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Bohner Our aim in this paper is to investigate some delay integral inequalities on time scales by using Gronwall’s inequality and comparison theorem.. The inequalities given here can be

Volume 2008, Article ID 831817, 13 pages

doi:10.1155/2008/831817

Research Article

Bounds for Certain Delay Integral

Inequalities on Time Scales

Wei Nian Li 1, 2

1 Department of Applied Mathematics, Shanghai Normal University, Shanghai 200234, China

2 Department of Mathematics, Binzhou University, Shandong 256603, China

Correspondence should be addressed to Wei Nian Li,wnli@263.net

Received 31 August 2008; Revised 21 October 2008; Accepted 22 October 2008

Recommended by Martin J Bohner

Our aim in this paper is to investigate some delay integral inequalities on time scales by using Gronwall’s inequality and comparison theorem Our results unify and extend some delay integral inequalities and their corresponding discrete analogues The inequalities given here can be used as handy tools in the qualitative theory of certain classes of delay dynamic equations on time scales Copyrightq 2008 Wei Nian Li This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

The unification and extension of differential equations, difference equations, q-difference equations, and so on to the encompassing theory of dynamic equations on time scales was initiated by Hilger 1 in his Ph.D thesis in 1988 During the last few years, some integral inequalities on time scales related to certain inequalities arising in the theory of dynamic equations had been established by many scholars For example, we refer the reader to literatures 2 8 and the references therein However, nobody studied the delay integral inequalities on time scales, as far as we know In this paper, we investigate some delay integral inequalities on time scales, which provide explicit bounds on unknown functions Our results extend some known results in9

Throughout this paper, a knowledge and understanding of time scales and time scale notation is assumed For an excellent introduction to the calculus on time scales, we refer the reader to monographes10,11

2 Main results

In what follows,R denotes the set of real numbers, R 0, ∞, Z denotes the set of integers,

N0  {0, 1, 2, } denotes the set of nonnegative integers, CM, S denotes the class of all continuous functions defined on set M with range in the set S,T is an arbitrary time scale,

Crd denotes the set of rd-continuous functions, R denotes the set of all regressive and rd-continuous functions, andR  {p ∈ R : 1  μtpt > 0, for all t ∈ T} We use the usual

conventions that empty sums and products are taken to be 0 and 1, respectively Throughout

this paper, we always assume that t0 ∈ T, T0  t0 , ∞T

The following lemmas are very useful in our main results

Lemma 2.1 see 9 Assume that p ≥ q ≥ 0, p / 0, and a ∈ R Then

a q/p ≤ q

p k q−p/p a  p − q



Lemma 2.2 Gronwall’s inequality 10 Suppose u, b ∈ C rd , m ∈ R, m ≥ 0 Then

ut ≤ bt 

t

implies

ut ≤ bt 

t

t0 e m

Lemma 2.3 comparison theorem 10 Suppose u, b ∈ C rd , a ∈ R Then

uΔt ≤ atut  bt, t ∈ T0 , 2.4

implies

ut ≤ ut0

e a

t, t0



t

t0 e a

Firstly, we study the delay integral inequality on time scales of the form

x p t ≤ at 

t

t0 bsx p sΔs  ct

t

t0



fsx q

τs gsx r s Δs, t ∈ T0 , E

with the initial condition

xt  ϕt, t ∈α, t0

∩ T,

ϕ

τt≤at1/p for t∈ T0 with τ t ≤ t0 , I where p, q, and r are constants, p / 0, p ≥ q ≥ 0, p ≥ r ≥ 0, τ : T0 → T, τt ≤ t, − ∞ < α 

inf{τt, t ∈ T0} ≤ t0, and ϕt ∈ Crdα, t0 ∩ T, R.

Theorem 2.4 Assume that xt, at, bt, ct, ft, gt ∈ CrdT0, R If at and ct are

nondecreasing for t ∈ T0, then the inequalityE  with the initial condition  I  implies

xt ≤ e b

t, t0

at  ct



Ft 

t

t0 e G

1/p

, 2.6

for any k > 0, t ∈ T0, where

Ft 

t

t0

b

s, t0 q/p

kp − q  qas

pk p−q/p gs



e b

s, t0 r/p

kp − r  ras

pk p−r/p Δs,

2.7

Gt  ct qfte b

t, t0 q/p

pk p−q/p rgt



e b

t, t0 r/p

pk p−r/p

t

t0 bsx p sΔs  ct

t

t0



fsx q

τs gsx r s Δs 1/p , t ∈ T0. 2.9

It is easy to see that zt is a nonnegative and nondecreasing function, and

Therefore,

x

τt≤ zτt≤ zt, for t ∈ T0 with τ t > t0 2.11

On the other hand, using the initial conditionI, we have

x

τt ϕτt≤at1/p ≤ zt, for t ∈ T0 with τ t ≤ t0 2.12 Combining2.11 and 2.12, we obtain

x

τt≤ zt, t ∈ T0 2.13

It follows from2.9, 2.10, and 2.13 that

z p t ≤ at 

t

t0 bsz p sΔs  ct

t

t0



Define a function wt by

ut 

t

t0



Then2.14 can be restated as

z p t ≤ wt 

t

Obviously, w∈ CrdT0, R, bt ≥ 0, b ∈ R UsingLemma 2.2, from2.17, we obtain

z p t ≤ wt 

t

t0 e b

Noting that wt is nondecreasing, from 2.18, we have

z p t ≤ wt  wt

t

t0 e b

t, σsbsΔs  wt

1

t

t0 e b

t, σsbsΔs

By10, Theorems 2.39 and 2.36i, we obtain

t

t0 e b

t, σsbsΔs  e b

t, t0



− e b t, t  e b

t, t0



− 1, t ∈ T0 2.20

It follows from2.19 and 2.20 that

z p t ≤ wte b

t, t0

 e b

t, t0

UsingLemma 2.1, from2.21, for any k > 0, we easily obtain

z q t ≤e b

t, t0

 q/p

at  ctutq/p

≤e b

t, t0

 q/p kp − q  qat

pk p−q/p qctut

pk p−q/p

z r t ≤e b

t, t0

 r/p

at  ctutr/p

≤e b

t, t0

 r/p kp − r  rat

pk p−r/p rctut

pk p−r/p

Combining2.16, 2.22, and 2.23, we have

ut ≤

t

t0

fse b

s, t0

 q/p kp − q  qas

pk p−q/p qcsus

pk p−q/p



 gse b

s, t0 r/p kp − r  ras

pk p−r/p  rcsus

pk p−r/p



Δs

 Ft 

t

2.24

where Ft and Gt are defined by 2.7 and 2.8, respectively Using Lemma 2.2, from

2.24, we have

ut ≤ Ft 

t

t0 e G

Therefore, the desired inequality2.6 follows from 2.10, 2.22, and 2.25 This completes the proof

Theorem 2.5 Suppose that all assumptions of Theorem 2.4 hold Then the inequalityE  with the

initial conditionI  implies

xt ≤e b

t, t0

at  ctFte G

t, t0 1/p

for any k > 0, t ∈ T0, where Ft and Gt are defined by 2.7 and 2.8, respectively.

nondecreasing for t∈ T0 Therefore, by10, Theorems 2.39 and 2.36i, we have

ut ≤ Fte G

t, t0

The desired inequality2.26 follows from 2.10, 2.22, and 2.27 The proof is complete

Corollary 2.7 Assume that xn, an, bn, cn, fn, gn are nonnegative functions defined

for n ∈ N0 If an and cn are nondecreasing in N0, and xn satisfies the following delay discrete inequality:

x p n ≤ an  n−1

s0

bsx p s  cnn−1

s0



with the initial condition

xn  ϕn, n ∈ {−ρ, , −1, 0}, ϕn − ρ ≤an1/p

{−ρ, , −1, 0}, then

xn ≤

n−1



s0



1 bs



an  cnHnn−1

s0



1 Js

1/p

, 2.28

for any k > 0, n ∈ N0,

Hn n−1

s0



t0



1 bt q/pkp − q  qas

pk p−q/p

gs

s−1

t0



1 bt r/pkp − r  ras

pk p−r/p



,

Jn  cn



s0



1 bs q/p

pk p−q/p  rgn

n−1

s0



1 bs r/p

pk p−r/p



2.29

Next, using the Chain Rule, we consider a special case of the delay integral inequality

E of the form

x p t ≤ C 

t

t0 bsx p sΔs 

t

t0 fsx p−1

τsΔs, t ∈ T0 , E

with the initial condition

xt  ϕt, t ∈α, t0

∩ T,

ϕ

τt≤ C 1/p for t∈ T0 with τ t ≤ t0 , I

where C and p ≥ 1 are positive constants, τt, α, and ϕt are defined as in  I

Theorem 2.8 Assume that xt, bt, ft ∈ CrdT0, R Then the inequality E with the initial

conditionI implies

xt ≤ C 1/p e b/p

t, t0



p1

t

t0 e b/p

Proof Define a function w t by

w p t  C 

t

t0 bsx p sΔs 

t

t0 fsx p−1

τsΔs, t ∈ T0 2.31

Using a similar way in the proof ofTheorem 2.4, we easily obtain that wt is a positive

and nondecreasing function, and

x

τt≤ wt, t ∈ T0 2.33 Differentiating 2.31, we obtain

pw p−1 θwΔt  btx p t  ftx p−1

where θ ∈ t, σt.

It follows from2.32–2.34 that

pw p−1 θwΔt ≤ btw p t  ftw p−1 t, t ∈ T0 2.35

Noting the fact that 0 < wt ≤ wθ and wΔt ≥ 0, from the above inequality, we have

pw p−1 twΔt ≤ btw p t  ftw p−1 t, t ∈ T0 2.36 Therefore,

wΔt ≤ bt p wt  ft p , t ∈ T0. 2.37

wt ≤ C 1/p e b/p

t, t0





t

t0 e b/p

t, σs fs

p Δs, t ∈ T0 2.38

Therefore, the desired inequality 2.30 follows from 2.32 and 2.38 This completes the proof ofTheorem 2.8

Corollary 2.9 Assume that xt, bt, ft ∈ CR, R If xt satisfies the following delay integral

inequality:

x p t ≤ C 

t 0

bsx p sds 

t 0

fsx p−1

with the initial condition

xt  φt, t ∈ β, 0,

φ

where C and p ≥ 1 are positive constants, ρt ∈ CR, R, ρt ≤ t, − ∞ < β  inf{ρt, t ∈ R} ≤

0, and φt ∈ Cβ, 0, R, then

xt ≤ C 1/pexp

t 0

bs



1

p

t 0

fs exp

t

s

bτ



Corollary 2.10 Assume that xn, bn, fn are nonnegative functions defined for n ∈ N0 If xn satisfies the following delay discrete inequality:

x p n ≤ C  n−1

s0

bsx p s  n−1

s0

with the initial condition

xn  ϕn, n ∈ {−ρ, , −1, 0},

where C and p ≥ 1 are positive constants, ρ and ϕn are defined as in  I1 , then

xn ≤ C 1/pn−1

s0



1bs p



 1p n−1

s0

fsn−1

is1



1bi p



Finally, we study the delay integral inequality on time scales of the form

x p t ≤ at  ct

t

t0



fsx q s  Ls, x

τs Δs, t ∈ T0 , E

with the initial conditionI , where p ≥ 1, 0 ≤ q ≤ p are constants, τt is as defined in the

inequalityE , and L : T0× R → Ris a continuous function

Theorem 2.11 Assume that xt, at, ct, ft ∈ CrdT0, R If at and ct are nondecreasing

for t ∈ T0, and

0≤ Lt, x − Lt, y ≤ Kt, yx − y, 2.41

for x ≥ y ≥ 0, where K : T0× R → Ris a continuous function, then the inequalityE with the

initial conditionI  implies

xt ≤

at  ct



Ht 

t

t0 e J

1/p

, 2.42

for any k > 0, t ∈ T0, where

Ht 

t

t0

fskp − q  qas

pk p−q/p  L



s, p − 1



Δs, 2.43

pk p−q/p  K



t, p − 1

ct

p . 2.44

zt 

t

t0



fsx q s  Ls, x

τs Δs, t ∈ T0 2.45

We easily observe that zt is a nonnegative and nondecreasing function, and  E can be restated as

UsingLemma 2.1, from2.46, we have

xt ≤at  ctzt1/p≤ p − 1

Therefore, for t∈ T0with τt ≥ t0, we obtain

x

τt≤ p − 1



τt



τtz

τt

p , 2.48

and for t∈ T0with τt ≤ t0, using the initial conditionI and 2.47, we get

x

It follows from2.48 and 2.49 that

x

τt≤ p − 1

Combining2.45, 2.46, and 2.50, byLemma 2.1, for any k > 0, we obtain

zt ≤

t

t0 fsas  cszs q/p Δs 

t

t0 L



s, p − 1

p as p cszs p



Δs

≤

t

t0 fs kp − q  qas

pk p−q/p qcszs

pk p−q/p

Δs



t

t0 L



s, p − 1

p as p  cszs p



− L



s, p − 1



 L



s, p − 1

≤

t

t0

fskp − q  qas

pk p−q/p  L



s, p − 1



Δs



t

t0

qcsfs

pk p−q/p  K



s, p − 1

p

cs

p

zsΔs

 Ht 

t

t0 JszsΔs, t ∈ T0,

2.51

where Ht and Jt are defined by 2.43 and 2.44, respectively

zt ≤ Ht 

t

t0 e J

Therefore, the desired inequality 2.42 follows from 2.46 and 2.52 The proof of

Noting Ht, defined by 2.43, is nondecreasing for t ∈ T0, we easily obtain the following result

Theorem 2.12 Suppose that all assumptions of Theorem 2.11 hold Then the inequalityE with

the initial conditionI  implies

xt ≤at  ctHte J

t, t01/p

for any k > 0, t ∈ T0, where Ht and Jt are defined by 2.46 and 2.47, respectively.

Corollary 2.14 Assume that xn, an, cn, fn are nonnegative functions defined for n ∈ N0.

If an and cn are nondecreasing in N0, and xn satisfies the following delay discrete inequality:

x p n ≤ an  cn n−1

s0



where p, q, and ρ are constants, p ≥ 1, p ≥ q ≥ 0, ρ ∈ N0, and L, K : N0× R → Rsatisfying

0≤ Ln, x − Ln, y ≤ Kn, yx − y, 2.54

for x ≥ y ≥ 0, then the inequality  E4  with the initial condition  I1  implies

xn ≤



an  cn Hnn−1

s0



1 Js 

1/p

for any k > 0, n ∈ N0, where



Hn  n−1

s0

fskp − q  qas

pk p−q/p  L



s, p − 1



,

Jn  qcnfn

pk p−q/p  K



n, p − 1

p

cn

2.56

3 Some applications

In this section, we present some applications of our results

Example 3.1 Consider the delay dynamic equation on time scales:

x p tΔ Mt, xt, xτt, t ∈ T0 , 3.1 with the initial condition

xt  ψt, t ∈α, t0

∩ T,

ψ

τt C 1/p for t∈ T0 with τt ≤ t0 , I

where M :T0× R2 → R is a continuous function, C  x p t0 and p > 0 are constants, α and

τt are as defined in the initial condition  I , and ψt ∈ Crdα, t0 ∩ T, R.

Theorem 3.2 Assume that

Mt,xt,xτt ≤ ftx q

τt   gtx r t, 3.2

where ft, gt ∈ C rdT0, R, q and r are constants, p ≥ q ≥ 0, p ≥ r ≥ 0 If xt is a solution of

3.1 satisfying the initial condition  I, then

xt ≤|C|  Ft 

t

t0 e G

1/p

for any k > 0, t ∈ T0, where

Ft 

t

t0 pk p−q/p  gs



kp − r  r|C|

pk p−r/p Δs, 3.4

pk p−q/p  rgt

pk p−r/p 3.5

equivalent delay integral equation on time scales

x p t  C 

t

t0 M

with the initial conditionI Noting the assumption 3.2, we have

x p t ≤ |C| t

t0



fs x q

τs   gsx r s Δs, t ∈ T0 , 3.7

with the initial condition I Therefore, byTheorem 2.4, from 3.7, we easily obtain the estimate3.3 of solutions of 3.1 The proof ofTheorem 3.2is complete

UsingTheorem 2.5, we easily obtain the following result

Theorem 3.3 Suppose that all assumptions of Theorem 3.2 hold If xt is a solution of 3.1

satisfying the initial conditionI, then

xt ≤ |C|  Fte G

t, t0 1/p

for any k > 0, t ∈ T0, where Ft and Gt are defined by 3.4 and 3.5, respectively.

3.1 satisfying the initial condition I in terms of the known functions for any k > 0, t ∈ T0, respectively

I3  with p  2, C  1/4, ρ  2, ϕn  1/2, n ∈ {−2, −1, 0}, bn  10−3n2, f n  10−4n,

using the result2.40 In our computations, we use E3 and 2.40 as equations and as we see inTable 1the computation values as inE3 are less than the values of the result 2.40

Table 1

solu-tion for some discrete inequalities The program is written in the programming Matlab 7.0

Acknowledgments

The author thanks the referee very much for his careful comments and valuable suggestions

on this paper This work is supported by the Natural Science Foundation of Shandong Province Y2007A08, the National Natural Science Foundation of China 60674026, 10671127, the Project of Science and Technology of the Education Department of Shandong ProvinceJ08LI52, and the Doctoral Foundation of Binzhou University 2006Y01

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with the initial condition

xt  φt, t ∈ β, 0,...

Table 1

solu-tion for some discrete inequalities The program is written in the