Journal of Applied Mathematics

Volume 2014 (2014), Article ID 826173, 9 pages

http://dx.doi.org/10.1155/2014/826173

Research Article

A Class of Volterra-Fredholm Type Weakly Singular Difference Inequalities with Power Functions and Their Applications

Yange Huang,¹ Wu-Sheng Wang,² and Yong Huang¹

¹Department of Mathematics and Computer Information Engineering, Baise University, Baise 533000, China

²School of Mathematics and Statistics, Hechi University, Yizhou, Guangxi 546300, China

Received 12 July 2014; Accepted 5 August 2014; Published 14 August 2014

Academic Editor: Junjie Wei

Copyright © 2014 Yange Huang 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 discuss a class of Volterra-Fredholm type difference inequalities with weakly singular. The upper bounds of the embedded unknown functions are estimated explicitly by analysis techniques. An application of the obtained inequalities to the estimation of Volterra-Fredholm type difference equations is given.

1. Introduction

Being an important tool in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equations, various generalizations of Gronwall inequalities [1, 2] and their applications have attracted great interests of many mathematicians [3–5]. Some recent works can be found in [6–28].

In 1981, Henry [12] discussed the following linear singular integral inequality:

In 2007, Ye et al. [18] discussed linear singular integral inequality

In 2014, Cheng et al. [28] discussed the following inequalities:

On the other hand, difference inequalities which give explicit bounds on unknown functions provide a very useful and important tool in the study of many qualitative as well as quantitative properties of solutions of nonlinear difference equations. More attentions are paid to some discrete versions of Gronwall-Bellman type inequalities (such as [29–50]).

In 2002, Pachpatte [36] discussed the following difference inequality:

In 2010, Ma [45] discussed the following difference inequality with two variables:

In 2014, Huang at el. [50] discussed the following linear singular difference inequality:

Motivated by the results given in [6, 11, 28, 36, 45, 49, 50], in this paper, we discuss the following inequalities:

2. Difference Inequalities with Two Variables

Throughout this paper, let , , and . For a function , its first-order difference is defined by . Obviously, the linear difference equation with the initial condition has the solution . For convenience, in the sequel, we complementarily define that .

Lemma 1. Assume that , , , and are nonnegative functions on . If and satisfies the following difference inequality:

then

Proof. Since is a constant. Let . From (10), we have

Since is nonnegative, we have

Let and in (13) and substituting and , successively, we obtain

From (14), we have

where . Substituting inequality (15) into (13), we get the explicit estimation (11) for .

Theorem 2. Assume that , , , , , and are nonnegative functions on and , , and are nondecreasing in both and . If

and satisfies the difference inequality (7), then

for all .

Proof. Fixing any arbitrary , from (7), we have

for all , where , , and are nondecreasing in both and .

Define a function by the right side of (18); that is,

for all . Obviously, we have

Using the difference formula and relation (20), from (21), we have

where we have used the monotonicity of in . From (22), we observe that

On the other hand, by the mean-value theorem for integrals, for arbitrarily given integers , with , , there exists in the open interval such that

From (23) and (24), we have

Let and in (25), and substituting and , successively, we obtain

It implies that

Using (20) and (21), from (27), we have

for all . Taking and in (28), we have

Since , are chosen arbitrarily, we replace and in (29) with and , respectively, and obtain that

for all . Applying the result of Lemma 1 to inequality (30), we obtain desired estimation (17).

Lemma 3 (see [39]). Let , , and . Then,

Theorem 4. Assume that , , , , , and are defined as in Theorem 2 and that and . If

and satisfies difference inequality (8), then

for all , where

and are arbitrary constants.

Proof. Define a function by

for all . Then, from (8), we have

Applying Lemma 3 to (38), we obtain

for all . Substituting (39) into (37), we obtain

for all , where , , and , are defined by (34), (35), and (36), respectively. Since , , and are nonnegative and nondecreasing in both and and by (34), (35), and (36), , , and are also nonnegative and nondecreasing in both and . Using Theorem 2, from (40), we obtain

for all . Substituting (41) into (38), we get our required estimation (33) of unknown function in (8).

3. Difference Inequality with Weakly Singular

For the reader’s convenience, we present some necessary Lemmas.

Lemma 5 (discrete Jensen inequality [47]). Let be nonnegative real numbers, a real number, and a natural number. Then,

Lemma 6 (discrete Hölder inequality [48]). Let    be nonnegative real numbers and , positive numbers such that . Then,

Lemma 7 (see [15, 49]). Let , , and . If , , and , then

where and is the well-known -function.

Now, we consider the weakly singular difference inequality (9).

Theorem 8. Let , , , , and . Assume that , , , , , , , and are nonnegative functions on and , , and are nondecreasing. If

and satisfies (9), then

where

and , , and , are arbitrary constants.

Proof. Applying Lemma 6 with , to (8), we obtain that

for all , , where is used. Applying Lemma 5 to (48), we have

for all , . By discrete Jensen inequality (42) with , , from (49), we obtain that

Applying Theorem 4 to (50), we have

This is our required estimation (46) of unknown function in (9).

4. Applications

In this section, we apply our results to discuss the boundedness of solutions of an iterative difference equation with a weakly singular kernel.

Example 9. Suppose that satisfies the difference equation

where , , , , , , , , and are nonnegative functions on , and , , and are nondecreasing. From (52), we have

Let , , and , are arbitrary constants, and

If

Applying Theorem 8 to (53), we obtain the estimation of the solutions of difference equation (52)

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Project no. 11161018), the Guangxi Natural Science Foundation of China (Projects nos. 2012GXNSFAA053009, 2013GXNSFAA019022), and the Scientific Research Foundation of the Education Department of Guangxi Autonomous Region (no. 2013YB243).

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