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Abstract and Applied Analysis
Volume 2012, Article ID 538247, 21 pages
http://dx.doi.org/10.1155/2012/538247
Research Article

Some Delay Integral Inequalities on Time Scales and Their Applications in the Theory of Dynamic Equations

1School of Science, Shandong University of Technology, Shandong, Zibo 255049, China
2School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China

Received 29 July 2011; Accepted 1 December 2011

Academic Editor: Agacik Zafer

Copyright © 2012 Qinghua Feng et al. 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.

Abstract

We establish some delay integral inequalities on time scales, which on one hand provide a handy tool in the study of qualitative as well as quantitative properties of solutions of certain delay dynamic equations on time scales and on the other hand unify some known continuous and discrete results in the literature.

1. Introduction

During the past decades, with the development of the theory of differential and integral equations as well as difference equations, a lot of integral and difference inequalities have been discovered (e.g., see [113] and the references therein), which play an important role in the research of boundedness, global existence, stability of solutions of differential and integral equations as well as difference equations. On the other hand, Hilger [14] initiated the theory of time scales as a theory capable to contain both difference and differential calculus in a consistent way. Since then many authors have expounded on various aspects of the theory of dynamic equations on time scales including various inequalities on time scales (e.g., see [1524], and the references therein). However, delay integral inequalities on time scales have been paid little attention so far. Recent results in this direction include the works of Li [25] and Ma and Pečarić [26] to our best knowledge.

In this paper, we will establish some new delay integral inequalities on time scales, which unify some known continuous and discrete results in the literature. New explicit bounds for unknown functions concerned are obtained due to the presented inequalities. Some applications will be presented for the established results.

Throughout this paper, denotes the set of real numbers and , while denotes the set of integers. For two given sets , we denote the set of maps from to by .

2. Some Preliminaries on Time Scales

A time scale is an arbitrary nonempty closed subset of the real numbers. In this paper, denotes an arbitrary time scale. On we define the forward and backward jump operators and such that , .

Definition 2.1. The graininess is defined by .

Remark 2.2. Obviously, if , while if .

Definition 2.3. A point is said to be left-dense if and , right-dense if and , left-scattered if , and right-scattered if .

Definition 2.4. The set is defined to be if does not have a left-scattered maximum, otherwise it is without the left-scattered maximum.

Definition 2.5. A function is called rd-continuous if it is continuous in right-dense points and if the left-sided limits exist in left-dense points, while is called regressive if . denotes the set of rd-continuous functions, while denotes the set of all regressive and rd-continuous functions, and .

Definition 2.6. For some , and a function , the delta derivative of at is denoted by (provided it exists) with the property such that for every there exists a neighborhood of satisfying

Remark 2.7. If , then becomes the usual derivative , while if , which represents the forward difference.

Definition 2.8. For and a function , the Cauchy integral of is defined by where .

The following two theorems include some important properties for delta derivative and the Cauchy integral on time scales.

Theorem 2.9 (see [27]). If , and , then(i)(ii)If are delta differentials at t, then is also delta differential at t, and

Theorem 2.10 (see [27]). If , and , then(i), (ii), (iii), (iv), (v), (vi)if for all , then .

Definition 2.11. The cylinder transformation is defined by where Log is the principal logarithm function.

Definition 2.12. For and , the exponential function is defined by

Remark 2.13. If , then for , . If , then for and ,

The following two theorems include some known properties on the exponential function.

Theorem 2.14 (see [28]). If , then the following conclusions hold:(i), and ,(ii),(iii),where .

Theorem 2.15 (see [28]). If , and fix , then the exponential function is the unique solution of the following initial value problem:

For more details about the calculus of time scales, we advise to refer to [29].

3. Main Results

In the rest of this paper, for the sake of convenience, we denote , where , and always assume .

Lemma 3.1 (see [30, Theorem  2.2]). Let and be continuous at , where , with . Assume that is rd-continuous on . If for any , there exists a neighborhood of , independent of , such that where denotes the derivative of with respect to the first variable, then implies

Theorem 3.2. Suppose with for , and is nondecreasing. with , and . . are constants, and . If for , satisfies the following inequality: with the initial condition where , then, where with , and is the inverse of .

Proof. Assume . Denote the right side of (3.4) by . Then If , for , since , then , and from (3.7) we have If , from (3.5) we obtain So from (3.8) and (3.9) we always have Furthermore, that is, According to [29, Theorem  1.90], considering , we have Combining (3.12) and (3.13), we obtain Setting in (3.14), an integration with respect to from to yields Since , then (3.15) implies Fix , and let . Then, Denote by . Then, that is, On the other hand, for , if , then If , then where lies between and . So from (3.21) and (3.22) we always have Combining (3.20) and (3.23), we deduce Setting in (3.24), an integration with respect to from to yields Considering , and is increasing, then we obtain Combining (3.7), (3.18), and (3.26), we have Setting in (3.27), considering that is selected arbitrarily, after substituting with , we get the desired result.
If , then we carry out the process above with replaced by , where , and after letting , we also get the desired result. So the proof is complete.

Remark 3.3. If we take , then Theorem 3.2 reduces to [31, Theorem  2.1]. If , , , , then Theorem 3.2 reduces to [32, Theorem  1]. If , , ,, , then Theorem 3.2 reduces to [33, Theorem  3(a6)]. If we take , , , , , then Theorem 3.2 reduces to [33, Theorem  6(b6)].

Theorem 3.4. Suppose , and is nondecreasing. are defined as in Theorem 3.2. , where denote the delta derivative of with respect to the first variable. If for , satisfies the following inequality: with the initial condition then provided that and for , where

Proof. Assume . Fix , and let . If the right side of (3.28) is , then and similar to the process of (3.8)–(3.10), we have Furthermore, by Lemma 3.1 and Theorem 2.9(ii) that is, where are defined in (3.31).
Considering , from (3.35), we deduce Let . Then, by Theorem 2.9(ii), , and (3.36) implies On the other hand, since , then . So , and . By Theorem 2.14(i), we have . Furthermore, by a combination of Theorem 2.9(ii), Theorems 2.15, and 2.14, we obtain Combining (3.37) and (3.38), we deduce Setting in (3.39), an integration with respect to from to yields Considering , it is then followed by and furthermore Combining (3.32) and (3.42), we obtain Setting in (3.43), since is selected arbitrarily, after substituting with , we get the desired result.
If , then we carry out the process above with replaced by , and after letting , we also get the desired result. So the proof is complete.

Remark 3.5. If we take , then Theorem 3.4 reduces to [34, Corollary  2.5]. If , , , , where is a nonnegative constant, and are replaced by , , , then Theorem 3.4 reduces to [35, Theorem  1]. If we take , where is a nonnegative constant, and , , are replaced by , , , then Theorem 3.4 reduces to [35, Theorem  5].
Next we will study the delay integral inequality on time scales with the following form where are defined as in Theorem 3.2, and .

Lemma 3.6. Suppose are defined as in Theorem 3.2, and , then for , implies

Proof. Denote the right side of (3.46) by . Then, , and Let . Then, , and . Furthermore, Since , by [28, Theorem  5.4], we have . So, Using , an integration for (3.49) from to yields which confirms the desired inequality. So the proof is complete.

Theorem 3.7. If for , satisfies the inequality (3.44) with the initial condition (3.5), then provided , where

Proof. Let where is an arbitrary small constant. Then, and similar to (3.8)–(3.10) Furthermore, Using (3.13), we obtain . So (3.56) implies Considering , an integration for (3.57) from to yields where , and is defined in (3.52). Then, Denote . Then, A suitable application of Lemma 3.6 to (3.60) yields Combining (3.54) and (3.61), we obtain After letting , we get the desired result. So the proof is complete.

Remark 3.8. If we take , , , , , then Theorem 3.7 reduces to [33, Theorem  1(a2)]. If we take , , , , , then Theorem 3.7 reduces to [33, Theorem  4(b2)].
Now we present a more general inequality than in Theorem 3.7.

Theorem 3.9. Suppose are defined as in Theorem 3.2, and . If for , satisfies the following inequality: with the initial condition (3.5), then for , where is defined as in Theorem 3.7, and with , and is the inverse of .

Proof. Let the right side of (3.63) is , is an arbitrary small constant, and is defined as in Theorem 3.7. Then, , and similar to the process of (3.53)–(3.58), we obtain Fix , and let . Considering is nondecreasing, we deduce that Denote the right side of (3.66) by . Then, , and Denote . Then, , and Furthermore, that is, On the other hand, similar to (3.21)–(3.23), we have So combining (3.71) and (3.72), we deduce Considering , an integration for (3.73) from to yields Since is increasing, then Combining (3.69) and (3.75), we obtain Setting in (3.76), since is selected from arbitrarily, then in fact (3.76) holds for all , that is, Considering , an integration for (3.77) from to yields which implies Setting in the above inequality, considering is selected from arbitrarily, after replacing with and letting , we get the desired result. So the proof is complete.

Remark 3.10. If we take , then Theorem 3.9 reduces to [33, Theorem  3(a7)]. If we take , then Theorem  3.9 reduces to [33, Theorem  6(b7)].
Following in a similar manner as the proof of Theorem 3.7 and 3.9, then we present two more theorems as follows, the special cases of which unify the continuous [33, Theorems  1(a3),  3(a8)] and discrete inequalities [33, Theorems  4(b3),  6(b8)].

Theorem 3.11. Suppose are defined as in Theorem 3.9. If for , satisfies the following inequality: with the initial condition (3.5), then provided , where , and is defined as in Theorem 3.7.

Theorem 3.12. Suppose are defined as in Theorem 3.9. If for , satisfies the following inequality: with the initial condition (3.5), then where is defined as in Theorem 3.2, and is defined as in Theorem 3.7.

4. Some Applications

In this section, we will present some applications for the results which we have established above and apply them to qualitative and quantitative analysis of solutions of certain delay dynamic equations on time scales.

Example 4.1. Consider the following delay dynamic integral equation on time scales: with the initial condition where , with , , , , and is a constant with .

Theorem 4.2. Suppose is a solution of (4.1)-(4.2), and , where , then

Proof. In fact, from (4.1) we have where . Then a suitable application of Theorem 3.2 (with , ) to (4.4) yields where . Using the expression of in (4.5), we obtain the desired result, and the proof is complete.

Example 4.3. Consider the following delay dynamic differential equation on time scales with the initial