Discrete Dynamics in Nature and SocietyVolume 2011 (2011), Article ID 320794, 12 pagesdoi:10.1155/2011/320794

Research Article

Nonlinear Dynamical Integral Inequalities in Two Independent Variables and Their Applications

Yuangong Sun

School of Mathematics, University of Jinan, Jinan, Shandong 250022, China

Received 25 May 2011; Accepted 17 July 2011

Academic Editor: Hassan A. El-Morshedy

Copyright © 2011 Yuangong Sun. 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.

In this paper, we investigate some nonlinear dynamical integral inequalities involving the forward jump operator in two independent variables. These inequalities provide explicit bounds on unknown functions, which can be used as handy tools to study the qualitative properties of solutions of certain partial dynamical systems on time scales pairs.

1. Introduction

Theory of dynamical equations on time scales, which goes back to Hilger's landmark paper [1], has received considerable attention in recent years. For example, see the monographs [2, 3] and the references cited therein. Since dynamical integral inequalities usually can be used as handy tools to study the qualitative theory of dynamical equations on time scales, many researchers devoted to the study of different types of integral inequalities on time scales. We refer the readers to [4–19].

To the best of our knowledge, the theory of partial dynamic equations on time scales has received less attention [20–24]. The main purpose of this paper is to investigate several nonlinear integral inequalities in two independent variables on time scale pairs, which can be used to estimate explicit bounds of solutions of certain partial dynamical equations on time scales. Unlike some existing results in the literature (e.g., [12]), the integral inequalities considered in this paper involve the forward jump operator and on a pair of time scales and , which results in difficulties in the estimation on the explicit bounds of unknown functions for and . As an application, we study the qualitative property of certain partial dynamical equations on time scales.

Throughout this paper, a knowledge and understanding of time scales and time scale notations is assumed. In what follows, and are two unbounded time scales, and . is the set of right-dense continuous functions on . For an excellent introduction to the calculus on time scales, we refer the reader to monographs [2, 3].

2. Problem Statements

Before establishing the main results of this paper, we first present two useful lemmas as follows.

Lemma 2.1. Let , , and . Then, for any , holds, where .

Proof. Set . It is not difficult to see that obtains its maximum at and This completes the proof of Lemma 2.1.

Lemma 2.2. Let with for . Then implies where and .

Proof. Note that , we have that is, By Theorem  6.1 [2, page 255], we get that Lemma 2.2 holds.

Consider the following nonlinear integral inequalities in two independent variables on time scales : where , and ( are nonnegative right-dense continuous functions on , () are constants.

The reason for studying inequalities of type (2.7)–(2.9) is that sometimes we may need to estimate the solutions of the following partial dynamical equation in the form with boundary conditions , , and , where is right-dense continuous, , and is a constant. Integrating (2.10) yields Therefore, the study on the integral inequalities of type (2.7)–(2.9) can provide explicit bounds of solutions of system (2.10) in some cases.

3. Main Results

Now, let us present the main results of this paper.

Theorem 3.1. If there exists a positive function , such that then inequality (2.7) implies where

Proof. Define a function by Then, for , is nondecreasing with respect to and , and A delta derivative of with respect to yields By Lemma 2.1, we have It follows from (3.5), (3.6), and (3.7) that Notice the definitions of , , and , we have Since , by Lemma 2.2 we get Then, (3.5) and (3.10) imply (3.2).

Theorem 3.2. If there exist positive functions , such that is Δ-differentiable with respect to , , and then inequality (2.8) implies where

Proof. Set Then, is nonnegative and nondecreasing with respect to and on , and By Lemma 2.1, we have Substituting (3.15) into (3.16), we get Note that Integrating by parts, we have Therefore, it follows from (3.17) and (3.19) that This together with Lemma 2.2 and (3.15) yields (3.12).

Theorem 3.3. If there exist positive functions , such that are -differentiable with respect to , and then inequality (2.9) implies where and .

Proof. Let the nonnegative and nondecreasing function be defined by Then, Based on the same arguments as in Theorem 3.2, we have Notice that By (3.26) and (3.27), we have Using the fact , Lemma 2.2 and (3.25), we get that (3.22) holds.It is worthy to mention that although some additional assumptions such as and are imposed in Theorems 3.1–3.3, they are easy to be satisfied by choosing appropriate adjusting functions and .

4. Applications

We now consider some applications of the main results in the partial dynamical system (2.10) under the boundary condition Denote . We have the following corollaries.

Corollary 4.1. Let , , and Then, the solution of system (2.10) under the boundary condition (4.1) satisfies for .

Proof. For , it follows from (2.11) and (4.2) that Let be a constant. A straightforward computation yields Since , we get (4.3) by Theorem 3.1.

Corollary 4.2. Let , and Then, the solution of system (2.10) under the boundary condition (4.1) satisfies where is defined as in Corollary 4.1 for .

Proof. For , it follows from (2.11) and (4.6) that holds for . Let and . A straightforward computation yields Hence, . By Theorem 3.2, we have that (4.7) holds.For the case when satisfies on , the solution of system (2.10) under the boundary condition (4.1) can be similarly estimated by Theorem 3.3. We omit it here.

Acknowledgment

This paper was supported by the Natural Science Foundations of Shandong Province (ZR2010AL002, JQ201119) and the National Natural Science Foundation of China (61174217).

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