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Boundary value problem of a third-order mixed type differential-difference equation
li NOg~- HOJAND
B o u n d a r y V a l u e P r o b l e m of a Third-Order M i x e d T y p e Differential-Difference E q u a t i o n Wang Peiguang
Department of Mathematics Hebei University Baoding, 071002 People', 's Republic of China
ABSTRACT In tlhis paper, the existence and uniqueness of the solution for the boundary value problem of a third order mixed differential-difference equation is given by using Schauder's fLxed point theorem, and an approximate solution is given by using the Picard's iterative methods. © Elsevier Science Inc., 1996
1.
INTRODUCTION
Due to the applications of the differential-difference equations in the fields of biological mathematics, mechanics, and control theory, people are getting more and more interested in the boundary-value problem. Wang and Wang [1, 2] have discussed the existence and uniqueness of solutions for second-order equation and second-order mixed boundary-value problems, respectively, and have given the estimation of the error between the solution and the approximate solution. In this paper, by using Schander's fixed point theorem, the same problem is studied for third-order mixed differentialdifference equations. Wang and Luo [3] present a special case of the problem discussed in this paper.
APPLIED MATHEMATICS AND COMPUTATION 80:273-286 (1996) © Elsevier Science Inc., 1996 0096-3003//96/$15.00 655 Avenue of the Americas, New York, NY 10010 SSDI 0096-3003(95)00303-7
274
W. PEIGUANG
We consider the boundary value problem V"( t) - cl y ' ( t - ~ ) - ~ y ' ( t + ~ ) ---f(t;y(t),~l(t-~-l),~i(t+~-2) y( t) = ¢b( t)
a-
y( t) = ~ ( t)
b <.< t <. b + ~
)
(a
(1.1)
~'1 <~ t <~ a, y'( a + O) = m (1.2)
where ~(t)= (y(t), y'(t), y"(t)); ~(t + A ) = ( y ( t + A), y'(t + A), y"(t + h)), ( h = - ~'i, ~'2), ~'i, ~'2 are positive numbers, ¢ ~ C2([ a - ~'1, a], R), 0 ~ C~([ b, b + ~'2], R) and f is continuous in t, and assume
Ic~l DEFINITION 1.
+
Ic21 <
(1.3)
1.
A function y(t) is called a solution of (1.1)-(1.2), if (i)
y(t) ~ C [ a - vl, b + ~2] ~ C:[ a, b], (ii) y"(t) on [a,b] has only some limited discontinuous points of the first kind, (iii) y(t) satisfies the condition (1, 2), (iv) Equation (1.1) holds for t ~ [ a, b], and t ~ a + k~-t, t ~: b -
k~'2 (k ~ Z+). For convenience, we make a similar change of the variable as [3]
z( t) = a( t)dp( t) + ~( t)~( t) + ~/( t) + y( t).
(1.4)
Then the problem in transformed into
z'( t) - clz"( t - ~'1) - c2z"( t + "r2) --F(t;~(t),~(t-~'l),~.(t+~'2) ) z(t)=0
t~[a-Tl,
a]U[b,b+r2] ,
(a
(1.5)
z'(a+0)=0,
(1.6)
where ~.(t) = (z(t), z'(t), z"(t)); z(t + A) = ( z ( t + A), z'(t + A), z"(t + A)), (A
-~- - - T1 , '/'2) ,
275
Third-Order Differential-Difference Equation
Constructing the Green function of problems (1.5)-(1.6) gives - ( ( t - a)( b - s ) / ( b - a ) ) 2 / 2 + ( t -
c ( t, s) =
I
a
s)2/2
a))2/2
0
a <~ t <. s <~ b a-'q
0
a-71<~s<~aorb<.s
Using the Green function, we can obtain the following lemma.
LEMMA 1 [3].
Let x(t) ~ K = { h(t); h ' ( t) on [ a, b] has only some limited discontinuous points of the first, kind, h(t) = 0, t ~ [ a - ~'1, a] U [b, b + r2], h ' ( a + 0) ~ 0}. Then Ix(k)(t)l < C3, k(b - a ) 3 - k M ( k = 0,1,2) in which M = maxa~ t< b] Y(3)(t)l, C3,o = 2/81, C3,1 = 1/6, C3, 2 = 2/3.
LEMMA 2. Let Q = maxnlFI, (t; ~,(t), ~,(t - ~'1), ~(t + ~'2)) e 12 [ a, b] × R 9. If z( t) is a solution of (1.5)-(1.6) then max
Iz'( t)] ~ Q/(1 - t c 1] -]c2])
c
(1.7)
a~ t~ b
LEMMA 3 [4]. Let B be a Banach space and let r > 0, r ~ R, S(x o, r) = { x e B: 1 i x - xol] < r}. Map S( x o, r) into B and (i) for all x, y ~ S(x o, r): i l T x - TyOi <~ a t l x (ii) r o = ( 1 ko) -1 × l I T x o - x o l ] ~ < r.
yll, (0 ~< a < 1),
Then
(1) T has a fixed point x* in S( Xo, ro); (2) x* is the unique fixed point of T in S( x o, r); (3) the sequence{xm} defined by xm+ 1 = Tx m. m = O, 1 ... converges to x* with ILx* - xmll < k~ro. 2.
E X I S T E N C E AND UNIQUENESS
Now we give the existence and uniqueness of the solution to the problem (1.5)-(1.6).
276
W. PEIGUANG
THEOREM 1.
Suppose that
(i) k i > 0 (i = 0, 1, 2) are given constants, l-I is a closed and bounded subset of [a, b] × C9[a, b] and IFI <~ Q, in which Q = max iFI,
a = {a <~ t <~ b, z ( t ) ~ C[a - vl, b + ~'2] N C2[a, b], ]z(i)(t)l ~< ki, I z ( ° ( t - T1) [ <
ki,
]z(i)(t-[-
T2)I <
ki};
(ii) the following inequality holds b - a < min{(cko/QC3,o) i/s, ( ck~/QC a ,),/2, ( c k j Q C 3 , 2 ) } ( c = 1 - I c , I - Ic21)
Then the boundary value problems (1.5)-(1.6) have at least one solution.
PROOF. Denote E = {z(t): z(t) ~ C[a - ~'1, b + r2] A C2[a, b], z(t) = O, t ~ [ a - T1, a] [,.J [b, b + ~'2]; z'(a + O) = O, maxa~t~blz(3)(t) ] ~< kj, j = 0, 1, 2, z ' ( t ) on [a, b] has only some limited discontinuous points of the first kind}; then E is a closed convex subset of the Banach space C2[ a, b]. We define an operator T as follows.
Tz(t) --/b÷~2 G(t, s){F(s; ~(s), ~ ( s - ~,), ~(s + ~2)) a--~- 1
+ [ e l z ' ( s - ~,) + c2z'(s + ~2)1} ds (a-
T, ~< t~< b + r2)
(2.1)
By the condition (ii) and L e m m a 1 and L e m m a 2, we have
]( Tz)(')( t)l < ( QC3 ,( b -
a)S-')/c
~ k,.
It is easy to see T maps E onto itself and is completely continuous. It then follows from Schauder's fixed point theorem that T has a fixed point z*(t). The z*(t) is a solution of (1.5)-(1.6).
Third-Order Differential-Difference Equation
277
COROLLARY. Assume the function F(t; ~(t), ~(t - ~'1), ~(t + r2)) on [a, b] × R 9 satisfying, 2
[El ~ h --~ E {hj+l[ z(J)( t)l °t{j) "+- lj+l[ z(J)( t - T1)I fl(j) j=0 + m.,I z(J)(t + ~2)I V(j)
(2.2)
where h, hi+l, lj+l, mj+ 1 are nonnegative constants, 0 ~ a ( j ) <. 1,0 ~ ( j ) < 1, 0 ~< T(J) ~< 1. Then the boundary value problem (1.5)-(1.6) has at least one solution, provided that
{(
O=
E {hi+l+ /j+l +
j=O
PROOF.
mj+l}Ca,j(b-
a) a-j
)/
c
< 1
(2.3)
Let
Zj+,(.(j)) = { ohi+ 1 ~j+l(~(J)) = { 0lj+~ mj+l(f~(J)) ---- {
ol(j) = 1 o<.(j)
mj+ l
v( J)
0
0 < ~(j) <1.
=
1
Using Theorem 1, we have 2
Q ~ h-~ E {(hj+l --hj+l)ka(J) -[- (lj+l
-
-
-lj+l ) k~(J)
j=o
"I'-(mj+ 1 -- -mj+l)k "y(j)} -{- I( ko, kl, k2) (2,4)
278
W. PEIGUANG
in which 2
I(k0, kl, ks) = 2 {~j+l + ~j+l + ~+~)kj.
(2.5)
j=O
Taking
kj = c a , / ( q , o ( ~ -
a) j)
( j = o, 1,2),
then the following inequality holds
I( ko, k1, ks)
Similarly, we have
I( ko, kl, k2) .
-hi+ 1 +-Ij+ 1 +Nj+,}Ca, i ( b - a ) a-j k a ( 0 3 , 1 ( b - a ) 2)
I( ko, kl, k~)
when a ( j ) = fl(j) = y(j) = 1, the coefficient of k "(j), k t~(j) and k ~'(j) are equal to zero; then by choosing kj sufficiently large and using (2.3), we have
Q<~ cko/( C3,o( b - a)3), q<~ ck2/( ca 2( b -
Q<~ ckt/( Ca,,( b - a)2),
~) ).
(2.6)
Third- Order Differential-Difference Equation
279
From (2.0, it is easy to get
b-
a < min{(ck0/QC3,0)l/3), (ckl/QC3,1)1/2), (ck2/QC3,2))};
(2.7)
Theorem 1 conditions are satisfied, so the corollary holds.
REMARK. Theorem 1 is a local existence theorem, whereas a corollary does not require any condition on the length of the interval.
THEOREM 2.
Assume the function F on [ a, b] × R 9 satisfies the follow-
ing condition 2
IFI < h + E {hj+ 11 zCJ)(t)[ + zj÷ll z(J (t- 1)1 ÷ mj÷ll z'J'(t ÷ j=O
(2.8) Then problems (1.5)-(1.6) have one solution, provided that
PROOF. From (2.8), we can easily get that z(t) -~ 0 is a solution of the problem (1.5)-(1.6); in order to prove the theorem, we only need to prove problems (1.5)-(1.6) have no nontrivial solutions. Supposing the problems (1.5)-(1.6) have nontrivial solutions, using Lemma 1, we have
IZ"(t)l ~
[2
E {hj+l "~- lj+l "~- mj+l}C3, j( b - a) 3-j j=o
1
max b I z"( t) l × a<~
280
W. PEIGUANG
then
max I z"( t)l < o a ~ t<~ b
Iz"( t)l a ~ t<~ b
by 0 < 0 < 1, we can get m a x ~ t~ b ]Z"(t)l = 0, then z ' ( t ) = 0. But this means t h a t z(t) is a nonzero linear function. This is impossible because we have z(a) = z(b) = 0, z ' ( a + 0) = 0 from the condition (1.6); therefore,
z(t) - O.
THEOREM 3. Let f on [ a, b] × R 9 be differential with respect to y(J) ( j = 0, 1, 2), and satisfy
l a f / 3 y(J)( t)l <~ hi+ 1, ] ~ f / 3 y(J)( t - T1)I
< lj+l, IOf/OY(J)(t + ~2)1 <~ mj+l. Then the boundary value problems (1.1)-(1.2) have a unique solution, provided that
hi+ 1 + lj+ l + mj+ 1} C3, j( b - a) 3-j
0 =
c < 1.
PROOF. Suppose yl(t), y2(t) are solutions of (1.1)-(1.2), respectively. Let z(t) = Yl(t) - Y2(t),
IOf/c~Y(i)( t)] = fj, [Of/OY(J)( t - ~'1)1 = gj, [Of/c~Y(J)( t + ¢2)J = Pj; then z(t) satisfies zm(t)
-- C1 Z m ( t -- T1) -- C2 z m ( t "~- '1"2)
---- f ( t ; ~ l ( t ) , ~ll(t -- T1) , ~ l ( t + 72) ) - f ( t ; ~lz(t), ~12(t + T1) ,
~2(t + a'2) )
Third-Order Differential-Difference Equation
281
and z(t) = 0 t ~ [a - ~-, a] U [b, b + ~'2] z'(a + O) = O. Let
aj(t) = fol fj(t; ~2(t) + s z ( t ) , 7]2(t - "Q), ~2(t + ~'2)) ds bj( t )
= £ 1 gj(t; ~:(t), ~:(t- ~) + ~z(t- ~), ~ ( t + ~)) d~
e.( t) -~- fo~pj(t; Y2(t), y 2 ( t -
T1) , ~]2( t Jr" T2) + sz( t-~- 'T2) ) ds.
Then ~ " ( t ) - ~ z " ( t - ~-,) - c~ z " ( t + ~-~)
2 E {aj(t) z(2(t) + bj(t) z(J)( t - "Q) + ej(t) z(J)(t + "r2)}. j=O Noticing [aj(t)] <<.hi+l, Jbj(t)] <~ lj+ 1, lej(t)l ~< mj+l, then the Theorem 2 condition is satisfied; therefore z(t) -- 0. 3.
PICARD'S ITERATIONS
We know it is very difficult to obtain the solutions of differential-difference equations, so we now introduce the approximate solution and discuss an estimation of the error between the approximate solution and the solution. Because problems (1.1)-(1.2) are equivalent to (1.5)-(1.6), we can discuss problems (1.5)-(1.6).
DEFINTION 2. A function z(t) is called an approximate solution of problems (1.5)-(1.6), if there exist e(6 > 0), such that
max I z " ( t ) - c l z " ( t a<~t<~b
.Q) - c 2 z " ( t + 72)
- F ( t; -~( t), -z( t - ~,), -z( t + ~-~))] < ~
(a.1)
282
W. PEIGUANG
and z( t) ~ 0 t e [ a - 71, a] U [ b, b + ~'2] z'( a + 0) - 0. In fact, the approximate solution z(t) can be expressed as
z( t) = fab+'~ a( t, s){F(s; ~( s), ~( s - r~), ~( s + r2) "{'[ClZm(8 -
T1) +
in which rl(t) = z"(t) - q z " ( t T1) , 7z(t + 72)).
DEFINITION 3.
C 2 Z m ( 8 "["
72) ] + TI(8)}
d8
(3,2)
71) - c2z"(t + r~) + F(t; ~(t), ~ ( t -
The function f is said to be of Lipschitz class for all (t; ~ ( t ) , ~(t - r , ) , ~(t + T2)),
(t; ~( t), 7( t -
71), 7( t + ~'2)) e [ a, b] x D, ( D e
Rg).
The following inequality is satisfied f( t; ~ ( t ) , ~( t - T1) , "~( t "4" T2) ) -- f(t; 7 ( t ) , 7( t - r l ) , 7( t + r2))l
2 <~ E {hj+llu'J)( t) - v(J)( t)l + li+llu(J) ( t -
"rl) - v(J)( t -
71) ]
~'~)
7 )1}
j~ 0
+%+,lu(J)(t
+
-
v(')(t
+
m which hi+ 1, lj+ 1, mj+ 1 are nonnegative constants. Denote R,, = {z(t): z(t) ~ C[a - rl, b + 72 ] c~ C2[a, b]; z(t) = O, t [ a - rl, a ] U [ b , b + r 2], z ' ( a + 0 ) = 0 z ' ( t ) on [a,b] has only some limited discontinuous point of the first kind} For z(t) E R,,, we define the norm max Iz'J)(t)l, Hzll = ()
(C3,o(b-a)J/c3
'
j)max.z(')(t a,~t
It is easy to verify that R,, is a Banach space.
"Q)I, +'r2). }.
283
Third-Order Differential.Difference Equation
For Ra, we can obtain by simple calculus the following lemma. LEMMA 4.
For any x ~ R a, define an operator
(3.3)
To = f b + "2 G( t, s) x " ( s - vl) ds. a--T 1
T h e n l T o z ( t ) l < Aollxll, IT; z(t)l < Al(b - a)llxJl, T~' x(t)l <~ A2(b - a)2ll xll in which A o = 3, A 1 = 81/4, A 2 = 81/2.
LEMMA 5.
For any x ~ R a, define an operator
(3.4)
T1 = fb+ "2 G( t, s) x " ( s + "2) ds. Ca
-T
1
Then l Tl x( t)l • Aoli zll, IT~ x( t)l
~
AI( b -
a)ll xll, IT~' x( t)l
<<.A2( b
-
a)211xll
in which A o = 3, A 1 = 81/4, A 2 = 81/2.
THEOREM 4. With respect to problems (1.5)-(1.6), we assume that there exists an approximate solution z( t), and (i) the F is of Lipschitz class on [ a, b] × D, in which D = { ( ~ ( t ) , ~( t - ~'1), u( t + ~'2)): lu(J)(t) - z(J)( t)l <~ g c 3 , / C 3 , o ( b - a) j, lu(J)( t -
r~) - z(J)( t -
~'1)1
<<-NC3, J C a , o ( b -
a) j, ]u(J)( t + ~'2) - z(J)( t + ~'2)1
<<. NC3, j/C3,o( b -
a)J};
(ii) 2
0
-- ~: {h~+l ÷ lj+l ÷ mj+~}C3, j(b - a)3-¢÷ 3(Icll + Ic2i) < 1; j=0
(iii) ,e(1 -- 0)-1 Ca,o (b - a) 3 ~< N
(e ~ max I~(t)l). a~t.~b
284
W. PEIGUANG
Then (1) there exists a solution z*( t) of (1.5)-(1.6) in S( z, N); (2) z*(t) is the unique of (1.5)-(1.6) in S(z, N); (3) the sequence { zm( t)} converges to z*( t),
in which Zm+ l( t ) = fab+~ G( t, s){F( s; ~m( s), $m( S - 7.1), ~m(S + ~'2)) q-[ ClZ~( S -- T1) Jr C 2 4 ( S
Zo= Z(t )
"4- T2)]} d,8
m= 0,1,2...
and has
IIz*(t)
-
zmll ~< OmNo •
( N O = (1 --
o)-'llz,
- =o11).
PROOF. For mapping T, using Theorem 1 we can get T to map R a to itself, and let z(t) ~ S(z, N). By the norm of R~, we have (Y~(t), ~(t 7.1), Y~(t+ z2)) ~ D. Further, if z(t), y(t) ~ S( z, N), using Lemma 1 and Lemma 2, we have
I T(J)x(t) - T(J)y( t)l <<. C3, y( b - a) 3-j max IF(t; ~( t), ~c( t as~ t ~ b
~ ] ( t ) , ~](t - T1) , y ( t + 7./'2))1
-F(t;
+
le~l
+ Ic21
7.1), ~( t + 7.2))
fb+,2a_ .l
G( t, s)[ ~ttt( s - 7.1) - ~/t'( s - 7"1) ] d8
fb+'~G(t'a-~-i s)[ z"(s
+ 7.2) - y"(s + 7.2)]
dsI
2
<~ C3, y( b- a) a-j max
E {hj+llx(J)( t) - y(Y)(t)l
a
+ lj+ 11 x(J)( t +
+ lclllfb+'~ a-~, G( t' s)[ ~"( s -
xO)(t
7.1) +
-y"(s-7.1)1
Y(J)( t -
y('(t +
dsj
7.1)1
285
Third-Order Differential-Difference Equation
+ I~:t f"+'~ c(t, ,)[ ~"(, + ~-~) - ¢ ' ( , + ~-~)1 '~1 a--'r 1
2
max
E
{hj+l + lj+l -4- mj+l}C3, j ( b -
a
+ Aj(ICll +
a)3-J]]x - yH
Ic21)llz- yll.
Hence, ( C3,o( b - a)J/C3, j)l T(J)x(t) - T (j) y( t)l 2
-< E {hj+l + lj+l + mj+l}c~,j(b - a)~-Jllx - yll j=O
+ At(Ca,o(b - a)J/c3, j)Ocl] + Ic2[)llx- yll.
Similarly, we have (C3,0(b- a)J/c3, j)IT(J)x( t - ~1) - T(J)Y( t - ~'1)1 2
-< E {h~+~ + I~+1 + mj+l}c~ j(~ - a)3-Jll~-
Yll
j=O -~- Aj( C3,o( b - a) J// c3, j)(Ic11 + Ic21)ll x - yll
( C3,0( b - a)J/C3, j)LT(J)x( t + "r2) - T(J)Y( t + ~2)] 2 ~'< E {hj+l + lj+l -4- mj+l}C3, j(b - a)3-JHx - y]]
j=O
+ Aj(C3,o(b - a)J/c3, j)(]cl] + ]c2])Hx- yH.
W. PEIGUANG
286 Therefore,
IITx(~) - Ty(t)ll < O l l x - yll.
(4.3)
Further, from (3.2), we have
Tz( t) - z( t) = f b+~2 G( t, s)~?( s) ds. a-1"
(4.4)
1
Using Lemma 1, we obtain
[T(J)z( t) - z(J)( t)[ < 8C3, j( b - a)3-J; hence,
( C3,o( b - a)J/Ca, j)lT(J)z( t) - z(J)( t)[ <~ 8 C 3 , o ( b - a) 3. Similarly, we have
( Ca, o( b - a)J/Ca, i)tT(J)z( t - ~'1) - z(J)( t - T1)] ~ 8C3,o( b - a) 3 ( C3, o( b - a)J/C3, j)[T(i)z( t + v2) - z(i)( t + ~2)1 < eC3,o( b - a) 3 then
IITz( t) - z(t)ll < 6 C 3 , o ( b -
a) 3
by condition (iii) ( 1 - 0 ) - l l l T z ( t ) - z(t)[[ < N. Thus, the conditions of Lemma 3 are satisfied and the conclusions (1)-(3) follow. REFERENCES 1 H.Z. Wang, Iterative Methods for Differential-difference Equation, (in Chinese) J. Jilin University, 3:31-42 (1988). 2 L.L. Wang, On Boundary Value Problem and its Singular Perturbation Solutions for a Mixed Differential-Difference Equations, Chinese Anhui University, Master Thesis, 1990. 3 P.G. Wang and Y. L. Luo, Iterative Methods for the boundary value problem of a third order differential-difference equation, Ann. of Diff. Eqs. 11(1):88-94 (1995). 4 L. B. Rall, Computational Solutions of Nonlinear Operator, John Wiley, New York, 1969.
B o u n d a r y V a l u e P r o b l e m of a Third-Order M i x e d T y p e Differential-Difference E q u a t i o n Wang Peiguang
Department of Mathematics Hebei University Baoding, 071002 People', 's Republic of China
ABSTRACT In tlhis paper, the existence and uniqueness of the solution for the boundary value problem of a third order mixed differential-difference equation is given by using Schauder's fLxed point theorem, and an approximate solution is given by using the Picard's iterative methods. © Elsevier Science Inc., 1996
1.
INTRODUCTION
Due to the applications of the differential-difference equations in the fields of biological mathematics, mechanics, and control theory, people are getting more and more interested in the boundary-value problem. Wang and Wang [1, 2] have discussed the existence and uniqueness of solutions for second-order equation and second-order mixed boundary-value problems, respectively, and have given the estimation of the error between the solution and the approximate solution. In this paper, by using Schander's fixed point theorem, the same problem is studied for third-order mixed differentialdifference equations. Wang and Luo [3] present a special case of the problem discussed in this paper.
APPLIED MATHEMATICS AND COMPUTATION 80:273-286 (1996) © Elsevier Science Inc., 1996 0096-3003//96/$15.00 655 Avenue of the Americas, New York, NY 10010 SSDI 0096-3003(95)00303-7
274
W. PEIGUANG
We consider the boundary value problem V"( t) - cl y ' ( t - ~ ) - ~ y ' ( t + ~ ) ---f(t;y(t),~l(t-~-l),~i(t+~-2) y( t) = ¢b( t)
a-
y( t) = ~ ( t)
b <.< t <. b + ~
)
(a
(1.1)
~'1 <~ t <~ a, y'( a + O) = m (1.2)
where ~(t)= (y(t), y'(t), y"(t)); ~(t + A ) = ( y ( t + A), y'(t + A), y"(t + h)), ( h = - ~'i, ~'2), ~'i, ~'2 are positive numbers, ¢ ~ C2([ a - ~'1, a], R), 0 ~ C~([ b, b + ~'2], R) and f is continuous in t, and assume
Ic~l DEFINITION 1.
+
Ic21 <
(1.3)
1.
A function y(t) is called a solution of (1.1)-(1.2), if (i)
y(t) ~ C [ a - vl, b + ~2] ~ C:[ a, b], (ii) y"(t) on [a,b] has only some limited discontinuous points of the first kind, (iii) y(t) satisfies the condition (1, 2), (iv) Equation (1.1) holds for t ~ [ a, b], and t ~ a + k~-t, t ~: b -
k~'2 (k ~ Z+). For convenience, we make a similar change of the variable as [3]
z( t) = a( t)dp( t) + ~( t)~( t) + ~/( t) + y( t).
(1.4)
Then the problem in transformed into
z'( t) - clz"( t - ~'1) - c2z"( t + "r2) --F(t;~(t),~(t-~'l),~.(t+~'2) ) z(t)=0
t~[a-Tl,
a]U[b,b+r2] ,
(a
(1.5)
z'(a+0)=0,
(1.6)
where ~.(t) = (z(t), z'(t), z"(t)); z(t + A) = ( z ( t + A), z'(t + A), z"(t + A)), (A
-~- - - T1 , '/'2) ,
275
Third-Order Differential-Difference Equation
Constructing the Green function of problems (1.5)-(1.6) gives - ( ( t - a)( b - s ) / ( b - a ) ) 2 / 2 + ( t -
c ( t, s) =
I
a
s)2/2
a))2/2
0
a <~ t <. s <~ b a-'q
0
a-71<~s<~aorb<.s
Using the Green function, we can obtain the following lemma.
LEMMA 1 [3].
Let x(t) ~ K = { h(t); h ' ( t) on [ a, b] has only some limited discontinuous points of the first, kind, h(t) = 0, t ~ [ a - ~'1, a] U [b, b + r2], h ' ( a + 0) ~ 0}. Then Ix(k)(t)l < C3, k(b - a ) 3 - k M ( k = 0,1,2) in which M = maxa~ t< b] Y(3)(t)l, C3,o = 2/81, C3,1 = 1/6, C3, 2 = 2/3.
LEMMA 2. Let Q = maxnlFI, (t; ~,(t), ~,(t - ~'1), ~(t + ~'2)) e 12 [ a, b] × R 9. If z( t) is a solution of (1.5)-(1.6) then max
Iz'( t)] ~ Q/(1 - t c 1] -]c2])
c
(1.7)
a~ t~ b
LEMMA 3 [4]. Let B be a Banach space and let r > 0, r ~ R, S(x o, r) = { x e B: 1 i x - xol] < r}. Map S( x o, r) into B and (i) for all x, y ~ S(x o, r): i l T x - TyOi <~ a t l x (ii) r o = ( 1 ko) -1 × l I T x o - x o l ] ~ < r.
yll, (0 ~< a < 1),
Then
(1) T has a fixed point x* in S( Xo, ro); (2) x* is the unique fixed point of T in S( x o, r); (3) the sequence{xm} defined by xm+ 1 = Tx m. m = O, 1 ... converges to x* with ILx* - xmll < k~ro. 2.
E X I S T E N C E AND UNIQUENESS
Now we give the existence and uniqueness of the solution to the problem (1.5)-(1.6).
276
W. PEIGUANG
THEOREM 1.
Suppose that
(i) k i > 0 (i = 0, 1, 2) are given constants, l-I is a closed and bounded subset of [a, b] × C9[a, b] and IFI <~ Q, in which Q = max iFI,
a = {a <~ t <~ b, z ( t ) ~ C[a - vl, b + ~'2] N C2[a, b], ]z(i)(t)l ~< ki, I z ( ° ( t - T1) [ <
ki,
]z(i)(t-[-
T2)I <
ki};
(ii) the following inequality holds b - a < min{(cko/QC3,o) i/s, ( ck~/QC a ,),/2, ( c k j Q C 3 , 2 ) } ( c = 1 - I c , I - Ic21)
Then the boundary value problems (1.5)-(1.6) have at least one solution.
PROOF. Denote E = {z(t): z(t) ~ C[a - ~'1, b + r2] A C2[a, b], z(t) = O, t ~ [ a - T1, a] [,.J [b, b + ~'2]; z'(a + O) = O, maxa~t~blz(3)(t) ] ~< kj, j = 0, 1, 2, z ' ( t ) on [a, b] has only some limited discontinuous points of the first kind}; then E is a closed convex subset of the Banach space C2[ a, b]. We define an operator T as follows.
Tz(t) --/b÷~2 G(t, s){F(s; ~(s), ~ ( s - ~,), ~(s + ~2)) a--~- 1
+ [ e l z ' ( s - ~,) + c2z'(s + ~2)1} ds (a-
T, ~< t~< b + r2)
(2.1)
By the condition (ii) and L e m m a 1 and L e m m a 2, we have
]( Tz)(')( t)l < ( QC3 ,( b -
a)S-')/c
~ k,.
It is easy to see T maps E onto itself and is completely continuous. It then follows from Schauder's fixed point theorem that T has a fixed point z*(t). The z*(t) is a solution of (1.5)-(1.6).
Third-Order Differential-Difference Equation
277
COROLLARY. Assume the function F(t; ~(t), ~(t - ~'1), ~(t + r2)) on [a, b] × R 9 satisfying, 2
[El ~ h --~ E {hj+l[ z(J)( t)l °t{j) "+- lj+l[ z(J)( t - T1)I fl(j) j=0 + m.,I z(J)(t + ~2)I V(j)
(2.2)
where h, hi+l, lj+l, mj+ 1 are nonnegative constants, 0 ~ a ( j ) <. 1,0 ~ ( j ) < 1, 0 ~< T(J) ~< 1. Then the boundary value problem (1.5)-(1.6) has at least one solution, provided that
{(
O=
E {hi+l+ /j+l +
j=O
PROOF.
mj+l}Ca,j(b-
a) a-j
)/
c
< 1
(2.3)
Let
Zj+,(.(j)) = { ohi+ 1 ~j+l(~(J)) = { 0lj+~ mj+l(f~(J)) ---- {
ol(j) = 1 o<.(j)
mj+ l
v( J)
0
0 < ~(j) <1.
=
1
Using Theorem 1, we have 2
Q ~ h-~ E {(hj+l --hj+l)ka(J) -[- (lj+l
-
-
-lj+l ) k~(J)
j=o
"I'-(mj+ 1 -- -mj+l)k "y(j)} -{- I( ko, kl, k2) (2,4)
278
W. PEIGUANG
in which 2
I(k0, kl, ks) = 2 {~j+l + ~j+l + ~+~)kj.
(2.5)
j=O
Taking
kj = c a , / ( q , o ( ~ -
a) j)
( j = o, 1,2),
then the following inequality holds
I( ko, k1, ks)
Similarly, we have
I( ko, kl, k2) .
-hi+ 1 +-Ij+ 1 +Nj+,}Ca, i ( b - a ) a-j k a ( 0 3 , 1 ( b - a ) 2)
I( ko, kl, k~)
when a ( j ) = fl(j) = y(j) = 1, the coefficient of k "(j), k t~(j) and k ~'(j) are equal to zero; then by choosing kj sufficiently large and using (2.3), we have
Q<~ cko/( C3,o( b - a)3), q<~ ck2/( ca 2( b -
Q<~ ckt/( Ca,,( b - a)2),
~) ).
(2.6)
Third- Order Differential-Difference Equation
279
From (2.0, it is easy to get
b-
a < min{(ck0/QC3,0)l/3), (ckl/QC3,1)1/2), (ck2/QC3,2))};
(2.7)
Theorem 1 conditions are satisfied, so the corollary holds.
REMARK. Theorem 1 is a local existence theorem, whereas a corollary does not require any condition on the length of the interval.
THEOREM 2.
Assume the function F on [ a, b] × R 9 satisfies the follow-
ing condition 2
IFI < h + E {hj+ 11 zCJ)(t)[ + zj÷ll z(J (t- 1)1 ÷ mj÷ll z'J'(t ÷ j=O
(2.8) Then problems (1.5)-(1.6) have one solution, provided that
PROOF. From (2.8), we can easily get that z(t) -~ 0 is a solution of the problem (1.5)-(1.6); in order to prove the theorem, we only need to prove problems (1.5)-(1.6) have no nontrivial solutions. Supposing the problems (1.5)-(1.6) have nontrivial solutions, using Lemma 1, we have
IZ"(t)l ~
[2
E {hj+l "~- lj+l "~- mj+l}C3, j( b - a) 3-j j=o
1
max b I z"( t) l × a<~
280
W. PEIGUANG
then
max I z"( t)l < o a ~ t<~ b
Iz"( t)l a ~ t<~ b
by 0 < 0 < 1, we can get m a x ~ t~ b ]Z"(t)l = 0, then z ' ( t ) = 0. But this means t h a t z(t) is a nonzero linear function. This is impossible because we have z(a) = z(b) = 0, z ' ( a + 0) = 0 from the condition (1.6); therefore,
z(t) - O.
THEOREM 3. Let f on [ a, b] × R 9 be differential with respect to y(J) ( j = 0, 1, 2), and satisfy
l a f / 3 y(J)( t)l <~ hi+ 1, ] ~ f / 3 y(J)( t - T1)I
< lj+l, IOf/OY(J)(t + ~2)1 <~ mj+l. Then the boundary value problems (1.1)-(1.2) have a unique solution, provided that
hi+ 1 + lj+ l + mj+ 1} C3, j( b - a) 3-j
0 =
c < 1.
PROOF. Suppose yl(t), y2(t) are solutions of (1.1)-(1.2), respectively. Let z(t) = Yl(t) - Y2(t),
IOf/c~Y(i)( t)] = fj, [Of/OY(J)( t - ~'1)1 = gj, [Of/c~Y(J)( t + ¢2)J = Pj; then z(t) satisfies zm(t)
-- C1 Z m ( t -- T1) -- C2 z m ( t "~- '1"2)
---- f ( t ; ~ l ( t ) , ~ll(t -- T1) , ~ l ( t + 72) ) - f ( t ; ~lz(t), ~12(t + T1) ,
~2(t + a'2) )
Third-Order Differential-Difference Equation
281
and z(t) = 0 t ~ [a - ~-, a] U [b, b + ~'2] z'(a + O) = O. Let
aj(t) = fol fj(t; ~2(t) + s z ( t ) , 7]2(t - "Q), ~2(t + ~'2)) ds bj( t )
= £ 1 gj(t; ~:(t), ~:(t- ~) + ~z(t- ~), ~ ( t + ~)) d~
e.( t) -~- fo~pj(t; Y2(t), y 2 ( t -
T1) , ~]2( t Jr" T2) + sz( t-~- 'T2) ) ds.
Then ~ " ( t ) - ~ z " ( t - ~-,) - c~ z " ( t + ~-~)
2 E {aj(t) z(2(t) + bj(t) z(J)( t - "Q) + ej(t) z(J)(t + "r2)}. j=O Noticing [aj(t)] <<.hi+l, Jbj(t)] <~ lj+ 1, lej(t)l ~< mj+l, then the Theorem 2 condition is satisfied; therefore z(t) -- 0. 3.
PICARD'S ITERATIONS
We know it is very difficult to obtain the solutions of differential-difference equations, so we now introduce the approximate solution and discuss an estimation of the error between the approximate solution and the solution. Because problems (1.1)-(1.2) are equivalent to (1.5)-(1.6), we can discuss problems (1.5)-(1.6).
DEFINTION 2. A function z(t) is called an approximate solution of problems (1.5)-(1.6), if there exist e(6 > 0), such that
max I z " ( t ) - c l z " ( t a<~t<~b
.Q) - c 2 z " ( t + 72)
- F ( t; -~( t), -z( t - ~,), -z( t + ~-~))] < ~
(a.1)
282
W. PEIGUANG
and z( t) ~ 0 t e [ a - 71, a] U [ b, b + ~'2] z'( a + 0) - 0. In fact, the approximate solution z(t) can be expressed as
z( t) = fab+'~ a( t, s){F(s; ~( s), ~( s - r~), ~( s + r2) "{'[ClZm(8 -
T1) +
in which rl(t) = z"(t) - q z " ( t T1) , 7z(t + 72)).
DEFINITION 3.
C 2 Z m ( 8 "["
72) ] + TI(8)}
d8
(3,2)
71) - c2z"(t + r~) + F(t; ~(t), ~ ( t -
The function f is said to be of Lipschitz class for all (t; ~ ( t ) , ~(t - r , ) , ~(t + T2)),
(t; ~( t), 7( t -
71), 7( t + ~'2)) e [ a, b] x D, ( D e
Rg).
The following inequality is satisfied f( t; ~ ( t ) , ~( t - T1) , "~( t "4" T2) ) -- f(t; 7 ( t ) , 7( t - r l ) , 7( t + r2))l
2 <~ E {hj+llu'J)( t) - v(J)( t)l + li+llu(J) ( t -
"rl) - v(J)( t -
71) ]
~'~)
7 )1}
j~ 0
+%+,lu(J)(t
+
-
v(')(t
+
m which hi+ 1, lj+ 1, mj+ 1 are nonnegative constants. Denote R,, = {z(t): z(t) ~ C[a - rl, b + 72 ] c~ C2[a, b]; z(t) = O, t [ a - rl, a ] U [ b , b + r 2], z ' ( a + 0 ) = 0 z ' ( t ) on [a,b] has only some limited discontinuous point of the first kind} For z(t) E R,,, we define the norm max Iz'J)(t)l, Hzll = ()
(C3,o(b-a)J/c3
'
j)max.z(')(t a,~t
It is easy to verify that R,, is a Banach space.
"Q)I, +'r2). }.
283
Third-Order Differential.Difference Equation
For Ra, we can obtain by simple calculus the following lemma. LEMMA 4.
For any x ~ R a, define an operator
(3.3)
To = f b + "2 G( t, s) x " ( s - vl) ds. a--T 1
T h e n l T o z ( t ) l < Aollxll, IT; z(t)l < Al(b - a)llxJl, T~' x(t)l <~ A2(b - a)2ll xll in which A o = 3, A 1 = 81/4, A 2 = 81/2.
LEMMA 5.
For any x ~ R a, define an operator
(3.4)
T1 = fb+ "2 G( t, s) x " ( s + "2) ds. Ca
-T
1
Then l Tl x( t)l • Aoli zll, IT~ x( t)l
~
AI( b -
a)ll xll, IT~' x( t)l
<<.A2( b
-
a)211xll
in which A o = 3, A 1 = 81/4, A 2 = 81/2.
THEOREM 4. With respect to problems (1.5)-(1.6), we assume that there exists an approximate solution z( t), and (i) the F is of Lipschitz class on [ a, b] × D, in which D = { ( ~ ( t ) , ~( t - ~'1), u( t + ~'2)): lu(J)(t) - z(J)( t)l <~ g c 3 , / C 3 , o ( b - a) j, lu(J)( t -
r~) - z(J)( t -
~'1)1
<<-NC3, J C a , o ( b -
a) j, ]u(J)( t + ~'2) - z(J)( t + ~'2)1
<<. NC3, j/C3,o( b -
a)J};
(ii) 2
0
-- ~: {h~+l ÷ lj+l ÷ mj+~}C3, j(b - a)3-¢÷ 3(Icll + Ic2i) < 1; j=0
(iii) ,e(1 -- 0)-1 Ca,o (b - a) 3 ~< N
(e ~ max I~(t)l). a~t.~b
284
W. PEIGUANG
Then (1) there exists a solution z*( t) of (1.5)-(1.6) in S( z, N); (2) z*(t) is the unique of (1.5)-(1.6) in S(z, N); (3) the sequence { zm( t)} converges to z*( t),
in which Zm+ l( t ) = fab+~ G( t, s){F( s; ~m( s), $m( S - 7.1), ~m(S + ~'2)) q-[ ClZ~( S -- T1) Jr C 2 4 ( S
Zo= Z(t )
"4- T2)]} d,8
m= 0,1,2...
and has
IIz*(t)
-
zmll ~< OmNo •
( N O = (1 --
o)-'llz,
- =o11).
PROOF. For mapping T, using Theorem 1 we can get T to map R a to itself, and let z(t) ~ S(z, N). By the norm of R~, we have (Y~(t), ~(t 7.1), Y~(t+ z2)) ~ D. Further, if z(t), y(t) ~ S( z, N), using Lemma 1 and Lemma 2, we have
I T(J)x(t) - T(J)y( t)l <<. C3, y( b - a) 3-j max IF(t; ~( t), ~c( t as~ t ~ b
~ ] ( t ) , ~](t - T1) , y ( t + 7./'2))1
-F(t;
+
le~l
+ Ic21
7.1), ~( t + 7.2))
fb+,2a_ .l
G( t, s)[ ~ttt( s - 7.1) - ~/t'( s - 7"1) ] d8
fb+'~G(t'a-~-i s)[ z"(s
+ 7.2) - y"(s + 7.2)]
dsI
2
<~ C3, y( b- a) a-j max
E {hj+llx(J)( t) - y(Y)(t)l
a
+ lj+ 11 x(J)( t +
+ lclllfb+'~ a-~, G( t' s)[ ~"( s -
xO)(t
7.1) +
-y"(s-7.1)1
Y(J)( t -
y('(t +
dsj
7.1)1
285
Third-Order Differential-Difference Equation
+ I~:t f"+'~ c(t, ,)[ ~"(, + ~-~) - ¢ ' ( , + ~-~)1 '~1 a--'r 1
2
max
E
{hj+l + lj+l -4- mj+l}C3, j ( b -
a
+ Aj(ICll +
a)3-J]]x - yH
Ic21)llz- yll.
Hence, ( C3,o( b - a)J/C3, j)l T(J)x(t) - T (j) y( t)l 2
-< E {hj+l + lj+l + mj+l}c~,j(b - a)~-Jllx - yll j=O
+ At(Ca,o(b - a)J/c3, j)Ocl] + Ic2[)llx- yll.
Similarly, we have (C3,0(b- a)J/c3, j)IT(J)x( t - ~1) - T(J)Y( t - ~'1)1 2
-< E {h~+~ + I~+1 + mj+l}c~ j(~ - a)3-Jll~-
Yll
j=O -~- Aj( C3,o( b - a) J// c3, j)(Ic11 + Ic21)ll x - yll
( C3,0( b - a)J/C3, j)LT(J)x( t + "r2) - T(J)Y( t + ~2)] 2 ~'< E {hj+l + lj+l -4- mj+l}C3, j(b - a)3-JHx - y]]
j=O
+ Aj(C3,o(b - a)J/c3, j)(]cl] + ]c2])Hx- yH.
W. PEIGUANG
286 Therefore,
IITx(~) - Ty(t)ll < O l l x - yll.
(4.3)
Further, from (3.2), we have
Tz( t) - z( t) = f b+~2 G( t, s)~?( s) ds. a-1"
(4.4)
1
Using Lemma 1, we obtain
[T(J)z( t) - z(J)( t)[ < 8C3, j( b - a)3-J; hence,
( C3,o( b - a)J/Ca, j)lT(J)z( t) - z(J)( t)[ <~ 8 C 3 , o ( b - a) 3. Similarly, we have
( Ca, o( b - a)J/Ca, i)tT(J)z( t - ~'1) - z(J)( t - T1)] ~ 8C3,o( b - a) 3 ( C3, o( b - a)J/C3, j)[T(i)z( t + v2) - z(i)( t + ~2)1 < eC3,o( b - a) 3 then
IITz( t) - z(t)ll < 6 C 3 , o ( b -
a) 3
by condition (iii) ( 1 - 0 ) - l l l T z ( t ) - z(t)[[ < N. Thus, the conditions of Lemma 3 are satisfied and the conclusions (1)-(3) follow. REFERENCES 1 H.Z. Wang, Iterative Methods for Differential-difference Equation, (in Chinese) J. Jilin University, 3:31-42 (1988). 2 L.L. Wang, On Boundary Value Problem and its Singular Perturbation Solutions for a Mixed Differential-Difference Equations, Chinese Anhui University, Master Thesis, 1990. 3 P.G. Wang and Y. L. Luo, Iterative Methods for the boundary value problem of a third order differential-difference equation, Ann. of Diff. Eqs. 11(1):88-94 (1995). 4 L. B. Rall, Computational Solutions of Nonlinear Operator, John Wiley, New York, 1969.