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Journal of Applied Mathematics
Volume 2011, Article ID 659563, 18 pages
http://dx.doi.org/10.1155/2011/659563
Research Article

Some New Delay Integral Inequalities in Two Independent Variables on Time Scales

School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Received 9 August 2011; Accepted 10 October 2011

Academic Editor: C. Conca

Copyright © 2011 Bin Zheng 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

Some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales are established, which can be used as a handy tool in the research of boundedness of solutions of delay dynamic equations on time scales. Some of the established results are 2D extensions of several known results in the literature, while some results unify existing continuous and discrete analysis.

1. Introduction

In the research of solutions of certain differential and difference equations, if the solutions are unknown, then it is necessary to study their qualitative and quantitative properties such as boundedness, uniqueness, and continuous dependence on initial data. The Gronwall-Bellman inequality [1, 2] and its various generalizations, which provide explicit bounds, play a fundamental role in the research of this domain. During the past decades, much effort has been done for developing such inequalities (e.g., see [315] and the references therein). On the other hand, Hilger [16] 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 (e.g., see [1719] and the references therein). In these investigations, integral inequalities on time scales have been paid much attention by many authors, which play a fundamental role in the research of quantitative as well as qualitative properties of solutions of certain dynamic equations on time scales. A lot of integral inequalities on time scales have been established (e.g., see [2026]), which have been designed to unify continuous and discrete analysis. But to our best knowledge, the Gronwall-Bellman-type delay integral inequalities on time scales have been paid little attention in the literature so far. Recent results in this direction include the work of Li [27] and that of Ma and Pečarić [28]. Furthermore, nobody has studied the Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales.

The aim of this paper is to establish some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales, which provide new bounds for the unknown functions concerned. Some of our results are 2D extensions of many known inequalities in the literature, while some results unify existing continuous and discrete analysis. For illustrating the validity of the established results, we will present some applications of them.

First we will give some preliminaries on time scales and some universal symbols for further use.

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 .

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 1.1. The graininess is defined by .

Remark 1.2. Obviously, if while if .

Definition 1.3. A point is said to be left-dense if and , right-dense if and , left-scattered if , and right-scattered if .

Definition 1.4. The set is defined to be if does not have a left-scattered maximum; otherwise it is without the left-scattered maximum.

Definition 1.5. A function is called rd-continuous if it is continuous at right-dense points and if the left-sided limits exist at 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 1.6. 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
Similarly, for some and a function , the partial delta of with respect to is denoted by or and satisfies where and is a neighborhood of .

Remark 1.7. If , then becomes the usual derivative , while if , which represents the forward difference.

For more details about the calculus of time scales, see [29]. In the rest of this paper, for the convenience of notation, we always assume that , where and furthermore assume .

2. Main Results

We will give some lemmas for further use.

Lemma 2.1. Suppose is a fixed number, and with respect to , , then implies where : and is the unique solution of the following equation

The proof of Lemma 2.1 is similar to [26, Theorem 5.6].

Lemma 2.2. Under the conditions of Lemma 2.1, and furthermore assuming is nondecreasing in for every fixed , , then one has

Proof. Since is nondecreasing in for every fixed , then from Lemma 2.1 we have On the other hand, from [29, Theorems 2.39 and 2.36 (i)] we have . Then collecting the above information, we can obtain the desired inequality.

Lemma 2.3 (see [11]). Assume that , and ; then, for any

Lemma 2.4. Let be continuous and nondecreasing in the second variable, and assume is a fixed number in . Suppose and satisfy the dynamics inequalities: Then for some implies for all .

The proof of Lemma 2.4 is similar to [26, Theorem 5.7].

Theorem 2.5. Suppose , and are nondecreasing. is a constant, and . . . . If for , satisfies the following inequality: with the initial condition then where

Proof. Fix , and . Let Then If and , then , and If or , then from (2.9) we have From (2.14) and (2.15) we always have Moreover From Lemma 2.3, we have So Then applying Lemma 2.1 to (2.19), we obtain So Setting in (2.21), it follows that Replacing with in (2.22), we obtain the desired inequality.

Remark 2.6. Theorem 2.5 is the 2D extension of [27, Theorem 1]. For its special case , the established bound for in (2.10) is a new bound compared with the result in [12, Theorem 2.2].

Remark 2.7. Assume in Theorem 2.5. If we apply Lemma 2.2 instead of Lemma 2.1 to (2.19) in the proof of Theorem 2.5, then we obtain another bound for as follows:

Now we will establish a more general inequality than that in Theorem 2.5.

Theorem 2.8. Suppose are the same as in Theorem 2.5, and . are constants, and , , . If for , satisfies the following inequality:

with the initial condition (2.9), then where

Proof. Fix , and . Let Then Similar to (2.14)–(2.16), we obtain So From Lemma 2.3, we have Combining (2.30) and (2.31) we get that Applying Lemma 2.1 to (2.32) yields Then Setting in (2.34) yields Considering is arbitrary and replacing with in (2.35), we obtain the desired inequality.

Remark 2.9. Assume in Theorem 2.8. If we apply Lemma 2.2 instead of Lemma 2.1 to (2.32) in the proof of Theorem 2.8, then we obtain another bound for as follows:

Remark 2.10. Theorem 2.8 is the 2D extension of [27, Theorem 3].

Theorem 2.11. Suppose are the same as in Theorem 2.5, and is a constant. If for , satisfies the following inequality: with the initial condition then

Proof. Let the right side of (2.37) be . Then For , if , then , and from (2.40) we have If or , from (2.38) we have So from (2.41) and (2.42), we always have Similarly, when and , then , and from (2.40) we have When or , considering , from (2.38) it follows that Combining (2.44) and (2.45), we always have By (2.43) and (2.46), we obtain Let the right side of (2.47) be . Then Considering , and , from (2.49) it follows that An integration of (2.50) with respect to from to yields .
Considering , it follows that Then combining (2.40), (2.48), and (2.51), we obtain and the proof is complete.

Remark 2.12. If we take , then Theorem 2.11 becomes the extension of the known Ou-Iang's inequality [13] to the 2D case.

The following theorem provides a more general result than Theorem 2.11.

Theorem 2.13. Suppose is a positive integer, and . Under the conditions of Theorem 2.11, if satisfies with the initial condition then

The proof of Theorem 2.13 is similar to Theorem 2.11. As long as we notice a delta d ifferentiable function , the following formula [26, Equation (6.2)] holds: Then following a similar manner as in Theorem 2.11, we can deduce the desired result.

Theorem 2.14. Suppose are the same as in Theorem 2.5, is nondecreasing, and are constants with . Furthermore, define a bijective function such that . If for , satisfies the following inequality: with the initial condition then where .

Proof. Fix , and . Let Then Similar to (2.14)–(2.16), we obtain Moreover, Let be the solution of the following problem: Considering and is nondecreasing and continuous, then from (2.63), (2.64), and Lemma 2.4, we have On the other hand, from the definition of we have . Then an integration with respect to from to yields that is, Combining (2.61), (2.65), and (2.67), we have Setting in (2.68), we get the desired result.

Remark 2.15. If we take , then Theorem 2.14 reduces to [14, Theorem 2.1], while Theorem 2.14 reduces to [15, Theorem 2.1] if we take .

Theorem 2.16. Suppose are the same as in Theorem 2.5, and furthermore, is delta differential on with respect to , . is nondecreasing, and is submultiplicative, that is, , for all . is a constant. is a bijective function such that . If for , satisfies the following inequality: with the initial condition then where and is the unique solution of the following equation:

Proof. Fix , and . Let Then Similar to (2.14)–(2.16), we can obtain Furthermore we have Let . Then from (2.76) it follows that Considering is nondecreasing in , by applying Lemma 2.2 to (2.77), we obtain On the other hand, Let be the solution of the following equation: Considering and is nondecreasing and continuous, then from (2.79), (2.80), and Lemma 2.4, we have From the definition of and (2.80), we have . Then similar to (2.66) and (2.67), we obtain Combining (2.74), (2.78), and (2.82), we have Setting in (2.83), we obtain Replacing with in (2.84) yields the desired inequality (2.71).

Theorem 2.17. Under the conditions of Theorem 2.16, if are constants with , , and for , satisfies the following inequality: with the initial condition (2.58), then where is defined as in Theorem 2.14, , , and is the unique solution of the following equation:

The proof of Theorem 2.17 is similar to that of Theorem 2.16, and we omit it here.

3. Some Simple Applications

In this section, we will present some examples to illustrate the validity of our results in deriving explicit bounds of solutions of certain delay dynamic equations on time scales.

Example 3.1. Consider the following delay dynamic integral equation: with the initial condition where , are the same as in Theorem 2.8, and . Furthermore, assume , where , and are the same as in Theorem 2.8.
From (3.1) we have Then according to Theorem 2.8, we can obtain the following estimate: where

Example 3.2. Considering the following delay dynamic integral equation: with the initial condition where , and .

Assume , where , then from (3.6) we have According to Theorem 2.13 (p = 3), we can reach the following estimate:

4. Conclusions

In this paper, we established some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales. As one can see, the presented results provide a handy tool for deriving bounds for solutions of certain delay dynamic equations on time scales. Furthermore, the process of constructing Theorems 2.5, 2.8, 2.14, 2.16 and 2.17 can be applied to the situation with independent variables.

Acknowledgments

This work is supported by the Natural Science Foundation of Shandong Province (ZR2010AZ003) (China). The authors thank the referees very much for their careful comments and valuable suggestions on this paper.

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