) ds ~ D o m ( G -1), xo
for all x lying in the parallelepiped x ~ < x < x* in f2.
dsl
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
579
(ii) If
U(X)
naii + _r xp(s)( ju(s)+
i= 1
xO
n
/
x
ai(xi) +
i=1
p(y)H(u(y))dy ds, xo
for x 6 f2 then for x ~ _< x _< x**
U(X) < E
)
<_E
p ( s ) W -1
[
)
W
ai(si) + al (x O)
xO
i=1
s1
I
al(Yl)
( ~2i_3ai(si)+a2 n (xO)+al(Yl))
+ gO (~'i_3ai(si)+a2(x~
X
]
+ f p(y) dy ds, xo
where
r
W (r) --
t + H (t)'
r > 0, ro > 0,
ro
and W -1 is the inverse function of W, and x** is chosen so that
Xl
I
al(s1)
-Jr" _ (~-~n=3ai(xi) + a2(xO) + al(s1)) _.l_H~.,~--~i=3ai(xi) _l_ a2(x2) 0 .+.
dsl
x
+ / p(s) ds ~ D o m ( W -1), xo
for all x ~ f2 lying in the parallelepiped x ~ < x < x** in S2.
5.12.11 Singare and Pachpatte (1981) Let ~(x), a(x), on f2, and let
b(x) and c(x) be nonnegative continuous functions u(x) be a positive continuous function defined on f2.
defined
580
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
(i) Let H(r) be a positive, continuous, strictly increasing, convex and submultiplicative function for r > 0, l i m r ~ H ( r ) - c~. Let ct(x),/~(x) be positive continuous functions defined on ~2 with or(x)+/~(x) = 1. If
u(s) > dp(x) - a(s)H -1 I ~ h(tr)H (q~(tr)) dcr
+ / S b(~r)
c(~)H(q~(~)) d~
)1
dcr ,
x
is satisfied for x < s; x, s 6 f2, then
u(s) > ct(s)H -1 [~
{ 1 + f(s)H(a(s)f1-1 (s)) f b(tr) x
l(s))b(~) q- c(~)] d~
x exp
for x < s; x, s 6 ~2. (ii) Let G(r) be
)}'1 do
,
a positive, continuous, strictly increasing, subadditive and submultiplicative function for r > 0 with limr~o~ G(r)= cxz, and let G -1 denote the inverse function of G. If i"e
U(S) > dp(X)-a(s)G -1 I ~fb(cr)G(~(cr)) dcr
Vx
for x < s; x, s 6 fl, then
u(s) > G -1
[6(r {
x exp
for x <_s; x, s 6 S2.
s
1 + 6(a(s)) f
(j
b(o-)
x
[b(~)G(a(s)) + c(~)] d~
)}l1
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
581
5.12.12 Pachpatte (unpublished manuscript) Let u(x), b(x) and c(x) be nonnegative continuous functions defined on f2. Let a(x) > 1 for x 6 f2 be a continuous and nondecreasing function in x. Let H(u) and H'(u) be as in Theorem 5.7.8. If
j ( j
u(x) < a(x) +
b(y)H
u(y) -t-
xo
c(s)H(u(s))ds
)
dy,
x0
for x ~ f2, then for x ~ f21 C f2,
u(x) < a(x)Q(x), where
Y
])
(1) + I[b(s) + c(s)] ds
Cds
H(s)
(**)
x0
x0
G(r)-
dy,
,
r > O, ro > O,
ro
G -1 is the inverse function of G and X
G(1)+f
[b(s) + c(s)] ds 6 Dom(G -1),
xo
for all x ~ f21 C f2.
5.12.13 Pachpatte (unpublished manuscript) Let u(x), b(x), c(x) and q(x) be nonnegative continuous functions defined on f2 and k > 1 is a constant. Let H(u), H'(u) be as in Theorem 5.7.8. Let g(u) be a continuously differentiable function defined for u > O, g(u)> 0 for u > 0 and g'(u) > 0 for u > 0 and g(u) is submultiplicative. If
X
u(x) <_k + ;_ b(y)H
( u(y) + / c(s)H(u(s)) as) dy
t.]
xo X i-J
+ / q(y)g(u(y)) dy, xo
x0
582
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for x ~ ~2, then for x ~ ~"~1 C ~~,
u(x) < Q(x)E -1
(k) +
q(y)g(Q(y))dy , xo
where
Q(x)
is defined in (**) with G, G -1 as involved therein, r
E(r) --
g(s) '
r > O, ro > O,
ro
E -1 is the inverse function of E, and x
G(1) +
f tb,s) + c s)]d
~
Dom(G-1),
xo x
E(k) + f q(y)g(Q(y)) dy ~ D o m ( E -1), xo
for x E ~21 C ~ .
5.12.14 Pachpatte (unpublished manuscript) Let u(x),
b(x) and c(x) be nonnegative continuous functions defined for a(x)> 1 for x ~ ~2 be a continuous and nondecreasing funcLet H (u) and H ' (u) be as in Theorem 5.7.8 and k(x, y, u), W(x, u)
x ~ ~2. Let tion in x.
x(j)b y
be as in Theorem 5.7.6. If
U(X)
< a ( x ) + [_ J xo
+W
+
c(s)H(u(s)) ds dy
xo
x, j k(x, y, u(y)) dy) , xo
for x ~ g2, then for x ~ ~"21 C ~"2,
u(x) < Q(x)[a(x) + W(x, r(x))],
where
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
583
Q(x)
is a
is defined as in (**) with G, G -1 involved therein and
r(x)
solution of the integral equation x
r(x) - / k ( x , y, Q(x)[a(y) + W(y, r(y))])dy, xo
existing on f2.
5.12.15 Pachpatte (unpublished manuscript) Let
u(x)
a(x) H(u)
and b(x) be nonnegative continuous functions defined on f2. Let _> 1 for x ~ f2 be a continuous and nondecreasing function in x. Let and
H'(u)
X[
be as in Theorem 5.7.8. If dy,
u(x) < a(x) + f b(y) u(y) + xo
for x ~ f2, then for x ~ f21 C f2,
u(x) < a(x)Qo(x), where
x [
Y
E
Qo(x)- 1 + [_ b(y) 1 + / b ( s ) G -1 G(1
dy, ( * * * )
tl
G(r) --
xo
xo
xo
s -( ~ (s~
r > O, ro > O,
G -1 is the inverse of G and x
G ( 1 ) + / b(~r) dcr ~ Dom(G -1), x0
for all x s f21 C f2.
5.12.16 Pachpatte (unpublished manuscript) Let u(x), b(x) and q(x) be nonnegative continuous functions defined on f2 and k _ 1 be a constant. Let H(u) and H'(u) be as in Theorem 5.7.8. Let
584
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
X[
(r
g(u) and g'(u) be as in Theorem 5.2.3. If Y
u(x) < k + [_ b(y) u(y) + f b(s)
]
b(o")H(u(o"))do" ds dy
Or Xo
x0
xo
X
+ / q(y)g(u(y)) dy, xo
for x ~ f2, then for x ~ f2~ C f2,
xo
where
Qo(x) is defined as in ( , , ,) with G, G -1 as involved therein, r
E(r)-
g-(s)' r > O, ro > O, ro
E -1
is the inverse function of E, and x
G(1) + f b(o.) do" ~ Dom(G -1), xo x
E(k) + / q(y)g(Qo(y)) dy ~ Dom(E -1), xo
for all x ~ ~1 C ~2.
5.12.17 Pachpatte (unpublished manuscript) Let Let Let
u(x), and b(x) be nonnegative continuous functions defined for x ~ ~. a(x) > 1 for x ~ f2 be a continuous and nondecreasing function in x. H(u) and H'(u) be as in Theorem 5.7.8 and k(x, y, u) and W(x, u) be
as in Theorem 5.7.6. If
dy
u(x) < a(x) + [_ b(y) u(y) J x0
+W
x0
,
k(x, y,u(y))dy xo
x0
,
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
585
for x ~ f2, then for x ~ ~"~1 c ~'-~,
u(x) < Qo(x)[a(x) + W(x, r(x))], where Qo(x) is defined as in ( , ,
, ) with G, G -1 as involved therein, and
r(x) is a solution of the integral equation x
r(x) - f k(x, y, Qo(y)[a(y) + W(y, r(y))l) dy, xo
existing on ~.
5.12.18 Singare and Pachpatte (1982) Let u(x), a(x), b(x), f ( x ) and g(x) be nonnegative continuous functions defined on ~2, G(u) be a continuous
strictly increasing convex and
G(u) = c~, for all x 6 f2; let
submultiplicative function for u > 0, l i m u ~
c~(x), /3(x) be positive continuous functions defined on a domain ~ , and c~(x) +/3(x) = 1. Let v(s; x) and e(s; x) be the solutions of the characteristic initial value problems
(-
1)nvsl...s n
(S; X)
--
[ f (s) + g(s)~(s)G(b(s)~ -1 (s))]v(s; x) = 0 in g2,
v(s; x ) = l on s i - - x i ,
1 < i < n,
and
(-- 1)n esl...~. (s; x) + f (s)e(s; x) = 0 in f2, e(s; x ) -
1 on
Si-
1 < i < n,
Xi,
respectively and let D + be a connected subdomain of fl containing x such that v > 0, e > 0 for all s ~ D +. I f D C D + and
u(x) < a(x) + b(x)G -1
(j (J f (s)
x0
))
g(cr)G(u(cr)) dcr
xo
ds
,
586
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for all x e D, then
u(x) < a(x) + b(x)G-a [ f e(s; x) ( f (s) ( f ot(o)G(a(tr)ot-l(o))g(o) xo
xo
as)] for all x e D.
5.12.19 Shinde and Pachpatte (1983) Let
u(x), v(x), a(x), b(x), p(x), hi(x)
(i = 1, 2, 3, 4) be nonnegative
continuous functions defined on S2, and h(x)=max{[hl(x)+h3(x)], [h2 (x) -t- ha (x)] }. Let G(r) be a continuous, strictly increasing convex and submultiplicative function for r > 0, limr--,~ G(r) - cxz for all x in f2, G -1 be the inverse function of G; or(x), /3(x) are positive continuous functions defined on f2 and or(x)+/3(x) = 1. Let w(s; x) be the solution of the characteristic initial value problem (--1)nwsl...Sn
w(s; x)
(S; X) --
-- 1 on
~(s)G(p(s)~-l(s))h(s)w(s; x) - 0 in
Si - - X i ,
g2,
1 < i < n.
Let D + be a connected subdomain of f2 containing x such that w > 0 for allseD
+.IfDCD
+and r
U(X)
< a(x) at- p(x)G -1 [ f hi (s)G(u(s)) ds l xo
xO
]
Isil hz(s)G(v(s))ds ,
+aexpf i=1
v(x) < b(x) + p(x)G -1
[r
exp
xO
+fh4(s)a(v(s))ds] xo
Isil h3(s)G(u(s))ds
-# i=1
,
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
587
for x ~ D, w h e r e / z is a nonnegative constant, then
E
u(x) < G -1 exp /x
]
Ixil Q(x) , i=1
v(x) <_ G-I[Q(x)],
for x ~ D, where
Q(x)-f(x)+fl(x)G(p(x)fl-l(x))(fw(s;x)h(s)f(s)ds), xo
in which f (x) -- exp
-I~ Z
Ixi[
~(x)G(a(x)~ -~ (x))
+ ol(x)a(b(x)o1-1 (x)),
i=1
for x ~ D.
5.12.20 Pachpatte (1993, 1996d) Let
u(x)> O, a(x)>_ O, b(x)>_ O, pi(x)> 0 for i - 1 ,
2.....
continuous functions defined for x ~ R~_ and q" R~_ x R+ ~
n-
1, be
R+ be a contin-
uous function which satisfies the condition
0 < q(x, V1)
--
q(x, V2)
~
k(x,
V2)(V 1 -- V2),
for x 6 R~_ and V1 >__ V2 >__ 0, where k" R~_ x R+ --+ R+ is a continuous function. Let the notation M[x, p, q] be as defined in Section 5.9. (i) If
u(x) < a(x) + b(x)m[x, p, q(y, u(y))], for all x 6 R~_, then
u(x) < a(x) + b(x)M[x, p, q(y, a(y))] exp(M[x, p, k(y, a(y))b(y)]), for all x ~ R~_.
F(u) be a continuous, strictly increasing, convex, submultiplicaF(u) = c~, F -1 denotes the inverse function of F and or(x), fl(x) be continuous and positive functions defined for x ~ R~_ and c t ( x ) + fl(x) = 1. If (ii) Let
tive function for u > O, l i m u _ ~
u(x) <_a(x) + b(x)F -l(M[x, p, q(y, F(u(y)))]),
588
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
for x ~ R~_, then u(x) < a(x) + b(x)F -1 (M[x, p, q(y, a(y)F(a(y)ot -1 (y)))] x exp(M[x, p, k(y, ot(y)F(a(y)ot -1 (y)))fl(y)F(b(y)f1-1 (y))])), for x 6 R~_. (iii) Let g(u), g'(u), f2, ~-1 be as in Theorem 5.2.1. If u(x) < a(x) + b(x)M[x, p, q(y, g(u(y))], for x 6 R~_, then for 0 _< x < x*, u(x) <_ a(x) + b(x)f2-~[~(c(x)) + M[x, p, k(y, g(a(y)))g(b(y))]], where c(x) -- M[x, p, q(y, g(a(y)))], and x* ~ R~_ is chosen so that f2(c(x)) + M[x, p, k(y, g(a(y)))g(b(y))] ~ Dom(f2 -1), for all x ~ R~_ such that 0 < x < x*.
5.13 Notes The results given in Section 5.2 deal with some basic nonlinear generalizations of Wendroff's inequality given by Bondge and Pachpatte (1979b, 1980a). Theorems 5.2.1 and 5.2.2 are taken from Bondge and Pachpatte (1979b) and (1980a) respectively. Theorems 5.2.3 and 5.2.4 are new and motivated by certain applications. The inequalities dealt with in Section 5.3 are due to Pachpatte (1995e) and Bondge and Pachpatte (1979b,c). Theorem 5.3.1 was recently established by Pachpatte (1995e). Theorems 5.3.2 and 5.3.3 are given by Bondge and Pachpatte (1979b) and Theorems 5.3.4 and 5.3.5 are given by Bondge and Pachpatte (1979c). Section 5.4 contains some useful generalizations of the inequalities given by Pachpatte (1974d, 1975h). Theorem 5.4.1 is due to Bondge and Pachpatte (1979b). Theorems 5.4.2, 5.4.3 and 5.4.4 are due to Bondge and Pachpatte (1980a).
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
589
Sections 5.5 and 5.6 are devoted to the inequalities recently investigated by Pachpatte (1980c,d, 1988c, 1993, 1996d). Theorems 5.5.1 and 5.5.2 are taken from Pachpatte (1988c). Theorem 5.5.3 is new and motivated by certain applications. Theorem 5.5.4 is taken from Pachpatte (1993, 1996d). Theorems 5.6.1 to 5.6.4 are taken from Pachpatte (1980c) and Theorems 5.6.5 and 5.6.6 are taken from Pachpatte (1980d). The results given in Section 5.7 deal with the integral inequalities in many independent variables established by various investigators (Beesack, 1975; Headley, 1974; Pachpatte, 1981c; Pelczar, 1963). Theorems 5.7.1 and 5.7.2 are due to Pelczar (1963), Theorem 5.7.3 is taken from Beesack (1975) and Theorem 5.7.4 is taken from Headley (1974). Theorems 5.7.5-5.7.9 are due to Pachpatte (1981 c). The inequalities given in Sections 5.8-5.10 were recently discovered by Pachpatte (1991, 1992, 1994a,b,d, 1996c, 1997, in press d, unpublished manuscript). Theorems 5.8.1 and 5.8.2 are taken from Pachpatte (in press d). Theorems 5.8.3, 5.8.4, 5.8.5 are taken from Pachpatte (1997), (unpublished manuscript) and (1996c), respectively. Theorems 5.9.1 to 5.9.4 are taken from Pachpatte (1994a), while Theorem 5.9.5 is taken from Pachpatte (1994b). The results in Theorems 5.10.1 and 5.10.2 are taken from Pachpatte (1994d) and Theorem 5.10.3 is new. Theorems 5.10.4 and 5.10.5 are taken from Pachpatte (1994d). Theorems 5.10.6 and 5.10.7 are taken from Pachpatte (1991). Theorems 5.10.8 and 5.10.9 are taken from Pachpatte (1992). Section 5.11 is devoted to the applications of some of the inequalities given in earlier sections to the various properties of the solutions of certain partial differential, integral and integro-differential equations. Section 5.12 contains some miscellaneous inequalities which can also be used in some new applications.
459
460
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
5.2 Generalizations of Wendroff's Inequality The fundamental role played by Wendroff' s inequality and its generalizations and variants in the development of the theory of partial differential and integral equations is well known. In this section we present some basic nonlinear generalizations of Wendroff's inequality established by Bondge and Pachpatte (1979b, 1980a) and some new variants, which can be used as tools in the study of certain partial differential and integral equations. Bondge and Pachpatte (1979b) proved the following useful nonlinear generalization of Wendroff's inequality. Theorem 5.2.1 Let u(x, y) and p(x, y) be nonnegative continuous functions defined for x, y ~ R+. Let g(u) be a continuously differentiable function defined for u > O, g(u) > 0 for u > 0 and g' (u) > 0 for u > O. If x
u(x, y) < a(x) + b(y) +
y
ff 0
(5.2.1)
t)g(u(s, t)) ds dt,
0
for x, y ~ R+, where a(x) > O, b(y) > O, a'(x) > O, b'(y) > 0 are continuous functions defined for x, y ~ R+, then for 0 < x < Xl, 0 < y < Yl, u(x, y)
<
-
f2(a(0) + b(y)) +
~"2- 1
x
y
+ffp(s,t)dsdt 0
0
]
g(a(s) + b(0))
ds
0
(5.2.2)
,
where
ds
f2(r) --
g(s)
r > 0, ro > 0,
(5.2.3)
ro
g2-1 is the inverse function of f2 and Xl, yl are chosen so that x
f2(a(0) + b(y)) +
x
y
i g(a(s)a'
0
for all x, y lying in the subintervals 0 < x <
0 Xl,
0 ~ y < Yl of R+. Vl
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
461
Proof: We note that since g'(u) > 0 on R+, the function g(u) is monotonically increasing on (0, 0r Define a function z(x, y) by the fight-hand side of (5.2.1), then z(x, O) = a(x) 4- b(O), z(O, y) -- a(O) 4- b(y), and Zxy(X,
y ) - p(x, y)g(u(x, y)).
(5.2.4)
Using u(x, y) < z(x, y) in (5.2.4) and the fact that z(x, y) > 0, we observe that
Y) < p(x, y). g(z(x, y)) Zxy( x,
(5.2.5)
From (5.2.5) and by using the facts that Zx(X, y ) > 0, z(x, y) > O, g'(z(x, y)) > 0 for x, y 6 R+, we observe that Zxy( X, Y)
< p(x, y) + g(z(x, y)) -
Zy(X, y ) >
0,
Zx(x, y)g' (z(x, y))Zy (x, y) [g(z(x, y))]2
i.e.
( Zx(X, y) ) < p(x, y). Oy k,g(-~, y))_ -
(5.2.6)
Keeping x fixed in (5.2.6), we set y - t; then, integrating with respect to t from 0 to y and using the fact that z(x, O) - a(x) + b(O), we have y
Zx(X, y) < -t- f p(x, t) dt. a'(x) g(z(x, y)) - g(a(x) 4- b(O))
(5.2.7)
0
From (5.2.3) and (5.2.7) we observe that
a Ox
m f2 (z(x, y ) )
--
Zx(X, y) g(z(x, y)) y
a'(x) / < 4- p(x, t) dt. - g(a(x) 4- b(0))
(5.2.8)
0
Keeping y fixed in (5.2.8), set x = s; then, integrating with respect to s from 0 to x and using the fact that z(O, y) - a(O) + b(y) we have x
f2(z(x, y)) < ~(a(0) 4- b(y)) 4- f g(a(s)a'(s) 4- b(0)) ds 0 x y
+ffp(s,t)dsdr 0 0
(5.2.9)
462
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Now substituting the bound on z(x, y) from (5.2.9) in u(x, y) < z(x, y), we obtain the desired bound in (5.2.2). The subintervals for x and y are obvious. II R e m a r k 5.2.1 From the proof of Theorem 5.2.1, it is easy to observe that, in addition to (5.2.2), we can conclude that
b'(t) u(x, y) < f2 -1 I f2(a(x) + b(0)) + / g(a(O) + b(t)) dt
xy
]
0
0 0
where the expression in the square bracket on the fight-hand side of (5.2.10) belongs to the domain of ~2-1 . We also note that the above conclusion applies to the following Theorems 5.2.3 and 5.2.4 and also some of the results given in our subsequent discussion. A Bondge and Pachpatte (1980a) gave the following generalization of Wendroff' s inequality. Theorem 5.2.2 Let u(x, y), a(x, y), b(x, y) and c(x, y) be nonnegative
continuous functions defined for x, y ~_R+. Let g(u), h(u) be continuously differentiable functions defined for u > O, g(u) > O, h(u) > 0 for u > 0 and g'(u) > 0, h'(u) > Ofor u > O, and let g(u) be subadditive and submultiplicative for u > O. If u(x,y)
(5.2.11)
0
for x, y ~ R+, then for 0 < x < x2, 0 < y <_ Y2, r"
u(x, y) < a(x, y) + b(x, y)h ( G -1 IG(A(x, y))
rrxY + - - - - c(s, 0 0
\
L
t))ds dt
])
,
(5.2.12)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
463
where x
y
ff fds
A(x, y ) -
0
t))dsdt,
(5.2.13)
0
F
G(r)-
g(h(s))
ro
,
r>0,
r0>0,
(5.2.14)
G -1 is the inverse function of G and x2, y2 are chosen so that x
y
G(A(x, y)) + f f c(s, t)g(b(s, t)) ds dt ~ Dom(G -1), 0
0
for all x, y lying in the subintervals 0 <_x <_x2, 0 <_ y <_ y2 of R+. P r o o f : From the hypotheses on g and h, we note that the functions g and h are monotonically increasing on (0, cx~). Define a function z(x, y) by x
Z(X, y) =
y
f f c(s 0
ds dt.
(5.2.15)
0
From (5.2.15) and using the fact that u(x, y) < a(x, y) + b(x, y)h(z(x, y)) from (5.2.11) and the hypotheses on g we have x
z(x, y) < A(x, y) +
y
f f c(s. 0
t))g(h(z(s, t)))as dt,
(5.2.16)
0
for x, y 6 R+, where A(x, y) is defined by (5.2.13). Now fix c~,/~ 6 R+ such that 0 < x _< oe, 0 < y
y
z(x, y) < A(c~, ~) + f f c(s, t)g,b(s, t))g(h(z(s, t)))ds dt, 0
(5.2.17)
0
for 0 < x < c~, 0 < y
y) -- c(x, y)g(b(x, y))g(h(z(x, y))) < c(x, y)g(b(x, y))g(h(v(x, y))).
(5.2.18)
464
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Now first assume that A(ct,/~) > 0; then from (5.2.18) we observe that 1;xy(X, Y)
< c(x, y)g(b(x, y)). g(h(v(x, y))) -
(5.2.19)
As in the proof of Theorem 5.2.1, from (5.2.19) we observe that
O < -c-(-x(g(h(v(x, 'Vy )xg( X( b' Y( x)y))) ')y ) ) o y-
(5.2.20)
By keeping x fixed in (5.2.20), setting y = t, and then integrating with respect to t from 0 to ~ we have
vx,x,o,
#
Vx(X, ~) < g(h(v(x, ~))) - g(h(v(x, 0)))
+f
c(x, t)g(b(x, t)) dt.
(5.2.21)
o
From (5.2.14) and (5.2.21) we observe that /,b
0
G(v(x, [3))< -ff-G(v(x,0)) + I c(x,t)g(b(x,t))dt. Ox Ox J
(5.2.22)
o
Now keeping y fixed in (5.2.22), setting x = s, and then integrating with respect to s from 0 to c~ we get
(5.2.23)
G(v(ct, f l ) ) < _ G ( A ( ~ , # ) ) + f f c ( s , t ) g ( b ( s , t ) ) d s d t , 0 o
Since z(ct, fl) < v(c~, fl) and ct, fl 6 R+ are arbitrary from (5.2.23) we have
[
z(x, y) < G -1 G(A(x, y)) +
:: o o
c(s, t)g(b(s, t))dsdt
]
,
(5.2.24)
for 0 _< x < X1, 0 ~ y _< Yl. The desired bound in (5.2.12) follows by using (5.2.24) in u(x, y) <_a(x, y) + b(x, y)h(z(x, y)). If A(ct,/~) in (5.2.17) is nonnegative, we carry out the above procedure with A(c~,/~) + E instead of A(ct,/~), where E > 0 is an arbitrary small constant, and subsequently pass to the limit as E - , 0 to obtain (5.2.12).
I
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES 465 The following theorem provides another useful generalization of Wendroff' s inequality. Theorem 5.2.3 Let u(x, y), c(x, y) and p(x, y) be nonnegative continuous functions defined for x, y ~ R+. Let g(u), g' (u), a(x), a'(x), b(y) and b'(y) be as in Theorem 5.2.1 and g(u) be submultiplicative on R+. If x
y
u(x,y)<_a(x)+b(y)+ffc(s,t)u(s,t)dsdt 0 x
0
y
+ffp(s,t)g(u(s,t))dsdt, 0
(5.2.25)
0
for x, y ~ R+, then for 0 < x < x3, 0 < y < Y3, u(x, y) < q(x, y) [~2- 1
j
f2(a(O) + b(y)) +
xy
+
f f p(s t)g(q(s, t))ds dt
]]
a'(s) g(a(s) + b(O)) ds
0
,
(5.2.26)
q'xy) exp(joc'st)
(5.2.27)
0
0
where
ds dt) ,
and f2, g2-1 are as defined in Theorem 5.2.1 and x3, Y3 are chosen so that x
a'(s) g(a(s) + b(0)) ds
f2(a(0) + b(y)) + 0 x
+
y
ff 0
t)g(q(s, t))ds dt ~ Dom(f2 -1),
0
for all x, y lying in the subintervals 0 < x < x3, 0 <_ y < y3 of R+.
D
Proof: We note that, since a'(x) > O, b'(y) > O, g'(u) > 0 for x, y, u R+, the functions a(x), b(y), g(u) are monotonically increasing on (0, c~).
466
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
Define a function m(x, y) by x
m(x, y) -- a(x) + b(y) +
y
f f p(s, t)g(u(s, t)) ds dt,
(5.2.28)
0 0
then m(x, O) - a(x) + b(O), m(O, y) - a(O) + b(y) and (5.2.25) can be restated as x
y
u(x,y)<_m(x,y)+ffc(s,t)u(s,t)dsdt.
(5.2.29)
0 0
Since m(x, y) is positive and nondecreasing in each variable x, y ~ R+, by applying Theorem 4.2.2 to (5.2.29) we have
(5.2.30)
u(x, y) < m(x, y)q(x, y),
where q(x, y) is defined by (5.2.27). From (5.2.28) and (5.2.30) we have
mxy(X, y ) -
p(x, y)g(u(x, y))
< p(x, y)g(m(x, y)q(x, y)) < p(x, y)g(m(x, y))g(q(x, y)), i.e.
mxy(X,
y)
< p(x, y)g(q(x, y)). g(m(x, y)) -
(5.2.31)
Now by following the same arguments as in the proof of Theorem 5.2.1 below the inequality (5.2.5) we get
m(x, y) < S2-~
~(a(O) + b(y)) +
g(a(s) + b(O)) 0
xy +ff.(s,
t)g(q(s, t ) ) d s dt
1
.
(5.2.32)
0 0
Using (5.2.32) in (5.2.30) we get the required inequality in (5.2.26). II A slight variant of Theorem 5.2.3 is given in the following theorem.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
467
Theorem 5.2.4 Let u(x, y), c(x, y), p(x, y), g(u), g'(u), a(x), a'(x), b(y) and b'(y) be as in Theorem 5.2.3. If u(x, y) < a(x) + b(y) +
j
iJ
c(s, y)u(s, y)ds +
0
p(s, t)g(u(s, t)) ds dt,
0 0
(5.2.33) for x, y ~ R+, then for 0 < x < x4, 0 < y < Y4,
[[
u(x, y) < f (x, y) S2-1
j
f2(a(O) + b(y)) +
g(a(s) + b(O))
as
0
xy p(s, t)g(F(s, t))dsdt
+ fl
1] ,
(5.2.34)
0 0
where
y, exp( c s y,ds) for x, y ~ R+, f2, so that
~'2- 1 , a r e
as defined in Theorem 5.2.1 and
(5.2.35)
X4,
24 are chosen
a'(s) ds g(a(s) + b(0))
f2(a(0) + b(y)) + 0 x y
+fJ'p(s,t)g(F(s,t))dxdt6Dom(f2-1), 0 0
for all x, y lying in the subintervals 0 < x <
X4,
0 < y < Y4 of R+. D
The proof of this theorem can be completed by following the proof of Theorem 5.2.3, with suitable changes. The details are omitted here. R e m a r k 5.2.2 We note that, if the inequality (5.2.33) in Theorem 5.2.4 is replaced by y
x
y
u(x,y)<_a(x)+b(y)+lc(x,t)u(x,t)dt+ffp(s,t)g(u(s,t))dsdt, 0
0 0
468
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
then the bound obtained in (5.2.34) is replaced by
[[
u(x, y) < Fo(x, y) ~2-1
f2(a(0) + b(y)) +
]]
xy
ja s,
g(a(s) + b(0)) ds
0
0 0
where
o'x y' exp(/oc'x"dt) for x, y 6 R+, and the expression in the inner square bracket on the righthand side belongs to the domain of f2 -1. A
5.3 Wendroff-type Inequalities During the past twenty years or so many authors have developed extensions and variants of Wendroff's inequality and exhibited applications in partial differential and integral equations. In this section we shall deal with the Wendroff-type inequalities investigated by Pachpatte (1995e) and Bondge and Pachpatte (1979b,c) which can be used in the study of certain partial differential and integral equations. Pachpatte (1995e) established the Wendroff-type inequalities in the following theorem. Theorem 5.3.1 Let u(x, y), a(x, y) and b(x, y) be nonnegative continuous functions defined on R2+ and L" R3+ -+ R+ be a continuous function which satisfies the condition 0 < L(x, y, v ) - L(x, y, w) < M(x, y, w ) ( v - w),
(L)
for x, y ~ R+ and v > w > O, where M" R 3 --+ R+ is a continuous function.
(i) if x
y
u~ y, <_a~x,y, + b~x y, / f L~s , u~s ,,, ~ d, 0 0
(5.3.1)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES 469
for x, y ~ R+, then u(x, y) < a(x, y) + b(x, y)A(x, y)exp ( / / M ( s ,
t, a(s, t))b(s, t)dsdt) , (5.3.2)
for x, y ~ R+, where x
y
A(x,y)=ffL(s,t,a(s,t))dsdt, 0
(5.3.3)
0
for x, y ~ R+. (ii) Let F(u) be a continuous, strictly increasing, convex and submultiplicative function for u > 0, limu_+~ F(u) -- oe, F -1 denotes the inverse function of F, u(x, y), ~(x, y) be continuous and positive functions defined on R 2 and or(x, y) + ~(x, y) = 1. If u(x, y) < a(x, y) -t- b(x, y)F-l ( / /
L(s, t, F(u(s, t))) ds dt) ,
(5.3.4)
for x, y ~ R+, then u(x, y) < a(x, y) + b(x, y)F -1
L(s, t, ~(s, t)
x F(a(s, t)ot-1 (s, t))) ds dt) /
•
exp ( / / M ( s ,
t, or(s, t)F(a(s, t)c~-1 (s, t)))
xfl(s,t)F(b(s,t)fl-l(s,t))dsdt)], for x, y ~ R+.
(5.3.5)
470
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
(iii) Let g(u), g'(u), h(u), h'(u) be as in Theorem 5.2.2. If (5.3.6)
u(x, y) < a(x, y) + b(x, y)h (fo fo L(s, t, g(u(s, t)))dsdt ) , for x, y E R+, then for O < x < xl, O < y < yl, /
u(x, y) < a(x, y) + b(x, y)h
(G -1 \
[G(B(x, I"
Y))
I..
xy
])
+//M(s,t,g(a(s,t)))g(b(s,t))dsdt
,
(5.3.7)
0 0
where
x y
B(x,y)-/fL(s,t,g(a(s,t)))dsdt,
(5.3.8)
0 0
G, G -1 be as defined in Theorem 5.2.2 and xl, yl be chosen so that x y
G(B(x, y)) +
ff
M(s, t, g(a(s, t)))g(b(s, t)) ds dt ~ Dom(G -1),
0 0
for all x, y lying in the subintervals 0 < x < Xl, 0 < y < Yl of R+. IS]
Proof:
(i) Define a function z(x, y) by x
y
z(x,y)-f/L(s,t,u(s,t))dsdt. 0
(5.3.9)
0
From (5.3.9) and using the fact that u(x, y) <_ a(x, y) + b(x, y)z(x, y) and the conditions on the function L we observe that x
y
z(x,y)
= A(x, y ) +
,, a(s, t) + b(s, ,)z(s, t)) 0 0
- L(s, t, a(s,
t))] ds dt
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES x
471
y
< A(x, Y) + / / M ( s ,
t, a(s, t))b(s, t)z(s, t)dsdt, (5.3.10)
t i l l
0
0
where A(x, y) is defined by (5.3.3). Clearly A(x, y) is nonnegative and nondecreasing in each variable x, y e R+. Now an application of Theorem 4.2.2 yields
z(x, y)
(5.3.11)
< a(x, y) + b(x, y)z(x, y) we get the required
u(x, y) < or(x, y)a(x, y)ot-1 (x, y)+ ~(x, y)b(x, y)fl-1 (x, y) x F -1
(/0/
)
L(s, t, F(u(s, t)))ds dt .
0
(5.3.12)
Since F is convex, submultiplicative and monotonic, from (5.3.12) we see that
F(u(x, y)) < or(x, y)F(a(x, y)ot-1 (x, y)) + 13(x, y)F(b(x, y)fl-l(x, y)) x
y
xJ'fL(x,t,F(u(s,t)))dsdt. 0
(5.3.13)
0
The estimate given in (5.3.5) follows by first applying the inequality established in (i) to (5.3.13) and then applying F -1 to both sides of the resulting inequality. (iii) Define a function z(x, y) by x P
y
P
Z(X, y)= I l L ( s , t, g(u(s, t))) ds d,. t.i
qt/
0
0
(5.3.14)
From (5.3.14) and using the fact that u(x, y) < a(x, y)+ b(x, y)h(z(x, y)) and, as noted in the proof of Theorem 5.2.2, that the functions g and h are monotonically increasing on (0, c~), and using the conditions on the function
472
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
L we observe that x
y
z(x, y) < / f
L(s, t, g(a(s, t)+ b(s, t)h(z(s, t))))ds dt
0 0 x y
ff
t, g(a(s, t))+ g(b(s, t))g(h(z(s, t))))ds dt
0 0 x y
t, g(a(s, t)) + g(b(s, t))g(h(z(s, t))))
= B(x, y) + 0 0
- L(s, t, g(a(s, t)))] ds dt x y
< B(x, y) +
ff
g(a(s, t)))g(b(s, t))
0 0
x g(h(z(s, t))) ds dt,
(5.3.15)
where B(x, y) is defined by (5.3.8). The rest of the proof follows by the same arguments as in the proof of Theorem 5.2.2 below the inequality (5.2.16), with suitable changes. Further details are omitted here. II The next two theorems proved by Bondge and Pachpatte (1979b) can be used more effectively in certain situations. Theorem 5.3.2 Let u(x, y), Ux(X, y), uy(x, y)and U~y(X, y)be nonnegative
continuous functions defined for x, y ~ R+, u(x, 0) = u(O, y) = O, and p(x, y) > 1 be a continuous function defined for x, y ~. R+. Let g(u), g'(u), a(x), a'(x), b(y), b'(y) be as in Theorem 5.2.1. If Uxy(X, y) < a(x) + b(y) + M [u(x, y) W f /o p(s, t)g(Ust(S, t)) ds dt] (5.3.16)
for x, y ~ R+, where M > 0 is a constant, then for 0 < x < x2, 0 < y < Y2, Uxy(X, y) < H -1
(a(0) + b(y)) +
a(s) + b(O) + g(a(s) + b(0)) 0
xy
0
0
]
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
where
473
r
H(r) --
f
r > 0, r0 > 0,
s + g(s)
(5.3.18)
ro
H -1 is the inverse function of H, and x2, y2 are chosen so that x
H(a(O) + b(y)) +
f
a'(s) ds a(s) + b(O) + g(a(s) + b(O))
0 x
y
+Mffp(s,t)dsdt
E Dom(H-1),
0 0
for all x, y lying in the subintervals 0 <_x <_x2, 0 <_ y < y2 of R+. D Proof: We note that since g'(u) >_ 0 on R+, the function g is monotonically increasing on (0, oo). Define a function z(x, y) by the right-hand side of (5.3.16), then z(x, O) = a(x) + b(O), z(O, y) = a(O) + b(y), Uxy(X, y) <_ z(x, y) and
Zxy(X, y) = M[uxy(X, y) + p(x, y)g(Uxy(X, Y))I < M p(x, y)[z(x, y)4-g(z(x, Y))I, i.e.
Zxy( X,
Y)
< M p(x, y). z(x, y) + g(z(x, y)) -
(5.3.19)
Now by following the same arguments as in the proof of Theorem 5.2.1 below the inequality (5.2.5) with suitable modifications we obtain the estimate for z(x, y) such that
z(x, y) < H -1 -
[o
(a(0) + b(y)) +
xy
]
r a(s) + b(O) +a g(a(s) s, + b(O)) ds 0
0 0
Using (5.3.20) in uxy(x, y) < z(x, y) we obtain the required bound in (5.3.17). II
474
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Theorem 5.3.3 Letu(x, y), Uxy(X, y), p(x, y), g(u), g'(u), a(x), a'(x), b(y) and b' (y) be as in Theorem 5.3.2. If x y
lgxy(X, y) < a(x) + b(y) +
f f p(s,t)g(u(s, t) + Ust(S, t)) ds dt,
(5.3.21)
0 0
for x, y e R+, then for 0 < x < x3, 0 < y < Y3,
Uxy(X, y) < a(x) + b(y) +
xy(
f f p(st)g
n -1
(a(0) + b(t))
0 0 s
+f
a'(sl) dSl a(sl) + b(O) + g(a(sl) + b(0))
+ j fo p(sl, tl)dsl dtl]
(5.3.22)
dsdt,
where H, H -1 are as defined in Theorem 5.3.2 and x3, y3 are chosen so that x
a'(sl) a(sl) + b(O) + g(a(sl) + b(0))
H(a(O) + b(y)) +
ds1
0 x y
+/fp(sl,
tl)dsldtl 6 Dom(H-1),
0 0
for all x, y lying in the subintervals 0 < x < x3, 0 < y < Y3 of R+. Proof: Since g'(u)>_ 0 on R+, the function g(u) is monotonically increasing on (0, oo). Define a function z(x, y) by the right-hand side of (5.3.21), then z(x, O) - a(x) + b(O), z(O, y) - a(O) + b(y),
Uxy(X, y) < z(x, y),
(5.3.23)
and Zxy(X, y ) -
p(x, y)g(u(x, y ) + Uxy(X, y)).
(5.3.24)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES From (5.3.23) and using
475
u(x, 0) - u(0, y) - 0, it is easy to observe that x y
(5.3.25)
u(x,y)<_ffz(s,t)dsdt. 0 0
Using (5.3.23) and (5.3.25) in (5.3.24) we get
Zxy(X,
( j/
y) < p(x, y)g z(x, y)+
z(s, t)ds dt
0 0
Define a function
)
.
(5.3.26)
v(x, y) by x
y
v(x,y)-z(x,y)+ffz(s,t)dsdt. oo From (5.3.27) and using the facts that Zxy(X, y)< p(x, y)g(v(x, (5.3.26) and z(x, y) < v(x, y) from (5.3.27) we observe that Vxy(X,
y)
= Zxy(X,
(5.3.27)
y)) from
y) + z(x, y)
< p(x, y)[v(x, y)+ g(v(x, y))], i.e.
Vxy(X, Y) < p(x, y). v(x, y) + g(v(x, y)) -
(5.3.28)
Now by following the same arguments as in the proof of Theorem 5.2.1 below the inequality (5.2.5) with suitable modifications we obtain the estimate for v(x, y) such that
v(x, y) < H -~
IH
(a(0) +
b(y)) +
xy
] tl)dS,
/
a'(sl) a(sl) + b(O) + g(a(sl) + b(0))
dsl
0
(5.3.29)
0 0
Using (5.3.29) in (5.3.26), and first keeping x fixed, setting y = t and integrating from 0 to y, then keeping y fixed, setting x - s in the resulting
476
MULTIDIMENSIONALNONLINEARINTEGRALINEQUALITIES
inequality and integrating from 0 to x we obtain
xy
Z(X, y) < a(x) + b(y) +
f f p(,,t)g 0
( H
-1
H(a(0) + b(t))
0
s
a'(sl) dsl a(sl) 4- b(0) -t- g(a(sl) + b(0))
+
])
st 0
(5.3.30)
0
Using (5.3.30) in (5.3.23) we get the desired inequality in (5.3.22). II
The following two theorems established by Bondge and Pachpatte (1979c) can be used in certain applications. Theorem 5.3.4 Let q~(x, y), a(x, y), b(x, y) be nonnegative continuous functions defined for x, y ~ R+, and u(x, y) be a positive continuous function defined for x, y ~ R+. Let F(u), t~(x, y), fl(x, y) be as in Theorem 5.3.1 part (ii). If
u(s, t) > q~(x, y) - a(s, t)F-l ( f f b(sl, tl)F(q~(Sl, tl))dsl d t l ) , \x
y
(5.3.31) for O < x < s < oo, O< y < t < oo, then
U(S, t) > Or(S,t)F -1 (o~-1 (s, t)F(qb(x, y)) exp (-fl(s, t)F(a(s, t)f1-1 (s, t))
x//b Sltl, sldtm x
(5.3.32)
y
for O < x < s < oo, 0 < y < t < c~. [3
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
477
Proof: Rewrite (5.3.31) as
~(x, y) < or(s, t)u(s, t)ot -1 (s, t) + ~(s, t)a(s, t)/~ -1 (s, t) xf-1
( f f b(sl, tl)f(qb(Sl, t l ) ) d s l d t l ) . \x
(5.3.33)
y
Since F is convex, submultiplicative and monotonic, from (5.3.33) we have
or(s, t)F(u(s, t)ot -1 (s, t)) > F(~(x, y)) - ~(s, t)F(a(s, t)~ -1 (s, t)) s
t
•
tl)F(dp(Sl, tl))dsldtl. x
(5.3.34)
y
Now an application of Theorem 4.9.4 part (i) with n = 2 yields the desired bound in (5.3.32).
II Theorem 5.3.5 Let u(x, y), a(x, y) and b(x, y) be as in Theorem 5.3.4. Let g(u) and g'(u) be as in Theorem 5.2.1. If s
t
u(s,t)>_ u ( x , y ) - a ( s , t ) f f b ( c r , x
o)g(u(a,o))dado,
(5.3.35)
y
for O < x < s < c~, O < y < t < c~, then for O < x < s < sl, O <_ y <_ t <_ tl, u(s, t) > f2 -1
[
f2(u(x, y)) - a(s, t)
JJ x
where f2,
~"2-1 are
]
b(cr, O)dcr do ,
(5.3.36)
y
as defined in Theorem 5.2.1 and s
t
.>
f2(u(x, y)) - a(s, t ) f f x
forO < x ~ s ~ s1,
O ~
d. ~ Dom(f2 -1 ),
y
y < t < tl. D
47'8
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
P r o o f : Since g'(u)> 0 on R+, the function g(u) is monotonically increasing on (0, c~). Rewrite (5.3.35) as
u(x, y ) < u ( s , t ) + a ( s , t ) f f b ( t r , x
rl)g(u(tr, 0)) dcr d0.
(5.3.37)
y
For fixed s and t in R+ we define for 0 < x < s, 0 < y < t, s
z(x, y) -- u(s, t) + a(s, t)
t
ff x
b(tr, o)g(u(tr, 0))dcr do,
(5.3.38)
y
then z(x, t) - z(s, y) - u(s, t), u(x, y) < z(x, y) and
Zxy(X, y ) = a(s, t)b(x, y)g(u(x, y)) < a(s, t)b(x, y)g(z(x, y)), i.e.
y) < a(s, t)b(x, y). g(z(x, y)) Zxy(X,
(5.3.39)
From (5.3.39) we observe that
Zxy(X, y) g' (Z(X, y))Zy (X, Y)Zx(X, y) < a(s, t)b(x, y) + g(z(x, y)) [g(z(x, y))]2 9
(5.3.40)
Here we note that Zx(X, y) and Zy(X, y) are nonpositive, which implies that Zx(X, y)zy(X, y) is nonnegative and hence (5.3.40) is true. Now (5.3.40) is equivalent to
0 (Zx(X'Y)) Oy g(z(x, y))
(5.3.41)
By keeping x fixed in (5.3.41), setting y - r/and integrating from y to t we have
Zx(X, t) g(z(x, t))
t
Zx(X, Y) a(s, t) / b(x, r/)do. g(z(x, y)) i 1
i
(5.3.42)
e,j
Y
From (5.2.3) and (5.3.42) we observe that t
-~x S2(z(x, t)) - -~x ~ ~2(z(x' y)) < a(s, t) f b(x, 0) dr/. Y
(5.3.43)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
479
Now keeping y fixed in (5.3.43), setting x - cr and integrating from x to s we have
f2(Z(X, y)) <_f2(u(s, t)) + a(s, t) f f b(cr, ~) dcrdo, x y which implies
f2(u(s,t))>~(u(x, y))-a(s,t)ffb(tr,
0) dcr d0.
(5.3.44)
x y The desired bound in (5.3.36) follows from (5.3.44). The subintervals of s and t in R+ are obvious.
B
5.4 Generalizations of Pachpatte's Inequalities Pachpatte (1974d, 1975h) investigated some nonlinear Bihari-type integral inequalities which are applicable in certain general situations. In this section we present some basic two independent variable generalizations of the certain inequalities in Pachpatte (1974d, 1975h) obtained by Bondge and Pachpatte (1979b, 1980a) which can be used as tools in the study of certain partial integro-differential and integral equations. Bondge and Pachpatte (1979b) established the following generalization of the inequality given by Pachpatte (1974d). Theorem 5.4.1 Let u(x, y) and p(x, y) be nonnegative continuousfunc-
tions definedfor x, y ~ R+. Let g(u), g' (u), a(x), a'(x), b(y) and b' (y) be as in Theorem 5.2.1. If XY
u(x,y)
(
u(s,t)
0 0
- I -t fl f)op) d( ssll,d ttl)g(u(sl, l)o
dsdt,
(5.4.1)
480
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
for x, y ~ R+, then for 0 <_x <_Xl, 0 5 Y <_ Yl,
xy
u(x, y) < a(x) + b(y) +
f f p~s,t ) . ,
(a(0) + b ( t ) )
0 0 s
f
a'(sl)
a(s1) -+- b(0) + g(a(sl) + b(0))
q-
st
dsl
]
+ J j,,,:s, ,,, :,s, ~,1 ,:,s,t
(5.4.2)
0 0
where H(r) -- f
ds s + g(s)'
(5.4.3)
r > 0, r0 > 0,
ro
H -1 is the inverse function of H, and Xl, Yl are chosen so that x
dsl H(a(O) + b(y)) + f a(sl) + b(0) a'(sl) + g(a(sl) + b(0)) 0
x y
+ffp(sl,
tl)dSldtl 6 Dom(H-1),
0 0
for all x, y lying in the subintervals 0 < x < Xl, 0 <__y < Yl of R+. D Proof: Since g'(u)> O, the function g(u) is monotonically increasing on (0, oo). Define a function z(x, y) by the fight-hand side of (5.4.1), then z(x, O) - a(x) + b(O), z(O, y) - a(O) + b(y), u(x, y) < z(x, y) and
Zxy(X,
y) = p(x, y)
(
u(x, y) +
JJ
p(sl, tl)g(u(s1, tl))ds1 dtl
0 0
< p(x, y)
( iJ Z(X, y) +
0 0
p(s1, tl)g(z(s1, tl))dsldtl
)
)
.
(5.4.4)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES 481 If we put x
v(x, y) -- Z(X, y) +
y
f f p(sl, tl)g(Z(Sl, tl)) dsl dtl, 0 0
then v(x, O) - a(x) + b(O), v(O, y) = a(O) + b(y), z(x, y) < v(x, y) and y)
Vxy(X,
Zxy(X, y) < p(x, y)v(x, y),
y) + p(x, y)g(z(x, y))
-- Zxy(X,
< p(x, y)(v(x, y)-t-g(v(x, y))), i.e.
Vxy(X' Y) < p(x, y). v(x, y) + g(v(x, y)) -
(5.4.5)
Now by following the same arguments as in the proof of Theorem 5.2.1 below (5.2.5) with suitable modifications and in view of (5.4.3) we obtain the estimate for v(x, y) such that
v(x, y) < H -1
IH
(a(0) + b(y)) +
xy
]
j
a'(sl) a(s1) + b(0) + g(a(sl) + b(0)) dsl
0
+// Sltl, ldta
(5.4.6)
0 0
Using (5.4.6) in (5.4.4), and first keeping x fixed, setting y = t and integrating from 0 to y, then in the resulting inequality keeping y fixed, setting x - s and integrating from 0 to x we obtain xy
z(x, y) < a(x) + b(y) +
f f p(s,t ) n -1
(a(0) + b(t))
0 0 s
-t-
f
a'(sl) ds1 a(sl) + b(O) -I- g(a(sl) + b(O))
0 s
t
+ffp(sl, 0 0
tl)dsldtl
]
dsdt.
(5.4.7)
482
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
Now using (5.4.7) in u(x, y) < z(x, y) we obtain the desired bound in (5.4.2). The subintervals for x and y are obvious. II
Bondge and Pachpatte (1980a) established the following three theorems which deal with the two independent variable generalizations of certain inequalities given by Pachpatte (1974d, 1975h). Theorem 5.4.2 Let u(x, y), a(x, y) and b(x, y) be nonnegative continuous functions defined for x, y ~ R+. Let g(u), g'(u) be as in Theorem 5.2.1 and in addition g(u) be subadditive on R+. If u(x, y) < a(x, y) + - - - - b(s, t)
u(s, t ) +
00
b(sl, tl)
00
• g(u(sl, tl))dsl dtl] dsdt,
(5.4.8)
xy{[ /
for x, y ~ R+, then for 0 < x < x2, 0 < y < Y2,
00 + f fo p(sl, tl)dsl dtl] ) where
xy
(
00 s1 tl
t2)g(a(s2, t2))ds2dt2
+ 0
(5.4.9)
I
dsl dtl,
(5.4.10)
0
H, H -1 are as defined in Theorem 5.4.1 and x2, y2 are chosen so that
xy
H(A(x,y))+ffp(sl, 0
t l ) d s l d t l 6 Dom(H-1),
0
for all x, y lying in the subintervals 0 < x < x2, 0 <_ y < y2 of R+.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES Theorem
483
5.4.3 Let u(x, y), a(x, y), b(x, y) and c(x, y) be nonnegative
continuous functions defined for x, y ~ R+. Let g(u) and g'(u) be as in Theorem 5.4.2. If
xy
IJJ
f f ~s o .~s.,~+
u(x, y) < a(x, y) +
s
O0
C~Sl.tl)U(S1, tl)dSl O0
+ flybys1, ,,~ + c~,, tl))g(u(sl, tl))dsl dtl 0
] dsdt, (5.4.11)
0
for x, y ~ R+, then for 0 < x <
X3,
0 < y < Y3,
.y st 0
where
dtl
0
0
{
H-1
H(B(x, y))
]}
(5.4.12)
0
xy [
s/j
B(x,y)=//,.s.. ,..
a(s,, tl) +
O0
b(s., t.)c(s.,t.)ds, dt. O0
s1 tl
+ ff~s. 0
,.~ + c~s. t2))g(a(s2, t2))ds2 dt2 ds1 dtl,
J
0
(5.4.13)
H, H -1 are as defined in Theorem 5.4.1 and x3, Y3 are chosen so that x
H(B(x, y))9-
y
ff~b~Sl, tl) -k-c(s1, tl))dsl dtl E Dom(H-1), 0
0
for all x, y lying in the subintervals 0 < x < x3, 0 < y < Y3 of R+. [2 Theorem
5.4.4 Let u(x, y), a(x, y), b(x, y), c(x, y) and k(x, y) be nonneg-
ative continuous functions defined for x, y ~ R+. Let g(u), g' (u) be as defined
484
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
in Theorem 5.2.2. If
xy
u(x, y) < a(x, y) + b(x, y)
f f c<~,t)g
( u(s, t) + b(s, t)
0 0
s,
) (5.4.14)
0 0
for x, y ~ R+, then for 0 <_x < x4, 0 <_ y <_ Y4, u(x, y) < a(x, y) + b(x, y)
(x, y) +
r
0 0
{[
Jr x. b slta Sldta]})
X
where
~-2-1
f2(L(x, y)) +
xy L(x,y)=ffc(sa, oo
c(s, t)g
(C(S1,
b(s, t)
tl) + k(sl, tl))
0 0
( tl)g
(
(5.4.15)
jtj a(sl, t l ) W b ( s l , tl)
k(s2, t2) oo
• g(a(s2, t2))ds2 dr2) ds1 dtl,
(5.4.16)
~, ~'2-1 are as defined in Theorem 5.2.1 and x4, y4 are chosen so that x y
~,~x, y~ + ff~c~sl, ta~ + ~sa, , 1 ~ ~ S l , t,~ ~s, ~,,~ ~om~ 1~, 0 0
for all x, y lying in the subintervals 0 < x < x4, 0 < y < y4 of R+.
r-q The details of the proofs of Theorems 5.4.2-5.4.4 follow by arguments similar to those in the proofs of Theorems 5.2.2 and 5.4.1 in view of the results given in Pachpatte (1974d, 1975h). Here we leave the details to the reader.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
485
In concluding, we note that the inequalities established in Sections 5.2-5.4 can be extended very easily to n (n > 3) independent variables. The precise formulations of these results are very close to those of the results given in Sections 5.2-5.4 with suitable modifications. It is left to the reader to fill in the details where needed.
5.5 Pachpatte's Inequalities I In view of wider applications, Wendroff' s inequality given in Beckenbach and Bellman (1961) has been generalized and extended in various directions. The present section is devoted to the Wendroff-like inequalities investigated by Pachpatte (1988c, 1993, 1996d) in order to apply them in the study of certain higher order partial differential equations. In what follows we shall use the notations and definitions as given in Section 4.5 without further mention. Pachpatte (1988c) established the Wendroff-like inequalities in the following two theorems. Theorem 5.5.1 Let u(x, y) and h(x, y) be nonnegative continuous functions defined for x, y ~ R+. Let a(x), b(y), p(x), q(y) be positive and twice continuously differentiable functions defined for x, y ~ R+, a' (x), b' (y), p' (x), q' (y) are nonnegative for x, y ~ R+ and define c(x, y ) - a(x) + b(y) + yp(x) + xq(y), for x, y ~ R+. Let g be a continuously differentiable function defined on R+ and g(u) > 0 on (0, cx~), g'(u) > 0 on R+ and
u(x, y) < c(x, y ) + a [ x , y,h(sl, tl)g(u(sl, tl))],
(5.5.1)
holds for x, y ~ R+. (i) If a"(x), p"(x) are nonnegative for x >_ O, then for 0 < x < Xl, O
[
X
+
cxO )
~(c(O, y)) + x k,g(c(O, y))
S
giC-~li-~
dslds
0 0 "1
+ A[x, y, h(sl, tl)]] ,
]
(5.5.2)
486
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
where
j( ds f2(r) --
g(s) '
r > 0, ro > 0,
(5.5.3)
ro ~'2-1 is the inverse of g2 and xl, Yl are chosen so that
(Cx(O,y) ) //a"(Sl)+yp"(Sl) x
f2(c(O, y)) + x
g(c(O, y))
s
+
dsl ds
g(C-~l~-~)
0 0
+ A[x, y, h(sl, tl)] E Dom(f1-1), for all x, y lying in the subintervals 0 <_x <_Xl, 0 ~ y <_ Yl of R+. (ii) If b"(y), q"(y) are nonnegative for y >_ O, then for 0 < x < x2, O
u(x, y) < ~2-1
+ffb"(tl)Wxq"(tl)dtldt+A[y,x,h(sl, g(c(O, tl ))
tl)] ]
(5.5.4)
o o
where f2, ~"2-1 are as defined in (i) and X2, 22 are chosen so that
(Cy(X,O))i/'"<,,,+xq",,,, y
a(c(x, 0)) + y k,g-(c--~,, 0))
t
g(c(O, tl))
+
dt~ dt.
0 0
+ A[y, x, h(sl, tl)] E Dom(~ -1), for all x, y lying in the subintervals 0 < x < X2, 0 ~ y < 22 of R+. [2
Theorem 5.5.2 Let u(x, y), h(x, y), a(x), b(y), p(x), q(y), a'(x), b'(y), if(x), q'(y), c(x, y), g(u) and g'(u) be as in Theorem 5.5.1 and u(x, y) <_ c(x, y ) + A[x, y, h(s1, tl)[U(S1, tl) + A[Sl, tl, h(s2, t2)g(u(s2, t2))]]], holds for x, y ~ R+.
(5.5.5)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
487
(i) If a" (x) and p" (x) are nonnegative for x ~ R+, then for 0 < x < x3, O< y < y3,
u(x,y)<_c(O,y)+xcx(O,y)+ff[a"(sl)+yp"(Sl)]dslds 0
0
+ A[x, y, h(sl, tl)Q1 (S1, tl)],
(5.5.6)
in which
Q1 (x, y) - H -1 x
[.
( cx.o.y. ) c(O, y) + g(c(O, y))
$2
//
"1-
(c(O, y)) + x
00
a"(s3)+yp"(s3) ds3 ds2 + A[x, y, h(s3, t3)]] , (5.5.7) C($3, O) -Jr-g(C(S3, 0))
where ds
H(r) --
r > 0, ro > 0,
s + g(s)
(5.5.8)
ro
H -1 is the inverse function of H and x3, Y3 are chosen so that x
H (c(O, y)) + x (
$2
Cx(O,y) ) . f+f a " ( s 3 ) + y p " ( s 3 ) ds3 ds2 c(O, y) -Jr-g(c(O, y)) c(s3, O) + g(c(s3, 0)) 0
0
+ A[x, y, h(s3, t3)] E D o m ( H -1), for all x, y lying in the subintervals 0 < x < x3, 0 < y < Y3 of R+. (ii) If b"(y), q"(y) are nonnegative for y ~ R+ then for 0 < x < x4, O< y < y4, y
u(x, y) < c(x, O) + yCy(X, O) -~-
t
f f tb"(tl)+ xq" (tl)] dtl dt 0
0
+ A[y, x, h(sl, tl)Q2(Sl, tl)],
(5.5.9)
488
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
in which
a2(x, y) - n -1
+ff
( c(x, O)cy,xo, ) + g(c(x, 0))
(c(x, 0)) + y
y t2
0 0
b"(t3)+xq"(t3) C(0, t3) + g(c(O, t3))
dt3 dt2
(5.5.10)
+ A[y, x, h(s3, t3)]] ,
where H, H -1 are as defined in (i) and x4, y4 are chosen so that y t2
(Cy(X,O) )+ffff'(t3)+xq"(t3) H(c(x, 0)) + y c(x, O) + g(c(x, g(c(O, 0))
c(0, t3) +
t3))
dt3 dt2
0 0
+ A[y, x, h(s3, t3)] E Dom(H-1), for all x, y lying in the subintervals 0 < x < x4, 0 < y < y4 of R+. D P r o o f of T h e o r e m 5.5.1" Since g'(u) >_ O, the function g(u) is monotonically increasing on (0, c~). Define a function z(x, y) by
z(x, y) = c(x, y) + A[x, y, h(sl, t l ) g ( u ( s 1 , t l ) ) ] .
(5.5.11)
From (5.5.11) it is easy to observe that z(0, y) -- c(0, y) -- a(0) + b(y) + yp(O),
(5.5.12)
z(x, O) -- c(x, O) = a(x) + b(O) + xq(O),
(5.5.13)
zx(O, y) -- cx(O, y) -- a'(O) + yp'(O) + q(y),
(5.5.14)
Zxx(X, O) -- Cxx(X, O) = a"(x),
(5.5.15)
Zxxy(X, O) -- Cxxy(X, O) "- p"(x),
(5.5.16)
and Zxxyy(X, y) -- h(x, y)g(u(x, y)).
(5.5.17)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
489
Using u(x, y) < z(x, y) in (5.5.17) we have
Zxxyy(X, y) < h(x, y)g(z(x, y)). From (5.5.18)and using the facts that z(x, y) > 0, g'(u(x, y)) > 0 for x, y ~ R+, we observe that
(5.5.18)
Zy(X, y) > O, zxxy(X, y) > O,
Zxxyy(X, y) < h(x, y)4- Zxxy(X, y)g'(z(x, y))Zy(X, y) g(z(x, y)) [g(z(x, y))l 2 i.e.
O___(Zx~y(x,y) ) ay \ g(z(x, y))
(5.5.19)
By keeping x fixed in (5.5.19), we set y = tl and then integrating with respect to tl from 0 to y and using (5.5.13) and (5.5.16) we have y
Zxxy(X, y) < p"(x) / g(z(x, y)) - g(c(x, 0)) -t- h(x, tl)dtl. 0
(5.5.20)
Again as above, from (5.5.20) and using the facts that z(x, y) > O, Zy(X, y) > O, Zxx(X, y) > 0 for x, y ~ R+, we observe that y
8 (Zxx(x,y)) Oy \ g - ~ , y))
< p"(x) 4- / h(x, tl ) dtl. - g(c(x, 0))
(5.5.e~)
0
By keeping x fixed in (5.5.21), setting y - t and then integrating with respect to t from 0 to y and using (5.5.13) and (5.5.15) we have y
g(z(x, y)) -
g(c(x, 0))
t
4-
h(x, tl) dtl dt.
(5.5.22)
0 0 As above, from (5.5.22) and using the facts that z(x, y) > O, Zx(X, y) > 0 for x, y ~ R+, we observe that
i9 ( z x ( x , y ) -~x \ g ( - ~ , y))
y
< a" (x) + yp" (x) g(c(x, 0))
t
(5.5.23) 0
0
490
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Now keeping y fixed in (5.5.23), setting x - sl and then integrating with respect to Sl from 0 to x and using (5.5.12) and (5.5.14) we have
Zx(X,y) < cx(O, y) + / a"(Sl) + yp"(Sl) g(z(x, y)) - g(c(O, y)) g(C~-liO-)) dsl 0 x
y
t
+fffh(sl,
tl)dtldtdsl.
(5.5.24)
000 From (5.5.3) and (5.5.24) we have x
f2(z(x, y)) = Zx(X, y) < cx(O, y) + f a"(Sl)q-- yp"(sl) -~x g(z(x, y)) - g(c(O, y)) gi-C-~ll-0-)) dsl x y t
+J'ffh(sl,
tl)dtldtds1.
(5.5.25)
0 0 0
Now keeping y fixed in (5.5.25), setting x - - s and then integrating with respect to s from 0 to x and using (5.5.12) we obtain
a (z(x, y)) <_ a (c(O, y)) + x g-@-~iy))
-~-
f f a " ( S l )g(c~-~l + y p "i (~-~i Sl)
dslds
0 0
+ A[x, y, h(sl, tl)].
(5.5.26)
The desired bound in (5.5.2) now follows by substituting the bound on z(x, from (5.5.26) in u(x, y) < z(x, y). The subdomains for x, y are obvious. Rewriting (5.5.11) in the form
y)
z(x, y) - c(x, y) + A[y,x, h(sl, tl)g(u(sl, tl))], since A[x, y, h(sl, tl)g(u(sl, tl))] -- A[y,x, h(sl, tl)g(u(sl, tl))] and by following the same arguments as in the proof of the inequality (5.5.2) given above with suitable modifications we obtain the required inequality in (5.5.4).
II
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
491
The proof of Theorem 5.5.2 follows by an argument similar to that in the proof of Theorem 5.5.1 and by closely looking at the proof of Theorem 4.5.2, with suitable modifications. The details are omitted here. We next establish the following inequality which can be used in the study of certain partial differential and integral equations. Theorem 5.5.3 Let u(x, y), f (x, y) and p(x, y) be nonnegative continuous functions defined for x, y ~ R+. Let g(u), g' (u) be as in Theorem 5.5.1 and g(u) be submultiplicative on R+. If xc c"
u(x, y) <_ c + / / tl
f ( s l , y)u(sl, y)dsl ds +A[x, y, p(sl,
tl)g(u(s1,
tl))],
tl
0 0
(5.5.27)
for x, y ~ R+, where c > 0 is a constant, then for 0 < x < x5, 0 < y < Y5, u(x, y) < Q(x, y){f2-1[f2(c)+ A[x, y, p(sl, tl)g(Q(Sl, tl))]]},
(5.5.28)
where
y, exp(jj ,sl y,dsls)
(5.5.29)
\0 0 ~, ~ - 1 are as defined in Theorem 5.5.1 and xs, Y5 are chosen so that
f2(c) + A[x, y, p(sl, tl)g(Q(sl, tl))] E Dom(f2 -1), for all x, y lying in the subintervals 0 < x < x5, 0 < y < Y5 of R+. D
Proof: Since g'(u)> 0 on R+, the function g(u) is monotonically increasing on (0, ec). We assume that c > 0 and that the standard limiting argument can be used to treat the remaining case. Define a function m(x, y) by
m(x, y) -- c + A[x, y, p(sl, tl)g(u(sl, tl))].
(5.5.30)
Then (5.5.27) can be restated as X
S
u(x, y) < m(x, y) + f / f (sl, y)u(sl, y) dsl ds. 0 0
(5.5.3~)
492
MULTIDIMENSIONALNONLINEARINTEGRALINEQUALITIES
Clearly m(x, y) is positive and nondecreasing in both the variables x, y 6 R+. From (5.5.31) we observe that
u(x, y) < 1 + f~ f m(x, y) --
y________~)dsl U(S1, y) ds. f (sl, Y)m(sl,
(5.5.32)
0 0
Define a function z(x, y) by the fight-hand side of (5.5.32); then
z(O, y) and
--
1,
u(x, y) m(x, y)
< z(x, y)
u(x, y) Zxx(X, y ) - f (x, y ) ~ < f (x, y)z(x, y) m(x, y) -
i.e.
Zxx(X, y) < f(x, y). z(x, y)
(5.5.33)
Now by following the proof of Theorem 5.5.1 we obtain
z(x, y) < exp
f (sl, y)dsl ds
- Q(x, y).
(5.5.34)
Using (5.5.34) in (5.5.32) we have
u(x, y) < Q(x, y)m(x, y).
(5.5.35)
From (5.5.30) and (5.5.35) we have mxxyy(X,
y ) - p(x, y)g(u(x, y)) < p(x, y)g(Q(x, y)m(x, y)) < p(x, y)g(Q(x, y))g(m(x, y)).
(5.5.36)
Now by following the proof of Theorem 5.5.1 we obtain
m(x, y)
<
~-1 [~"2(c)-~-A[x,y, p(sl, tl)g(Q(Sl, tl))]].
(5.5.37)
By using (5.5.37) in (5.5.35) we get the required inequality in (5.5.28). 1
The inequalities given in the following theorem have been recently established by Pachpatte (1993, 1996d) and are motivated by the study of certain higher order partial differential equations.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
493
Theorem 5.5.4 Let u(x, y), a(x, y) and b(x, y) be nonnegative continuous functions defined for x, y ~ R+ and h" R3+ --+ R+ be a continuous function which satisfies the condition 0 < h(x, y, V1) -- h(x, y, 1;2) ~ k(x, y, V2)(V1 -- V2), for x, y ~ R+ and Vl > (i) If
V2
(H)
>_ 0 where k" R3+ --+ R+ is a continuous function.
u(x, y) < a(x, y) + b(x, y)B[x, y, h(s, t, u(s, t))],
(5.5.38)
for x, y ~ R+ then u(x, y) < a(x, y) + b(x, y)p(x, y)exp(B[x, y, k(s, t, a(s, t))b(s, t)]), (5.5.39) for x, y ~ R+, where p(x, y) -- B[x, y, h(s, t, a(s, t))],
(5.5.40)
for x, y E R+. (ii) Let F(u) be a continuous, strictly increasing, convex, submultiplicative function for u > O, l i m u ~ F(u) -- o0, F -1 denote the inverse function of F, and ~(x, y), fl(x, y) be continuous and positive functions for x, y ~ R+ and or(x, y) + fl(x, y) -- 1. If u(x, y) <_ a(x, y) + b(x, y)F -1 (B[x, y, h(s, t, F(u(s, t)))]),
(5.5.41)
for x, y ~ R+, then u(x, y) < a(x, y) + b(x, y)F -l(B[x, y, h(s, t, t~(s, t)F(a(s, t)ct -1 (s, t)))]
x exp(B[x, y, k(s, t, ~(s, t)F(a(s, t)c~-1 (s, t))) x fl(s, t)F(b(s, t)fl -l(s, t))])),
(5.5.42)
for x, y ~ R+. (iii) Let g(u) be a continuously differentiable function defined for u >_ O, g(u) > 0 for u > 0 and g'(u) > 0 for u > 0 and g(u) is subadditive and submultiplicative for u > O. If u(x, y) < a(x, y) + b(x, y)B[x, y, h(s, t, g(u(s, t)))],
(5.5.43)
494
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for x, y ~ R+, then for 0 < x < x0, 0 < y < Yo, u(x, y) < a(x, y ) + b(x, y)f2-1[f2(q(x, y)) d- B[x, y, k(s, t, g(a(s, t)))g(b(s, t))]],
(5.5.44)
where q(x, y) = B[x, y, h(s, t, g(a(s, t)))],
(5.5.45)
g2, f2 -1 are as defined in Theorem 5.5.1 and xo, Yo are chosen so that f2(q(x, y)) + B[x, y, k(s, t, g(a(s, t)))g(b(s, t))] 6 Dom(f2 -1), for all x, y lying in the subintervals 0 < x < xo, 0 < y < Yo of R+. 0 Proof: (i) Define a function z(x, y) by
z(x, y) - B[x, y, h(s, t, u(s, t))].
(5.5.46)
From (5.5.46) and using u(x, y) < a(x, y) + b(x, y)z(x, y) and the condition (H), we observe that
z(x, y) < B[x, y, h(s, t, a(s, t) + b(s, t)z(s, t))] -- p(x, y) + B[x, y, {h(s, t, a(s, t) + b(s, t)u(s, t)) - h ( s , t, a ( s ,
t))}]
< p(x, y) + B[x, y, k(s, t, a(s, t))b(s, t)z(s, t)],
(5.5.47)
where p(x, y) is defined by (5.5.40). Clearly p(x, y) is nonnegative for x, y R+. It is sufficient to assume that p(x, y) is positive, since the standard limiting argument can be used to treat the remaining case. Now since p(x, y) is positive and monotonic nondecreasing in x, y 6 R+, from (5.5.47) we observe that
z(x, y) < l + B [x, y, k(s, t, a(s, t))b(s, t) z(s, t) ] p(x, y) p(s, t) "
(5.5.48)
Define a function v(x, y) by the fight-hand side of (5.5.48), then it is easy to observe that
D~D~v(x, y) -- k(x, y, a(x, y))b(x, y) z(x, y_____~) p(x, y) <_ k(x, y, a(x, y))b(x, y)v(x, y).
(5.5.49)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
495
The rest of the proof follows by the same arguments as in the proof of Theorem 4.5.4 with suitable changes, and hence further details are omitted. (ii) The proof follows by the same arguments as in the proof of Theorem 5.3.1 part (ii), applying the inequality given in part (i) above. The details are omitted here. (iii) Define a function z(x, y) by
z(x, y) = B[x, y, h(s, t, g(u(s, t)))].
(5.5.50)
From (5.5.50) and using u(x, y) < a(x, y) + b(x, y)z(x, y) and following the same arguments as in the proof of inequality (5.5.47) in part (i) with suitable changes we have
z(x, y) < q(x, y) + B[x, y, k(s, t, g(a(s, t)))g(b(s, t))g(z(s, t))],
(5.5.51)
where q(x, y) is defined by (5.5.45). The rest of the proof can be completed by following the proofs of Theorems 5.2.2 and 4.5.4 with suitable changes. We leave the details to the reader. II
5.6 Pachpatte's Inequalities II In the past few years, the inequality given by Snow (1971) has attracted much attention and a number of its generalizations and their applications have appeared in the literature. In this section we present some basic nonlinear integral inequalities in two independent variables investigated by Pachpatte (1980c,d) which can be used in the analysis of various problems in the theory of partial integro-differential and integral equations. Pachpatte (1980c) gave the following useful inequalities in two independent variables. Theorem 5.6.1 Let u(x, y), a(x, y), b(x, y), c(x, y), p(x, y) and q(x, y) be nonnegative continuous functions defined on a domain D. Let Po(xo, Yo) and P(x, y) be two points in D such that (x - xo)(y - Yo) > 0 and R be the rectangular region whose opposite corners are the points Po and P. Let G(r) be a continuous, strictly increasing, convex and submultiplicative function for r > 0, l i m r ~ G(r) -- c~ for all (x, y) in D, ~(x, y), fl(x, y) be positive continuous functions defined on a domain D, and or(x, y) + ~(x, y) = 1. Let
496
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
v(s, t; x, y) and w(s, t; x, y) be the solutions of the characteristic initial value problems L[v]
[p(s, t) -+- fl(s, t)G(b(s, t)f1-1 (s, t))(c(s, t)
= Vst -
+ q(s, t))]v = 0,
(5.6.1)
v(s, y) = v(x, t) = 1, and M[w]
-- Wst -
[fl(s, t)G(b(s, t)/3-1 (s, t))c(s, t) - p(s, t)]w "- O,
w(s, y) = w(x, t) = 1,
(5.6.2)
respectively and let D + be a connected subdomain of D which contains P and on which v > 0 and w > 0 (Figure 4.1). If R C D + and u(x, y) satisfies u(x, y) < a(x, y) + b(x, y)G -1 [xf / c(s, t)a(u(s, t)) ds dt
+
ff
,,
dsdt
,(5.6.3)
X [C(Sl, tl) + q(sl, tl)]V(Sl, tl; s, t)dsl dtl } dsdt] ,
(5.6.4)
q(Sl, t l ) G ( U ( S l , tl))ds1 dtl \xoYo
xoYo
then u(x, y) also satisfies u(x, y) <_a(x, y) + b(x,
y)G-1 [~~ w(s, t; x, y)
x { ol(s, t)G(a(s, t)c~-1 (s, t))c(s, t) s t "~-P(S,t) f/Ol(Sl,
tl)a(a(sl,
tl)ol-l(sl,
tl))
xo yo
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
497
Proof: Rewrite (5.6.3) as u(x, y) < a(x, y)a(x, y)ct -1 (x, y ) + fl(x, y)b(x, y)fl-1 (x, y) x G -1
c(s, t)G(u(s, t)) ds dt +
JJ
p(s, t)
xo Yo
x
q(sl, tl)G(U(Sl, tl))dsldtl
) ] dsdt
.
(5.6.5)
Since G is convex, submultiplicative and monotonic, from (5.6.5) we have G(u(x, y)) <_ ix(x, y)G(a(x, y)ct -l(x, y)) + fl(x, y)G(b(x, y)fl-l(x, y)) x
[jj
c(s, t)G(u(s, t))ds dt +
i.xo Yo
x
r
p(s, t)
xo Yo
q(sl, tl)G(u(sl,
tl))ds1 dtl
) ] dsdt
.
(5.6.6)
The estimate given in (5.6.4) follows by first applying Theorem 4.8.2 to (5.6.6) with a(x, y ) = t~(x, y)G(a(x, y)a-l(x, y)), b(x, y ) = 13(x, y)G(b(x, y)13-1 (x, y)) and u(x, y) = G(u(x, y)) and then applying G -1 to both sides of the resulting inequality. II Theorem 5.6.2 Let u(x, y), a(x, y), b(x, y), c(x, y), p(x, y), q(x, y), Po(xo, Yo) and P(x, y) be as in Theorem 5.6.1. Let H(r) be a positive, continuous, strictly increasing, subadditive and submultiplicative function for r > O, limr~c~, H (r) -- c~, H -1 is the inverse function of H. Let v(s, t; x, y) and w(s, t; x, y) be the solutions of the characteristic initial value problems L[v]
=
Vst -
[p(s,
t ) --[-
H(b(s, t))(c(s, t) + q(s, t))]v = 0,
v(s, y) -- v(x, t) = 1,
(5.6.7)
and M[w] -- Wst - [H(b(s, t))c(s, t) - p(s, t)]w -- O, w(s, y) - w(x, t) -- 1,
(5.6.8)
498
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
respectively and let D + be a connected subdomain of D which contains P and on which v > 0 and w > 0 (Figure 4.1). If R C D + and u(x, y) satisfies u(x, y) < a(x, y) + b(x, y)H-l [x~~ C(S, t)H(u(s, t))dsdt
xy (jj
+ ff.(s.,~ x0 Y0
)]
q(Sl, tl)n(u(s1, tl))ds1 dtl \xo
dsdt ,(5.6.9)
Y0
then u(x, y) also satisfies u(x, y) < H-1 [H(a(x, y)) q-H(b(x, y)) [xfo ~w(s,t; x, y )
•
{
H(a(s, t))c(s, t) + p(s, t)
JJ
H(a(sl, tl))
xo Yo
• [C(Sl, tl)+q(sl,tl)]V(Sl, tl;S,t)dsldtl}dsdt]] . (5.6.10) D Proof: Since H is subadditive, submultiplicative and monotonic, from (5.6.9) we have
H(u(x, y)) < H(a(x, y)) + H(b(x, y)) [xfo ~ c(s, t)H(u(s, t))dsdt
+ f f ~s.,~
q(sl, tl)H(U(Sl, tl))dsl dtl
dsdt .
xo Yo
(5.6.11) The desired bound in (5.6.10) follows by first applying Theorem 4.8.2 to (5.6.11) with a(x, y) - H(a(x, y)), b(x, y) - H(b(x, y)) and u(x, y) H(u(x, y)) and then applying H -1 to both sides of the resulting inequality. II
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
499
Pachpatte (1980c) also gave the following two inequalities which can be used in some applications. Theorem 5.6.3 Let u(x, y), a(x, y), b(x, y), c(x, y), p(x, y), q(x, y) and
Po(xo, Yo), P(x, y) be as in Theorem 5.6.1. Let G(r), t~(x, y), fl(x, y) be as in Theorem 5.6.1. Let v(s, t; x, y) and w(s, t; x, y) be the solutions of the characteristic initial value problems L[v] -- v~t - fl(s, t)G(b(s, t)~ -1 (s, t))[c(s, t) + p(s, t) + q(s, t)]v = O,
(5.6.12)
v(s, y) - v(x, t) - 1, and M[w]
--
Wst
-
~(s,
t)G(b(s, t)~ -1 (s, t))c(s, t)w - O, (5.6.13)
w(s, y) -- w(x, t) -- 1, respectively and let D + be a connected subdomain of D which contains P and on which v > 0 and w > 0 (Figure 4.1). If R C D + and u(x, y) satisfies u(x, y) < a(x, y) + b(x, y)G-l [xfo ~ c(s, t)G(u(s, t)) ds dt
+ // xo
P(S, t) (G(u(s, t)) + ~(s, t)G(b(s, t)~-l (s, t)) YO
//q sl l o U Sl l sld l)dsd] xo Yo
then u(x, y) also satisfies u(x, y)
x { ~(s, t)G(a(s, t)ot-l(s, t))[c(s, t) + p(s, t)] + ~(s, t)G(b(s, t)~ -1 (s, t))p(s, t)
6
500
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
s t
•
ff Ct(Sl,
tl)G(a(s1,
tl)~-l(sl, tl))
xo yo
• [C(Sl, tl) + p(sl, tl) + q(Sl, tl)] • V(S1,
tl;S, t)dsl dtl } ds dt] .
(5.6.15)
U] Theorem 5.6.4 Let u(x, y), a(x, y), b(x, y), c(x, y), p(x, y), q(x, y),
Po(xo, Yo) and P(x, y) be as in Theorem 5.6.1. Let H and H -1 be the same functions as defined in Theorem 5.6.2. Let v(s, t; x, y) and w(s, t; x, y) be the solutions of the characteristic initial value problems L[v] = Vst - H(b(s, t))[c(s, t) + p(s, t) + q(s, t)]v = O,
(5.6.16)
v(s, y) = v(x, t) = 1, and M[w] = Wst - H(b(s, t))c(s, t)w -- O,
(5.6.17)
w(s, y) -- w(x, t) = 1, respectively and let D + be a connected subdomain of D which contains P and on which v > 0 and w > 0 (Figure 4.1). If R C D + and u(x, y) satisfies
u(x, y) < a(x, y) + b(x, y ) H - l [ f / c(s, t)H (u(s, t)) ds dt
xy
(
t.xo YO
st)]
xo yo
xo Yo
(5.6.18)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
501
then u(x, y) also satisfies u(x, y) < H-l [H(a(x, Y)) + H(b(x, Y)) [xfo f w(s, t; x, Y) I"
• ~ H(a(s, t))[c(s, t) + p(s, t)] + n(b(s, t))p(s, t)
[
s
•
ff
t
H(a(sl, tl))[C(Sl, tl) -I- p(sl, tl) + q(sl, tl)]
xo Yo
x v sl tl S t Ol tll S t]]
(5.6.19) [3
The proofs of Theorems 5.6.3 and 5.6.4 can be completed by following similar arguments to those in the proofs of Theorems 5.6.1 and 5.6.2 and applying Theorem 4.8.3. The details are omitted here. R e m a r k 5.6.1 Note that in the special case when p(x, y) = q(x, y) = O, Theorems 5.6.1-5.6.4 reduce to the further generalizations of the integral inequality established by Snow (1971); see also the results given in Bondge and Pachpatte (1980b). In the special case when c(x, y) = 0, Theorems 5.6.1-5.6.4 reduce to the new inequalities which can be used in certain applications. A Pachpatte (1980d) established the following inequalities, which can be used in the analysis of a class of nonlinear non-self-adjoint hyperbolic partial integro-differential and integral equations. Theorem 5.6.5 Let u(x, y), a(x, y), b(x, y), c(x, y), p(x, y), q(x, y),
r(x, y), h(x, y) and g(x, y) be nonnegative continuous functions defined on a domain D. Let Po(xo, Yo) and P(x, y) be two points in D such that (x - xo)(y - Yo) > 0 and R is the rectangular region whose opposite corners are the points Po and P. Let G, G -1, a, ~ be as in Theorem 5.6.1. Let V (s, t; x, y) be the solution of the characteristic initial value problem M[V] = 0,
(5.6.20)
where M is the adjoint operator of the operator L defined by L[~] = ~st + al~s + a2~rt q-a3~,
(5.6.21)
502
MULTIDIMENSIONALNONLINEARINTEGRALINEQUALITIES
in which a l - - ~ G ( b ~ - l ) c q , a2--3G(b~-l)cp and a 3 = - [ g + 3G(b3-1)c(r + h)]. Let W(s, t;x, y) be the solution of the characteristic initial value problem N[W] - O, (5.6.22) where N is the adjoint operator of the operator T defined by
T[q~] -
~st --I- bl~)s qt_ b2~) t +
b3~,
(5.6.23)
in which bl - -~G(b~ -1)cq, b2 - -~G(b~ -1)cp, and b3 - -~G(b~ -1) x c ( r - h). Let D + be a connected subdomain of D which contains P and on which V > 0 and W > 0 (Figure 4.5). If R C D + and u(x, y) satisfies
[r
u(x, y) < a(x, y) + b(x, y)G -1 p(x, y)
c(s, y)G(u(s, y))ds
xo
y
+ q(x, y) / c(x, t)G(u(x, t))dt yo x y
+r(x,
y)ffc(s,t)G(u(s,t))dsdt xo Yo
xy
xo Yo
x G(u(sl, tl))dsl then u(x, y) also satisfies
(jj xo Yo
dtl)dsdt] ,
[j
u(x, y) < a(x, y) + b(x, y)a -1 p(x, y)
(5.6.24)
c(s, y)G(u(s, y))ds
xo
y
+ q(x, y) / c(x, t)G(u(x, t))dt + r(x, y)Qo(x, y) yo xy
+ ~x, y ~ / / ~ s , xo Yo
]
~o~s, ~ dsd~ ,
(5.6.25)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
503
where Qo(x, y) is defined by the right member of Q(x, y) -
xy
f f w (s,,; ~, y)c(s, t)
{ a(s, t) + b(s, t)h(s, t)
xo Yo
(ffV(Sl,tl;s,t)a(sl,tl)C(Sl,tl)ds~dtl)}dsdt, xo Yo
(5.6.26) by replacing a(x, y) by ct(x, y)G(a(x, y)ot-l(x, y)) and b(x, y) by fl(x, y) • G(b(x, y)fl-l(x, y)). Further, if q(x, y)--O, then u(x, y) <_a(x, y) + b(x, y)G -1 Jr(x, y)Qo(x, y) x
y
+h(x, y)fJ'g(s,t)Qo(s,t)dxdt xo Yo
+ p(x, y ) / c ( s , y)fo(s,
y)exp
c(sl, y)
xo
x fl(Sl, y)G(b(sl, y)fl-l(Sl, y))p(sl,
y)dsl) ds] , (5.6.27)
where f o(x, y) is defined by the right-hand side of
f (x, y)----a(x, y)+ b(x, y) [r(x, y)Q(x, y)
xy
]
(5.6.28)
xo Yo
replacing a(x, y) by ol(x, y)G(a(x, y)ct-l(x, y)), b(x, y) by fl(x, y)G(b(x, y) • fl-1 (x, y)) and Q(x, y) by Qo(x, y). Again, if p(x, y) - O, then
504
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
u(x, y) < a(x, y) + b(x, y)G -1 Jr(x, y)Qo(x, y) x
y
y
+h(x, y)//g(s,t)Qo(s,t)dsdt+q(x, y)/c(x,t)fo(x,t) xo Yo
Yo
x e x p ( / c ( x , tl)fl(X, tl)G(b(x, tl)fl-l(x, tl))
(5.6.29) where Qo(x, y) and f o(x, y) are as defined above. I-7 Proof: Rewrite (5.6.24) as
u(x, y) < or(x, y)a(x, y)ot-1 (x, y) + ~(x, y)b(x, y)~-I (x, y)
[j
xG -1 p(x, y) c(s, y)G(u(s, y))ds xo
y
+ q(x, y) / c(x, t)G(u(x, t))dt yo X
y
+ r(x, y ) f f c,s xo yo
xy
+h(x,y,ffg(s,O xO YO
t))ds dt
(jj C(Sl, tl) xo Yo
• G(u(sl, tl))ds1 dtl) dsdt I .
(5.6.30)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
505
Since G is convex, submultiplicative and monotonic we have
G(u(x, y)) < or(x, y)G(a(x, y)ot-1 (x, y)) at- ~(x, y)G(b(x, y),6 -1 (x, y)) x
[j
p(x, y)
c(s, y)G(u(s, y))ds xo
Y
+ q(x, y) / c(x, t)G(u(x, t))dt yo x
y
+r(x,y)ffc(s,t)G(u(s,t))dsdt
xy (jr
xo Yo
c s ,ta x0 Y0
XXo Y0
x G(u(sl, tl))dsl dtl) dsdt] .
(5.6.31)
The estimate in (5.6.25) follows by first applying Theorem 4.8.8 with a(x, y) - o~(x, y)a(a(x, y)c~ -1 (x, y)), b(x, y) -- fl(x, y)G(b(x, y)~-I (x, y)), and u(x, y) - G(u(x, y)) and then applying G -1 to both sides of the resulting inequality. The rest of the proof when q(x, y) = 0 and p(x, y) = 0 follows by a similar argument to that in the last part of the proof of Theorem 4.8.8, in view of the proof of the first part of this theorem, with suitable modifications. The details are omitted here. l
Theorem 5.6.6 Let u(x, y), a(x, y), b(x, y), c(x, y), p(x, y), q(x, y),
r(x, y), h(x, y), g(x, y), Po(xo, yo) and P(x, y) be as in Theorem 5.6.5. Let H, H -1 be as in Theorem 5.6.2. Let V(s, t; x, y) be the solution of the characteristic initial value problem (5.6.20) in which M is the adjoint operator of the operator L defined by (5.6.21) with a l - - -H(b)cq, a2 = - H ( b ) c p and a3 = - [ g + H(b)c(r + h)]. Let W(s, t; x, y) be the solution of the characteristic initial value problem (5.6.22) in which N is the adjoint operator of the operator T defined by (5.6.23) with bl = - H (b)cq, b2 = - H (b)cp, and
506
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
b3 - - H (b)c(r - h). Let D + be a connected subdomain of D which contains P and on which V > 0 and W > 0 (Figure 4.5). IfR C D + and u(x, y) satisfies
[j
u(x, y) < a(x, y) + b(x, y)H -1 p(x, y)
c(s, y)H(u(s, y))ds xo
Y
+ q(x, y) / c(x, t)H(u(x, t))dt Yo x
y
x
y
+r(x,y)f./'c(x,t)H(u(s,t))dsdt+h(x,y)ffg(s,t) xo Yo
xo Yo
then u(x, y) also satisfies u(x, y) < H -1 [n(a(x, y)) + H(b(x, y)) [p(x, y ) f c(s, y)H(u(s, y))ds xo Y
+ q(x, y) / c(x, t)H(u(x, t))dt + r(x, y)Ql(X, y) Yo
xy -t-h(x, y)//g(s,t)Ql(s,t)dsdt
1] ,
(5.6.33)
xo Yo
where Q1(x, y) is defined by the right-hand side of (5.6.26) by replacing a(x, y) by H(a(x, y)) and b(x, y) by H(b(x, y)). Further, if q(x, y) - O, then u(x,y)
[/c(s,y)fl(s,Y) xo
xexp(fs C(Sl'y)H(b(sl'Y))p(sl'y)dsl)ds]] ' (5"6"34)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
507
where f l(X, y) is defined by the right-hand side of (5.6.28) by replacing a(x, y) by H(a(x, y)), b(x, y) by H(b(x, y)), and Q(x, y) by Ql(X, y). Again, if p(x, y)= O, then
[
u(x, y) < H -1 f l(x, y)+ H(b(x, y))q(x, y)
[/
c(x, t)f l(x, t)
x e x p ( / c ( x , tl)H(b(x, tl))q(x, tl)dtl)dt]] , (5.6.35) where Q1 (x, y) and f l(x, y) are as defined above. D Proof: Since H is subadditive, submultiplicative and monotonic, from (5.6.32) we have
[ /
H(u(x, y)) < II(a(x, y)) + H(b(x, y)) p(x, y)
c(s, y)H(u(s, y))ds
xo Y P
+ q(x, y) / c(x, t)H (u(x, t)) dt t,i
Yo x
y
+r(x, y)f/c(s,t)H(u(s,t))dsdt
xy
xo Yo
xo Yo
(/r \ xo Yo
X H(U(Sl, tl))dsl dtl) dsdt] .
(5.6.36)
The desired bound in (5.6.33) follows by first applying Theorem 4.8.8 to (5.6.36) with a(x, y)= H(a(x, y)), b(x, y) = H(b(x, y)) and u(x, y)= H(u(x, y)) and then applying/4 -1 to both sides of the resulting inequality. Further, by setting q(x, y ) = 0 and p(x, y ) = 0 in (5.6.32) and applying Theorem 4.8.8 we obtain the desired bounds in (5.6.34) and (5.6.35).
I
508
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES R e m a r k 5.6.2 We note that the functions V(s, t; x, y) and W(s, t; x, y)
involved in Theorems 5.6.5 and 5.6.6 are the well-known Riemann functions relative to the point P(x, y). The existence and continuity of the Riemann function is well known and may be demonstrated by the method of successive approximations (Counant and Hilbert, 1962). A
5.7 Inequalities in Many Independent Variables The integral inequalities involving functions of many independent variables which provide explicit bounds on unknown functions play a fundamental role in the development of the theory of partial differential equations. The last few years have witnessed a great deal of research concerning such inequalities and their applications in the theory of partial differential equations. This section deals with some basic inequalities established in Pelczar (1963), Headley (1974), Beesack (1975) and Pachpatte (1981c) which provide a very useful and important device in the study of many qualitative properties of the solutions of various types of partial differential, integral and integrodifferential equations. Pelczar (1963) initiated the study of some inequalities for a broad class of operators. In order to present the main results in Pelczar (1963) we need the following definitions found there. We call a set P partly ordered if for some pairs of elements x, y ~ P a relation x < y is defined in such a way that: (a) for each x ~ P, x < x, (b) i f x < y and y < x, then x = y and (c) i f x < y and y < z, then x < z. Let P be a partly ordered set and Q c P; we call z the upper bound of
Q in P if z ~ P and if x ~ Q, then x < ~. We call ~ the supremum of the set Q (abbreviated sup Q) if ~ is an upper bound of Q in P and if x is an upper bound of Q in P, then x < ~. Each partially ordered set can have at most one supremum. The set P will be said to satisfy the condition (II) if the difference x - y ~ P is defined for each x, y ~ P in such a way that (d) if x < y, then for each x ~ P is x - z < y - z, (e) there exists an element 0 ~ P, such that for each x ~ P is x - 0 - x and (f) x - y if and only if x - y = 0. The set P will be said to satisfy the condition (II*) if, for each x, y ~ P, there exists in P, z = sup{x, y}.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
509
The main result established by Pelczar (1963) is embodied in the following theorem. T h e o r e m 5.7.1 A s s u m e that: (al)
The set P is not empty, partly ordered and fulfils the conditions (II) and
(a2)
the functions W (x) and L(x) are defined in the set P, a n d are such that
(ii*), W ( P ) C P and L ( P ) C P,
(a3)
if x <_ L(x), then x < O,
(a4)
if x < y, then W (x) < W (y),
(as)
if 0 < W (x) - W (y), then W (x) - W (y) < L ( x - y),
(a6)
w
(a7)
v
is a solution o f the equation w = w (w),
(5.7.1)
v 5_ W ( v ) .
(5.7.2)
~ P is such that
Then we have
v < w.
(5.7.3) [2
Proof: Let z be the supremum of the set {w, v}. Then w
and
v
From (5.7.4), condition (c) and assumption w=W(w)
and
(5.7.4)
(a4) it follows that
v
(5.7.5)
From (5.7.5) and the definition of z as the sup {w, v} it follows that z < W (z). From condition (d) we have z - w _< W(z) - w - W(z) - W(w). Then, from (5.7.4) and the assumption (as) it follows that z - w < L(z - w).
In view of the assumption
(5.7.6)
(a3), the inequality (5.7.6) implies that z - w - 0.
Hence z - w, which means (5.7.3) holds. II
510
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
R e m a r k 5.7.1 It is easy to see that we can assume that the function L(x) is defined only for 0 < x (x 6 P). It is also easy to see that under the assumptions of Theorem 5.7.1, equation (5.7.1) can have at most one solution in P. A As an application, consider the following equation u(x) = f ( x ) + f
F(x, y, u(y))dy,
(5.7.7)
t /
E
where x - (Xl . . . . . Xn), y = (Yl . . . . . Yn) and E is an n-dimensional set, and the inequality
v(x) < f (x) + / " F(x, y, v(y)) dy.
(5.7.8)
t.i
E
By making use of Theorem 5.7.1, Pelczar (1963) proved the following important result. Theorem 5.7.2 Assume that (bl) (b2) (b3) (b4)
F (x, y, z) is defined and continuous in E x E x R, R -- (-oo, oo ), f (x) is defined and continuous in E, if z < 2, then F(x, y, z) < F(x, y, z), IF(x, y, z) - F(x, y, z)l < l(x, y, Iz - zl), where the function l(x, y, z) is defined in E x E x R and such that if w(x) < f l (x, y, w(y)) dy, E
then
w(x) <_o, (bs) u(x) is a solution of equation (5.7. 7) in E, (b6) v(x) is a continuous function defined in E and fulfils the inequality (5.7.8). Then in the set E we have v(x) < u(x).
(5.7.9)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
511
Proof: It is easy to see that the set P of all continuous functions w(x) defined for x 6 E fulfils the assumptions of Theorem 5.7.1 concerning the set P. Now, for u 6 P, we define W(u) as the function which at the point u has the value at the point z defined by the formula
z(x) -- f (x) + f F (x, y, u(y)) dy. E I I
By L(u) we denote the function defined in P, which at the point u 6 P has the value at the point w defined by the formula
w(x) -- f l (x, Y, u(y)) dy. E tI
It is easy to see that the functions W(u) and L(u) fulfil all assumptions of Theorem 5.7.1. Hence for each solution w of the equation w-
W(w),
and each function, which fulfils the inequality
v <_ W(v), we have v < w. This means that for each solution of the equation (5.7.7) and for each function v(x) which fulfils the inequality (5.7.8), we have
v(x) <_ w(x), and the proof is complete. II R e m a r k 5.7.2 In particular if we put l (x, y, z) = kz and assume that the function F(x, y, z) is bounded and that k < / z ( E ) , where/z(E) is the measure of the set E, then all the assumptions of Theorem 5.7.1 are satisfied. A In order to establish the next theorem, given by Headly (1974), we require the following result, given in Beesack (1975, p. 88). T h e o r e m 5.7.3 Let G be an open set in R N, and for x, y ~ G let
G(x, y) -- {z ~ R N " zj - )Uxj + ( 1 - )U)yj,
O < )~j < 1,1 < j < N},
~12
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
denote the rectangular parallelepiped with one diagonal joining the points x, y. Let the points x ~ y ~ G be such that Go = G(x ~ y) C G, and let the functions a(x), k(x, t, z) be real-valued and continuous on Go and on Gr x R respectively, where Gr -- {(x, t ) ' x ~ Go, t ~ G(x ~ x)}. Suppose also that k is nondecreasing in z for each (x, t) ~ GT and that Ik(x, t, z)l ~ h(t)g(Izl),
(5.7.10)
for (x,t,z) ~ Gr x R, where h ~ L(Go) and g is continuous and nondecreasing on R+ with f l ds/g(s) - c~. Then the integral equation u(x)
= a(x) +
f
k(x, t, u(t))dt,
(5.7.11)
q.r
G(x~
has a solution which is continuous on Go. Moreover, if {En} is a strictly decreasing sequence with lim En = 0, and if Un is a continuous solution on Go of the integral equation Un(X) - a(x) + En -3t- l
k(x, t, Un(t))dt,
(5.7.12)
t.i
G(x~
then U(x) = lim Un(x) exists uniformly on Go, and U(x) is the maximal solution of (5.7.11). D In case x ~ _< y (that is, x ~ < Yi for 1 < i _< N), so that x ~ Go - G(x ~ y) implies x ~ x < y and t ~ G(x~ implies x~ t _< x, this result is a special case of results in Walter (1970), namely Theorem II, p. 131 (with n - 1, H i ( x ) - G(x~ together with Remark VI(/3), pp. 136-7, with n -- 1, and no /~s) and Theorem VII, p. 139. Cases where x ~ 2~ Y can be reduced to this case by appropriate change of independent variables. For example, if y~ < x~o for certain subscripts ct while x~ _< y~ for the remaining subscripts 13, then the change of variables ~ = - x ~ x ~ - x~, y---~= - y ~ , y ~ - y~, ~ - - t ~ , ~ - t~, and ~ - - x ~ , ~ - x~ reduces the integral equations (5.7.11), (5.7.12) to equations over Go - G(x-~ Y) having x-~ < y, to which Walter's results do apply.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
513
As noted in Remark VI(/3) of Walter (1970, p. 136), if hypothesis (5.7.10) is deleted, then equation (5.7.11) may have only a local continuous solution, that is a solution defined in G(x ~ yO) for some yO ~ G(xO, y). In this case, if G(x ~ yO) is also a common domain of existence of solutions of (5.7.12), then the final conclusion of the theorem still holds, but only on G(x ~ yO). Headley (1974, Theorem 1) considered the case of Theorem 5.7.3 having
k = k(t,z) continuous on Go x R and nondecreasing in z. Hypothesis (5.7.10) was overlooked in Headley (1974) and in the case k = k(t, z) of the following theorem given by Headley (1974, Theorem 2). T h e o r e m 5.7.4 Let G, Go = G(x ~ y) and the functions a, k be as in
Theorem 5.7.3, and let the function b be continuous on Go and satisfy the inequality. f v(x) < a(x) + ]
k(x, t, v(t)) dt,
x 6 Go.
(5.7.13)
tl
G(x~ Then v(x) < U(x),
x ~ Go,
(5.7.14)
where U is the maximal solution of (5.7.11) on Go. rq Proof: We may assume that x ~ ~ yi for 1 _< i _< N, since otherwise U(x) - a(x) and (5.7.13) is v(x) <_ a(x). For n _> 1, let Un be a continuous solution of the integral equation (5.7.12) on Go, where En -- n -1 for example. Then
v(x) < Un(X) for x 6 Go,
n >_ 1.
(5.7.15)
For, if this were false for some n and some x ~ Go, then because
1 v(x ~ < a(x ~ < a(x ~ + - - u. (x~ n
it would follow from the continuity of v and
Un
that for some point z ~ Go
with z ~ x ~ we would have v(x) < Un (x) for x ~ G(x ~ z ) -
Un(z). But then v(Z) < a(z) +
f [ J
G(x~
k(z, t, v(t)) dt
{z} but v(z) -
514
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
'
/
< a(z) + - + n
k(z,
t, U n ( t ) )
dt
- - Un ( z ) .
G(x~
It follows that (5.7.15) must hold. Taking limits as n ~ oe, (5.7.14) now follows from Theorem 5.7.3.
II R e m a r k 5.7.3 If hypothesis (5.7.10) is deleted, then (5.7.14) will hold on any subset G(x ~ yO) C Go such that equations (5.7.11) and (5.7.12) have continuous solutions on the common domain G(x ~ yO), if such y 0 6 Go exists. This was the case considered in Headley (1974), and for N = 2 also by Rasmussen (1976). A In the further discussion, some useful integral inequalities in n independent variables given by Pachpatte (1981 c) are presented, which are motivated by a well-known integral inequality due to Wa2eski (1969). We use the same notation as given in Section 4.9 without further mention. Pachpatte (1981c) gave the following general version of Wa2eski's inequality given in Wa2eski (1969). Theorem 5.7.5 Let u(x) and a(x) be nonnegative continuous functions defined on ft. Let k(x, y, z) and W (x, z) be nonnegative continuous functions defined on f22 • R and g2 x R, respectively, and nondecreasing in the last variables, and k(x, y, z) be uniformly Lipschitz in the last variable. If
u(x) < a(x) + W
(j) x,
k(x, y, u(y))dy
,
(5.7.16)
xo
then u(x) <_ a(x) + W(x, r(x)),
(5.7.17)
for x ~ f2, where r(x) is the solution of the equation x
r(x) -- / k(x, y, a(y) + W (y, r(y))) dy,
(5.7.18)
xo
existing on f2. 5
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
515
Proof: Define a function z(x) by x
z(x) - / k(x, y, u(y))dy,
(5.7.19)
,J
xo
then (5.7.16) can be restated as
u(x) < a(x) + W(x, z(x)).
(5.7.20)
Using the monotonicity assumption on k and (5.7.20) in (5.7.19) we obtain x
z(x) < / k(x, y, a(y) + W(y, z(y)))dy.
(5.7.21)
x0
Now a suitable application of Theorem 5.7.2 or Theorem 5.7.4 to (5.7.18) and (5.7.21) yields z(x) < r(x), (5.7.22) where r(x) is the solution of equation (5.7.18). Now, using (5.7.22) in (5.7.20) we obtain the desired bound in (5.7.17). II The following inequality established by Pachpatte (1981c) combines the features of two inequalities, namely, the n independent variable generalization of Wendroff's inequality (Bondge and Pachpatte, 1979a) and the integral inequality given by Headley (1974, Theorem 2), and can be used more effectively in the theory of certain integral and integro-differential equations involving n independent variables. Theorem 5.7.6 Let u(x), f (x), g(x), q(x), c(x) be nonnegative continuous
functions defined on f2, with f (x) > 0 and nondecreasing in x and q(x) > 1. Let k(x, y, z) and W(x, z) be nonnegative continuous functions defined on ~"22 • R and f2 x R, respectively; k(x, y, z) is nondecreasing in x and z and is uniformly Lipschitz in z and W (x, z) is nondecreasing in both x and z. If u(x) < f (x) + q(x)
[/
g(y)u(y) dy +
xo
+W(x,
fk(x,y,u(y))dy), x0
/
xo
g(y)q(y)
(/) ] c(s)u(s) ds
dy
xo
(5.7.23)
516
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for x E f2, then u(x) < Eo(x)[f (x) + W(x, r(x))],
(5.7.24)
for x ~ f2, where
EJ
Eo(x) -- q(x) 1 +
(J
g(y)q(y) exp
q(s)[g(s)+ c(s)] ds
xo
xo
)] dy
,
(5.7.25)
and r(x) is the solution of the equation x
(5.7.26)
r(x) = f k(x, y, Eo(y)[f (y) + W(y, r(y))])dy, xo
existing on f2. [3
Proof: Define a function
re(x) by
m ( x ) = f ( x ) + W ( x , fk(x,y,u(y))dy),
(5.7.27)
x0
then (5.7.23) can be restated as
u(x) < m(x) + q(x)
[j
g(y)u(y) dy +
x0
Since
m(x) is positive,
J (J) ] g(y)q(y)
c(s)u(s) ds dy .
xo
nondecreasing and
xo
q(x)
(5.7.28) > 1, we observe from (5.7.28)
that U(X)
m(x) -< q(x) I 1 f+
xo
c(s) u(s) ds dy . g(Y)m(y) dy + g(y)q(y) m(s) ) ] u(y) j (f x0
xo
(5.7.29) Define a function
z(x)
such that
g(Y)rnr dy +
z(x) = 1 + xo
z(x) -
1 on Xj = X jo,
u(s)
g(y)q(y) xo
l <_j < n , _
c(s)m(O xo
dy,
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES then
517
x
D 1 . . . DnZ(X) -- g(x) U(~'mlx)+ g(x)q(x) f c(s) m(s)U(S)ds, xo
which in view of (5.7.29) implies
D1. . . DnZ(X) <_ g(x)q(x)
[ j Z(X) +
c(s)q(s)z(s) ds
xO
]
.
(5.7.30)
If we put x
v(x) -- z(x) + / c(s)q(s)z(s) ds, (5.7.31)
x0
v(x) = z(x) on xj -- xj0 . . l <.j < .n then (5.7.32)
D1. . . Dnv(X) = D1. . . DnZ(X) -'[-c(x)q(x)z(x). Using the facts that D1 . . . D n z ( X ) < g(x)q(x)v(x) z(x) < v(x) from (5.7.31) in (5.7.32) we obtain
from (5.7.30) and
D1. .. Dnv(X) < q(x)[g(x) + c(x)]v(x).
(5.7.33)
From (5.7.33) and using the facts that Dnv(x) >_ 0, D 1 . . . Dn-lV(X) >__0 and v(x) > 0, we observe that
D1. . .Dnv(x) (Dn v ( x ) )(D1 9 9 9 D n - 1 v ( x ) ) < q(x)[g(x) + c(x)] + v(x) v2(x) i.e.
Dn ( DI ""v--~) Dn-l V(X))
< q(x)[g(x) + c(xl].
(5.7.34)
By keeping X1 . . . . . Xn-1 fixed in (5.7.34), setting x , , - sn and then integrating with respect to Sn from x n0 to xn we have x
D1. . .Dn-lV(X)
v(x)
< / q ( x l , . . . , Xn-1, Sn)[g(Xl . . . . , Xn-1, Sn ) xo "+" r
.....
X n - 1 , Sn )]
dSn.
(5.7.35)
518
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Again as above, from (5.7.35) we observe that x
Dn-1
(ol'''on-2V(X))
v--~)
<_
/
q(xl
....
, X n - 1 , Sn
)[g(Xl . . . . , X n - 1 , Sn )
xo X n - 1 , Sn)] dsn.
"1-" C ( X 1 , . . . ,
By keeping Xl . . . . , X n - 2 and X n fixed in the above inequality, setting X n - 1 - sn-1 and then integrating with respect to Sn-1 from Xn_ 1 ~ to Xn-1 we have Xn- 1 Xn 9
q(Xl . . . .
<
v(x)
o
Xn- 1
, X n - 2 , S n - 1 , Sn)
gO
x [g(x1 . . . . .
X n - 2 , S n - 1 , Sn) X n - 2 , S n - 1 , Sn)] dsn d s n - 1 .
+ C(X1,...,
Continuing in this way we have x2
Xn
DlV(X)
v(x)
< f " " f q(xl , s2, . . . , Sn )[g(xl , s2, . . . , Sn )
x~
xO
-]- C(X1, $2 . . . . .
Sn)] d s n . . ,
ds2.
Now keeping x2 . . . . , Xn fixed in the above inequality, we set Xl - - s l and then integrating with respect to Sl from x ~ to Xl we have
v(x)
Substituting this bound on v(x) in (5.7.30), setting Xn - - Yn and then integrating both sides with respect to Yn from x n0 to Xn, then setting Xn-1
Yn-1 and integrating with respect to Yn-1 from X nO_ 1 t o Xn-1, and continuing in this way, finally setting Xl - Yl and then integrating with respect to yl from x ~ to Xl, we obtain
X
( j )
z(x) < 1 + / g ( y ) q ( y ) e x p x0
q(s)[g(s) + c(s)] ds x0
dy.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
~19
Substituting this bound on z(x) in (5.7.29) we have
u(x) < Eo(x)m(x),
(5.7.36)
where Eo(x) is defined by (5.7.25). From (5.7.27) and (5.7.36) we have
u(x) < Eo(x) [ f (x) + W (x, f k(x, y, u(y)) dy) ] .
(5.7.37)
xo
Now a suitable application of Theorem 5.7.5 yields the desired bound in (5.7.24).
II R e m a r k 5.7.4 We note that in the special case when c ( x ) - 0, the inequality established in Theorem 5.7.6 reduces to another interesting inequality which can be used in some applications. A Another interesting and useful integral inequality given by Pachpatte (1981c) in n independent variables involving two nonlinear functions on the fight-hand side of the inequality is embodied in the following theorem. Theorem 5.7.7 Let u(x), f (x), g(x), q(x), k(x, y, z), W (x, z) be as in
Theorem 5.7.6 and furthermore f (x) > 1. Let H" R+ --+ R+ be a continuously differentiable function with H(u) > 0 for u > O, H'(u) > 0 for u > 0 and satisfies (1/v)H (u) < H (u/v) for v > 1, u > 0 and H (u) is submultiplicative for u > O. If
X
)
u(x) < f (x) + q(x) ;ag(y)H(u(y))dy + x0
x,
k(x, y, u(y))dy
,
xo
(5.7.38)
for x ~ f2, then for x ~ f21 C f2, u(x) < El(x)[f (x)-b W(x, r(x))], where
(5.7.39)
El(x)-q(x)G-l[G(1)Wfg(y)H(q(y))dy], (5.7.40)
xo
520
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
in which
fas v
G(v) =
H(s)'
v > 0, v0 > 0,
(5.7.41)
v0
G -1 is the inverse of G and x
G(1) + / g ( y ) H ( q ( y ) ) d y ~ Dom(G-1), xo Or
for all x ~ ~, and r(x) is a solution of the equation x D
r(x) = / k ( x , y, E~ (y)[f (y) + W(y, r(y))]) dy,
(5.7.42)
t.I
x0
existing on ~2. D Proof: We note that since H'(u) >_0 on R+, the function H is monotonically increasing on (0, ~ ) . Define a function m(x) as in the proof of Theorem 5.7.6 given by (5.7.27); then (5.7.38) can be restated as x D
u(x) < re(x) + q(x) / g(y)H(u(y))dy.
(5.7.43)
L I
x0
Since re(x)> 1 and nondecreasing and q(x)> 1, then from (5.7.43) we observe that u(x)
Define a function z(x) by x
z(x)=l+fg(y)H(u(Y>) xo
z(x) -- 1 on
Xj
=
0
X j,
then D1. . . Dnz(X) -- g(x)H
l <_j_ < n _,_
u(x) ) m(x) '
(5.7.44)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
521
which in view of (5.7.44) and the submultiplicative character of H implies (5.7.45)
D1. . . DnZ(X) < g(x)H(q(x))H(z(x)).
From (5.7.45) and using the facts that DnH(Z(X)) > O, D1...Dn-lZ(X) >_ 0 and H ( z ( x ) ) > 0, we observe that D1. . . Dn Z(X)
n (z(x) )
[Dn H (Z(X) )] [ D 1
< g(x)H(q(x)) +
Dn-1Z(X)]
9 9 9
[H (z(x) )] 2
i.e.
Dn ( D I ' ' ' D n - l z ( X ) ) it-(-~ff)-)
<_ g ( x ) H ( q ( x ) ) .
Now by following an argument similar to that in the proof of Theorem 5.7.6, with suitable modifications, we obtain X2
< H(z(x)) -
Xn
... gO
g ( x l , Y2 . . . . , y n ) H ( q ( x l , Y2 . . . . .
yn))dyn..,
dy2.
gO
(5.7.46) From (5.7.41) and (5.7.46), we observe that X2
Xn
D 1 G ( u ( x ) ) < . f . . . f g(xl , y2 . . . . . Yn ) gO gO x H ( q ( x l , Y2 . . . . .
Yn)) d y n . . ,
dy2.
By keeping x2 . . . . , Xn fixed in the above inequality, setting Xl -- yl and then integrating with respect to Yl from x ~ to Xl we have
u(x) < G -~
[ r G(1)+
xo
g(y)H(q(y))dy
]
.
The rest of the proof is immediate by analogy with the last argument in the proof of Theorem 5.7.6. The subdomain ~"~1 of ~ is obvious. II Pachpatte (1981c) gave the following inequality, which can be used in more general situations.
522
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
5.7.8 Let u(x), f (x), g(x), k(x, y, z) and W(x, z) be as in Theorem 5.7. 7. Let H ' R + --+ R+ be a continuously differentiable function with H (u) > 0 for u > O, H'(u) > 0 for u > 0 and satisfying (1/v)H (u) < H (u/v), for v > 1, u > O. If Theorem
j(j (/)
u(x) < f (x) +
g(y)
u(y) +
xo
+ W
x,
g(s)H(u(s))ds
)
dy
x0
k(x, y, u(y))dy
,
(5.7.47)
xo
for x ~ ~2, then for x ~ ~'22 C ~'2, u(x) < Ez(x)[f (x) + W(x, r(x))],
(5.7.48)
where x
[ / ] y
E2(x) -- 1 + f g(y)F -1
g(s)ds
F(1) +
x0
dy,
(5.7.49)
x0
in which o"
F(cr)
-
/
ds , s + H (s)
cr > 0, cro > 0,
(5.7.50)
o-0
F-1 is the inverse function of F and x
F(1) + / g ( s ) d s
Dom(F-1),
t] x0
for all x ~ f2e, and r(x) is a solution of the equation x
F(X)
/ k(x, y, E e ( y ) [ f (y) -F W(y, r(y))])dy,
(5.7.51)
tJ Xo
existing on f2. D The details of the proof of this theorem follow by an argument similar to that in the proof of Theorem 5.7.7, together with the proof of Theorem 5.4.1, and the details are omitted here.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
523
Pachpatte (1981c) has established the following generalization of the integral inequality given by Young (1973). Theorem 5.7.9 Let u(x), a(x), b(x), c(x) and g(x) be nonnegative continuous functions defined on f2. Let f (x), k(x, y, z) and W(x, z) be as in Theorem 5.7.6. Let v(y; x) and e(y; x) be the solutions of the characteristic initial value problems (-- l)nvyl...y n (y; x) - [a(y)b(y) + a(y)g(y) + c(y)lv(y; x) - 0 in f2,
(5.7.52) v(y; x ) -
1 on Y i - xi,
1 <_ i <_ n,
and (-- 1)neyl...y n (y; x) - [a(y)b(y) - c(y)]e(y; x) -- 0 in f2,
e(y, x ) -
1 on Yi - xi,
(5.7.53)
1 < i < n,
respectively, and let D + be a connected subdomain of f2 containing x such
]
that v > O, e > 0 for all y ~ D +. If D C D + and
X (j)
u(x)
xo
g(s)u(s) ds
dy
x0
(5.7.54)
xo
for x ~ S2, then
(5.7.55)
u(x) < E3(x)[f (x) + W(x, r(x))], where
E3 (x) -- 1 + a(x)
IJ{
e(y; x)
b(y) + c(y)
xo
xv(s;y)ds)}dy]
(J
[b(s) + g(s)]
xo
,
(5.7.56)
524
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
and r(x) is a solution of the equation x
r(x) - i k(x, y, E3(y)[f (y) + W(y, r(y))])dy,
(5.7.57)
xo
existing on f2. D Proof: Define a function m(x) as in the proof of Theorem 5.7.6 given by (5.7.27); then (5.7.54) can be restated as
u(x) < m(x) + a(x)
[:
]
i (:)
b(y)u(y) dy +
xo
c(y)
xo
g(s)u(s) ds d y .
xo
(5.7.58) Since m(x) is positive nondecreasing, we observe from (5.7.58) that
u(x) re(x) -< 1 + a(x)
[i
u(y)) dy + b(Y)m(y
xo
:(i c(y)
xo
)]
g(s) u(s) m(s) ds dy . xo
(5.7.59) Define a function z(x) such that
z(x) -- f b ( y ) u(y)...,,, ~ dy + i c(y) ( : x0
z(x) = 0 on Xi
x0
g ( s )u(s) ~ ds dy,
x0
l
- - X O,
then we obtain
D1. . .Dnz(X) = b(X)m(x) + c(x)
(:U"') g(s)~
ds ,
xo
which in view of (5.7.59) implies
Da...Dnz(x)
(5.7.60) Adding c(x)z(x) to both sides of the above inequality we have
D1. . . Dnz(X) + c(x)z(x) < b(x)[1 + a(x)z(x)] + C(X) [Z(X) + : g(s)[ 1 + a(s)z(s)] ds] . x0
(5.7.61)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
525
If we put x
q/(x) -- z(x) + f
g(s)[ 1 -I- a(s)z(s)] ds, (5.7.62)
xo
~p(x) -- z(x) -- 0 on
Xi
-"
X O,
1 < i < n,
then we obtain D1.
Using
. . Dn ~(x)
-- D1.
. . DnZ(X)
D1...Dnz(x)
-k- g ( x ) [ 1
+
(5.7.63)
a(x)z(x)].
from
+a(x)z(x)]+c(x)q/(x)
(5.7.61)
and
z(x) < q/(x) from (5.7.62) in (5.7.63) we have
L[ap] -- D 1 . . . Dn q/(x) - [a(x)b(x) + a ( x ) g ( x ) + c(x)]q/(x) < [b(x) + g(x)]. Furthermore,
all
pure
Xl,...,Xi-l,Xi+l,...,Xn
mixed
(5.7.64) derivatives
of
~
with
respect
to
up to order n - 1 vanish on xi = x ~ 1 < i < n.
If w is a function which is n times continuously differentiable in D, then n
w L ~ - ~pMw -- Z
( - 1)k-lDk[(DoD1 999Dk-1 w ) ( D k + l . . . DnDn+l ~)],
k=l
(5.7.65)
where M w = ( - 1 ) n D 1 . . . D n w ( X ) - [a(x)b(x) + a ( x ) g ( x ) + c(x)]w(x),
with Do -- Dn+l --- I the identity. By integrating (5.7.65) over D, using s as the variable of integration, and noting that ap vanishes together with all its mixed derivatives up to order n - 1 on Sk = x ~ 1 < k < n, we then obtain n
S
z<-,,'-' i
D
k=l
Sk__Xk
(5.7.66) where d s ' - d s 1 . . , dsk-1 dSk+l . . . dsn. Now let w be chosen as the function v satisfying (5.7.52). Since v -
1
on Sk = xk, 1 <_ k <_ n, it follows that D1 . . . D k - l V(S; x) -- 0 on Sk -- Xk for
526
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
2 < k < n. Thus (5.7.66) becomes
f v(s; x)L~(s) ds - / v(s;x)D2... Dn~(y) dy' - ~(x). O
(5.7.67)
Sl = X l
By the continuity of v and the fact that v - 1 on s -- x, there is a domain D + containing x on which v > 0. Now multiplying (5.7.64) throughout by v and using (5.7.62) and (5.7.67), we obtain
g(s)]v(s; x)ds.
7t(x) < f [ b ( s ) + x0
Now substituting this bound on 7t(x) in (5.7.61) we obtain
Lz -- D1... Dnz(X)- [a(x)b(x)- c(x)]z(x)
<_b(x)+c(x)(f[b(s)+g(s)]v(s;x)ds). X { (j )} x0
Again, by following the same argument as above we obtain the estimate for z(x) such that
z(x) < / e(y; x) b(y) + c(y) xo
[b(s) +
g(s)]v(s; y)ds
dy.
xo
Now substituting this bound on
z(x) in
(5.7.59) we obtain
u(x) < E3(x)m(x), where E3 (x) is defined by (5.7.56). From the definition of
(5.7.68)
m(x) and
(5.7.68)
we have
u(x) < E3(x) [f (x) + W (x, f k(x, y, u(y))dy) ] . xo
Now a suitable application of Theorem 5.7.5 yields the desired bound in
(5.7.55). II
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
527
In concluding this section we note that there is no essential difficulty in establishing the further generalizations of the integral inequalities established by Pachpatte (1979a, 1980b,c,d) and many other theorems given in earlier sections in the set-up of Theorem 5.7.9. Since the details of these results are very close to those given in the proof of Theorem 5.7.9 with suitable modifications, they are left to the reader to fill in where needed.
5.8 Pachpatte's Inequalities III Pachpatte (1996c, 1997, in press d, unpublished manuscript) investigated some useful integral inequalities in two independent variables which claim their origin in the inequalities given in Theorems 3.4.1 and 3.4.3 respectively by Ou-Iang (1957) and Dafermos (1979). In this section we shall give some basic inequalities investigated in Pachpatte (1996c, 1997, in press d, unpublished manuscript) which can play a vital role in the study of certain new classes of partial differential and integral equations. Here we shall use the notation and definitions as given in Section 4.5 without further mention. The following inequality established by Pachpatte (in press d) deals with the two independent variable generalization of the inequality given by OuIang in Theorem 4.3.1. Theorem 5.8.1 Let u(x, y) and p(x, y) be nonnegative continuous functions defined for x, y c R+. If x
U2(X, y ) < c
2
y
+2ff
p(s, t)u(s, t) ds dt,
(5.8.1)
0 0
for x, y ~ R+, where c > 0 is a constant, then x y
u xy c+fJ
p(s, t) dt ds,
(5.8.2)
0 0
for x, y ~ R+. D
Proof: It is sufficient to assume that c is positive, since the standard limiting argument can be used to treat the remaining case. Let c > 0 and define a function z(x, y) by the fight-hand side of (5.8.1), then
528
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
z(x, 0) -- z(0, y)
=
r
U(X, y) <_ ~/Z(X, y) and D2Dlz(x, y) < 2p(x, y). ~/z(x, y)
(5.8.3)
From (5.8.3) and the facts that ~/z(x, y) > 0, Dlz(x, y) > 0, D2z(x, y) > 0 for x, y 6 R+, we observe that
D2Dlz(x, y) [Dlz(x, y)][D2~/z(x, y)] < 2p(x, y)+ ~/Z(X, y) [~/Z(X, y)]2 i.e.
Dlz(x, y)) < 2p(x, y). ~/z(x, y) -
D2
(5.8.4)
By keeping x fixed in (5.8.4), setting y - t and integrating with respect to t from 0 to y, and using the fact that DlZ(X, 0) = 0, we have y
Dlz(x, y) < 2 [ p(x, t)dt. ~/z(x, y) - J
(5.8.5)
0
Now keeping y fixed in (5.8.5) and setting x = s and integrating with respect to s from 0 to x we have x
V/z(x, y)
y
< c + / / p(s, t ) dt ds.
(5.8.6)
0 0
Using (5.8.6) in u(x, y) < ~/z(x, y), we get the required inequality in (5.8.2). II Another useful integral inequality involving functions of two independent variables established by Pachpatte (in press d) is given in the following theorem. Theorem 5.8.2 Let u(x, y), p(x, y) and c be as in Theorem 5.8.1. Let
g(u) be a continuously differentiable function defined on R+ and g(u) > 0 for u > O, g'(u) >_0 for u >_O. If x
y
a~
a~
u2(x, y) <_c 2 + 2 / / p ( s , 0
0
t)u(s, t)g(u(s, t))dtds,
(5.8.7)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
529
for x, y ~ R+, then for 0 <_ x <_ Xl, 0 ~ y <_ Yl, u(x, y)
[ jj
< ~2 - 1
~'~(c)
+
p(s, t)dt ds
0 0
]
,
(5.8.8)
where ds g(s) '
~2(r) -
(5.8.9)
r > 0, r0 > 0,
ro ~'~-1
is the inverse of f2 and
Yl are chosen so that
Xl,
x
y
f2(c)+ffp(s,t)dtdseDom(S2-1), 0 0
for all x, y lying in the subintervals 0 < x < Xl, 0 <_~y < Yl of R+. D
Proof: Since g'(u) _> 0 on R+, the function g is monotonically increasing on (0, c~). As in the proof of Theorem 5.8.1, we assume that c > 0 and define a function z(x, y) by the right-hand side of (5.8.7); then z(x, O ) z(O, y) - c 2, u(x, y) < x/z(x, y) and it is easy to observe that
D2Dlz(x, y) ~/z(x, y)
< 2p(x, y)g(v/z(x, y)).
(5.8.10)
Now by following the same arguments as in the proof of Theorem 5.8.1 below (5.8.3) up to (5.8.6), from (5.8.10) we get x
V/Z(X, y ) < c +
y
S l "s ,.,.+z
(5.8.11)
0 0
Now a suitable application of Theorem 5.2.1 to (5.8.11) yields
V/z(x, y)
< ~'2- 1
[ ij ~'2(c)Jr-
0 0
p(s, t)dt ds
]
.
(5.8.12)
Using (5.8.12) in u(x, y) < ~/z(x, y), we get the required inequality in (5.8.8). The subintervals for x and y are obvious. II
530
MULTIDIMENSIONALNONLINEARINTEGRALINEQUALITIES
The following theorem deals with fairly general inequalities established by Pachpatte (1997) which can be used in some applications. Theorem 5.8.3 Let u(x, y), f (x, y), g(x, y) and h(x, y) be nonnegative continuous functions defined for x, y E R+ and c be a nonnegative constant. (al) If
U2(x,y)
x,S1,1,d,ldSl)h s ,./, ,.] d,as for x, y ~ R+, then
u(x, y) <_ p(x, y) [1 +
xy[, (jj // ,~s,~,exp
0
0
\0
(5.8.13)
,,,~Sl ,1,
0
Sl] d,o]
(5.8.14)
for x, y ~ R+, where x
y
p(x,y)-c+ffh(s,t)dtds, 00 for x, y ~ R+. (a2) If
xy[
u2(x, y) < c2-t- 2 f f 0
f (s, t)u(s, t)
0
+ h(s, t)u(s, t)] dt ds,
(jj \0
(5.8.15)
g(sl, tl)U(Sl, tl)dtl dsl
)
0
(5.8.16)
MULTIDIMENSIONAL
NONLINEAR INTEGRAL INEQUALITIES
531
for x, y ~ R+, then u(x,y)
f ( s , t ) ( S f f g(sl, tl)dtldsl)dtds), 0
\0
0
(5.8.17)
for x, y ~ R+, where p(x, y) is defined by (5.8.15). D Proof: (al) We first assume that c > 0 and define a function z(x, y) by the fight-hand side of (5.8.13). Then z(x, O)- z(O, y ) - c2, u(x, y)< ~/z(x, y) and D2Dlz(x,y)<2v/z(x,Y)[f(x,y)(v/z(x,Y)
+
) ]
xy tl)dZ(S,,,l)dtldSl -t-h(x, y) ffg(sl,
.
(5.8.18)
0 0
From (5.8.18) and using the facts that D2z(x, y) > 0 for x, y 6 R+, we observe that
OlZ,X y,) _
~/z(x, y)> O, Dlz(x, y)>0,
y,( z,x y,
D2 \ ~ ~ , y)
x y
Jr f f g ( s l , tl)V/Z(Sl, tl)dtl 0 0
ds1
) ]
-+-h(x, y) . (5.8.19)
Now by following the same arguments as in the proof of Theorem 5.8.1 below (5.8.3) up to (5.8.6), from (5.8.19) we get
y,+jj o ,s st
)
+ff~,s,,,,VZ,Sl,l,d,,ds, 0 0
d, ds
(5.8.20)
532
MULTIDIMENSIONAL N O N L I N E A R I N T E G R A L INEQUALITIES
where p(x, y) is defined by (5.8.15). Since p(x, y) is positive and monotonic nondecreasing in x, y 6 R+, by applying Theorem 4.4.2, we get
[ jj (JJ tljdtlSl)dto]
V/Z(X, y) < p(x, y) 1 +
f (s, t) exp
0 0
[f(s1, tl)
\0
0
(5.8.21)
Now by using (5.8.21) in u(x, y) < ~/z(x, y) we get the required inequality in (5.8.14). If c > 0 we carry out the above procedure with c -t- e instead of c, where E > 0 is an arbitrary small constant, and subsequently pass to the limit as E ~ 0 to obtain (5.8.14). (a2) The proof follows by the same arguments as those given in the proof of (al) above with suitable modifications, and hence the details are omitted here. II A generalization of Theorem 5.8.1 in the other direction given by Pachpatte (unpublished manuscript) is contained in the following theorem. Theorem 5.8.4 Let u(x, y), f (x, y) and g(x, y) be nonnegative contin-
uous functions defined for x, y E R+ and c be a nonnegative constant. Let L" R3+ --+ R+ be a continuous function which satisfies the condition 0 < L(x, y, v) - L(x, y, w) < k(x, y, w)(v - w),
(L)
for x, y ~ R+ and v > w > O, where k" R3+ --+ R+ is a continuous function. If x
U2(X, y )
< c2
y
+ 2 f f [f (s, t)u(s, t)L(s, t, u(s, t)) 0
0
+ g(s, t)u(s, t)] dt ds
(5.8.22)
for x, y ~ R+, then u(x, y) < p(x, y) + q(x, y) exp ( fo fo f (s, t)k(s, t, p(s, t)) dt ds ) , (5.8.23)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
a33
for x, y ~ R+, where x
y
(5.8.24)
p(x,y)-c+ffg(s,t)dtds, 0 x
0
y
q(x, y) -- f / f (s, t)L(s, t, p(s, t)) dt ds, o o
(5.8.25)
for x, y ~ R+. V]
Proof: It is sufficient to assume that c is positive, since the standard argument can be used to treat the remaining case. Let c > 0 and define a function z(x, y) by the fight-hand side of (5.8.22), then z(x, 0) = z(0, y) = c 2, u(x, y) < ~/z(x, y) and
D2Dlz(x, y) < [ f (x, y)L(x, y, v/z(x, y))+ g(x, y)]. ~/z(x, y)
(5.8.26)
Now by following the same arguments as in the proof of Theorem 5.8.1 below (5.8.3) up to (5.8.6) we get x
y
~/z~x, y~ ~_~ , y~ + / / i ~ s . , ~ s . , .
,/z~s. ,~,,,s
~ ~~
, . 1 1 1
0
Define
0
v(x, y) by x
y
y)-//f(s,
v(x,
t)L(s, t, V/Z(S, t))dt ds.
(5.8.28,
0 0
From (5.8.28) and using the fact that hypothesis (L) we observe that x
v(x, y) <
~/z(x, y) < p(x, y)+ v(x, y) and the
y
f f e(s,,~(s ,, p(s,,~ +v(s, t))dtds 0
0 x
y
= q(x, y) + / f f (s, t){L(s, t, p(s, t) 0
0
+ v(s, t)) - L(s, t, p(s, t))} dt ds
534
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES x
< q(x, y ) +
ff
y
f(s, t)k(s, t, p(s, t))v(s, t)dtds.
(5.8.29)
0 0
Clearly q(x, y) is nonnegative and monotonic nondecreasing for x, y 6 R+. Now an application of Theorem 4.2.2 to (5.8.29) yields
v(x, y) < q(x, y)exp (fo ! f(s,t)k(s,t, p(s,t))dtds ) .
(5.8.30)
The desired inequality in (5.8.23) now follows by using (5.8.30) in (5.8.27) and the fact that u(x, y) < ~/z(x, y).
II Pachpatte (1996c) has established the inequalities in the following theorem which can be used more effectively in certain applications. Theorem 5.8.5 Let u(x, y), f (x, y) and h(x, y) be nonnegative continuous functions defined for x, y ~ R+ and c be a nonnegative constant. (bl) If U2 (X,
y)
< c 2 -+- 2B[x,
y, f (s, t)u 2(s, t) + h(s, t)u(s, t)].
(5.8.31)
for x, y ~ R+, then u(x, y) < p(x, y)exp(B[x, y, f (s, t)]),
(5.8.32)
for x, y ~ R+, where p(x, y) -- c + B[x, y, h(s, t)],
(5.8.33)
for x, y ~ R+. (b2) Let g(u) be a continuously differentiable function defined on R+ and g(u) > O f or u > O, g'(u) > 0 for u > O. If uZ(x, y) < c 2 + 2B[x, y, f (s, t)u(s, t)g(u(s, t)) + h(s, t)u(s, t)], for x, y ~ R+, then for 0 < x <
X1,
(5.8.34)
0 ~ y < Yl,
u(x, y) < f2-1[f2(p(x, y ) ) + B[x, y, f (s, t)]],
(5.8.35)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
where p(x, y) is defined by (5.8.33) and f2, 5.8.2 and Xl, Yl are chosen so that
~'2-1
535
are as defined in Theorem
f2(p(x, y)) + B[x, y, f (s, t)] ~ Dom(f2 -1), for all x, y lying in the subintervals 0 < x < Xl, 0 <_ y < Yl of R+. (b3) Let L and k be as in Theorem 5.8.4. If uZ(x, y) < c 2 + 2B[x, y, f (s, t)u(s, t)L(s, t, u(s, t)) (5.8.36)
+ h(s, t)u(s, t)], for x, y ~ R+, then u(x, y) < p(x, y ) + q(x, y)exp(B[x, y, f (s, t)k(s, t, p(s, t))]),
(5.8.37)
for x, y ~ R+, where p(x, y) is defined by (5.8.33) and q(x, y) -- B[x, y, f (s, t)L(s, t, p(s, t))],
(5.8.38)
for x, y ~ R+. U]
Proof: (bl) Assume that c > 0 and define a function z(x, y) by the fighthand side of (5.8.31); then u(x, y) <_ ~/z(x, y) and Dm 2 D n1z(x, y) _< 2[f(x, y)v/z(x, y ) + h(x, y)]. ~/z(x, y)
(5.8.39)
using the facts that z(x, y ) > O, D2z(x, y ) > 0 , D~-lD'~z(x, y) > O, for x, y 6 R+, we observe that (see Theorem 4.5.4) From
(5.8.39)
and
m--1 n
( D z _ DlZ(X, Y) D2 \ ~/z(x, y)
) <_ 2[f(x, y)v/z(x, y ) + h(x, y)].
(5.8.40)
Now by following the same arguments as in the proof of Theorem 4.5.4, below the inequality (4.5.48) up to (4.5.53), with suitable modifications, we get X Sn-2
S1
y
tm-1
tl
OlZ(X, y)< 2 / f . . . J r f ...f[f(s,t)v/Z(S,t) 4 z ( x , y) 0 0 00 0 0 + h(s, t)] dt d t l . . , dtm-1 ds dsl 999 dsn-2, (5.8.41)
536
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
Now keeping y fixed in (5.8.41), setting x - - S n - 1 and then integrating with respect to s,,-1 from 0 to x and using the fact that ~/z(0, y) - c, we have
v/z(x, y) <_ p(x, y)+ B[x, y, f (s, t)v/z(s, t)],
(5.8.42)
where p(x, y) is defined by (5.8.33). Clearly p(x, y) is positive and monotonic nondecreasing in x, y ~ R+. By applying Theorem 4.5.4 part (i) we get
V/z(x, y) <_ p(x, y)exp(B[x, y, f (s, t)]).
(5.8.43)
Now using (5.8.43) in u(x, y) <_ ~/z(x, y), we get the required inequality in (5.8.32). If c _ 0, we carry out the above procedure with c + E instead of c, where E > 0 is an arbitrary small constant, and subsequently pass to the limit as E --+ 0 to obtain (5.8.32). (ba) Assume that c > 0 and define a function z(x, y) by the right-hand side of (5.8.34), then u(x, y) <_ ~/z(x, y) and
D~D'~z(x, y) _< 2[f(x, y)g(v/z(x, y))+ h(x, y)]. ~/z(x, y)
(5.8.44)
Now by following the same steps as in the proof of part (bl) up to (5.8.42) we get
v/z(x, y) <_ p(x, y)+ B[x, y, f (s, t)g(v/z(s, t))],
(5.8.45)
where p(x, y) is defined by (5.8.33). The rest of the proof can be completed by following the ideas used in the proof of Theorem 5.2.2 and Theorem 4.5.4 with suitable changes. Here we omit the further details. (b3) Assume that c > 0 and define a function z(x, y) by the fight-hand side of (5.8.36), then u(x, y) <_ ~/z(x, y) and
D'~D~z(x, y) _< 2[f(x, y)L(x, y, V/Z(X, y)) + h(x, y)]. ~/z(x, y)
(5.8.46)
Now by following the same steps as in the proof of part (bl) up to (5.8.42) we get
v/z(x, y) <_ p(x, y) + B[x, y, f (s, t)L(s, t, V/Z(S, t))],
(5.8.47)
where p(x, y) is defined by (5.8.33). Define a function v(x, y) by
v(x, y) = B[x, y, f (s, t)L(s, t, V/z(s, t))].
(5.8.48)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
537
From (5.8.48) and using the fact that ~/z(x, y) < p(x, y ) + v(x, y), and the hypothesis (L) we have
v(x, y) < B[x, y, f (s, t)L(s, t, p(s, t) + v(s, t))] = q(x, y) + B[x, y, f (s, t){L(s, t, p(s, t) + v(s, t)) - L ( s , t, p ( s ,
t))}]
< q(x, y) + B[x, y, f (s, t)k(s, t, p(s, t))v(s, t)],
(5.8.49)
where q(x, y) is defined by (5.8.38). Clearly q(x, y) is nonnegative and monotonic nondecreasing in each variable x, y 6 R+. Now an application of Theorem 4.5.4 part (i) yields
v(x, y) <_ q(x, y)exp(B[x, y, f (s, t)k(s, t, p(s, t))]).
(5.8.50)
Using (5.8.50) in (5.8.47) and then u(x, y) < ~/z(x, y), we get the desired inequality in (5.8.37). The proof of the case when c is nonnegative can be completed as mentioned in the proof of part (bl).
II
5.9 Pachpatte's Inequalities IV Inspired by the study of certain new classes of differential and integral equations, Pachpatte (1994a,b) established some useful integral inequalities in one and more than one independent variables. It seems that the inequalities given in Pachpatte (1994a,b) will quite likely be a useful source for future work. In this section, we offer some basic inequalities given in Pachpatte (1994a,b) which claims their origin to be in the inequality given by Ou-Iang (1957). This discussion uses the notation and definitions as given in Section 4.5 and also the following notation and definitions without futher mention. Let R~_ denote the product R+ • ... • R+ (n times). A point (xl, Xn) in R~_ is denoted by x and we denote d x ~ . . . d x l by dx. We define the operators lj recursively by I o u ( x ) = u(x), I j u ( x ) = bj(x)Dju(x), j 1, 2 . . . . . n with bn (x) - 1, where u(x) and bj(x) are some functions defined for x 6 R~_ and Dj = O/Oxj. For x 6 R~_ and some functions b j ( x ) > O, j = 1, 2 . . . . . n - 1, n > 2 an integer, and k(x) defined for x 6 R~_, we set . . . ,
M[x, b, k(y)] = M[xl, . . . , X n , bl, . . . , bn-1, k(y)]
538
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES Xl
Xn-1 ,.,
0
bl (Yl, X2,
. . . , Xn ) 0
bn-
1 (Yl,
99 9
Y n - 1, Xn )
Xn
x f k(y) dy. 0
Pachpatte (1994a) investigated the following useful inequality. Theorem 5.9.1 Let F(x, y) and g(x, y) be nonnegative continuous func-
tions defined for x, y ~ R+ and let p > 1 be a constant. If FP(x, y) < c + B[x, y, g(s, t)F(s, t)],
(5.9.1)
for x, y ~ R+, where c > 0 is a constant, then F(x, y)
< [c ( p - 1 ) / p - ~ - ( ( p -
1)/p)B[x, y, g(s,
t)]] 1 / ( p - l ) ,
(5.9.2)
for x, y ~ R+. D Proof: We first assume that c > 0 and define a function
z(x, y) by the
fight-hand side of (5.9.1), then F(x, y) < (z(x, y) and
D~D~z(x, y) < g(x, y)(//z(x, y).
(5.9.3)
From (5.9.3) and the facts that z(x, y) is positive and D2(P4'z(x, y)) and D~-lD~z(x, y) are nonnegative for x, y 6 R+, we observe that
D'~D~z(x, y) [D2(~z(x, y))][D~-ID~z(x, y)] <_g(x, y)+ (z(x, y) [(z(x, y)]2 i.e.
(D~-ID'~z(x, y) ) D2 ~k --P~~xl~ < g(x, y).
(5.9.4)
By keeping x fixed in (5.9.4), we set y - t and then, by integrating with respect to t from 0 to y and using the fact that D~-lD~z(x, O ) - O , m 2, 3 . . . . . we have
D2m - 1 D ~ z ( x , Y) I "Y < [ g(x, t)dr. ~z(x, y) - J
(5.9.5)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
539
Again as above, from (5.9.5) and the facts that z(x, y) is positive and D z ( ( z ( x , y)) and D m-2 e D nlZ(X, y) are nonnegative for x, y ~ R+, we observe that
(
D'~-2D'~z(x' Y) P~Z(X, y)
D2
)J <
g(x, t) dt.
(5.9.6)
-
o
By keeping x fixed in (5.9.6), setting y - tl, then integrating with respect to tl from 0 to y and using the fact that D m 2 - 2 D nlz(x, O) -- O, m -- 3, 4 . . . . , we have m-2
De
n
Y tl
DlZ(X,y)
0
Continuing in this way we obtain tl
Y tm-1
D~z(x,y) <
S S ...S ,(x,,)dtd,,d,m_,.
(z(x, y) -
0
0
(5.9.7)
0
From (5.9.7) and the facts that z(x, y) is positive and Dl((Z(X, y)) and DT-lz(x, y) are nonnegative for x, y ~ R+, we observe that y tm-1
tl
D~z(x, y ) < S . f "" S g(x' t)dt dtl . .. dtm-1 ~z(x, y) 0
0
+
0
[DI( P~/Z(X,y))][DT-lz(x, y)] [ pv/z(x, y ) ] 2
i.e.
D1
D1 -lz(x' y) ~Z(X, y)
< -
i 0
0
. ..
g(x, t) dt dtl .. 9 dtm-1.
(5.9.8)
0
By keeping y fixed in (5.9.8), setting x - s, then integrating with respect to s from 0 to x and using the fact that DT-lz(0, y) - 0 , for n -- 2, 3 , . . . , we have
D~-lz(x, y)
x
y tm-1
0 0
0
tl
(z(x, y) 0
541)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Continuing in this way, we obtain x Sn-2
Sl y tm-I
tl
DlZ(X'Y)
0 0
0
0
(5.9.9) Now by keeping y fixed in (5.9.9), setting x - Sn-1, then integrating with respect to Sn-1 from 0 to X and using the fact that z(0, y) - c, we have
(~/Z(X, y))p-1 _ ( p ~ ) p - 1 < ((p _ 1)/p)B[x, y, g(s, t)].
(5.9.10)
From (5.9.10) and using the fact F(x, y) <_ ( z ( x , y), we observe that F ( x , y) ~ [c (P-1)/p q- ( ( p -
1)/p)B[x, y, g(s, t)]] 1/(p-l).
This is the required inequality in (5.9.2). If c >_ 0, we carry out the above procedure with c + e instead of c, where e > 0 is an arbitrary small constant, and subsequently pass to the limit E --+ 0 to obtain (5.9.2). II Another useful inequality established by Pachpatte (1994a) is given in the following theorem. Theorem 5.9.2 Let u(x, y), v(x, y), hi(x, y) for i - 1, 2, 3, 4 be nonnegative continuous functions defined for x, y ~ R+ and let p > 1 be a constant. If c l, c2 and lz are nonnegative constants such that
uP(x, y) < Cl -Jr B[x, y, hi (s, t)u(s, t)] -b B[x, y, hz(s, t)~(s, t)], (5.9.11)
Vp (X, y) < C2 -[- B[x, y, h3(s, t)u(s, t)] + B[x, y, h4 (s, t)v(s, t)], (5.9.12)
for x, y ~ R+, where -~(x, y) - exp(-p/~(x + y))u(x, y) and ~(x, y) = exp(p/x(x + y))v(x, y ) f o r x, y ~ R+, then u(x, y) <_ exp(/z(x + y))[(2p-l(cl + r
(p-1)/p -[" 2 P - I ( ( p - 1)/p)
x B[x, y, h(s, t)]] 1~(p-l), V(X, y) <___[(2P-I(c1 -'[-r
(p-1)/p A t - 2 p - I ( ( p -
• B[x, y, h(s, t)]] 1~(p-l),
(5.9.13) 1)/p) (5.9.14)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
541
for x, y ~ R+, where h(x, y) - max{[hl (x, y) -+- h3(x, y)], [h2(x, y) -+- h4(x, y)]},
(5.9.15)
for x, y ~ R+.
Proof: We first multiply (5.9.11) by exp(-p/z(x + y)) and observe that exp(-p/z(x + y))uP(x, y) < Cl -t- B[x, y, hi (s, t)-u(s, t)] + B[x, y, h2(s, t)v(s, t)].
(5.9.16)
Define F(x, y) -- exp(-/x(x + y))u(x, y) + v(x, y).
(5.9.17)
By taking the pth power on both sides of (5.9.17) and using the elementary inequality f (al -k- a2) r ~ 2 r-1 (a~ + a2), (where al, a2 > 0 are reals and r > 1) and (5.9.16), (5.9.12) we observe that FP(x, y) < 2P-l[exp(--plz(x + y))uP(x, Y) + vP(x, y)]
< 2P-1 [Cl -t--C2 -k- B[x, y, [hi (s, t) -at- h3(s, t)]u(s, t)] + B[x, y, [hz(s, t) + h4(s, t)]v(s, t)]].
(5.9.18)
Now using the fact that exp(-p/z(x + y)) < exp(-/z(x + y)) and (5.9.15) in (5.9.18) we observe that FP(x, y) < 2P-I(cl -+- c2) -+-B[x, y, 2p-lh(s, t)F(s, t)].
(5.9.19)
The bounds in (5.9.13) and (5.9.14) follow from an application of Theorem 5.9.1 to (5.9.19) and splitting. II
The inequalities in the following two theorems involving functions of n independent variables given by Pachpatte (1994a) can be used in the study of certain new classes of partial differential and integral equations. Theorem 5.9.3 Let F(x) > O, k(x) > 0 and bi(x) > 0 for i = 1, 2 . . . . . n - 1, be continuous functions defined for x ~ R~_ and let p > 1 be a constant. If FP(x) < c + M[x, b, k(y)F(y)], (5.9.20)
542
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
f o r x ~ R~_, where c > 0 is a constant, then F(x) < [r
_~_ ( ( p _
1)/p)M[x, b, k(y)]] 1~(p-l)
(5.9.21)
for x ~ R~_. [B Theorem 5.9.4 Let u(x) > O, v(x) > O, bi(x) > O, i - 1, 2 . . . . . n - 1, and hj(x) > O, i = 1, 2, 3, 4, be continuous functions defined for x ~ R~_ and let p > 1 be a constant. If c l, c2 and I~ are nonnegative constants such that
uP(x) < Cl + M[x, b, hi (y)u(y)] + M[x, b, h2(y)~(y)],
(5.9.22)
Vp (X) <_ C2 -I- M[x, b, h3 (y)~(y)] + M[x, b, h4 (y)v(y)],
(5.9.23)
for x ~ R~_, where -~(x) - exp(-plzEin__lXi)U(X) for x ~ R~_ and ~(x) exp(plzEn=lxi)v(x) for x E R~_, then u(x) < exp
~
xi
[(2p-l(cl +
c2)) (p-1)/p
i=1 -t- 2p-1 ((p -- 1)/p)M[x, b, h(y)]] 1~(p-l). v(x) <__[(2p-l(cl
(5.9.24)
-1-c2)) (p-1)/p
-I--2 p-1 ((p -- 1)/p)M[x, b, h(y)]] 1~(p-l),
(5.9.25)
for x ~ R~_, where h(x) = max{[hl(x) + h3(x)],
[h2(x) -'t- h4(x)]},
(5.9.26)
for x ~ R~_.
IS] The proofs of Theorems 5.9.3 and 5.9.4 proceed much as the proofs of the theorems given above and follow by closely looking at the proofs of the main results given in Sections 5.7 and 3.10. The details are left to the reader. The following theorem, which is a slight variant of the results established by Pachpatte (1994b) involving functions of n independent variables, can be proved by using similar arguments to those mentioned above. Theorem 5.9.5 Let u(x), f (x) and h(x) be nonnegative continuous functions defined for x ~ R~_ and c be a nonnegative constant. Let bj(x) > O, j - 1, 2 , . . . , n - 1, n > 2 an integer, be continuous functions defined for x~R~_.
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
543
(i) If U2 (X) < C2 -Jr-2M[x, b, f (y)u 2(y) + h(y)u(y)],
(5.9.27)
f o r x E R~_, then u(x) < p(x) exp(M[x, b, f (y)]),
(5.9.28)
p(x) -- c + M[x, b, h(y)],
(5.9.29)
for x ~ R~_, where
f o r x E R~. (ii) Let g(u), g'(u) be as in Theorem 5.8.2. If U2(X) < C2 -[-- 2M[x, b, f (y)u(y)g(u(y)) + h(y)u(y)],
(5.9.30)
f o r x E R~_, then f o r 0 <_ Xi ~ X~, i "- 1, 2 . . . . . n, u(x) <__~-l[~(p(x)) --1-M[x, b, f(y)]],
(5.9.31)
where p(x) is defined by (5.9.29), f2, f2 -1 are as defined in Theorem 5.8.2 and x ~ R~_ be chosen so that f2(p(x)) + M[x, b, f (y)] 6 Dom(f2 -1), for all X i lying in the subintervals 0 < Xi <__X~, i -- 1, 2 . . . . . n of R+. (iii) Let L" R~_ x R+ --+ R+ be a continuous function satisfying the condition 0 < L(x, v) - L(x, w) < k(x, w)(v - w), (5.9.32) for x ~ R~_ and v > w > O, where k" R~_ x R+ --+ R+ is a continuous function. If U2(x) <_ C2 + 2M[x, b, f (y)u(y)L(y, u(y)) + h(y)u(y)],
(5.9.33)
for x E R~_, then u(x) < p(x) + q(x) exp(M[x, b, f (y)k(y, p(y))]),
(5.9.34)
f o r x E R~_, where p(x) is defined in (5.9.29) and q(x) - M[x, b, f (y)L(y, p(y))],
(5.9.35)
f o r x 6 R~_. [2
544
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
5.10 Pachpatte's Inequalities V Pachpatte (1991, 1992, 1994d) investigated a number of new inequalities involving functions of two independent variables which claim their origins in the inequalities used by Haraux (1981) and Engler (1989). This section presents some basic inequalities given in Pachpatte (1991, 1992, 1994d) which can be used as tools in the study of certain new classes of partial differential and integral equations. We shall use the same notation and definitions as given in Section 4.5 without further mention. Pachpatte (1994d) investigated the inequalities given in the following two theorems. Theorem 5.10.1 Let u" E --+ R1, p" E --+ R+ be continuous functions and c > 1 be a constant, where E -- [0, ct] x [0,/3], ot > 0,/5 > 0, R1 = [1, c~) and R+ -- [0, c~). If x
y
u(x,y)<_c+ffp(s,t)u(s,t)logu(s,t)dtds, 0
(5.10.1)
0
for (x, y) ~ E, then u(x, y) <_ c Q(x'y),
(5.10.2)
Q(x, y) = exp ( fo fo p(s, t) dt ds ) ,
(5.10.3)
for (x, y) ~ E, where
for (x, y) ~ E. D Proof: Define a function v(x, y) by the fight-hand side of (5.10.1), then u(x, y) <_v(x, y) and
D2D1v(x, y ) - p(x, y)u(x, y)log u(x, y) <_ p(x, y)v(x, y)log v(x, y).
(5.10.4)
From (5.10.4) and the facts that v(x, y) > 0, DlV(X, y) > 0, D2v(x, y) > 0 for (x, y) 6 E, we observe that
D2D1v(x, y) D1v(x, y)D2v(x, y) < p(x, y)log v(x, y)+ v(x, y) Iv(x, y)]2 '
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
545
i.e.
y) ) < p(x, y)log v(x, y). v(x, y) -
D1 v(x,
D2
(5.10.5)
By keeping x fixed in (5.10.5), setting y = t, integrating from 0 to y and using the fact that DlV(X, 0) -- 0, we get y
D1 v(x, y) < / p(x, t) log v(x, t) dt, v(x, y) -
(5.10.6)
0
Keeping y fixed in (5.10.6) and setting x using the fact that v(0, y) - c, we obtain x
log v(x, y) < log c 4-
s, integrating from 0 to x and
y
ff
p(s, t)log v(s, t)dt ds.
(5.10.7)
0 0
Now a suitable application of the inequality given in Theorem 4.2.1 (see Remark 4.2.1) yields
logv(x, y) < (logc)exp ( / f o = log c Q~x'Y).
(5.10.8)
From (5.10.8) we observe that
v(x, y)
< c Q(x'y).
(5.10.9)
Now using (5.10.9) in u(x, y ) < v(x, y) we get the required inequality in (5.10.2).
II Theorem 5.10.2 Let u" E -~ R1, p" E ~ R+ be continuous functions and c >_ 1 be a constant, where E and R1 are as defined in Theorem 5.10.1. Let g(u) be a continuously differentiable function defined on R+ and g(u) > 0 on (0, c~) and g'(u) > 0 on R+. If x
y
u(x, y) < c + f / p ( s , 0 0
t)u(s, t)g(log u(s, t)) dt ds,
(5.10.10)
546
MULTIDIMENSIONAL
NONLINEAR
for (x, y) ~ E, then for (x, y) u(x, y) < exp
INTEGRAL
~ E1 C E,
( [ JJ ~-1
INEQUALITIES
S2(logc) +
p(s, t) dt ds
0 0
])
,
(5.10.11)
where f2(r)- f
ds
g(s) '
(5.10.12)
r > O, ro > 0 ,
ro
~'2 - 1
is the inverse of f2 and (x, y) x
~ E1 C
E is chosen so that
y
~2(logc)+ffp(s,t)dtds~Dom(S2-1), 0
for (x, y)
0
6 E1 C E.
D Proof: Since g'(u)>_ 0 on R+, the function g(u) is monotonically increasing on (0, c~). Define a function v(x, y) by the fight-hand side of (5.10.10). By following the arguments as in the proof of Theorem 5.10.1 up to the inequality (5.10.7) we obtain x
log v(x, y) < log c +
y
f f p(s, 0
(5.10.13)
t)g(log v(s, t))dt ds.
0
Now a suitable application of Theorem 5.2.1 yields
log v(x, y) < ~_l lf2(log c) + ~ ~o p(s, t) dt ds]
(5.10.14)
From (5.10.14) we observe that
v(x, y) < exp
( [ jj ~-I
~(log c) +
0 0
p(s, t) dt ds
l)
.
(5.10.15)
The desired inequality in (5.10.11) now follows by using (5.10.15) in u(x, y) < v(x, y). The subdomain E1 of (x, y) in E is obvious. II
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
547
Another interesting and useful inequality is embodied in the following theorem. Theorem 5.10.3 Let u" E --+ R1, p, q" E --+ R+ be continuous functions and c > 1 be a constant, where E and R1 are as defined in Theorem 5.10.1. Let g(u), g'(u) be as defined in Theorem 5.10.2 and furthermore we assume that g(u) is submultiplicative on R+. If
u(x, y) <_ c + ;;,-~- - -Y U(S, t) [ p(s, t)log u(s, t) 0 0 "1
+ q(s, t)g(log u(s, t))] dt ds,
J
(5.10.16)
for (x, y) ~ E, then for (x, y) ~ E2 C E,
[ { [ J/
u(x, y) < exp Q(x, y)
f2 -1
f2(log c ) +
q(s, t)g(Q(s, t))dt ds
0 0
]}] ,
(5.10.17) where Q(x, y) is defined by (5.10.3), ~2, g2-1 are as defined in Theorem 5.10.2 and (x, y) ~ E2 C E is chosen so that x
y
g2(log c) + / / q ( s , 0
t)g(Q(s, t)) dt ds ~ Dom(g2 -1),
0
for (x, y) E E2 C E. D Proof: Since g'(u)> 0 on R+, the function g(u) is monotonically increasing on (0, cxz). Define a function v(x, y) by the fight-hand side of (5.10.16). Then by following the arguments as in the proof of Theorem 5.10.1 up to the inequality (5.10.7) we have x y
log v(x, y) < log c +
ff
p(s, t)log v(s, t)dt ds
0 0 x
y
+f/q(s,t)g(logv(s,t))dtds. 0 0
(5.10.18)
548
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
Define a function m(x, y) by x
m(x, y ) = log c +
y
ff 0
(5.10.19)
q(s, t)g(log v(s, t))dt ds.
0
Then (5.10.16) can be restated as x
log v(x, y) < m(x, y) +
y
ff 0
(5.10.20)
p(s, t)log v(s, t)dt ds.
0
Since m(x, y) is positive and nondecreasing in both the variables x and y, by applying Theorem 4.2.2 to (5.10.20) we have log v(x, y) < Q(x, y)m(x, y),
(5.10.21)
where Q(x, y) is defined by (5.10.3). From (5.10.19) and (5.10.21) we observe that x
m(x, y) < log c +
y
ff 0
0 x
< log c +
ff 0
q(s, t)g(Q(s, t)m(s, t)) dt ds
y
q(s, t)g(Q(s, t))g(m(s, t))dt ds. (5.10.22)
0
Now a suitable application of Theorem 5.2.1 to (5.10.22) yields
[
m(x, y) < f2-1 f2(log c) +
JJ
]
q(s, t)g(Q(s, t)) dt ds .
0
(5.10.23)
0
Using (5.10.23) in (5.10.21) we get log V(X, y) < Q(x, y) { ~--~-1 [f2(1ogC) xy
0
0
]}
(5.10.24)
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES 549 From (5.10.24) we get
I ]}]
V(X, y) < exp [Q(x, y) { f2-1 f2 (log c) x
0
y
(5.10.25)
0
Using (5.10.25) in u(x, y) < v(x, y) we get the desired inequality in (5.10.17). The subdomain E2 of (x, y) in E is obvious.
II The inequalities in the following theorems are also established by Pachpatte (1994d) and can be used in some applications. Theorem 5.10.4 Let u" E --+ R1, p, q" E --+ R+ be continuous functions and c > 1 be a constant, where E and R1 are as defined in Theorem 5.10.1.
if
xy
[
j) q(Sl, tl)
logu(s,,, + O0
O0
x logu(sl, tl)dtl dsl] dtds,
(5.10.26)
for (x, y) ~ E, then u(x, y)
(5.10.27)
< c Q~
for (x, y) ~ E, where
Qo(x, y)
--
1 +
tl) -+-
----
0 0
\0
0
tl)] dtl ds1
at ds,
(5.10.28)
for (x, y) ~ E. D
551} MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES Theorem 5.11).5 Let u(x, y), p(x, y), c, g(u) and g'(u) be as in Theorem 5.10.2. If
xy
u(x, y) < c +
ff
[
jj
p(s, t)u(s, t) log u(s, t ) +
O0
p(sl, tl) O0
(5.10.29)
x g(logu(sl, t l ) ) d t l ds1] dtds,
for (x, y) ~ E, then for (x, y) ~ E3 C E,
] ]
st 0
(5.10.30)
0
where
ds
G(r)=
s + g(s)
,
r>0, ro>0,
(5.10.31)
ro
G -1 is the inverse function of G and (x, y) ~ E3 C E is chosen so that x
y
G(logc)+/fp(sl,
tl)dtadSl ~ Dom(G-1),
0 0
for (x, y) ~ g 3 C g .
D The proofs of Theorems 5.10.4 and 5.10.5 follow by the same arguments as in the proofs of Theorems 5.10.1 and 5.10.2, using Theorems 4.4.1 and 5.4.1 respectively, with suitable modifications. The details are omitted. The following theorems established by Pachpatte (1991) deals with the two independent variable generalizations of the inequalities given in Theorems 3.9.4 and 3.9.6. Theorem 5.10.6 Let u" E --> R1, p" E -~ (0, c~), q" E --> R+ be continuous functions and c >_ 1 be a constant, where E and R1 are as defined in
MULTIDIMENSIONAL NONLINEARINTEGRALINEQUALITIES 551 Theorem 5.10.1. If x y
u(x,y)<_c+ffp(s,t)u(s,t) 0 0
(
f
f q(sl, tl)log u(sl, tl)dtl dsl) dtds (5.10.32)
00
for (x, y) ~ E, then u(x, y) <
(5.10.33)
c Ql(x,y),
for (x, y) ~ E, where Ql(X,y)=exp (~o ~o p(s,t) ( ~ / oq ( s l , tl)dtldsl) dtds)
(5.10.34)
for (x, y) ~ E. 7q Theorem 5.10.7 Let u(x, y), p(x, y), q(x, y) and c be as in Theorem 5.10.6. Let g(u), g'(u) be as in Theorem 5.10.2. If x y U(X,y)<_c+ffp(s,t)u(s,t) 0 0
)
q(sl, tl)g(logu(sl, tl))dtl dsl dtds,
(5.1o.35)
for (x, y) E E, thenfor (x, y) ~ E4 C E,
+ f fxpy ( s , t ) 0 0
( ~ 0 ~0
q(sl, tl)dtldSl ) d t d]] s , (5.10.36)
552
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
where g2, f2-1 are as defined in Theorem 5.10.2 and chosen so that
y)~ E4 C E is
(x,
f2(logc) + ~ j o p(s, t) \o ( j ~ o q(sl, tl)dtl dsl) dtds E for (x, y) ~ E4 C E. 7q Proofs of Theorems 5.10.6 and 5.10.7: The details of the proof of Theorem 5.10.7 only are given; the proof of Theorem 5.10.6 can be completed similarly. Since g'(u) > 0 on R+, the function g(u) is monotonically increasing on (0, c~). Define a function z(x, y) by the fight-hand side of (5.10.35). Then
u(x, y) < z(x, y) and D2Dlz(x, y) <_p(x, y)z(x, y) X
q(Sl,
tl)g(logZ(Sl, tl))dtldSl
.
(5.10.37)
> 0,
Dlz(x, y) >_0,
0
From
(5.10.37) and using the D2z(x, y) >__0, we observe that
D2(Dlz(x'-Y))\z(x,y)
facts
that
z(x, y)
-
) "
(5.10.38) By keeping x fixed in (5.10.38), setting y - t, integrating from 0 to y and using the fact that Dlz(x, 0) - 0 we get
y (jj
Dlz(x, Y) < z(x, y) - f_ p(x, t)
q(sl, tl)g(logZ(Sl, tl))dtl
ds1
)
dt.
tJ
0
\0
0
(5.10.39) Now keeping y fixed in (5.10.39), setting x - s, integrating from 0 to x and using the fact that z(0, y) = c, we obtain
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES x
553
y
logz(x,y)<_logc+ffp(s,t) 0 0
x (~o~q(sl, l ) ) d t l dtl)g(logz(sl, s l ) d t d S . ot Define a function
(5.10.40)
v(x, y) by the right-hand side of (5.10.40).
Then it is easy
to observe that
1 D 2y)D l D2D1 ~p(x,
\
V(X,y)) -- q(x, y)g(log z(x, Y)) < q(x, y)g(v(x, y)).
From
(5.10.41)
and
using
the
facts
that
Dl((1/p(x, y))DzDlV(X,y)) > O, we observe D2
(5.10.41)
g'(u)>O, D2v(x,y)>O,
that
Dl((1/p(x, y))D2DlV(X,y))) < q(x, y). g(v(x, y)) -
(5.10.42)
By keeping x fixed in (5.10.42), setting y = tl, integrating from 0 to y and using the fact that Dl((1/p(x,
O))D2DlV(X,0))
- - 0 , we obtain
Dl((1/p(x, y))D2DlV(X,y)) < / q(x, tl)dtl. g(v(x, y))
(5.10.43)
As above from (5.10.43) we observe that y
DI( (1/p(x'y))D2Dlv(x'y))g(v(x, y)) _
(5.10.44)
0
By keeping y fixed in (5.10.44), setting x - sl, integrating from 0 to x and using the fact that (l/p(0,
y))D2Dlv(O,y) -- O, we have
D2DlV(X,y) <_p(x, y) ( ~ f q(sl, tl)dtldsl) . g(v(x, y))
(5.10.45)
554
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
Again as above from (5.10.45) we observe that
D2 (DlV(X'Y))
0
q(sl, tl)dtldSl
.
(5.10.46)
Keeping x fixed in (5.10.46), setting y = t, integrating from 0 to y and using the fact that D1 v(x, 0) = 0, we have
DlV(X,Y)
<
g(v(x, y)) -
Y (iJ q(sl,
i_ p(x, t) 0
\0
tl) dtl
aS')
dt.
(5.10.47)
0
From (5.10.12) and (5.10.47) we observe that 0
D1v(x,Y)
Ox
g(v(x, y)) -
- - f2 ( v ( x , y ) ) =
<
Y (Ji
[_ p(x, t) 0
\0
q(sl, tl ) dtl
) dsl
dt.
0
(5.10.48) Now keeping y fixed in (5.10.48), setting x = s, integrating from 0 to x and using the fact that v(0, y) = log c, we have
f2(v(x, y)) < f2(logc) + ~ i o p(s, t) ( i ~ qo( s l , tl)dtl dSl) (5.10.49) Using the bound on
v(x, y)
from (5.10.49) in (5.10.40) we have
I-
log z(x, y) < f2 -1 /S2(log c) i_
rrxy (ijqs"'ld'lds')
+jjp(s,t) 0 0
]
dtds .
\0
(5.10.50)
0
From (5.10.50) we get
Z(X, y) < exp
[~"2-1 [Q(log c)
+iiP(S't) (jf q(sl'tl)dtlds1)dtds]] 9 0 0
\0
0
(5.10.51)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
555
The desired inequality in (5.10.36) now follows by using (5.10.51) in u(x, y) < z(x, y). The subdomain E4 of (x, y) in E is obvious. II Pachpatte (1992) established the inequalities given in the following theorems, which can be used in certain applications. Theorem 5.10.8 Let u" E --+ R1, h" E ~ R+ be continuous functions and c > 1 be a constant, where E and R1 are as defined in Theorem 5.10.1.
If u(x, y) < c + B[x, y, h(s, t)u(s, t) log u(s, t)],
(5.10.52)
for (x, y) 9 E, then u(x, y)
<
cexp(B[x,y,h(s,t)]),
(5.~0.53)
for (x, y) 9 E. D
Theorem 5.10.9 Let u" E --+ R1, h" E --+ R+ be continuous functions and c > 1 be a constant, where E and R1 are as defined in Theorem 5.10.1. Let g(u), g'(u) be as in Theorem 5.10.2. If u(x, y) < c + B[x, y, h(s, t)u(s, t)g(log u(s, t))].
(5.10.54)
for (x, y) 9 E, then for (x, y) 9 Eo C E, u(x, y) <_ exp(f2-1[f2(log c) -4- B[x, y, h(s, t)]]),
(5.~0.55)
where f2, f2 -1 are as defined in Theorem 5.10.2 and (x, y) 9 Eo C E is chosen so that
f2(log c) + B[x, y, h(s, t)] 9 Dom(f2 -1), for (x, y) 9 Eo C E.
89 The proofs of Theorems 5.10.8 and 5.10.9 can be completed by following the ideas used in the proof of Theorem 5.10.7 and Theorem 5.8.5 and looking closely at the proof of Theorem 4.5.4, with suitable changes. The details are omitted here. In concluding this section, we note that the inequalities established in Theorems 5.10.1-5.10.9 can be extended very easily to n > 3 independent
556
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
variables. The precise formulations of these results are very close to those of the results mentioned above, with suitable modifications. It is left to the reader to fill in the details where needed. We note that the inequalities given in Chapters 4 and 5 are recently established and still admit various generalizations and extensions in different directions. For various other inequalities related to the inequalities given in Chapters 4 and 5 and their applications, the interested reader is referred to Apartsin and Men (1979), Beesack (1985), Cheung (1993), Conlan and Daiz (1963), Corduneanu (1982, 1983, 1987), DeFranco (1976), Fink (1981), Gutowski (1978), Westphal (1949), Yang (1984, 1985) and Young (1982).
5.11 Applications In this section we present applications of some of the inequalities given in earlier sections to study the qualitative behaviour of the solutions of certain partial differential and integro-differential equations. Most of the inequalities given here are recently investigated and can be used as tools in the study of certain new classes of partial differential, integral and integro-differential equations.
5.11.1 Hyperbolic Partial Differential Equations In this section we first present an application of Theorem 5.2.4 to obtain bounds on the solutions of a nonlinear hyperbolic partial differential equation of the form
Uxy(X, y) = (b(x, y)u(x, y))y -1- f (x, y, u(x, y)),
(5.11.1)
with the boundary conditions
u(x, O) = a(x),
u(O, y) = r(y),
a(O) = r(O),
(5.11.2)
where a, z" R+ ~ R, b" R 2 ~ R, f " R2+ • R --+ R are continuous functions and Ib(x, Y)I _< c(x, y),
(5.11.3)
If(x, y, u)l _< p(x, y)g(lul),
(5.11.4)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
557
where c(x, y), p(x, y), g(r) are as defined in Theorem 5.2.4. It is easy to observe that the problem (5.11.1)-(5.11.2) is equivalent to the integral equation x
u(x, y) = a(x, y)+ /b(s, y)u(s, y)ds 0 x
y
+ f f f (s, t, u(s, t)) ds dt,
(5.11.5)
0 0
where
u(x, y) is
a solution of (5.11.1) with (5.11.2), x
a(x, y) = a(x) 4- z(y) - a(O) - / b(s, O)a(s)ds. 0
We assume that
la(x, Y)I _< k,
(5.11.6)
where k _> 0 is a constant. Using (5.11.3), (5.11.4) and (5.11.6) in (5.11.5) we have x
x
y
lu(x,y)l <_k+fc(s,y)lu(s,y)lds+ffp(s,t)g(lu(s,t)l)dsdt. 0
0 0
(5.11.7) Now a suitable application of Theorem 5.2.4 to (5.11.7) yields
lu(x, y)l < F(x, y) [~2_l [f2(k) + ~ jo p(s, t)g(F(s, t)) dsdt] ] (5.11.8) where F(x, y), g2, f2 -1 are as defined in Theorem 5.2.4. If the fight-hand side in (5.11.8) is bounded then we obtain the boundedness of the solutions of (5.11.1)-(5.11.2). . We next apply Theorem 5.5.3 to obtain the bound on the solution of a nonlinear fourth-order partial differential equation of the form
Uxxyy(X,y) 4- (a(x, y)u(x, y))yy = F(x, y, u(x, y)),
(5.11.9)
558
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
with the given boundary conditions u(x, o) = Oo(x),
bty(X, O) = ~1 (X),
u(O, y) - Oo(y),
ux(O, y) -- ~r1(y),
(5.11.10)
where a" R 2 --+ R, F" R 2 x R--+ R are continuous functions and q~o(x), ~O(y),~l(X),~rl(y) are real-valued twice continuously differentiable functions for x, y 6 R+ and ~0(0) -- ~/ro(O), ~1 (0) -- ~r6(O), ~6(0) -- ~r 1 (0), ~tl (0) -- ~rtl (0). (5.11.11) It is easy to observe that the problem (5.11.9)-(5.11.11) is equivalent to the integral equation X
S
u(x,y)-k(x,y)-f/a(sl,
y)u(sl, y)dslds
0 0
(5.11.12)
-t- A[x, y, F ( s 1 , tl, u(s1, tl)],
where u(x, y) is a solution of (5.11.9) with (5.11.10)-(5.11.11) and the notation A[x, y, F] is defined as in Theorem 5.5.3,
k(x, y) - ~o(x) + Oo(Y) -t- x~/rl(y) + Y~I (x) --
~b0(0 ) -- ~b6(0)X
- y~l(O) - xydYl(O)-k- f j a(sl, O)q~O(Sl)dsl ds. (5.11.13) 0 0
We assume that
Ik(x, y)l < c,
(5.11.14)
la(x, Y)I _< f (x, y),
(5.11.15)
IF(x, y, u)l < p(x, y)g(lul),
(5.11.16)
where c, f (x, y), p(x, y), g(r) are as defined in Theorem 5.5.3. Using (5.11.14)-(5.11.16) in (5.11.12) we have x
s
lu(x, y)l < c + / f f (sl, y)lU(Sl, y)l dsl dS 0 0
+ A[x, y, p(sl, tl)g(lU(Sl, tl)l)].
(5.11.17)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
559
Now an application of Theorem 5.5.3 yields lu(x, y)l _< Q(x,
y){f2-1[~2(c)+A[x,
y,
p(sl, tl)g(O(Sl, tl))]]}, (5.11.18)
where Q(x, y), f2, f2 -1 are as defined in Theorem 5.5.3. If the fight-hand side of (5.11.18) is bounded, then we obtain the bound on the solutions of (5.11.9)-(5.11.11). For similar applications, see Pachpatte (1979a, 1980d, 1988c)
5.11.2 Hyperbolic Partial Integro-differential Equations This section presents an application of Theorem 5.4.2 to obtain bounds on the solutions of a nonlinear hyperbolic partial integro-differential equation of the form Uxy(X, y) -- f
0
, y, u(x, y),
//
k(x, y, s, t, u(s, t)) ds d t )
(5.11.19)
0 0
with the given boundary conditions u(x, O) -- or(x),
u(O, y) -- r(y),
u(0, 0) -- 0,
(5.11.20)
where or, r ' R + --+ R, k" R 4 x R --+ R, f " R2+ x R x R --+ R are continuous functions. The problem (5.11.19)-(5.11.20) is equivalent to the integral equation
xy(
s,
st
0 0
) dsdt,
(5.11.21)
0 0
where u(x, y) is a solution of (5.11.19) with (5.11.20). We assume that
I~(x) + r(y)l ~ a(x, y),
(5.11.22)
Ik(x, y, s, t, u)l < b(s, t)g(lul),
(5.11.23)
If(x, y, u, v)l < b(x, y)[lu[ + Ivl],
(5.11.24)
560
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
where a(x, y), b(x, y), g(r) are as defined in Theorem 5.4.2. Using (5.11.22)-(5.11.24) in (5.11.21) we have
xy
(
,u~xy~,~_a~xy~+//~s,~ ,~st~,
st
0 0
)
0 0
Now an application of Theorem 5.4.2 to (5.11.25) yields
xy
{
,u~xy~ ~_a~xy~+a~xy~+//~s,~ ~1 0 0
+//b~Sltl,dsld,1 0 0
n(A(s,t))
]}
~sd,
(5.11.26)
where A(x, y), H, H -1 are as defined in Theorem 5.4.2. The fight-hand side of (5.11.26) gives the bound on the solution u(x, y) of (5.11.19)-(5.11.20). The inequalities established in Theorems 5.5.1 and 5.5.2 can be used to study the problems of boundedness of the solutions of nonlinear fourth-order partial differential and integro-differential equations of the forms Uxxyy(X,
y)- f (x, y, u(x, y)),
(5.11.27)
and Uxxyy(X,
y)- f (x, y, u(x, y), X $2
Y
t2
fff/k(x,y,
\
s3, t3, u(s3, t3))dt3dt2ds3ds2)
(5.11.28)
0 0 0 0
respectively with the given boundary conditions (5.11.10)-(5.11.11), and under some suitable conditions on the functions involved in (5.11.27) and (5.11.28). For more details, the reader is referred to Pachpatte (1979c, 1988c).
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
561
5.11.3 Higher Order Hyperbolic Partial Differential Equations In this section we present an application of Theorem 5.10.8 given by Pachpatte (1992), to obtain the bound on the solution of the following higher order hyperbolic partial differential equation of the form
D~D~u(x, y ) = h(x, y)u(x, y)log lu(x, Y)l,
(5.11.29)
with the given boundary conditions
OJu(x, o) OyJ
=otj(x),
0
= fli(Y),
O
.
.
.
.
.
(51130)
oiu(o, y) Oxi
(5.11.31)
where h ~ C[R2+, R], aj ~ c(n)[R+, R], fli ~ c(m)[R+, R] and
cxj(i) ( 0 ) - - r ' iB (j) .
. 0 < . j < .m - . 1
0 .< i <. n - 1
.
(5 11.32)
It is easy to observe that the problem (5.11.29)-(5.11.32) is equivalent to the integral equation
u(x, y) = q(x, y) + B[x, y, h(s, t)u(s, t) log lu(s, t)l],
(5.11.33)
where u(x, y) is a solution of (5.11.29) with (5.11.30)-(5.11.32), the notation B[x, y, f] is defined as in Theorem 5.10.8 and
xi_ 1 q(x, y ) -
i=l
-~
m
---------~fli-l(y) ( i - 1) + E
"
n
j=l
Xi_ 1 ~ ( i - 1)v
i--1
yj-1 ( j _ 1)vaJ -l(x)
"
"
y j - 1 0 t J -/)(0)" ( j - 1)v
"=
(5 11.34)
"
From (5.11.33) we observe that
lu(x, y)l < Iq(x, y)l + B[x, y, Ih(s, t)llu(s, t)ll log lu(s, t)ll].
(5.11.35)
From (5.11.35) we observe that 1 + lu(x, Y)I ___ 1 + Iq(x, Y)I + B[x, y, Ih(s, t)l(1 + lu(s, t)l)
• log(1 + lu(s, t)l)].
(5.11.36)
562
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
We assume that
1 + Iq(x, Y)I ~ c,
(5.11.37)
where c > 1 is a constant. Using (5.11.37) in (5.11.36) we have
1 + lu(x, Y)I <~ c + n[x, y, Ih(s, t)l(1 + lu(s, t)l) x log(1 + lu(s, t)l)].
(5.11.38)
Now an application of Theorem 5.10.8 to (5.11.38) yields 1 + lu(x, y)l ~
cexp(B[x'y'lh(s't)l]).
(5.11.39)
From (5.11.39) we observe that lu(x, Y)I ~
[cexp(B[x'y'lh(s't)l])- 1],
(5.11.40)
for (x, y ) 6 E, where E is defined as in Theorem 5.10.8. The inequality (5.11.40) gives the bound on the solution u(x, y) of (5.11.29)-(5.11.32) in terms of the known functions.
5.11.4 Multivariate Hyperbolic Partial Integro-differential Equations In this section we present applications of Theorem 5.7.6 to the boundedness and behavioural relationships of the solutions of some nonlinear hyperbolic partial integro-differential equations given by Pachpatte (1981c). As a first application, we obtain a bound on the solution of a nonlinear hyperbolic partial integro-differential equation of the form
D1. . .Dnu(X) -- A
(X / , u(x),
B(x, y, u ( y ) ) d y
)
+ F(x, u(x)),
(5.11.41)
xo
with the conditions prescribed on xi - x~ 1 < i < n, where F" f2 x R ~ R, B: f22 x R ~ R, A: f2 x R 2 ~ R are continuous functions. We assume that IB(x, y, u)l _< c(y)lul,
(5.11.42)
iA(x, u, v)[ _< g(x)(lul + Iv]),
(5.11.43)
IF(x, u)l <_ k(x, [ul),
(5.11.44)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
563
where c(y), g(x), k(x, y, 4)) - k(y, ~) are as defined in Theorem 5.7.6. Let the boundary conditions be such that the given equation (5.11.41) is equivalent to the integral equation
X (/ yu(y),
-- h(x) + [_ A
hi(X)
B(y,
Su(s)) ds
) / dy +
F(y, u(y)) dy,
d
x0
x0
x0
(5.11.45) where h(x) depends on the given boundary conditions. We assume that Ih(x)l ~ f(x),
(5.11.46)
were f(x) is defined as in Theorem 5.7.6. Using (5.11.42)-(5.11.44) and (5.11.46) in (5.11.45) we have
x
lu(x)l <_f (x) + [_g(y)lu(y)l dy
+ [~x(j --
c(s)lu(s)lds dy
t /
x0
x0
x0
+ / k(y, lu(y)l) dy.
(5.11.47)
J
x0
Now a suitable application of Theorem 5.7.6 with q(x) -- 1, w(x, ~) - 4) and k(x, y, ~) - k(y, ~) yields (5.11.48)
lu(x)l _< E; (x)[f (x) + r(x)], where
E~(x) -- 1 + [--x g(y) exp (
j [g(s) +) c(s)] ds
dy,
(5.11.49)
Or
x0
x0
and r(x) is a solution of the equation X
Y(X)
/ k(y, E; (y)[f (y) + r(y)]) dy. Or
x0
If the right-hand side in (5.11.48) is bounded then we obtain the boundedness of the solution u(x) of (5.11.41). Our second application deals with the behavioural relationship between the solutions of (5.11.41) with the conditions prescribed on x i - x~
564
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
1 < i < n, and the nonlinear hyperbolic partial integro-differential equation of the form
(X j
D1...Dnv(X) --Ao
, v(x),
Bo(x, y, v ( y ) ) d y
)
,
(5.11.50)
xo
with the conditions prescribed on xi - x~ 1 < i < n, where B" ~22 x R -+ R, Ao" ~2 x R 2 ~ R are continuous functions. We assume that IB(x, y, u) - Bo(x, y, v)l < c(y)lu - vl,
(5.11.51)
IA(x, u,-a) - Ao(x, v, v)l _< g(x)[lu - vl + lu - vl]
(5.11.52)
IF(x, u)l ~ k(x, lul),
(5.11.53)
where c(y), g(x) and k(x, ~) are as explained above. Equations (5.11.41) and (5.11.50) are equivalent to the integral equation (5.11.45) and
X ( / /
v(x) -- h(x) +
Ao
y, v(y),
x0
)
Bo(y, s, v(s))ds
dy,
(5.11.54)
x0
where h(x) depends on the given boundary conditions. From (5.11.45) and (5.11.54) we have u - v -- h(x) - h(x) +
X{( / / A
y, u(y),
xo
- Ao
( j y, v(y),
)
B(y, s, u(s))ds
xo
Bo(y, s, v(s)) ds
xo
)} J dy +
F(y, u(y)) dy.
x0
(5.11.55)
Using (5.11.51)-(5.11.53) and lul - Ivl ~ In - vl and assuming that Ih(x) h(x)l < f ( x ) , and that the solution v(x) of (5.11.50) is bounded by a constant M > 0 in (5.11.55), where f ( x ) is as defined in Theorem 5.7.6, we have lu - vl < f (x) +
/ ( / g(y)
x0
lu - vl +
c(s)lu(s) - v(s)l ds
)
dy
xo
x
+ /k(y, xo
M + lu(y) - v ( y ) l ) d y .
(5.11.56)
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
565
Now a suitable application of Theorem 5.7.6 yields lu - vl < E; (x)[f (x) + r(x)],
(5.11.57)
where E*(x) is defined by (5.11.49) and r(x) is a solution of the integral equation x ari
r(x) - / k ( y , M
+ E~(y)[f (y) + r(y)]) dy.
(5.11.58)
.J
xo
If the right-hand side of (5.11.57) is bounded then we obtain the relative boundedness of the solution u(x) of (5.11.41) and the solution v(x) of (5.11.50). If f ( x ) in (5.11.57) is small enough and, say, less than ~, where E > 0 is arbitrary, if equation (5.11.58) admits only an identically zero solution, and if E~(x) in (5.11.57) is bounded and E--+ 0, then we obtain l u ( x ) - v(x)l ~ 0, which gives the equivalence between the solutions of (5.11.41) and (5.11.50).
5.12 Miscellaneous Inequalities 5.12.1 Dragomir and Ionescu (1987, 1989) Let u, a, b, be nonnegative continuous functions defined on R2+ and L" R3+ -+ R+ be a continuous function which satisfies the condition
0 < L(x, y, v) - L(x, y, w) < M(x, y, w)(v - w),
(L)
for x, y 6 R+ and v > w > 0, where M is a nonnegative continuous function defined on R3+. (i) If x y
U(X, y ) < a ( x ,
y)+b(x,
y)ffL(s,t,u(s,t))dtds, 0 0
for x, y 6 R+, then
ux,
ax, +
[ex
0)
-1],
(*)
566
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for x, y 6 R+, where P(x, y) is given by
P(x, y) - [{L(x, y, a(x, y))}2 at- {M(x, y, a(x, y))}2b2(x, y)]l/2, for x, y 6 R+. (ii) If u(x, y) satisfies the inequality (.) above, then x y
u(x, y) < a(x, y) + b(x, y)Q(x, y ) / / L ( s ,
t, a(s, t))dt ds,
0 0
for x, y 6 R+, where x y
Q(x,y)-l+ffM(s,t,a(s,t))b(s,t) 0 0
x exp
(
ffM(sl,
tl, a(sl, tl))b(sl,
tl)dtl d s l ) d t d s ,
0 0
for x, y 6 R+.
5.12.2 Pachpatte (in press k) Let u(x, y), p(x, y), q(x, y) and r(x, y) be nonnegative continuous functions defined for x, y 6 R+. Let k(x, y, s, t) and its partial derivatives el (x, y, s, t) = (O/Ox)k(x, y, s, t), k2(x, y, s, t) = (O/Oy)k(x, y, s, t), kl2(X, y, s, t) = (O2/OxOy)k(x, y, s, t) be nonnegative continuous functions for 0 < s < x < c~, 0 _< t < y < oe and L" R 3 ~ R+ be a continuous function satisfying the condition
0 < L(x, y, v) - L(x, y, w) < M(x, y, w)(v - w), for x, y 6 R+, v > w > O, where M'R3+ --+ R+ is continuous.
(i) If x y
u(x,y)<_
p(x,y)+q(x,y)ffk(x,y,s,t)[r(s,t)u(s,t) 0 0
+ L(s, t, u(s, t))] dt ds,
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES 567 for x, y 6 R+, then
u(x, y) < p(x, y) + q(x, y) ( f f A(Sl, tl)dtl dSl) \0
\0
0
0
for x, y 6 R+, where
A(x, y) -- k(x, y, x, y)[r(x, y)p(x, y) + L(x, y, p(x, y))] x
-Jr- f
kl (x,
y, s, y)[r(s, y)p(s, y)+ L(s, y, p(s, y))] ds
0 y
-k- f k2(x, y, x, t)[r(x, t)p(x, t) + L(x, t, p(x, t))] dt 0
x y +ffkl2(x,y,s,t)[r(s,t)p(s,t)+L(s,t,p(s,t))]dtds, o o
B(x, y) --k(x, y, x, y)[r(x, y)4-M(x, y, p(x, y))]q(x, y) x
+ / k l ( x , y, s, y)[r(s, y)+ M(s, y, p(s, y))]q(s, y)ds o y
+ / k2(x, y, x, t)[r(x, t) 4- M(x, t, p(x, t))]q(x, t) dt 0 x y
+
ff
kl2(X, y, s, t)[r(s, t) + M(s, t, p(s, t))]q(s, t)dt ds,
0 0
for x, y 6 R+. (ii) Let F, F -1, oe,/~ be as in Theorem 5.3.1, part (ii). If
u(x,y)
0
+ L(s, t, F(u(s, t)))] dt ds) ,
568
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for x, y 6 R+, then
u(x,y)
x exp
[(fo fo -A(sl,tl)dtldSl )
(OJOJ)] B(Sl, tl) dtl ds1
,
for x, y 6 R+, where A(x, y) and B(x, y) are defined by the fight-hand sides of A(x, y) and B(x, y) defined above by replacing p(x, y) by c~(x, y)F(p(x, y)c~-l(x, y)) and q(x, y) by fl(x, y)F(q(x, y)fl-l(x, y)) for
x, y6R+.
5.12.3 Pachpatte (1979b) Let u(x, y), b(x, y) be nonnegative continuous functions defined for x, y R+ and a(x, y) > 0 for x, y 6 R+ be a continuous function. Let g(u) be a continuously differentiable function defined for u >_O, g(u) > 0 for u > 0 and g'(u) > 0 for u >_ O. (i) If ax(X, y), ay(X, y), axy(X, y) exist and are nonnegative continuous functions defined for x, y 6 R+ and
axy(X, y) < q(x, y)g(a(x, y)), for x, y 6 R+, where q(x, y) is a nonnegative continuous function defined for x, y 6 R+, and suppose further that
x y
b(s, t)g(u(s, t)) dt ds,
u(x,y)<_a(x,y)+ff o o for x, y 6 R+; then for 0 < x _
Xl,
[
0 ~ y < Yl,
u(x, y) < f2-1 f2(a(O, y)) +
J
as(s,O)
g(a(s, 0))
0
x y
+
/f [q(s, t) + b(s, 0 0
t)] dt dsJ
ds
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
569
where f2, ~ - 1 are as defined in Theorem 5.2.1 and X l, Yl are chosen so that x
x y
as(s,O) ~(a(O, y)) + f g(a(s, 0)) d s + f f t q ~ s , t ) + b ~ s , t ~ l a t d s ~ D o m ~ _ l ) 0
0 0
for all x, y lying in the subintervals 0 _< x _< Xl, 0 < y _< yl of R+. (ii) If
ax(X, y), ay(X, y), axy(X, y)
exist and are nonnegative continuous
functions defined for x, y 6 R+ and
axy(X, y) < q(x, y)[a(x, y) + g(a(x, q(x, y)
for x, y ~ R+, where
y))],
is nonnegative continuous function defined for
x, y ~ R+ and suppose further that
xy
(
u(x,y)<_a(x,y)+ffb(s,t)
u(s,t)
s,
0 0
+ffb(sl, o o
) tl)g(u(sl, t l ) ) d t l d S l
dtds,
for x, y ~ R+, then for 0 _< x < x2, 0 < y < y2, x y
u(x, y) < a(x, O) + a(O, y) - a(O, O) +
f f q(s, t)[a(s, t) + g(a(s,
t))] dt as
0 0 xy
+ ff
b(s, t)H -1
0 0
+
asl(Sl'O) a(sl, O) + g(a(sl, 0))
(a(O, y)) +
dsl
0
s, s,, fftq
t l ) + b(sl, tl)]
dtl dsl
1
a t ds,
o o
where H, H - 1 are as defined in Theorem 5.4.1 and x2, Y2 are chosen so that x
H(a(O, y)) +
f
asl (S1, 0) a(sl, O) + g(a(sl, 0))
dsl
0 x y
+
fftq
tl)-!- b(sl, tl)] dtl ds1 c
Dom(H-1),
0 0
for all x, y lying in the subintervals 0 _< x _< x2, 0 _< y < Y2 of R+.
570
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
5.12.4 P a c h p a t t e (unpublished m a n u s c r i p t ) Let u(x, t) and p(x, t) be nonnegative continuous functions defined for x I1 -- [0, 1], t E IT -- [0, T]. Let r h(t), f ( t ) be positive and continuous functions defined for x ~ I1, t ~ IT and define c(x, t) - r + xh(t) + f (t) for x ~ I1, t ~_ IT and let g(u), g'(u) be as in Theorem 5.5.1 and x
s
t
u(x,t)
tl)g(u(sl, tl))dtldSldS,
(i) If r is twice continuously differentiable function such that r > 0, r > 0 for x ~ 11, and f(t), h(t) be continuously differentiable functions such that f ' ( t ) ___0, h'(t) > 0 for t ~ IT, then for 0 < x < Xl,
O
[
( Cx(O, t) ~"2(C(0,t)) -!- X \g(c(O, t))
U(X, t) ~ ~"2-1 x
s
)]J d-
~bt'(Sl)
g(C(S1, 0))
dslds
0 0
t
1
-Jrfffp(sl, tl)dtldsldS,J ooo where ~, f2 -1 are as defined in Theorem 5.5.1 and Xl, tl are chosen so that x
ff2(c(O, t)) + x \g(c(O, t))
+
s
if g(C(S1,0)) r
o o x
s
t
+//fp(sl, ooo
tl)dtldslds~Dom(f2-1),
for all x, t lying in the subintervals 0 < x < xx of I1 and 0 < t < tl of It. (ii) If r f(t), h(t) be continuously differentiable functions such that r >_ O, f ' ( t ) >_ O, h'(t) >_ 0 for x ~ I1, t E IT, then for 0 _< x <_ x2, O_
[
u(x, t) < ~,~-1 ~(c(x, 0)) -+-
-
-~fffp(sl, o o o
J o
[ f ' ( t l ) + xht(tl)] dtl
g(c(O, tl))
]
tl) dSldSdtl ,
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES where g2,
~"2- 1 a r e
as defined in Theorem 5.5.1 and
X2,
571
t2 are chosen so that
t
~(c(x, 0)) -+- /
[ f ' ( t l ) -i- xh'(tl)]
g(c(O, tl ))
0 t
x
dtl
s
+fffp(sl, tl,dsldsdtl E Dom(f2-1), 0
0
0
for all x, t lying in the subintervals 0 _< x _< X2 of 11 and 0 _< t _< t2 of I t .
5.12.5 Pachpatte (unpublished manuscript) Let u(x, t) and p(x, t) be nonnegative continuous functions defined for x 11 -- [0, 1], t ~ Ir -- [0, T]. Let ~b(x), h(t), f ( t ) be positive and continuous functions defined for x ~ 11, t ~ Ir and define c(x, t) = ~(x) + xh(t) + f (t) for x ~ I1, t ~ It. Let g(u), g'(u) be as in Theorem 5.5.1 and
x st( sfs/tj U(x.t)~C(X.t)-J-//f.(SI. tl) U(SI.tl)~p(S3. t3) 0
0
0
0
0
0
x g(u(s3, t3))dr3 ds3 ds2) dtl ds1 ds,
holds for x ~ I1, t E I t . (i) If ~b(x) is a twice continuously differentiable function such that ~b'(x) >_ 0, q~"(x) >_ 0 for x ~ I1 and f (t), h(t) are continuously differentiable functions such that f'(t) >__0, h'(t) >_ 0 for t ~ I t , then for 0 < x <_ x3, 0
xs u(x,t)
s
000
t
0
572
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
in which
(
Q1 (x, t) - H -1 x
+
cx,o,t, )
(c(O, t)) 4- x c(O, t) + g(c(O, t)) $2
C($3, 0) + g ( c ( s 3 , 0 ) )
0 0 x
s
]
t
+fffp(s3,
t3)dt3ds3ds ,
000
H, H -1 are as defined in Theorem 5.5.2 and x3, t3 are chosen so that
H (c(O, t)) 4- x
(
c(O, t) 4- g(c(O, t)) x
)
s2
+
C($3, 0) + g ( c ( s 3 , 0 ) )
0 0 x
s
ds3 ds2
t
+ffJ'p(s3, ooo
t3)dt3ds3ds~Dom(H-1),
for all x, t lying in the subintervals 0 < x < x3 of I1 and 0 < t _< t3 of IT. (ii) If 4~(x), f(t), h(t) are continuously differentiable functions such that tp'(x) >_ O, f'(t) >_O, h'(t) >_0 for x e I1, t e I t , then for 0 _< x _< X4,
O
f tf' (tl) + xh' (tl )] dtl
u(x, t) < c(x, O) +
0 t
x
s
+ f f f p(sl, tl)Q2(Sl, tl)dsl dsdtl, 0
0 0
in which H Q2(X, t) - - H - 1
(C(X, 0)) +
txs 0
0 0
f 0
[f'(t2) +xh'(t2)] C(0, t2) + g(c(0, t2)) dt2
]
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
~'/3
H, H -1 are as defined in Theorem 5.5.2 and X4, t4 are chosen so that t
H(c(x, 0)) +
f
[f'(t2) nt-xh'(t2)]
c(O, t2) + g(c(O, t2))
dt2
0 t
x
s
+ f f f p(s3, t3)ds3dsdt3 6 Dom(H-1), 0
0
0
for all x, t lying in the subintervals 0 < x < x4 of 11 and 0 < t < t4 of
IT.
5.12.6 Rasmussen (1976) Let P0(xo, Y0) and Pl(Xl, Yl) be points in the domain D such that (xl - x 0 ) (Yl -- Y0) > 0 and such that the closed rectangle R1 with opposite vertices P0 and P1 is contained in D. Let ~(x, y), g(x, y) and k(x, y, u) be functions, with the former two continuous on D and with the latter continuous on D x R nondecreasing in u, and satisfying the Lipschitz condition Ik(x, y, u) k(x, y, v)l < L l u - vl. If for all (x, y) in R1, x
y
c p ( x , y ) < _ g ( x , y ) + f f k ( t , s , ep(s,t))dsdt, xo yo then cp(x, y) < U(x, y) on R1, where U(x, y) is a maximal solution of the equation x P
y
P
u(x, y) -- g(x, y) + / / k ( t , ,.J
s, u(s, t))ds dt.
t,r
xo Yo
5.12.7 Pachpatte (unpublished manuscript) Let u(x, y), b(x, y), c(x, y) and p(x, y) be nonnegative continuous functions defined for x, y E R+. Let a(x, y) > 0 for x, y ~ R+ be a continuous and nondecreasing function in x. (i) Let g, g', h, h' be as in Theorem 5.2.2. If u(x,
y) < a(x, y)+ [_ 0
+h
( j
y) u(s, y)+
p(sl, y)u(sl, y)dsl
0
c(s, t)g(u(s, t)) dt ds , 0
)
ds
574
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for x, y 6 R+, then for 0 < x <
Xl,
0 _< y < Yl,
u(x,Y)<~(x,y)[a(x,y)+h(G-1 [G(fo fo c(s,t)g(a(s,t)ap(s,t))dtds) x
])]
y
0 0
where
X ~(x, y) -- 1 + f b(s,
y)exp
[b(sl,y)+ p(sl, y)] dsl
) ds,
0
G, G -1 are as defined in Theorem 5.2.2 and xl, yl are chosen so that
x
y
+ffc(s,t)g(Tt(s,t))dtds~Dom(G -1) oo for all x, y lying in the subintervals 0 < x < xl, 0 < y < Yl of R+. (ii) Let
k(x, y, u) be a nonnegative continuous
X ( j u(x, y) < a(x, y)+ f b(s, y) u(s, y)+
function defined on R 2 •
) p(sl, y)u(sl, y)dsl ds
R, which is nondecreasing in z and is uniformly Lipschitz in z. If
0 x
0
y
+ffk(s,t,u(s,t))dtds, 0 0
for x, y 6 R+, then
u(x, y) < 7t(x, y)[a(x, y)+ r(x, y)], for x, y 6 R+, where ap(x, y) is as defined above and the integral equation
r(x, y) is a solution of
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES x
575
y
r(x,y)-ffk(s,t,O(s,t)[a(s,t)+r(s,t)])dtds, oo existing for x, y 6 R+.
5.12.8 Pachpatte (unpublished manuscript) u(x, y),b(x, y) and c(x, y) be nonnegative continuous functions defined for x, y 6 R+. Let a(x, y) > 1 for x, y 6 R+ be a continuous and
Let
nondecreasing function in x. (i) Let g, g', h, h' be as in Theorem 5.2.2 and let H, H ' be as in Theorem 5.7.8. If
u(x, y) < a(x, y)+
j0 ( j0 b(s, y)
u(s, y)+
b(sl, y)H(u(sl, y ) ) d y
)
ds
for x, y 6 R+, then for 0 < x < xl, 0 < y < Yl,
U(x,Y)<--~I(X,Y)[a(x,Y)nt-h(a-1 [a (fo fo c(s,t)
) xy x g(a(s, t)Ol (s, t)) dt ds
+
o;c,s
where G, G -1 are as defined in Theorem 5.2.2,
[ j0
x
aPl(X, y)
-- 1 + f b(s, y)F -1 F ( 1 ) +
0
]
b(sl, y)dsl ds,
in which F, F - 1 a r e as defined in Theorem 5.7.8 and Xl, Yl are chosen so that x
F(1)
+ f b(sl, y)dsl ~ D o m ( F - 1 ) , 0
576
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
and
G x y
+
//
c(s, t)g(~l (s, t)) dt ds 6 Dom(G -1 ),
0 0
for all x, y lying in the subintervals 0 < x < Xl, 0 _< y _< Yl of R+. (ii) Let
k(x, y, u) be a nonnegative continuous function defined on R 2 x
X(j
R which is nondecreasing in u and is uniformly Lipschitz in z. If
u(x, y) < a(x, y) + / b(s, y)
u(s, y) +
0
b(sl, y)H(u(sl, y)) dsl
ds
0
x y
+ffk(s,t,u(s,t))dtds, 0
0
for x, y 6 R+, then for 0 < x < x2, 0 < y < y2,
u(x, y) < ~1 (x, y)[a(x, y)+ r(x, y)], where lPl (X, y) is as defined above and
r(x, y) is a solution of the integral
equation x
y
k(s, t, 7tl (s, t)[a(s, t) + r(s, t)]) dt ds, 0
0
existing for x, y 6 R+.
5.12.9 Pachpatte (1981b) Let
u(x, y), b(x, y), c(x, y), p(x, y) and q(x, y) be nonnegative continuous
functions defined on a domain D. Let g, g' be as in Theorem 5.2.3 and k > 0 is a constant. Let
Po(xo, yo) and P(x, y) be two points in D such that
( x - x o ) ( y - Yo) > 0 and let R C D be a rectangular region whose oppov(s, t; x, y) be the solution of the
site comers are the points P0 and P. Let
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
577
characteristic initial value problem
L[v]
--
Vst
b(s, t)[c(s, t) + p(s, t)]v -- O,
--
v(s, y) -- v(x, t) -- 1,
and let D + be a connected subdomain of D which contains P and on which v > 0 (Figure 4.1). If R C D + and
u(x, y) < k + b(x, y)
(Ji
+ ff
c(s, t)u(s, t)ds dt
xo Yo
xy
(jj
C(S, t)b(s, t)
o o x
u(x, y) satisfies
p(sl, tl)U(s1, tl)dSl dtl
)) as at
\o o
y
+ffq(s,t)g(u(s,t))dsdt, xo yo
for (x, y) 6 R, then for (x, y) 6 R1 C R,
[
u(x, y) < Q(x, y)S2 -1 ~2(k) +
ii
]
q(s, t)g(Q(s, t))ds dt ,
xo Yo
where
y,
y (x /{C,Sl [C($2, t2) -k- p(S2,
X
,,c,s ,, t2)]V(S2, t2; Sl, tl)ds2dt2
ds1 dtl
yo
for (x, y) 6 R, f2, ~ - I are as defined in Theorem 5.2.1 and xl, yl are chosen so that x
y
f2(k) + j" f q(s, t)g(Q(s, t)) ds dt ~ Dom(f2 -1), xo Yo
for all (x, y) 6 R1 C R.
,
578
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
5.12.10 Bondge and Pachpatte (1979a) Let
u(x) and p(x)
be nonnegative continuous functions defined on f2, where
! ai(xi) > 0, ai(xi) >_0 for 1 < i < n are continuous functions defined for xi > x ~ Let H" R+ ~ R+ be continuously differentiable function with H(u) > 0 for u > 0 and H'(u) > 0 for u > 0.
f2 is defined as in Section 4.9. Let
(i) If x
U(X)< ~-~ai(xi)-+i----1
f p(s)n(u(s))ds,
x0
for x ~ f2, then for x ~ < x < x*,
iO(aixi,+alxo,)i=
u(x) < G -1 x1
al(Sl)
J
+ .o H(y~in=3ai(xi)+a2(xO)+al(Sl))
ds1
X ]
+ f p(s) ds , xo
where
G(r)- f
ds H(s)'
r > 0, r0 > 0,
ro
and G -1 is the inverse of G, and x* be chosen so that
) xj
G
ai(xi) + a l ( x 0) i=2
+
,
al(s1) x0
n(~-~in=3ai(xi) + a2(x 0) -{- al (s1))
x
+ f p(s) ds ~ D o m ( G -1), xo
for all x lying in the parallelepiped x ~ < x < x* in f2.
dsl
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
579
(ii) If
U(X)
naii + _r xp(s)( ju(s)+
i= 1
xO
n
/
x
ai(xi) +
i=1
p(y)H(u(y))dy ds, xo
for x 6 f2 then for x ~ _< x _< x**
U(X) < E
)
<_E
p ( s ) W -1
[
)
W
ai(si) + al (x O)
xO
i=1
s1
I
al(Yl)
( ~2i_3ai(si)+a2 n (xO)+al(Yl))
+ gO (~'i_3ai(si)+a2(x~
X
]
+ f p(y) dy ds, xo
where
r
W (r) --
t + H (t)'
r > 0, ro > 0,
ro
and W -1 is the inverse function of W, and x** is chosen so that
Xl
I
al(s1)
-Jr" _ (~-~n=3ai(xi) + a2(xO) + al(s1)) _.l_H~.,~--~i=3ai(xi) _l_ a2(x2) 0 .+.
dsl
x
+ / p(s) ds ~ D o m ( W -1), xo
for all x ~ f2 lying in the parallelepiped x ~ < x < x** in S2.
5.12.11 Singare and Pachpatte (1981) Let ~(x), a(x), on f2, and let
b(x) and c(x) be nonnegative continuous functions u(x) be a positive continuous function defined on f2.
defined
580
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
(i) Let H(r) be a positive, continuous, strictly increasing, convex and submultiplicative function for r > 0, l i m r ~ H ( r ) - c~. Let ct(x),/~(x) be positive continuous functions defined on ~2 with or(x)+/~(x) = 1. If
u(s) > dp(x) - a(s)H -1 I ~ h(tr)H (q~(tr)) dcr
+ / S b(~r)
c(~)H(q~(~)) d~
)1
dcr ,
x
is satisfied for x < s; x, s 6 f2, then
u(s) > ct(s)H -1 [~
{ 1 + f(s)H(a(s)f1-1 (s)) f b(tr) x
l(s))b(~) q- c(~)] d~
x exp
for x < s; x, s 6 ~2. (ii) Let G(r) be
)}'1 do
,
a positive, continuous, strictly increasing, subadditive and submultiplicative function for r > 0 with limr~o~ G(r)= cxz, and let G -1 denote the inverse function of G. If i"e
U(S) > dp(X)-a(s)G -1 I ~fb(cr)G(~(cr)) dcr
Vx
for x < s; x, s 6 fl, then
u(s) > G -1
[6(r {
x exp
for x <_s; x, s 6 S2.
s
1 + 6(a(s)) f
(j
b(o-)
x
[b(~)G(a(s)) + c(~)] d~
)}l1
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
581
5.12.12 Pachpatte (unpublished manuscript) Let u(x), b(x) and c(x) be nonnegative continuous functions defined on f2. Let a(x) > 1 for x 6 f2 be a continuous and nondecreasing function in x. Let H(u) and H'(u) be as in Theorem 5.7.8. If
j ( j
u(x) < a(x) +
b(y)H
u(y) -t-
xo
c(s)H(u(s))ds
)
dy,
x0
for x ~ f2, then for x ~ f21 C f2,
u(x) < a(x)Q(x), where
Y
])
(1) + I[b(s) + c(s)] ds
Cds
H(s)
(**)
x0
x0
G(r)-
dy,
,
r > O, ro > O,
ro
G -1 is the inverse function of G and X
G(1)+f
[b(s) + c(s)] ds 6 Dom(G -1),
xo
for all x ~ f21 C f2.
5.12.13 Pachpatte (unpublished manuscript) Let u(x), b(x), c(x) and q(x) be nonnegative continuous functions defined on f2 and k > 1 is a constant. Let H(u), H'(u) be as in Theorem 5.7.8. Let g(u) be a continuously differentiable function defined for u > O, g(u)> 0 for u > 0 and g'(u) > 0 for u > 0 and g(u) is submultiplicative. If
X
u(x) <_k + ;_ b(y)H
( u(y) + / c(s)H(u(s)) as) dy
t.]
xo X i-J
+ / q(y)g(u(y)) dy, xo
x0
582
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for x ~ ~2, then for x ~ ~"~1 C ~~,
u(x) < Q(x)E -1
(k) +
q(y)g(Q(y))dy , xo
where
Q(x)
is defined in (**) with G, G -1 as involved therein, r
E(r) --
g(s) '
r > O, ro > O,
ro
E -1 is the inverse function of E, and x
G(1) +
f tb,s) + c s)]d
~
Dom(G-1),
xo x
E(k) + f q(y)g(Q(y)) dy ~ D o m ( E -1), xo
for x E ~21 C ~ .
5.12.14 Pachpatte (unpublished manuscript) Let u(x),
b(x) and c(x) be nonnegative continuous functions defined for a(x)> 1 for x ~ ~2 be a continuous and nondecreasing funcLet H (u) and H ' (u) be as in Theorem 5.7.8 and k(x, y, u), W(x, u)
x ~ ~2. Let tion in x.
x(j)b y
be as in Theorem 5.7.6. If
U(X)
< a ( x ) + [_ J xo
+W
+
c(s)H(u(s)) ds dy
xo
x, j k(x, y, u(y)) dy) , xo
for x ~ g2, then for x ~ ~"21 C ~"2,
u(x) < Q(x)[a(x) + W(x, r(x))],
where
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
583
Q(x)
is a
is defined as in (**) with G, G -1 involved therein and
r(x)
solution of the integral equation x
r(x) - / k ( x , y, Q(x)[a(y) + W(y, r(y))])dy, xo
existing on f2.
5.12.15 Pachpatte (unpublished manuscript) Let
u(x)
a(x) H(u)
and b(x) be nonnegative continuous functions defined on f2. Let _> 1 for x ~ f2 be a continuous and nondecreasing function in x. Let and
H'(u)
X[
be as in Theorem 5.7.8. If dy,
u(x) < a(x) + f b(y) u(y) + xo
for x ~ f2, then for x ~ f21 C f2,
u(x) < a(x)Qo(x), where
x [
Y
E
Qo(x)- 1 + [_ b(y) 1 + / b ( s ) G -1 G(1
dy, ( * * * )
tl
G(r) --
xo
xo
xo
s -( ~ (s~
r > O, ro > O,
G -1 is the inverse of G and x
G ( 1 ) + / b(~r) dcr ~ Dom(G -1), x0
for all x s f21 C f2.
5.12.16 Pachpatte (unpublished manuscript) Let u(x), b(x) and q(x) be nonnegative continuous functions defined on f2 and k _ 1 be a constant. Let H(u) and H'(u) be as in Theorem 5.7.8. Let
584
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
X[
(r
g(u) and g'(u) be as in Theorem 5.2.3. If Y
u(x) < k + [_ b(y) u(y) + f b(s)
]
b(o")H(u(o"))do" ds dy
Or Xo
x0
xo
X
+ / q(y)g(u(y)) dy, xo
for x ~ f2, then for x ~ f2~ C f2,
xo
where
Qo(x) is defined as in ( , , ,) with G, G -1 as involved therein, r
E(r)-
g-(s)' r > O, ro > O, ro
E -1
is the inverse function of E, and x
G(1) + f b(o.) do" ~ Dom(G -1), xo x
E(k) + / q(y)g(Qo(y)) dy ~ Dom(E -1), xo
for all x ~ ~1 C ~2.
5.12.17 Pachpatte (unpublished manuscript) Let Let Let
u(x), and b(x) be nonnegative continuous functions defined for x ~ ~. a(x) > 1 for x ~ f2 be a continuous and nondecreasing function in x. H(u) and H'(u) be as in Theorem 5.7.8 and k(x, y, u) and W(x, u) be
as in Theorem 5.7.6. If
dy
u(x) < a(x) + [_ b(y) u(y) J x0
+W
x0
,
k(x, y,u(y))dy xo
x0
,
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
585
for x ~ f2, then for x ~ ~"~1 c ~'-~,
u(x) < Qo(x)[a(x) + W(x, r(x))], where Qo(x) is defined as in ( , ,
, ) with G, G -1 as involved therein, and
r(x) is a solution of the integral equation x
r(x) - f k(x, y, Qo(y)[a(y) + W(y, r(y))l) dy, xo
existing on ~.
5.12.18 Singare and Pachpatte (1982) Let u(x), a(x), b(x), f ( x ) and g(x) be nonnegative continuous functions defined on ~2, G(u) be a continuous
strictly increasing convex and
G(u) = c~, for all x 6 f2; let
submultiplicative function for u > 0, l i m u ~
c~(x), /3(x) be positive continuous functions defined on a domain ~ , and c~(x) +/3(x) = 1. Let v(s; x) and e(s; x) be the solutions of the characteristic initial value problems
(-
1)nvsl...s n
(S; X)
--
[ f (s) + g(s)~(s)G(b(s)~ -1 (s))]v(s; x) = 0 in g2,
v(s; x ) = l on s i - - x i ,
1 < i < n,
and
(-- 1)n esl...~. (s; x) + f (s)e(s; x) = 0 in f2, e(s; x ) -
1 on
Si-
1 < i < n,
Xi,
respectively and let D + be a connected subdomain of fl containing x such that v > 0, e > 0 for all s ~ D +. I f D C D + and
u(x) < a(x) + b(x)G -1
(j (J f (s)
x0
))
g(cr)G(u(cr)) dcr
xo
ds
,
586
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
for all x e D, then
u(x) < a(x) + b(x)G-a [ f e(s; x) ( f (s) ( f ot(o)G(a(tr)ot-l(o))g(o) xo
xo
as)] for all x e D.
5.12.19 Shinde and Pachpatte (1983) Let
u(x), v(x), a(x), b(x), p(x), hi(x)
(i = 1, 2, 3, 4) be nonnegative
continuous functions defined on S2, and h(x)=max{[hl(x)+h3(x)], [h2 (x) -t- ha (x)] }. Let G(r) be a continuous, strictly increasing convex and submultiplicative function for r > 0, limr--,~ G(r) - cxz for all x in f2, G -1 be the inverse function of G; or(x), /3(x) are positive continuous functions defined on f2 and or(x)+/3(x) = 1. Let w(s; x) be the solution of the characteristic initial value problem (--1)nwsl...Sn
w(s; x)
(S; X) --
-- 1 on
~(s)G(p(s)~-l(s))h(s)w(s; x) - 0 in
Si - - X i ,
g2,
1 < i < n.
Let D + be a connected subdomain of f2 containing x such that w > 0 for allseD
+.IfDCD
+and r
U(X)
< a(x) at- p(x)G -1 [ f hi (s)G(u(s)) ds l xo
xO
]
Isil hz(s)G(v(s))ds ,
+aexpf i=1
v(x) < b(x) + p(x)G -1
[r
exp
xO
+fh4(s)a(v(s))ds] xo
Isil h3(s)G(u(s))ds
-# i=1
,
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
587
for x ~ D, w h e r e / z is a nonnegative constant, then
E
u(x) < G -1 exp /x
]
Ixil Q(x) , i=1
v(x) <_ G-I[Q(x)],
for x ~ D, where
Q(x)-f(x)+fl(x)G(p(x)fl-l(x))(fw(s;x)h(s)f(s)ds), xo
in which f (x) -- exp
-I~ Z
Ixi[
~(x)G(a(x)~ -~ (x))
+ ol(x)a(b(x)o1-1 (x)),
i=1
for x ~ D.
5.12.20 Pachpatte (1993, 1996d) Let
u(x)> O, a(x)>_ O, b(x)>_ O, pi(x)> 0 for i - 1 ,
2.....
continuous functions defined for x ~ R~_ and q" R~_ x R+ ~
n-
1, be
R+ be a contin-
uous function which satisfies the condition
0 < q(x, V1)
--
q(x, V2)
~
k(x,
V2)(V 1 -- V2),
for x 6 R~_ and V1 >__ V2 >__ 0, where k" R~_ x R+ --+ R+ is a continuous function. Let the notation M[x, p, q] be as defined in Section 5.9. (i) If
u(x) < a(x) + b(x)m[x, p, q(y, u(y))], for all x 6 R~_, then
u(x) < a(x) + b(x)M[x, p, q(y, a(y))] exp(M[x, p, k(y, a(y))b(y)]), for all x ~ R~_.
F(u) be a continuous, strictly increasing, convex, submultiplicaF(u) = c~, F -1 denotes the inverse function of F and or(x), fl(x) be continuous and positive functions defined for x ~ R~_ and c t ( x ) + fl(x) = 1. If (ii) Let
tive function for u > O, l i m u _ ~
u(x) <_a(x) + b(x)F -l(M[x, p, q(y, F(u(y)))]),
588
MULTIDIMENSIONALNONLINEAR INTEGRAL INEQUALITIES
for x ~ R~_, then u(x) < a(x) + b(x)F -1 (M[x, p, q(y, a(y)F(a(y)ot -1 (y)))] x exp(M[x, p, k(y, ot(y)F(a(y)ot -1 (y)))fl(y)F(b(y)f1-1 (y))])), for x 6 R~_. (iii) Let g(u), g'(u), f2, ~-1 be as in Theorem 5.2.1. If u(x) < a(x) + b(x)M[x, p, q(y, g(u(y))], for x 6 R~_, then for 0 _< x < x*, u(x) <_ a(x) + b(x)f2-~[~(c(x)) + M[x, p, k(y, g(a(y)))g(b(y))]], where c(x) -- M[x, p, q(y, g(a(y)))], and x* ~ R~_ is chosen so that f2(c(x)) + M[x, p, k(y, g(a(y)))g(b(y))] ~ Dom(f2 -1), for all x ~ R~_ such that 0 < x < x*.
5.13 Notes The results given in Section 5.2 deal with some basic nonlinear generalizations of Wendroff's inequality given by Bondge and Pachpatte (1979b, 1980a). Theorems 5.2.1 and 5.2.2 are taken from Bondge and Pachpatte (1979b) and (1980a) respectively. Theorems 5.2.3 and 5.2.4 are new and motivated by certain applications. The inequalities dealt with in Section 5.3 are due to Pachpatte (1995e) and Bondge and Pachpatte (1979b,c). Theorem 5.3.1 was recently established by Pachpatte (1995e). Theorems 5.3.2 and 5.3.3 are given by Bondge and Pachpatte (1979b) and Theorems 5.3.4 and 5.3.5 are given by Bondge and Pachpatte (1979c). Section 5.4 contains some useful generalizations of the inequalities given by Pachpatte (1974d, 1975h). Theorem 5.4.1 is due to Bondge and Pachpatte (1979b). Theorems 5.4.2, 5.4.3 and 5.4.4 are due to Bondge and Pachpatte (1980a).
MULTIDIMENSIONAL NONLINEAR INTEGRAL INEQUALITIES
589
Sections 5.5 and 5.6 are devoted to the inequalities recently investigated by Pachpatte (1980c,d, 1988c, 1993, 1996d). Theorems 5.5.1 and 5.5.2 are taken from Pachpatte (1988c). Theorem 5.5.3 is new and motivated by certain applications. Theorem 5.5.4 is taken from Pachpatte (1993, 1996d). Theorems 5.6.1 to 5.6.4 are taken from Pachpatte (1980c) and Theorems 5.6.5 and 5.6.6 are taken from Pachpatte (1980d). The results given in Section 5.7 deal with the integral inequalities in many independent variables established by various investigators (Beesack, 1975; Headley, 1974; Pachpatte, 1981c; Pelczar, 1963). Theorems 5.7.1 and 5.7.2 are due to Pelczar (1963), Theorem 5.7.3 is taken from Beesack (1975) and Theorem 5.7.4 is taken from Headley (1974). Theorems 5.7.5-5.7.9 are due to Pachpatte (1981 c). The inequalities given in Sections 5.8-5.10 were recently discovered by Pachpatte (1991, 1992, 1994a,b,d, 1996c, 1997, in press d, unpublished manuscript). Theorems 5.8.1 and 5.8.2 are taken from Pachpatte (in press d). Theorems 5.8.3, 5.8.4, 5.8.5 are taken from Pachpatte (1997), (unpublished manuscript) and (1996c), respectively. Theorems 5.9.1 to 5.9.4 are taken from Pachpatte (1994a), while Theorem 5.9.5 is taken from Pachpatte (1994b). The results in Theorems 5.10.1 and 5.10.2 are taken from Pachpatte (1994d) and Theorem 5.10.3 is new. Theorems 5.10.4 and 5.10.5 are taken from Pachpatte (1994d). Theorems 5.10.6 and 5.10.7 are taken from Pachpatte (1991). Theorems 5.10.8 and 5.10.9 are taken from Pachpatte (1992). Section 5.11 is devoted to the applications of some of the inequalities given in earlier sections to the various properties of the solutions of certain partial differential, integral and integro-differential equations. Section 5.12 contains some miscellaneous inequalities which can also be used in some new applications.