Abstract and Applied AnalysisVolume 2011 (2011), Article ID 584951, 24 pagesdoi:10.1155/2011/584951

Research Article

Some New Volterra-Fredholm-Type Discrete Inequalities and Their Applications in the Theory of Difference Equations

Bin Zheng¹ and Qinghua Feng1,2

¹School of Science, Shandong University of Technology, Shandong, Zibo 255049, China²School of Mathematical Sciences, Qufu Normal University, Shandong, Qufu 273165, China

Received 21 March 2011; Accepted 29 June 2011

Academic Editor: Martin D. Schechter

Copyright © 2011 Bin Zheng and Qinghua Feng. 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 Volterra-Fredholm-type discrete inequalities in two independent variables are established, which provide a handy tool in the study of qualitative and quantitative properties of solutions of certain difference equations. The established results extend some known results in the literature.

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. Many such generalized inequalities (e.g., see [3–30] and the references therein) have been established in the literature including the known Ou-Liang's inequality [3]. In [8], Ma generalized the discrete version of Ou-Liang's inequality in two variables to Volterra-Fredholm form for the first time, which has proved to be very useful in the study of qualitative as well as quantitative properties of solutions of certain Volterra-Fredholm-type difference equations. But since then, few results on Volterra-Fredholm-type discrete inequalities have been established. Recent results in this direction include the work of Ma [9] to our knowledge. We notice, in the analysis of some certain Volterra-Fredholm-type difference equations with more complicated forms, that the bounds provided by the earlier inequalities are inadequate and it is necessary to seek some new Volterra-Fredholm-type discrete inequalities in order to obtain a diversity of desired results.

Our aim in this paper is to establish some new generalized Volterra-Fredholm-type discrete inequalities, which extend Ma's work in [9], and provide new bounds for unknown functions lying in these inequalities. We will illustrate the usefulness of the established results by applying them to study the boundedness, uniqueness, and continuous dependence on initial data of solutions of certain more complicated Volterra-Fredholm-type difference equations.

Throughout this paper, denotes the set of real numbers , and denotes the set of integers, while denotes the set of nonnegative integers. Let , where and are two constants. are two constants, and , are all constants. If is a lattice, then we denote the set of all -valued functions on by and denote the set of all -valued functions on by . Finally, for a function , we have provided .

2. Main Results

Lemma 2.1 (see [15]). Assume that ,, and then for any

Lemma 2.2. Suppose that , is a constant. If is nondecreasing in the third variable, then, for , implies that

Lemma 2.3. Suppose that . If is nondecreasing in the first variable, then, for , implies that

Remark 2.4. Lemma 2.3 is a direct variation of [19, Lemma 2.5()], and we note here.

Theorem 2.5. Suppose that , , , , with nondecreasing in the last two variables. are nonnegative constants with , while are nonnegative constants with . If, for , satisfies then provided that , where

Proof. Denote Then, we have and, furthermore, from Lemma 2.1 we have So where , and is defined in (2.8). Then, using that is nondecreasing in every variable, we obtain where is defined in (2.11).Since are nondecreasing in the last two variables, then is also nondecreasing in the last two variables, and by a suitable application of Lemma 2.2 we obtain where is defined in (2.10). Furthermore, considering the definition of and (2.17) we have where is defined in (2.9). Then, Combining (2.17) and (2.19) we deduce Then, combining (2.13) and (2.20), we obtain the desired result.

Corollary 2.6. Suppose that with nondecreasing in every variable. . are defined as in Theorem 2.5. If, for , satisfies then provided that , where

The proof of Corollary 2.6 can be completed by setting ,,, ,,,,,,,,,,,,,,, in Theorem 2.5.

Corollary 2.7. Suppose that are defined as in Theorem 2.5. If, for , satisfies then provided that , where

Theorem 2.8. Suppose that , are defined as in Theorem 2.5. Furthermore, assume that is nondecreasing in the first variable. If, for , satisfies then provided that , where

Proof. Denote Then, we have Obviously is nondecreasing in the first variable. So by Lemma 2.3 we obtain where . Define Then, and, furthermore, by (2.39) and Lemma 2.1 we have where , and are defined in (2.29)–(2.31) respectively.Similar to the process of (2.15)–(2.20) we deduce where are defined in (2.32) and (2.33).Combining (2.39) and (2.41), we get the desired result.

Remark 2.9. If we set ,,,,,,,,,,,,,,,= ,,,,,,,, or set in Theorem 2.8, then immediately we get two corollaries which are similar to Corollaries 2.6 and 2.7, and we omit the details for them.

Theorem 2.10. Suppose that are defined as in Theorem 2.5. , satisfies for . If, for , satisfies then provided that , where

Proof. Denote Then and, furthermore, from Lemma 2.1 we have where = and are defined in (2.44) and (2.46) respectively.Similar to the process of (2.15)–(2.20) we deduce where are defined in (2.45) and (2.47) respectively.Combining (2.50) and (2.52), we get the desired result.

Theorem 2.11. Suppose that , are defined as in Theorem 2.5. Furthermore, assume is nondecreasing in the first variable. ,, are defined as in Theorem 2.10. If, for , satisfies then provided that , where

The proof for Theorem 2.11 is similar to the combination of Theorems 2.8 and 2.10, and we omit the details here.

Remark 2.12. If we take ,,, and ,, in Corollary 2.6, then Corollary 2.6 reduces to [9, Theorem 2.5]. If furthermore , then Corollary 2.6 reduces to [9, Theorem 2.1]. If we take ,, and , ,, in Theorem 2.10, then Theorem 2.10 reduces to [9, Theorem 2.7]. If furthermore , then Theorem 2.10 reduces to [9, Theorem 2.6].

3. Applications

In this section, we will present some applications for the established results above and show that they are useful in the study of boundedness, uniqueness, and continuous dependence of solutions of certain difference equations.

Example 3.1. Consider the following Volterra-Fredholm sum-difference equation: where , is an odd number, ,, are two integers defined the same as in Theorem 2.5.

Theorem 3.2. Suppose that is a solution of (3.1), and , ,,, where are nonnegative constants satisfying , and are nondecreasing in the last two variables; then one has provided that , where

Proof. From (3.1) we have Then a suitable application of Theorem 2.5 (with ) to (3.4) yields the desired result.

The following theorem deals with the uniqueness of the solutions of (3.1).

Theorem 3.3. Suppose that ,, hold for , where , with nondecreasing in the last two variables, and , where , and , then (3.1) has at most one solution.

Proof. Suppose that are two solutions of (3.1). Then, Treat as one variable, and a suitable application of Corollary 2.7 yields , which implies that . Since is an odd number, then we have , and the proof is complete.

Finally we study the continuous dependence of the solutions of (3.1) on the functions .

Theorem 3.4. Suppose that is a solution of (3.1), , , hold for , where , with nondecreasing in the last two variables, and, furthermore, where is a constant, and is the solution of the following difference equation: where ; then one has provided that , where

Proof. From (3.1) and (3.7) we deduce Then a suitable application of Corollary 2.7 yields the desired result.

Remark 3.5. We note that the results in [1–30] are not available here to establish the analysis above.

Acknowledgment

The authors thank the referees very much for their careful comments and valuable suggestions on this paper.

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