Finite difference inequalities in one variable

Chapter 4

F i n i t e d i f f e r e n c e i n e q u a l i t i e s in o n e variable 4.1 Introduction The theory of finite difference equations has gained increasing significance in the last decades as is apparent from the large number of publications on the subject. A great variety of methods and tools are available for handling such equations. In the study of many finite difference and sum-difference equations, one often needs some new and specific type of finite difference inequalities for proving various theorems or approximating functions. The desire to widen the scope of applications of such inequalities resulted in the necessity of discovering new finite difference inequalities which are directly applicable in the new situations. In this chapter, we offer various basic finite difference inequalities recently investigated in [35,37,39,44,45,53,55,57,67,68,70,73,75] which can be used as powerful tools in certain applications. Some fundamental applications are given to illustrate the usefulness of certain inequalities.

4.2 Fundamental finite difference inequalities In this section, we focus our attention on some basic inequalities established by Pachpatte in [57] (see also [42]) which provide explicit bounds on unknown functions and can be used as an effective tool in the development of the theory of finite difference equations and numerical analysis. We start with the following theorems which deals with the finite difference inequalities proved in [57]. 197

Finite difference inequalities in one variable

198 T h e o r e m 4.2.1.

Let u (n) , a (n) , b (n) , c (n) , p (n) E D (No, R+ ) and Ac (n) >

0 for n C No., If u(n)_
(

c(n)+

p(s) u(s) sz0

)

(4.2.1)

,

for n C No, then n-1

u(n)_
c(0) H [ l + b ( s ) p ( s ) ] s--0

n-1

n-1

+E[Ac(s)+a(s)p(s)] s=0

n

)

[l+b(a)p(a)]

,

(4.2.2)

a=s+l

for n C N0. Proof.

Define a function z ( n ) by n--1

(4.2.3)

p (~) ~ (~)

z (~) - ~ (~) + ~ s--0

Then z ( 0 ) - c(0) and (4.2.1) can be restated as (4.2.4)

u (n) < a (n) + b (n) z (n). From (4.2.3) and (4.2.4)we observe that Az (~) - Ac (~) + p (~) ~ (~)

<_ b (n) p (n) z (n) + [Ac (n) + a (n) p (n)].

(4.2.5)

Now by applying Theorem 1.2.1 given in [42, p. 11] to (4.2.5) we get n--1

z (n) < c (0) n

[1 + b (a) p (a)]

s--0 n--1

n-1

+E[Ac(s)+a(s)p(s)] s=0

H

[l+b(~)p(~)].

(4.2.6)

cr=s+l

Using (4.2.6) in (4.2.4) we get the required inequality in (4.2.2). R e m a r k 4.2.1. We note that in the special case when c(n) - 0 , the inequality given in Theorem 4.2.1 reduces to the inequality given by Pachpatte, see [42, Theorem 1.2.3, p. 13]. T h e o r e m 4.2.2.

Let u ( n ) , a ( n ) , b ( n ) , c (n), Ac (n) be as in Theorem 4.2.1.

Chapter 4

199

(al) Let L" No x R+ ---, R+ be a function such that 0_ y > 0, where defined for n E No, y E R+. If

M(n, y)

(

(4.2.7) is a real-valued nonnegative function

/

s=0

(4.2.8)

,

for n E No, then

u (n) _< a (n) + b (n)

(

c (0)

n--1 +E

[1 + M (s, a (s)) b (s)] z

[Ac(s) + L ( s , a ( s ) ) ]

s=0

n--1 ) I I [1 + M ( ~ , a ( c r ) ) b ( c T ) ] cr=s+l

,

(4.2.9)

for n E No. (a2) Let L" No x R+ ---, R+ be a function which satisfies the condition 0

(4.2.10)

for n E No, x _> y >_ 0, where M(n, y) is as in (al), r R+ ~ R+ is a continuous and strictly increasing function with r (0) - 0, r is the inverse function of r and

~)-1 (xy)

_~ r

(X)r

(y),

(4.2.11)

for x , y E R+. If

u(n)
(

c(n)+

)

L(s,u(s)) , s--O

(4.2.12)

for n E No, then

u (n) _< a (n) + b (n) r

t c (0) n-1 I I [1 + s--o

M (s, a (s))

(~-1 (b (8))]

n--1 n--1 / + E [Ac(s)+L(s'a(s))] I I [l+M(cr'a(c~))c-l(b(cr))] ' s=0 a=s+l for n E No.

(4.2.13)

Finite difference inequalities in one variable

200

(al) Define a function z(n) by

Proof.

n--1

z(~) - ~ ( ~ ) + ~

(4.2.14)

L (~, ~ (~))

s=O

Then z(O) -- c(O) and (4.2.8) can be restated as (4.2.15)

u (n) < a (n) + b (n) z (n). From (4.2.14), (4.2.15) and (4.2.7)we have Az (~) - zx~ (~) + L (,~, ~ (~))

< Ac (n) + L (n, a (n) + b (n) z (n)) - L (n, a (n)) + L (n, a (n)) (4.3.16)

_< M (n, a (n)) b (n) z (n) + [Ac (n) + L (n, a (n))]. Now by applying Theorem 1.2.1 given in [42] to (4.2.16) we get n--1

z(n) <_ c(O) I I [1 + hi (s,a(s))b(s)] s=0 n--1

+ E

n--1

[Ac (s) + L (s, a (s))] I I

s=0

(4.2.17)

[1 + AI (c~,a (or)) b (~)].

a=s+l

Using (4.2.17) in (4.2.15) we get the desired inequality in (4.2.9). (a2) Define a function z(it) by (4.2.14). Then z(0) - c(0) and (4.2.12) can be restated as (4.2.18)

u (n) < a (n) + b (n) r (z (n)). From (4.2.14), (4.2.18), (4.2.10) and (4.2.11)we have

Az (~) - / x ~ (~) + L (~, ~ (~))

< Ac(n)+L(n,a(n)+b(it)O(z(n)))-L(n,a(n))+L(it,

a(n))

<_ M (n, a (n)) r -1 (b (n) O (z (n))) + [Ac ( n ) + L (n, a (n))] <_ M (n, a (n))

r

(b (it))

Z (It) -nt-

[AC (It)

Jr_ L

(n, a (n))].

(4.2.19)

Now an application of Theorem 1.2.1 given in [42] to (4.2.19) yields n--1

z(n) < c(O) H [1 + ]~I (s,a(s))O -1 (b (s))] s=0 n--1

+E[Ac(s)+n(s'a(s))] s=0

n--1

H

[ l + A I ( ~ , a ( ~ ) ) O -l(b(~))].

~--s+l

Using (4.2.20)in (4.2.18) we get (4.2.13).

(4.2.20)

Chapter 4

201

c(n)

R e m a r k 4.2.2. If we take = 0 in Theorem 4.2.2, part (al), then we recapture the inequality given by Dragomir in [11] (see also [10]). We note that from Theorem 4.2.2, part (al), one can easily obtain the corollaries similar to that of given in [10] (see also [11]) which can also be used in certain applications. In the following theorems we present some useful generalizations of the inequalities given in [42, Theorem 2.3.1, Corollary 3.3.1]. Theorem

4.2.3.

Let

u (n),a (n), b (n),c (n), p (n) E D (No,R+).

(bl) Let Aa (n) > 0 for n E No; g 6 C (R+, R+) be a nondecreasing function with > 0 for u > 0. If

g(u)

n-1

u(n) < a(n) + Ep(s)g(u(s)),

(4.2.21)

s--0 for n E No, then for 0 _< n < rtl; rt, n l C No, ~t (~t) < G -1

[G (a (0)) + ~~=o( A a9 (( as()s ) )

+ p (s)

)1

(4.2.22)

where

T0

(4.2.23)

g(s),

r0 > 0 is arbitrary and G -1 is the inverse of G and n l c No is chosen so that G (a (0)) +

~(

Aa (s)

s=0

g(a(s))

+ p (8)) C Dolt ( G - l )

for all n E No lying in 0 _< n _< rtl.. (b2) Let A t ( n ) _> 0 for n E No; g, G, G-1 be as in part (bl) and suppose in addition, is subadditive and submultiplicative. If

g(u)

n -- 1

/

u(n)
,

(4.2.24)

s--O for n E No, then for 0 _< n _< n2; n, n2 E No, u (n) < a (n) + b (n) G -~ [G (c (0)) +

~(Ac(s)+p(s)g(a(s))) g ( i ii

1

(4.2.25)

,

Finite difference inequalities in one variable

202

and rt2 c No is chosen so that

a (c (o)) +

~(Ac

(s) + p (s) g (a (s))

s0

g

+ p (s) g (b ( s ) ) ) E Dora ( G - l ) ,

i;ii

for all n C No lying in 0 < n _< n2. Proof. (bl) Let a(n) > 0 for n E No and define a function z(n) by the right hand side of (4.2.21). Then z(O) - a(O), u (n) <_ z (n), z(n) > 0 and Az (n) - Aa (n) + p (n) g (u (n)) _< A a (n) + p (n) g (z (n)).

(4.2.26)

From (4.2.23), (4.2.26) and the fact that a (n) <_ z (n) we observe that

z(,~+l)

/

G (z (n + 1)) - G (z (n)) -

ds

z(~)

g(s)

< /',z (,~) - g (~ (,~)) Aa (n) + p (n) g (z (n))

< -

g

Aa(n) (a (n))

+p(~).

(4.2.27)

By taking n - s in (4.2.27) and summing up over s from 0 to n -

a (~ (~)) < a (z (o)) +

~(

Aa (s)

s:o

g (a (;i)

1 we get

+p(~))

which implies

(~) <

o -1

g ~ (~))

a (a (o)) +

(4.2.28)

s=O

Using (4.2.28) in u ( n ) <_ z (n) we get the required inequality in (4.2.22). If a (n) > 0 for n E No, we carry out the above procedure with a (n) + s instead of a (n), where s > 0 is an arbitrary small constant, and subsequently pass to the limit as e --+ 0 to obtain (4.2.22). The subdomain 0 < n < nl is obvious.

Chapter 4 (b2) Let

c(n) >

203

0 for n E No and define a function

z(n)

by

n-1

z(n) -- c(n) + Ep(s)g(u(s)).

(4.2.29)

s=0

Then z(0) - c(0),

z(n)

> 0 for n E No and (4.2.24) can be restated as (4.2.30)

u (n) _< a (n) + b (n) z (n). From (4.2.29) and (4.2.30)we have

ZXz (~) - zx~ (~) + p (~) g (~ (~)) _< Ac (n) + p (n) g (a (n) + b (n) z (n)) _< (Ac ( n ) + p ( n ) g ( a ( n ) ) ) + p ( n ) g ( b ( n ) ) g ( z ( n ) ) .

(4.2.31)

From (4.2.23), (4.2.31) the fact that c (n) < z (n) and following the proof of (bl) we obtain

z (~) < G -~ [G (~ (0)) +

~(Ac(s)+p(s)g(a(s)))+p(s)g(b(s))l, ~=0

(4.2.32)

g(~(~))

Using (4.2.32) in (4.2.30) we get (4.2.25). The proof of the case when c (n) _> 0 can be completed as mentioned in the proof of (bl).The subdomain 0 <_ n _< n2 is obvious. R e m a r k 4.2.3. If we take a(n) = k, a nonnegative constant, then the inequality given in Theorem 4.2.3, part (bl) reduces to the discrete version of the well known Bihar's inequality, see [42, p. 103]. For a detailed account on such inequalities, see [42] and also [85]. Theorem

4.2.4.

Let

u (n),a

(n), b (n) E D (No, R+).

(cl) Let Aa (n) > 0 for n E No. If n-1

u 2 (n) < a (n) + 2 ~

b ( s ) u (s),

(4.2.33)

s=0

for n C No, then

~(~) -< ~ for n E N0.

+

8--0

2~~)

+ ~(~) '

(4.2.34)

204

Finite difference inequalities in one variable

(c2) Let h E C (R+, R+) be a nondecreasing function with h(u) > 0 for u > 0. If n-1

u 2 (n) _~ a (n) + 2 Z

b (s) h (u (s)),

(4.2.35)

s--0

for n E No, then for 0 _~ n _~ n3; n, n3 E No, 1

u (n) < { H -1 [H (a (0)) + ~ ( s=o

Aa (s)

(4.2.36)

h(v/a(s) )

where

ds

H (r) -

(4.2.37)

h (x/~) 'r > O, ro

ro > 0 is arbitrary and H -1 is the inverse of H and n3 E No is chosen so that H (a (0)) +

~(

Aa (s)

+ 2 b ( s ) ) C Dora(H-I),

for all n E No lying in 0 _< n < rt3. Proof. (c 1) Let a(n) > 0 for n C No and define a function z(n) by the right hand side of (4.2.33). Then z (0) - a (0),u (n) _< V/z (n) and Az (n) - Aa (n) + 2b (n) u (n) < Aa ( n ) + 2b (n) V/Z (n).

(4.2.38)

Using the facts that v/z (n) > 0, Az (n) > 0, v/Z (n) < V/Z (n + 1), a (n) < z (n) for n E No and (4.2.40) we observe that (see [42, p. 212]) z(n+l)-z(n)

A (v/z(n)) - v/z(n + l) + v/z(n ) < -

2v/

< Aa (n) + 2b (n) v/z (n) /',a

+b(n), -

2v/a(

)

Chapter 4 which implies

v/z (~) _< v/a (0) +

205

~(~a(~) )

(4.2.39)

2 ~ i ~;) + b (~)

s--0

Using (4.2.39) in u (n) <_ V/Z (n) we get the required inequality in (4.2.34). The proof of the case when a (n) > 0 for n 9 No can be completed as in the proof of Theorem 4.2.3, part (bl). (c2) Let a(n) > 0 for n 9 No and define a function z(n) by the right hand side of (4.2.35). Then z(0) - a(0), z(n) > O, u (n) < v/z (n) and .

Az (n) < Aa (n) + 2b (n) h (v/z ( n ) ) .

(4.2.40)

As in the proof of Theorem 4.2.3, part (bl), from (4.2.37), (4.2.40) and the fact that a (n) _< z (n) we observe that

~XH (z (~)) <

<

ZXa(~)

Az(n)

+ 2b (n).

The rest of the proof can be completed by following the proof of Theorem 4.2.3, part (bl).We omit the details. R e m a r k 4.2.4. We note that the inequality given in Theorem 4.2.4, part (cl) can be considered as a generalization of the inequality in Corollary 3.3.1 given in [42], while the inequality in part (c2) is a slight variant of the special version of the inequality in Theorem 3.3.5 given in [42].

4.3 Some more finite difference inequalities Due to various motivations, several new finite difference inequalities which yield explicit estimates on unknown functions have been investigated and used extensively in the literature, see [42]. In this section, we offer some more finite difference inequalities recently established by Pachpatte in [35,45,55,68] which can be used as tool in certain new applications. Our first theorem deals with the finite difference inequalities proved in [68]. T h e o r e m 4.3.1. Let u (n),a (n) c D (No, R+); k (n, o), A l k (n, o-) C D (E, R+),whereE{(re, n) 9

Finite difference inequalities in one variable

206

(al)

Let g 6 C (R+, R+) be a nondecreasing function with g(u) > 0 for u > 0.

If n--1

u (n) <_c + E k (n, cr)g (u (cT)),

(4.3.1)

o'=0

for n E No, where c > 0 is a real constant, then for 0 < n < nl;n, nl C No,

]

(4.3.2)

A l k (rt, o-),

(4.3.3)

[ U ( n ) ~ G - 1 / G (c) +

H 8=0

where n--1

H (n) - k (n + 1, n) + E (7--0

a

-

g

> 0,

(4.3.4)

?~o

ro > 0 is arbitrary, G -1 is the inverse of G and nl E No is chosen so that n--1

G (c) + ~

H (s) ~ Dora (G - I )

s=0

for all n C No lying in 0 _< n < rtl. (a2) Let g, G, G -1 be as in (al) and suppose in addition g(u) is subadditive. If n--1

u (n) <_a (n) + ~

k (n, a)g (u (or)),

(4.3.5)

(7=0

for n E No, then for 0 _< n < n2; n, n2 E No,

u ( n ) < - a ( n ) + G - I [G(B(n))+~H(s)ls=o

(4.3.6)

where H(n)is given by (4.3.3), n-1

B (n) - E k (n, o)g (a (a)), v~--0

for n C No and n2 c No is chosen so that n--1

G (B (n)) + ~

H (s) E Dora ( G - l ) ,

s--0

for all n C No lying in 0 <_ n <_ n2.

(4.3.7)

Chapter 4

207

P r o o f . (al) Let c > 0 and define a function z(n) by the right hand side of (4.3.1). Then z(O) ~ c, u (n) < z (n), z(n) > 0 and rt--1

Az (n) = k (n + 1, n) g (u (n)) + ~ / ~ 1

~ (Tt, Or) g

(tt (if))

cry0

<_ H (~)g (z (~)), where H(n) is given by (4.3.3), see [42, p. 22]. The rest of the proof can be completed by following the similar arguments as in the proof of Theorem 4.2.3, part (bl). We omit the details. (a2) The proof follows by closely looking at the proof of (al) and the proof of Theorem 4.2.3, part (b2). Here we leave the details to the reader. The next theorem contains the inequalities established in [55]. T h e o r e m 4.3.2.

Let u (n), k (n, a ) , Alk (n, a) and c be as in Theorem 4.3.1.

(bl) If n--1

u 2 (n) < c + E

k (n, a)u (a),

(4.3.8)

r

for n E No, then n--1

1

(~) _< v~ + ~ Z H (~),

(4.3.9)

s=0

for n E No, where H(n) is given by (4.3.3). (b2) Let g(u) be as in Theorem 4.3.1, part (al).If n--1

u 2 (n) < c + Z

k (n, cr)u (~) g (u (~)),

(4.3.10)

6r--O

for n E No, then for 0 _< n _< n3; n, n3 E No,

(~) < a -~ a (v~) + ~ ~=0 where H(n) is given by (4.3.3), G, G -~ are as defined in Theorem 4.3.1, part (al) and n3 c No is chosen so that

s--0

for all n E No lying in 0 < n _< n3.

208

Finite difference inequalities in one variable

Proof. (bl) Let c > 0 and define a function z ( n ) by the right hand side of (4.3.8). Then z(0) - c, u (n) _~ v/z (n), z ( n ) is positive and nondecreasing for n E No and n

Az (n) - E

n-1

k (n + 1,a) u(a) - E

a=O n--1

+~

k (n + 1, a) u (a)

~=0 n--1

k (~ + 1, ~)~ (~) - Z k (~, ~)~ (~)

~--0

cr--O n--1

= k (n + 1, n ) u (n) + E

Alk (n, a)u (a)

(7--0 n--1

_< k (n + 1, n) V/Z (n) + E

Alk (n, a)v/z (a)

or=0

_< H (n) V/z (n).

(4.3.12)

The rest of the proof follows by using the similar arguments as in the proof of Theorem 4.2.4, part (Cl) below (4.2.40) with suitable changes. We omit the details. (b2) The proof follows by closely looking at the proof of part (bl) given above and the proof of Theorem 3.3.5 given in [42]. We omit it here to avoid repetition. The discrete inequalities established in [35,45] are embodied in the following theorems. Theorem

4.3.3.

Let u (n) , a (n) , b (n) , g (n) , h (n) E D (No, R+) and p > 1

be a real constant. (Cl)

If n--1

~, (~) < ~ (~) + b (~) ~

[g (~)~, (~) + h (~)~ (~)],

(4.3.13)

s=O

for n E No, then u(n) <

•

{

a(n)+b(n)E

g(s) a ( s ) + h ( s ) s=0

for n e No.

p-1 P

1_

nH lE l + b ( a ) ( g ( a ) + cr--s+l

(

}p P

(4.3.14)

Chapter 4

209

(c2) Let c(n) be a real-valued positive and nondecreasing function defined on

No. n-1

u p (n) < cp ( n ) + b (n) ~

[g (s)u p ( s ) + h (s)at (s)],

(4.3.15)

s--0

for n E No, then u (n) < c (n)

l + b (n) Z

[g (s) + h (s) c 1-p (s)]

8=0

x II

l+b(cr)

g(cr)+~

cr=s+l

(~)

(4.3.16)

P

'

for n E No. (c3) Let k (n, (7), nl]~ (/Z, 0") be as in Theorem 4.3.1. If n-1

u p (n) < a (n) + b (n) Z

k (n, s) [g (s) u p (s) + h (s) u (s)],

(4.3.17)

s---O

for n E No, then

(4.3.18) a-----O

r=a+l

for n ff No, where A(n)=k(n+l,n)b(n)

g(n)+

P

n-1

(4.3.19) s--0

[? (n) - k (n + 1, n ) ( g (n) a (n) + h (n) ( p n-1 q - ~

p for n r No.

(4.3.20)

Finite difference inequalities in one variable

210

(c 1) Define a function z(n) by

Proof.

n--1

z (n) - ~

[g (s)u p (s) + h (s)u (s)].

(4.3.21)

s=0

Then z ( 0 ) - 0

and (4.3.13) can be written as

u p (n) < a (n) + b (n) z (n).

(4.3.22)

From (4.3.22), as in the proof of Theorem 1.3.2, part (al) we obtain

p--1

u (n) <

--~ +

a(n))

+

b(n)

P

z (n).

(4.3.23)

P

From (4.3.21) and using (4.3.22),

(4.3.23) w~ get

(see [42, p. 13])

Az (n) <_ b (n) (g (n) + h (n) ) z ( + [g (n) a (n) + h (n) ( p -

+ a (n) )

.

(4.3.24)

Now a suitable application of Theorem 1.2.1 given in [42, p.ll] to (4.3.24) yields n--1

zl l <_Z [ ls, alsl+hls, (P l s=0

x

II

P

l+b(a)

cr=s+l

Using (4.3.25) in

(

(4.3.22)

g(a)+

.

(4.3.25)

P

we get the required inequality in (4.3.14).

(c2) Since c(n) is positive and nondecreasing function for n 6 No, from (4.3.15) we observe that

c(~i

<_ l + b (n) E

9 (s)

~-~

+ h (s) c l-p (s)

-~

.

s--0

Now an application of the inequality given in inequality in (4.3.16).

(C1) tO

(4.3.26) yields the desired

(c3) Define a function z(n) by n-1

z (~) - ~ s---O

k

(~, ~)[g (~)~ (~) +

h

(~)~(~)].

(4.3.27)

Chnpter 4

211

Then z(0) - 0 and as in the proof of part (c1), from (4.3.17) we see that the inequalities (4.3.22) and (4.3.23) hold. From (4.3.27) and using (4.3.22), (4.3.23) and the fact that the function z(n) is nondecreasing in n, we observe that

Az (n) -- k (n + 1, n)) [9 (n) up (n) + h (n) u(n)] n--1 nt- Z A l k (?'t, 8)[g (8)U p ( 8 ) + h (8)u(8)] 8=0 _< k (n + 1, n)[g (n) (a (n) + b (n) z (n))

n-1 -t- 2 A l k (1~' 8)[g (8) (a (S) + b (s) z (s)) 8--0

+h (s) (P -p 1 + a P(S)+ b(s)z(~) ) p

<_ ~ (~) z (~) + t? (,~).

(4.3.28)

Now a suitable application of Theorem 1.2.1 given in [42, p. 11] to (4.3.28) yields

n--1 n--1 z (n) _< E / ) (or) f l [1 + A (~-)]. a~--~0 T---aq-1

(4.3.29)

From (4.3.29)and (4.3.22)the desired inequality in (4.3.18)follows. Theorem

4.3.4.

Let u ( n ) , a ( n ) , b ( n ) , g (n) E D (No, R+) and p > 1 be a

real constant. (dl) Let L" No • R+ --+ R+ be a function such that 0
(4.3.30)


for n C No, x > y > 0, where M " No z R+ --, R+.If

n--1

up (n) < a (n) + b (n) Z L (s, u (s)),

(4.3.31)

8--0 for n E No, then

u(n)<-{ a ( n ) + b ( n ) p~ -Ll(ss= o ' P -+-~a(S))p

n--1 x

II

~=s+l for n E N0.

[I + M ( or, p - 1 P

+

a(cr))b(a)] .... P

-7

}p2 (4.3.32) '

Finite difference inequalities in one variable

212

(d2) Let L " No x R+ -+ R+ be a function which satisfies the condition 0 < L (n, x) - L (n, y) < AI (n, y)

1/)-1 (X -- y),

(4.3.33)

for n 6 No, x _> y > 0,where M 9 No x R+ --~ R+, ~ 9 R+ + R+ is a continuous and strictly increasing function with ~ (0) - 0, ~p-1 is the inverse function of and //)--1 (xy) < //)--1 (X)//)--1 (y) for x, y 6 R+. If

(4.3.34)

u P ( n ) < a ( n ) + b ( n ) ~ ( ~ 'L (\s ' us ( s=) ) ) o for n E No, then

u(n)<{a(n)+b(n)~p(~L(s,p -

+ a(S))p

ks=0 n--1

• cr=s+l

1+ M

(

1

or,

p-1

_~_

P

//)--1

p

(4.3.35)

p

for n 6 No. (d3) Let W ( r ) , G, G -1 be as in T h e o r e m 1.3.2, part (b3). If n--1

u p (n) <_a (n) + b (n) E g (s) W (u (s)),

(4.3.36)

s--O

for n 6 No, t h e n for 0 < n < rt4; rt, rt4 6 No, 1 U(?%) <

a(Tt) nt- b(?%)a -1

a (]~(Tt)) 2r-

g(8) W

T

, (4.3.3r)

s=0

where n-1 _]_

s=o

P

,

P

and rt4 6 No is chosen so t h a t

a (D

+Z g

w

-7

c Do.

s--0

for all n 6 No lying in 0 _< n _< rt 4. T h e proof follows by closely looking at the proofs of T h e o r e m 1.3.2 and 4.3.3, see also [42]. Here we omit the details.

Chapter 4

213

T h e o r e m 4.3.5. Let u ( n ) , f (n) E D(No, R+), h(n, cr) E D ( E , R + ) and c > 0, p > 1 are real constants and E is defined as in Theorem 4.3.1. (el) Let g, H, H -1 be as in Theorem 1.3.3. If

~,(~) < ~+

/ (~) g (~ (~)) + ~ s=0

h (~, ~) g (~ (~)) ,

(4.3.38)

or=0

for n E No, then for 0 < n < n5;n, n5 E No, u (n) < { H -1 [H (c) + F (n)] } ~',

(4.3.39)

where

F(~) =

(~)+ s=0

h(~,~)

,

(4.3.40)

a=0

and n5 E No is chosen so t h a t

H (c) + F (n) E Dora (H -1) for all n E No lying in 0 < n < ns. (e2) If

~ (~) < ~ + ~

f (~)~ (~) +

s=0

h (~, ~)~ (~) ,

(4.3.41)

~0

for n E No, then

u (n) <_ c ~

+

p

1

F (n)

,

(4.3.42)

for n E No, where F(n) is given by (4.3.40). (el) Let c > 0 and define a function z(n) by the right hand side of (4.a.aa). T h ~ z(0) = ~, ~(~) < (z(~)}5, z(~)is positive and nondecreasing

Proof.

for n E No and n-1

Az (n) -- f (n) g (u (n)) + ~

h (n, or) g (u (0))

cr----O

(4.3.43)

Finite difference inequalities in one variable

214

From (1.3.41) and (4.3.43) we observe that z(n+l)

H (~ (~ + 1)) - ~ / ( ~ (~)) -

(1)

z(n)

g s

<

s--1

< f (n)+ E

h (n, 0).

(4.3.44)

5r=0

The rest of the proof follows as in the proof of Theorem 4.3.2, part (bl) with suitable changes. We omit the details. (e2) The proof is similar to that of Theorem 1.3.4. We omit it here to avoid repetition.

4.4 Finite difference inequalities with iterated SUIIIS

The main concern of this section is to present some finite difference inequalities involving iterated sums, investigated by Pachpatte in [53, 67,73] which can be used as tools in the study of general classes of finite difference and sum-difference equations. Our first theorem deals with the inequalities proved in [53]. T h e o r e m 4.4.1. Let u ( n ) , f (n),a(n) e D(No, R+), k ( n , a ) , A l k ( n , cr) E D (E, R+) and c _> 0 be a real constant, where E - { (rn, n) E Ng" 0 < n < rn OC} . ( a l ) If

u(n) < c + E

f(s)

k ( s , o ) u(cr

,

(4.4.1)

[l+f(o)+A(cr)]

,

(4.4.2)

u(s)+

s--0

a=0

for n E No, then u(n)
1+

f(s) s=0

a--0

Chapter 4

215

for n E No, where n-1

A (n) - k (n + 1, n) + E

(4.4.3)

Alk (n, r ) ,

3"---0

for n E N0. (a2) If u(n)_
f(s)

u(s)+

s=0

(4.4.4)

k ( s , o ) u(o) , a=0

for n E No, then u(n) < a ( n ) + B ( n )

I+E

n-1

s-i~

f(s)

s=0

[l+f(cr)+A(cr)]

]

,

(4.4.5)

a=0

for n E No, where B(n)=E

[

f(s)

a(s)+

s---0

k(s,o) a(a or=0

,]

(4.4.6)

,

for n E No and A(n) is given by (4.4.3). P r o o f . (al) Define a function z(n) by the right hand side of (4.4.1). Then z ( 0 ) - c, u (n)5_ z (n) and Az (n) = f (n) u (n) +

k (s, o ) u (or) or--0

_
z(n)+

k(s, cr) z(o)

(4.4.7)

.

o'--0

Define a function v(n) by n-1

v (n) - z (n) + E

k (s, a) z (a).

(4.4.8)

or--0

Then v(0) - z(0) = c, z (n) < v (n), Az (n) < f (n)v (n), v(n) is nondecreasing for n E No, and n

Av (n) -- Az (n) + Z o=0

rt--1

k (n + 1, or) z (a) - Z o=0

k (n,a) z (a)

Finite difference inequalities in one variable

216

n-1

= Az (n) + k (n + 1, n) z (n) + E

Axk (n, a) z (a)

o'--0

< If (n) + d (n)] v (n),

(4.4.9)

where A(n)is given by (4.4.3). The inequality (4.4.9)implies (see [42, p. 12]) n--1

(4.4.10)

v (n) <_ c H [1 + f (a) + A(a)]. o'--0

Using (4.4.10)in Az (n) < f (n)v (n) we get n--1

(4.4.11)

Az (n) <_cf (n) H [1 + f (a) + A (a)]. o'--0

The inequality (4.4.11) implies the estimate (4.4.12)

z ( n ) < c [ l + ~ f ( s ) n s = o ~=0 [ l + f ( ~ ) + A ( c r ) ] ] . Using (4.4.12) in u (n) _< z (n) we get the desired inequality in (4.4.2).

(a2) The proof can be completed by closely looking at the proofs of (al) given above and Theorem 1.4.1, part (a2). We omit the details. R e m a r k 4.4.1. We note that the inequalities given in Theorem 4.4.1 are the discrete analogues of the inequalities given in Theorem 1.4.1. In the special case when k (n, or) = g (a), the inequality in (al) reduces to the inequality established earlier by Pachpatte, see [42, Theorem 1.4.1, p. 26]. For slight variants of the inequalities given in Theorem 4.4.1, see [42]. In the following theorems we present the inequalities established in [67]. Let u(n) E D(No, R+) , k ( n , a ) , A l k ( n , cr) E D(E,R+), h (n, s, a) , Alh (n, s, a) E D(F,R+) and c _> 0 be a real constant, where E -

T h e o r e m 4.4.2.

{(~, ~) ~ No~. 0 < ~ < ~ < ~ } , r - {(~,~,~) ~ N 3 0 _< ~ < ~ < ~ < ~ }

(bl) If (~) < ~ + ~

k (~, ~)~ (~) +

s=0

h (~, ~, ~ ) ~ (~) s=0

,

(4.4.13)

ka-0

for n E No, then n--1

u(n)
(4.4.14)

Chapter 4

217

for n E No, where n--1 P (n) - k (n + 1,n) + E h (n + 1,n,(7), or=0

(4.4.15)

(~ (n) -- E A l k (n, 7-) + A l h (n, 7-, (7) r=0 r=0 \ a = 0

(4.4.16)

,

for n E No (b2) Let

g 6 C (R+,R+) be a nondecreasing function with g(u) > 0 for u > 0.

If

(~) < ~ + ~

k (,~, ~) g (~ (~)) + E

s=0

h (~, ~, ~) g (~ (~))

s=0 \or=0

/

,

(4.4.17)

for n c No, then for 0 _< n <_ hi; r t, rtl C No, U(n) ~ a -1

a ( C ) q-

[ g ( 8 ) -4- Q(8)]

,

(4.4.18)

s=0

where P ( n ) ,

Q(n) are given by (4.4.15), (4.4.16),

G (r) - ]

- ~dt, r

> O,

(4.4.19)

ro

ro > 0 is arbitrary, G -1 is the inverse of G and n l C No be chosen so that n--1 (C) + E [P (S) -Jr-Q (S)] C 8=0

for all n E No lying in 0 < n <

DO?Tt

(~-1),

n l.

Proof. (51) Define a function z(n) by the right hand side of (4.4.13), then z(0) - c and u (n) _< z (n). From the hypotheses, we observe that z(n) is nondecrea~ing for n E No and Az (n) - y~. k (n + 1, s ) u (s) + E h (n + 1, s, a ) u (a) s=0 s=0 \ a = 0

s=0

s=0 \ a = 0 n--1

n--1

= k(n + 1, n)u(n)+ E k(n + 1, s) u ( s ) - ~ k(n,s)u(s) s----O

8zO

Finite difference inequalities in one variable

218

+ E h(n + 1, n, or)u (or)+ a=0

h(n + 1, s, a) u (a) s=0 \ a = 0

s=0 \ a = 0 n-1

= k (n + 1, n ) u (n) + ~

A l k (Tt, 8 ) u

(8)

s-0

+~h(n+l,n,cr)u(a)+~(~Alh(n,s,cr)u(a)) a=0

s=0 \ a = 0

n-1

< k (n + 1, n) z (n) + E

A1]~ (n, 8) Z

(8)

s=0

+~h(n+l,n,a)z(cr)+~(~ a=0

Alh(n's'r~)z(cr)) s=0 \ a = 0

< [P (n) + Q (n)] z (n).

(4.4.20)

Now a suitable application of the Corollary 1.2.2 given in [42, p. 12] to (4.4.20) yields n--1

(4.4.21)

z(n)
Using (4.4.21) in u (n) < z (n) we get the required inequality in (4.4.13). (b2) Let c > 0 and define a function z(n) by the right hand side of (4.4.17). Then z(O) = c, u (n) <_z (n), z(n) is positive and nondeereasing for n E No and by following the proof of (bl) with suitable modifications we get Az (n) < [P (n) + Q (n)] g (z (n)).

(4.4.22)

The rest of the proof can be completed by following the proof of Theorem 4.2.3, part (bl). Here we omit the details. T h e o r e m 4.4.3. Let u(n), k(n, s) , h (n, s, or), c be as in Theorem 4.4.2 and b e D (No, R+).

Chapter 4

219

(C1) If (~) < ~ + ~

b (~)~ (~) +

s=O

k (~, ~) ~ (~)

s=O \r=O

) (4.4.23)

s=0 \T--0 \a=0 for n E No, then

n-l[ u (n) _< c I I

~1 1 + b (s) +

s=0

~1(~ k (s, T) +

r=0

)] h (s, ~-, a)

,

(4.4.24)

r=0 \a=0

for n C No. (c2) Let g(u) be as in Theorem 4.4.2, part (b2). If

~(~) < ~+ ~

b(~)g (~(~)) + ~

s=0

k (~,~)g (~ (~))

s=0 \r=O (4.4.25)

for n E No, then for 0 < n < n2;n, n2 C No,

it (n) ~ G -1 [G (c) +

b(~)+

s=0

k(~,~) +

r=0

h (~,~,~)

,

(4.4.26)

T=0 \a=0

where G, G -1 are as in Theorem 4.4.2, part (b2) and n2 E No be chosen so that

n-l[ G (c) + E

s--1 b (s) +

s=0

~(~

)]

k (s, v-) +

T=0

h (s, T, a)

C Dora (G-I) ,

T=0 \a=0

for all n C No lying in 0 <_ n < n2. (c1) Define a function z(n) by the right hand side of (4.4.25). Then z(O) - c, u (n) <_ z (n), z(n) is nondecreasing for n e No and

Proof.

Az (~) - b (~) ~ (~) + ~

k (~, ~)~ (~) +

r=0 < b (~) z (~) + Z

r=0

h (~, ~, ~) ~ (~)

r=0 \a=0

k (~, ~)z (~) +

h (~, ~, ~) z (~)

r=0 \a=0

)

)

Finite difference inequalities in one variable

220

(4.4.27) Now a suitable application of Corollary 1.2.2 given in [42, p. 12] to (4.4.27) yields (4.4.28)

z ( n ) < - c"~ ~ I I [ l +sb ( s ) +T=o ~=k ( s " c ) + o~r=O " ( ~\~=0 h(s'7-'a))] Using (4.4.28) in u (n) _< z (n) we get the desired inequality in (4.4.24).

(c2) The proof can be completed by following the proof of (el) and closely looking at the proof of Theorem 4.4.2, part (b2). Here we omit the details. R e m a r k 4.4.2. We note that the inequalities in Theorems 4.4.2 and 4.4.3 parts (bl) and (Cl) provides the growth estimates on the discrete versions of the integral inequalities due to Bykov and Salpagarov [9] given in Theorem 1.4.2, while the inequalities in (b2) and (c2) provides the growth estimates on the general versions of the inequalities given in [9], which can be used conveniently in certain applications. The inequalities established in [73] are embodied in the following theorems.

--

In wbat follows, let Ji {(Ttl,...,Tti) " (Ttl,...,ni) E N~} for i - - 1,...,m . For any functions w(n) E D(No, R+), ki(nl,...,ni) E D(Ji,R+) for i - 1,...,m; first we give the following notations used to simplify the details of presentation:

B~ [~] (~) -

~(n~l nl=0 \n2=0

...

(n?o1 ~=

k, ( ~ , ..., ~ )

/

...

/

,

G[w](Tt)-- kl(Tt)w(?~)--~-~k2(Tt,Tt2)w(~t2)-nt~ (n~lk3(Tt~Tt2,Tt3) W(n3)l n2 --0

...

+

n2--O \ n 3 - - O

~ (n~l (n~_~-i n2=0 \n3=0 nm=O ...

k m ( ~ , , ~ , ..., ~ m ) ~ ( ~ m )

T h e o r e m 4.4.4. Let u(n),a(n) C D(No, for i - 1, ..., m and c > 0 is a real constant.

R+),

I I ....

ki(Ttl,...,Tti)

E D(Ji, R+)

(dl) If m

(~) < ~ + ~

i=1

B~ [~] ( ~ ) ,

(4.4.29)

Chapter 4

221

for n C No, then n-1

u(n) < c H

(4.4.30)

[l+G[1](nl)],

hi--0

for n E No.

(d2) Let a(n) be nondecreasing for n E No. If m

u (n) _< a (n) + E

(4.4.31)

Bi [u] (n),

i=1

for n E No, then n-1

[1 + G [1] (rt 1)],

u (n) _< a (n) H

(4.4.32)

nl--0

for n E No.

Proof.

(dl) Define a function z(n) by the right hand side of (4.4.29) i.e.,

Z(?"t)-- C--~~ kl(~l)U(~l)-t- ~ (n~lk2(nl,n2)u(n2)) nl=O

nl=0 \n2=0

nl=0 \n2=0

\n3=0

nl=0 \n2=0

\n3=0

+

...

k m ( ~ , , ~ , , ~ , ..., ~ ) ~ \

(~m)

....

n,,,=0

Then z(O) - c, u (n) <_ z (n), z(n) is nondecreasing for n e No and a ~ (~) - kl (~)~ (~)+

k~ (~, ~ ) ~ ( ~ ) + n2:0

+... + ~

...

,

k~ (~, ~ , ~ ) ~ ( ~ )

r~2 --0 \ n 3 --0

km ( ~ , ~ , ~ , - . . , ~ m ) ~ ( ~ m )

n2=0 \n3=0

[

k l (Tt) -~-

...

k 2 (Tt, 7t2) -~n2=0

+

n2=0 \n3--0

k 3 (Tt, r t 2 , / / 3 ) n2=0 \n3=0

/

k nm ---0

)

))1

..-

)

222

Finite difference inequalities in one variable

i.e., Az (n) _< G [1] (n) z (n).

(4.4.33)

Now a suitable application of Theorem 1.2.1 given in [42] to (4.4.33) yields n-1 z (n) < c H [1 + G [1] (Tt1)].

(4.4.34)

hi=0

Using (4.4.34) in u (n) < z (n) we get the desired inequality in (4.4.30). (d2) The proof can be completed by closely looking at the proof of Theorem 1.2.4 given in [42] and by making use of the inequality established in (dl) . We omit the details. Theorem

4.4.5.

Let u(n), ki (ftl, ...,Tti) for i - 1,...,m be as in Theorem

4.4.4. (el) Let r (n) E D (No, R+) and Ar (n) _> 0 for n E No. If m

u (n) < r (n) + E B~ [u] (n), i=1

(4.4.35)

for n E No, then n--1 n--1 n--1 [1 + G [1] (a)], (4.4.36) u (n) < r (0) H [1 + G [1] (Tt1)1 -+- E /~* (nl) H nl:0 rtl--0 cr=rtlnt-1

for n E N0. (e2) Let a (n), b (n) E D (No, R+). If m

u (n) <_ a (n) + b(n) E

Bi [u] (n),

(4.4.37)

Ttl-- 0

for n E No, then n-1 u (n) <_ a (n) + b (n) E G [a] (rt 1 ) nl--O

for n E No.

n-1

II

a--n1+1

[1 + G [b] (a)],

(4.4.38)

Chapter 4

223

P r o o f . (e l) F r o m the hypotheses on r (n) we observe that r (n) is nondecreasing for n E No. Define a function z(n) by the right hand side of (4.4.35). Then z (0) - r (0), u (n) <_ z (n), z (n) is nondecreasing for n E No a n d as in the proof of Theorem 4.4.4, part (dl) we have (4.4.39)

Az (n) <_ Ar (n) + G [1] (n) z (n).

Now a suitable application of Theorem 1.2.1 given in [42] to (4.4.39) yields n--1

n-1

[1 + G [11 (711)3-Jr- E

z (n) <_ q5(0) r I hi=0

n-1

A~(/t I ) II

nl=0

[1 + G [1] (or)]. (4.4.40)

o'=nl+l

Using (4.4.40) in u (n) < z (n) we get the required inequality in (4.4.36). (e2) Define a function z(n) by m

z (n) - E

Bi [u] (n).

(4.4.41)

i=0

Then as in the proof of Theorem 4.4.4, part (dl), z(O) - O, z(n) is nondecreasing for n c No; (4.4.37) can be restated as (4.4.42)

u (n) < a (n) + b (n) z (n), and

rt2 ---0

...

+

E

rt2 --0 \ n 3 = O

"'"

n2--O \ n 3 - - O

km(~'~2'?ta'""?'tm)~l'(~m) \

"'"

nrn --0

(4.4.43)

< G [a] (n) + G [b] (n) z (n). Now an application of Theorem 1.2.1 given in [42] to (4.4.43) yields n--1

z(n)_< E nl--O

n--1

G[a](nl)

1-I

[I+G[b](o)1.

(4.4.44)

a=nl-+-I

Using (4.4.44) in (4.4.42) we get the required inequality in (4.4.38). R e m a r k 4.4.3. The inequalities in Theorem 4.4.4 and 4.4.5 are motivated by the integral inequalities established by various investigators and given in [3, pp. 100-108]. For some useful singular finite difference inequalities, we refer the interested readers to the recent paper by Medvecl [27] and some of the references cited therein.

224

F i n i t e d i f f e r e n c e i n e q u a l i t i e s in o n e variable

4.5 Bounds on certain finite difference inequalities The main goal of this section is to present some specific type of finite difference ineualities investigated by Pachpatte in [37,39,44,54,70,75]. The inequalities given here can be used in the analysis of certain finite difference and sumdifference equations. Our first theorem deals with the finite difference inequalities proved in [70]. T h e o r e m 4.5.1.

Let u (n) , a (n) , b (n) , c (n) , f (n) , g (n) E D (No~,Z, R + ) .

(al) Suppose that Aa (n) 2 0 for n E N~,Z and n--1

u (n) <_ a (n)

+E

b (s) u (s)

+E

S--C~

c (s) u (s),

(4.5.1)

S--C~

for n E N~,Z. If /3

8--1

ql-~c(s)

(4.5.2)

H [ l + b ( T ) ] < 1,

S----OL

T--OL

then n--1

n--1

n--1

u (n) < NI H [1 + b (s)] + Z s=c~

Aa (s) H

8=c~

(4.5.3)

[1 + b (~)],

cr--s+l

for n E Nc~,r where

N 1 =

1 -ql

[

a (a) +

c (s) 8--c~

Aa (T)

]

[1 + b (cr)] . cr=rA-1

T--O~

(4.5.4)

(a2) Suppose that n--1

u (n) < a (n) + b (n) E

/3

f (s)u (s) + E

S--C~

g (s)u (s),

(4.5.5)

S--C~

for n E N~,Z. If q2-E

g(s) L2(s) < 1,

(4.5.6)

Chapter 4

225

u (n) <_ L~ (n) + N2L2 (n),

(4.5.7)

then

for n C N~,~, where n--1

Ll(n)-a(n)+b(n)E

n--1

f(s) a(s) H s=a n-1

L2(n)-c(n)+b(n)E

[l+f(~)b(a)],

(4 5.s)

[l+f(~)b(a)],

(4.5.9)

~=s+l n-1

f(s) c(s) H a=s+l

s=a

and 1 N2

1

-

-

(4.5.10)

q2 s e a g (8) L1 ( s ) .

(a3) L e t r ( n , s) , Ar(n,s) E D

( N a.Z, 2 R+)

n--1

f o r o ~ <_ 8 < n <_~ a n d _

/3

(4.5.11)

u (n) _< a (n) + Z r (n, s) u (s) + Z c (s) u (s), S--- 0~

for rt E N a , ~ .

S---O~

If s--1

q3--~c(s) S---O~

(4.5.12)

H [l+/3(m)] < 1, T~C~

then n--1

n--1

n--1

u (n) _< a (n) + N3 H [1 +/~ (s)] + Z ~ (s) H s=a

s=o~

[1 +/3 (a)], (4.5.13)

(7=8+1

for n E N~,Z, where n-1

(n) - r (n + 1, n)a (n) + ~ / ~ 1

?" (n,

(4.5.14)

S)a (s),

S'-'- O~

n--1

/3 (n)

- r

(n

-t-

1, n)

+ ~/~1

(4.5.15)

?~ (Tt, S ) ,

S--O~

and N3

1-

q3

~c(s) =

a(s)+ZA(7-) r=a

[l+/3(a)] a=r+l

.

(4.5.16)

Finite difference inequalities in one variable

226

(al) Define a function z(n) by the right hand side of (4.5.1). Then u (n) _< z(~), 3 z (a) - a (a) + ~ c (s)u (s),

(4.5.17)

8--C~

and Az (n) - Aa (n) + b (n)u (n) < Aa (n) + b (n) z (n).

(4.5.18)

Now a suitable application of Theorem 1.2.1 given in [42] to (4.5.18) and using the fact that u (n) <_ z (n) we have n-1

n-1

n-1

u(n)_
H

s=c~

[l+b(a)].

(4.5.19)

o-=8+1

From (4.5.17),(4.5.19) and in view of (4.5.2) we have z ( a ) < N1.

(4.4.20)

Using (4.5.20) in (4.5.19) we get the required inequality in (4.5.3). The proofs of (a2) and (a3) follows by closely looking at the proof of (al) and the proofs of Theorem 1.5.1, part (a2) and Theorem 1.5.2, part (bl). Here we omit the details. R e m a r k 4.5.1. By taking c(n) = 0 in (al) and N~,Z is replaced by No, we get the inequality given in Theorem 1.2.6 in [42]. The inequalities in (a2) and (a3) can be considered as the useful variants of the inequalities in Theorems 1.2.3 and 1.3.4 given in [42]. The next theorem contains the inequalities investigated in [54,75]. T h e o r e m 4.5.2.

Let u (n) E D (N~,9 , R+) and k > 0 be a real constant.

(bl) Let a (n, s) , b (n, s) , c (n, s) E D (E, R+ ) ; a(n, s), b(n, s) be nondecreasing in n for each s E N~,~ where E -

u (n) _< k + ~

a (n, s) u (s) +

{(n, s ) E N ~ , n a

c (s, a) u (a)

+

_
b (n, s) u (s), (4.5.21)

for n C N~,~. If /3

q (n) - E 8---C~

n-1

b (n, s) H [1 + B (n, ~)1 < 1, ~---O~

(4.5.22)

Chapter 4

227

for n E N~,Z, where

B(n,~)-a(n,~)

l+Ec(~,cr)

~-1

1

O ' - - C~

,

(4.5.23)

for (n,~) E E, then k

(~) <

n--1

1-q(n)

(4.5.24)

H[I+B(n'~)]'

(=c~

for n E N~,~. (b2) Let f (n), g (n), h (n) E D (N~,z, R+) and

(~) < k +

f (~) ~ (~) + S--OL

g ( ~ ) ~ (~) + ( 7 - - C~

h ( ~ ) ~ (~

,

(4.5.25)

0 " - - Ol

for n E Na,3. If /9

r - E

5r--1

h (a) H [1 + f (r) + g(r)] < 1,

(4.5.26)

then

n-1 u(n) < k -1-r

H[l+f(s)+g(s)]

(4.5.27)

S--C~

for n E N~,fl. Proof.

(bl) Fix any m E N~,Z, then for c~ _< n _< m, from (4.5.21) we have

u (n) _< k + E

[

a (m, s) u (s) + E

S--C~

]

c (m, or) u (or) +

0 " - - C~

b (m, s) u (s). (4.5.28) S---C~

Define a function z(n, m), c~ _< n <_ m by the right hand side of (4.5.28). Then for c~ _< n _< rn, u (n) _< z (n, m), z(n, m) is nondecreasing in n,

z (c~, m) - k + E

b (m, s) u (s),

(4.5.29)

S - - OZ

and

n-1 zxlZ (~..~) - a (.~. ~) ~ (~) + ~ ~--C~

] c (~. ~) ~ (~)

Finite difference inequalities in one variable

228

< a(m,n) i.e.,

[

1+

c(n,o 0"~0~

'1

z(n,m),

(4.5.30) for c~ _< n _< m. By setting n - ~ in (4.5.30) and subsituting ~ - ct, c~+1, ..., m - 1 successively, we obtain z (m, m) _< z (a, m)

l+a(m,~)

1+

c(~,~

~zOL

.

(4.5.31])

O'zC~

Since m is arbitrary, from (4.5.31) and (4.5.29) with m replaced by n and using u (n) _< z (n, n ) w e have u(n)
l+a(n,~)

1+

~'--Ot

c(~,a

(4.5.32)

,

0 " - - C~

where 3 z (a, n) - k + E b (n, s ) u (s),

(4.5.33)

S'--Ct

Using (4.5.32) on the right hand side of (4.5.33) and in view of (4.5.22) it is easy to observe that k

z (c~ n) < '

-

1 -

(4.5.34)

q (n)"

Using (4.5.34)in (4.5.32) and (4.5.23)we get (4.5.24). (b2) Define a function z(n) by the right hand side of (4.5.25). Then z (a) - k,, u (n) _< z (n) and Az (n) -- f (n)

u(n)+

g(~)u(a)§ E r - - C~

<_f(n)

, 0 " - - C~

Iz ( n ) + E n--1g ( ~ ) z ( ~ ) + E h ( 3( 7 ) z ( ~ )

1,

for n E N~,Z. Define a function v(n) by

n--1 v (~) - z (~) + Z O'~C~

g (~) z (~) + Z 0 " - - C~

h (~) ~ (~).

(4.5.35)

Chapter 4

229

Then z (n) _< v (n), Az (n) _< f (n)v (n), v (~) - k + Z

h (~) z (~),

(4.5.36)

O'--- C~

and Av (~) - / x ~ (~) + g (~) z (~)

_< [f (n) + g (n)] v (n).

(4.5.37)

Now a suitable application of Theorem 1.2.1 given in [42] to (4.5.37) yields n--1

v (n) < v (a) I I [1 + f ( s ) + g (s)].

(4.5.38)

8~OL

Using (4.5.38)in z (n) < v (n) we get n--1

z (n) < v (a) H [1 + f ( s ) + g (s)],

(4.5.39)

8~C~

for n E N~,Z. Using (4.5.39) on the right hand side of (4.5.36) and in view of (4.5.26) we observe that k v (a) < ~ . -1-r

(4.5.40)

Using (4.5.40) in (4.5.39) and the fact that u (n) _< z (n) we get the required inequality in (4.5.27). R e m a r k 4.5.2. We note that, if we take in Theorem 4.5.2, part (bl), c(n, s) 0, then we get the inequality established in [52, Theorem 2]. Furthermore, in the various special cases of Theorem 4.5.2, we get new inequalities which can be used conveniently in certain situations. In the following theorem, we present some of the inequalities established in [39,44]. T h e o r e m 4.5.3.

Let u ( n ) , a (n), b (n) E D (No, R+).

(c1) Let a(n) be nonincreasing for n E No. If u(n)_
E

b(s) u ( s ) ,

(4.5.41)

s=n+l

for n E No, then oo

u(n)_
H s--nq-1

for n E N0.

[l+b(s)],

(4.5.42)

Finite difference inequalities in one variable

230

(c2) Let c (n)E D (No, R+). If OO

(~) _< a (~) + b (~) ~

~ (~) ~ (~),

(4.5.43)

s=n+l

for n E No, then OO

u(n)_
l-I

[ l + c ( s ) b(s)],

(4.5.44)

s=n+l

for n E No, where OO

d (n) -

E

(4.5.45)

c (s) a (s) ,

s=n+l

for n E N0. Let L" No x R+ ~ R+ be a function which satisfies the condition

(C3)

0v>0,


(4.5.46)

whereM'N0xR+~R+.If (X)

u(n)
E

L(s,u(s)),

(4.5.47)

s=n+l

for n E No, then OO

u(n)
H

[l+M(s,a(s))b(s)],

(4.5.48)

s=n+l

for n E No, where OO

e(n)-

E

(4.5.49)

L(s,a(s)),

s=n+l

for n E N0 Proof.

(cl) Let a(n) > 0 for n E No, then from (4.5.41) it is easy to observe

that (X)

u(n) < 1 + a(n) -

E s=n+l

b(s) U(S) a(s)"

(4.5.50)

Define a function z(n) by the right hand side of (4.5.50), then ~u(n) < z (n) and z ( n ) - z (n + 1) - b(n + 1)

u ( n + 1) a ( n + 1)

231

Chapter 4 <_b(n+l) z(n+l).

(4.5.51)

From (4.5.51) we observe that z(n)_<[l+b(n+l)lz(n+l).

(4.5.52)

By setting n - s in (4.5.52) and then substituting s - n , n + 1 , . . . , m (m > n + 1 is arbitrary in No ) successively, we obtain the estimate

1

m

z (n) <_ z (m) I I

[1 + b (s)].

(4.5.53)

s=n+l

Noting that

lim ~Tt ----+ OO

z (m)-

1 and by letting m --+ oc in

(4.5.53)

we get

cx)

z (n) _< H

[1 + b(s)].

(4.5.54)

s=n+l

Using (4.5.54) in ~ < z (n) we get the desired inequality in (4.5.42). The proof of the case when a (n) _> 0 can be completed as mentioned in the proof of Theorem 4.2.3, part (bl). (c2) Define a function z(n) by oo

z (n) -

E

c (s) u (s),

(4.5.55)

s=n+l

for n E No Then (4.5.43) can be written as u (n) < a (n) + b (n) z (n).

(4.5.56)

From (4.5.55) and (4.5.56)we have (x)

d

+

(s) b (s) z (s),

(4.5.57)

s=n+l

where d(n) is given by (4.5.45). Clearly d(n) is real-valued, nonnegative and nonincreasing function for n E No. Now a suitable application of the inequality in part (Cl) to (4.5.57)yields (x)

z(n)
H

[ l + c ( s ) b(s)].

s=n+l

Using (4.5.58) in (4.5.56) we get the required inequality in (4.5.44).

(4.5.58)

Finite difference inequalities in one variable

232

(c3) Define a function z(n) by CK)

z (n) -

E

L (s, u (s)),

(4.5.59)

s=n+l

then from (4.4.47) we have u (n) _< a (n) + b (n) z (n).

(4.5.60)

From (4.5.59), (4.5.60) and the hypotheses on L, we observe that (x)

z(n)_<

E

[n(s,a(s)+b(s)z(s))-L(s,a(s))+L(s,a(s))]

s=n+l oo

< e (n) + ~

M (s, a (s)) b (s) z (s),

(4.5.61)

s=n+l

where e(n) is given by (4.5.49). Clearly c(n) is real-valued, nonnegative and nonincreasing function for n C No. Now an application of the inequality in part (Cl) to (4.5.61)yields CXD

z (n) _< e (n) 1-I [1 + M (s, a (s)) b (s)].

(4.5.62)

s=n+l

The desired inequality in (4.5.48)follows from (4.5.60)and (4.5.62). Our last theorem in this section gives the inequalities proved in [37]. T h e o r e m 4.5.4. constant.

Let u(n),a(n),b(n)

E D(No.R+) and p > 1 be a real

(dl) Let f (n), g (n) c D (No, R+). If (3O

Up ( n )

~ a (n) + b (n) E

If (s)u (s) + g (s)],

(4.5.63)

s=n+l

for n C No, then u (n) _< [a (n) + b (n) A (n) H s=n+l

1+

f (s)

,

(4.5.64)

P

for n C No, where A(n)-

E s:n+l

for n E N0.

f(s)

p

1 P

+ a(s)) + g (s)], P

(4.5.65)

233

Chapter 4

(d2) Let L , M be as in Theorem 4.5.3, part (c3) and the condition (4.5.46) holds. If (3O

u p (n) _~ a ( n ) + b (n) E

L (s, u (s)),

(4.5.66)

s=n+l

for n E No, then u (n) < [a (n) + b (n) B (n) x II

I+M

s, p-1

s=n+l

+ a(s)

P

P

b(s)

(4.5.67)

--~

'

for n E No, where B(n)-

E n s=n+l

s,

+

,

P

(4.5.68)

P

for n E No. Proof.

(dl) Define a function

z(n) by

oo

z(n)-

E

(4.5.69)

[f(s) u ( s ) + g ( s ) ] '

s=n+l

for n E No. Then (4.5.63) can be written as (4.5.70)

u p (n) <_ a (n) + b (n) z (n).

From (4.5.70) as in the proof of Theorem 1.3.1, part (al) we obtain u(n)_< p - 1 P

+ a ( n ) + b ( n ) z(n). P P

(4.5.71)

From (4.5.69) and (4.5.71)we have z(n)<

E

f(s)

s=n+l

(

p-1

+

a(s)

P

+

P

b(s) P

z

)

+ g

]

f (s) b (S) z (s),

= A (n) + E s=n+l

(4.5.72)

P

where A(n) is given by (4.5.65). Clearly A(n) is real-valued, nonnegative and nonincreasing function for n E No. Now an application of Theorem 4.5.3, part (c3) to (4.5.72)yields z(n)_
II s=n+l

l+f(s)

b(s)

.

P

The desired inequality in (4.5.64)follows from (4.5.70)and (4.5.73).

(4.5.73)

Finite difference inequalities in one variable

234

(d2) The proof can be completed by closely looking at the proof of (dl) and the proof of Theorem 4.5.3, part (c3). We omit the details.

4.6 Applications The inequalities given in earlier sections are recently investigated and used in various contexts. In this section we present applications of some of the inequalities to study basic properties of solutions of certain finite difference and sum-difference equations, which we hope will be a source for future work.

4.6.1 Perturbed difference equations Consider a system of finite difference equations

x(n+ 1)= A(n)x(n)+ f(n,x(n))+r(n),

x(O)=xo,

(4.6.1)

as a perturbation of the linear system y(n+l)=A(n)y(n),

y(O)=xo

(4.6.2)

where n c No, x, y, f, r are the elements of R ~, the m dimensional Euclidean space, A(n) is an m x m matrix with det A (n) ~ 0, the functions r and f are defined on No and No x R TM respectively and x0 is a given vector in R "~. The symbol I.] will denote some convenient norm on R m as well as a corresponding consistent matrix norm. We denote by Y(n) the fundamental solution matrix of the system (4.6.2) such that Y (0) = I, the identity matrix. It is known that the solution x(n) of (4.6.1) is equivalent to the sum-difference equation (see [42,

p. 55]) n--1

x(n) - Y ( n ) Y -l (O) x o + ~ - ~ Y ( n ) Y -l (s+ l ) { f ( s , x ( s ) ) + r ( s ) } .

(4.6.3)

s=0

We assume that the fundamental solution matrix Y(n) of (4.6.2) satisfies

]Y (n) y - 1 (s)J <_ M, 0 <_ s <_n; s, n E No,

(4.6.4)

where M is a positive constant. The following theorems illustrate the applications of Theorem 4.2.1 (see [57]).

Chapter 4 T h e o r e m 4.6.1.

235

Suppose that the function f in (4.6.1) satisfies (4.6.5)

If (n, x)l ~ p (n)Ixl,

for n E No, x E R m, where p (n) E D (No, R+).If x(n) is any solution of equation (4.6.1) for n E No, then

Ix(n)I < M

n-1 Ix01 + E (Ir(s)I + Ix~ s=0

n--1

II

}

[1 + Mp(a)]

, (4.6.6)

a=s+l

for n E No, where M is given as in (4.6.4). P r o o f . By using the variation of constants formula any solution x(n) of (4.6.1) is represented by (4.6.3). Using (4.6.4), (4.6.5) in (4.6.3) we obtain (4.6.7) \s=0

s=0

Now a suitable application of Theorem 4.2.1 to (4.6.7) yields the required estimation in (4.6.6). T h e o r e m 4.6.2.

Suppose that the function f in (4.6.1) satisfies

If (n, x) - f (n, Y)I -< P (n)Ix - Yl,

(4.6.s)

for n E No, x,y E R m, where p(n) E D(No, R+). Then the equation (4.6.1) has at most one solution on No. Proof.

Let xl (n) and x2 (n) be two solutions of (4.6.1) on No, then we have

n-1 X 1 (?Z)- X2 (/3,) -- E Y (?Z) ~/---1 (8 nt- 1) { f (8, X 1 (8)) -- f (s, x2 (s))}. (4.6.9) s=0

From (4.6.9), (4.6.4), (4.6.8)we obtain n--1 IXl (n) -- X2 (n)l ~ E IY ( n ) y - 1 8=0 n-1 < M E p (s)Ix1 (s) - x2 (s)l. s=0

(8 -nt- 1)l If (8, Xl (8)) - f (s, x2 (s))l

(4.6.10)

By a suitable application of Theorem 4.2.1 to (4.6.10)we have Ix I (rt) -- Z2 (n)[ ~__ 0.Therefore Xl (n) - x2 (n) i.e., there is at most one solution of the equation (4.6.1) on No.

236

Finite difference inequalities in one variable

4.6.2 Volterra type difference equations involving iterated sums In this section we present applications of the inequality in Theorem 4.4.2, part (bl) (see [67]) to study certain properties of solutions of nonlinear sum-difference equation of the form y (n) - f (n) + E

F (n, s, y (s)) + E

s=0

H (n, s, ~, y (~))

s=0 \ a = 0

)

,

(4.6.11)

for n E No, where y (n) E D (No, R) is an unknown function, f c D (No, R); F " E1 x R --, R, H " E2 x R ---, R in which/~1 -- { (~, 8) E N 3 90 ~ 8 ~ n < (x)}, E~ - { ( ~ , ~, ~) c x 3 . 0

~}.

_< ~ < ~ < ~ <

T h e o r e m 4 . 6 . 3 . Suppose that the functions f, F, H in equation (4.6.11) satisfy the conditions If (n)l < c,

(4.6.12)

IF (n, s, Y)I -< k (n, s)lyl,

(4.6.13)

(4.6.14) IH (n, s, a, Y)I <- h (n, s, or)lYl, where c _> 0 is a real constant and k (n, s) c D (El, R+), h (n, s, or) E D (E2, R+). If y(n) is any solution of equation (4.6.11) on No, then rt--1

[y (n)l < c H [1 + P (s) + Q (s)],

(4.6.15)

8=0

where P(n), Q(n) are given by (4.4.15), (4.4.16)in which A l k (n,s) C D (El, R+), Alh (~,8,0)e

D (E2,/~+) .

P r o o f . Let y(n) be a solution of equation (4.6.11). Using (4.6.12)-(4.6.14) in (4.6.11 ) we have lY (n)l < c + E

k (n, s)lY (s)l +

s=0

h (n, s, a)lY (a)l s=0 \ a = 0

)

9

(4.6.16)

Now an application of Theorem 4.4.2, part (bx) to (4.6.16) yields the required estimate in (4.6.15). Theorem 4.6.4. the conditions iF

(~, ~, y)

Suppose that the functions F, H in equation (4.6.11) satisfy - F

(~, ~, y)l _< k (~, ~)ly

- yl,

(4.6.17)

IH (n, s, a, y) - H (n, s, (7, Y)I <- h (n, s, a ) [ y - Yl , (4.6.18) where k ( n , s ) , h ( n , s , a ) are as in Theorem 4.6.3. Let p ( n ) , Q ( n ) be as in Theorem 4.6.3. Then the equation (4.6.11) has at most one solution on No.

Chapter 4

237

P r o o f . Let u(n) and v(n) be two solutions of equation (4.6.11) on No. Using this fact and the conditions (4.6.17), (4.6.18) we have n--1

s--0

+ ~ ( ~ h\(~n: '0s ' a )

lu(a)-v(a)l)

(4.6.19)

Now a suitable application of Theorem 4.4.2, part (bl) (when c - 0) to (4.6.19) yields u(n) - v(n) i.e., there is at most one solution of equation (4.6.11) on No

4.6.3 Volterra-Fredholm type sum-difference equations In this section we present applications of the inequality in Theorem 4.5.1, part (a2) to study certain properties of solutions of Volterra-Fredholm type sum-difference equation of the form n-1

(~) - ~ (~) + Z

r (~, ,, z (s)) + ~

S~O~

a (~, ~, z (~)),

(4.6.20)

S---C~

for n E N~,Z, where z(n) c D(N~,z,R)

is an unknown function, e(n) E

D (N~,~,R); F , G " E x R - + R in which E -

/ ( n ' s ) E N~,~'c~ < s < n < fl},

-%

%

J

see [70]. Theorem

4.6.5.

Suppose that the functions e, F, G in equation (4.6.20) sat-

isfy the conditions le (n)] _< a (n),

(4.6.21)

IF (n,s,z)] < b(n) f (s)]zl,

(4.6.22)

(4.6.23) where a (n), b (n) , c (n) , f (n) , g (n) E D (N~,z, R+ ) . Let q2 be as in (4.5.6), Theorem 4.5.1, part (a2).If z(n) is a solution of equation (4.6.20) on N~,Z, then

z (rt)l < L1 (rt) + N2L2 (n),

(4.6.24)

for n E N~,~,where L1 (n), L2 (n), N2 are as in Theorem 4.5.1, part (a2)

Finite difference inequalities in one variable

238

P r o o f . Let z(n) be a solution of equation (4.6.20) on N~,~. Using the fact that z(n) is a solution of equation (4.6.20) and (4.6.21)-(4.6.23) we have n--1

tz (n)l < a (n) + b (n) E

fl

f (s)Iz (s)l + c (n) E

8---O~

g (s)Iz (s)t.

(4.6.25)

8---C~

Now an application of the inequality in Theorem 4.5.1, part (a2) to (4.6.25) yields the required estimate in (4.6.24). Theorem 4.6.6. the conditions

Suppose that the funcrions F, G in equation (4.6.20) satisfy

IF (n, s, z) - F (n, s, z)l -< b (n) f (s)Iz - 21,

(4.6.26)

IG (n, s, z) - G (n, s, 2)1 < c (n) g (s)Iz - 2],

(4.6.27)

where b ( n ) , c (n), f (n), g (n) E D (N~,z, R+). Let q2, L1 ( n ) , L 2 (Tt), N2 b e as in Theorem 4.5.1, part (a2). Then the equation (4.6.20) has at most one solution on N,~,~. P r o o f . Let u(n) and v(n) be two solutions of equation (4.6.20) on N~,~.Using the facts that u(n) and v(n) are the solutions of equation (4.6.20) and (4.6.26), (4.6.27) we have n--1

In (n) - v (n)l < b (n) E

fl

f (s)[u (s) - v (s)l + c (n) E

8-'-C~

g (s)In (s) - v (s)l.

SzC~

(4.6.28) Now an application of the inequality given in Theorem 4.5.1, part (a2) (with a(n) - 0 which in fact implies L 1 (n) -- 0, N 2 - 0 ) t o (4.6.28) yields u(n) -- v(n) i.e., there is at most one solution of equation (4.6.20) on N~,~.

4.6.4 Fredholm type sum-difference equations In this section we present applications given in [75] of the special version of the inequality in Theorem 4.5.2, part (b2) to study the properties of solutions of the Fredholm type sum-difference equation

=

F

x

k

,

(4.6.29)

O'~Or

with the given initial condition

x(

)-x0

(4.6.30)

239

Chapter 4

where x, k, F are the elements of R TM an m-dimensional Euclidean space with norm I.I and k : E x R m ~ R m, F : N~,Z x R m x R TM ~ R TM, in which E-{(n }. , s) E N ~2, ~ ' ~ < s < n < _ 3 Theorem

4.6.7.

Assume that (4.6.31) (4.6.32)

IF (n, x, y)l -< f (n) (Ixl + lyl), where e (n), h (n), f (n) E D (N~,z, R+) and e (n) > 1. Let vr--1

r0-

E

h (0) H [1 + e ( r ) f (r)] < 1.

G--(~

(4.6.33)

T---C~

If x(n) is any solution of (4.6.29)-(4.6.30), then n-1

Ix (~)1< -

1]

Ix01 l-r0

[1 + e (s) f (s)],

(4.6.34)

8~C~

for n r N~,fl. Proof. The solution x(n) of (4.6.29)-(4.6.30) satisfies the following equivalent sum-difference equation

x(~)-x0+~F

~,x(,),

8---C~

k(~,~,x(~)) dr--

(3.6.35))

C~

Using (4.6.31), (4.6.32)in (4.6.35)we observe that

nl [x(n)[_<[x0[+E

( f(s)

--

nl < Ix~

[x(s)[+~e(s)h(o)lx(o)l O'=C~

( f ( s ) e(s)

8---O~

) )

Ix(s)l+Eh(o)lx(cr)l

.

(4.6.36)

O" - - - O~

Now a suitable application of Theorem 4.5.2, part (b2) (when g(n) = 0) to (4.6.36) yields (4.6.34). Theorem

4.6.8.

Let x(n), y(n), n e N~,Z be the solutions of (4.6.29) with

initial conditions x (c~) = x0,

(4.6.37)

y (c~) = y0,

(4.6.38)

Finite difference inequalities in one variable

240

respectively.Suppose that the functions k and F in equation (4.6.29) satisfy the conditions

Ik (~, ~, x ) - k (~, ~, y)l _< ~ (~)h ( ~ ) I x - yl,

(4.6.39)

IF (n, x, y) - F (n, 2, Y)I < f (n) (Ix - 21 + lY - Yl),

(4.6.40)

where e(n),h(n), f(n) are given as in Theorem 4.6.7. (4.6.33). Then

Let r0 be as given in

n-1

Ix (n) - y (n)l < Ix~ - Y01 H -

.

-

[1 + e (s) f (s)],

(4.6.41)

T0

for n E N~,3. P r o o f . Using the facts that x(n), y(n) are the solutions of (4.6.29)-(4.6.37), (4.6.29)-(4.6.38) respectively, we have

x(~)-

y(~) - ~0 - y0 +

r

~,~(~),

S--C~

-r

~, y (~),

k (~, ~, y (~))

k (~,~,x (~)) 0 " - - OL

(4.6.42)

.

O'zO~

Using (4.6.39), (4.6.40), (4.6.42)we observe that

Ix (n) - y (n)l _< IXo - Y o l + ~ f (s)e (s)

x (s) - y (s)l

S---Og

+ ~

h ( ~ ) I x (~) - y (~)1

9

(4.6.43)

O'----C~

Now a suitable application of Theorem 4.5.2, part (b2) (when g(n) = 0) to (4.6.43) yields the desired estimate in (4.6.41), which shows the continuous dependence of solutions of equation (4.6.29) on given initial data. Finally, we note that a variety of new methods and tools are developed by various investigators to study different types of finite difference equations. The inequalities and applications given above are recently investigated and further progress is expected.

Chapter 4

241

4.7 Notes Owing to the considerable applications,recently some new finite difference inequalities are developed to widen the scope of their applications. This chapter presents some basic finite difference inequalities recently developed in the literature. Sections 4.2-4.5 are devoted to the variety of new finite difference inequalities investigated by Pachpatte in [35,37,39,44,45,53,54,55,57,67,68,70,73 ,75]. I think that these inequalities places a new stepping stone to the vast literature on the subject and inspire further work in this area.In section 4.6,some applications are discussed to illustrate,how some of these inequalities can be used to study various types of finite and sum-difference equations.The number of applications of the inequalities given here is considerable and those presented in section 1.6 are taken from some of the above noted references.