Journal of Applied Mathematics

Volume 2014 (2014), Article ID 236965, 9 pages

http://dx.doi.org/10.1155/2014/236965

Research Article

A Class of Iterative Nonlinear Difference Inequality with Weakly Singularity

Chunmiao Huang,¹ Wu-Sheng Wang,¹ and Xiaoliang Zhou²

¹School of Mathematics and Statistics, Hechi University, Yizhou, Guangxi 546300, China

²Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524088, China

Received 27 March 2014; Accepted 28 April 2014; Published 8 May 2014

Academic Editor: Junjie Wei

Copyright © 2014 Chunmiao 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 new nonlinear weakly singular difference inequality, which is solved by change of variable, discrete Hölder inequality, discrete Jensen inequality, the mean-value theorem for integrals and amplification method, and Gamma function. Explicit bound for the unknown function is given clearly. Moreover, an example is presented to show the usefulness of our results.

1. Introduction

Being an important tool in the study of qualitative properties of solutions of differential equations and integral equations, various generalizations of Gronwall inequalities and their applications have attracted great interests of many mathematicians (such as [1–21]). Gronwall-Bellman inequality [22, 23] can be stated as follows: if and are nonnegative and continuous functions on an interval satisfying

for some constant , then

In 1981, Henry [2] discussed the following linear singular integral inequality:

In 2007, Ye et al. [20] discussed linear singular integral inequality:

In 2011, Abdeldaim and Yakout [21] studied a new integral inequality of Gronwall-Bellman-Pachpatte type

On the other hand, many physical problems arising in a wide variety of applications are governed by finite difference equations. The theory of difference equations has been developed as a natural discrete analogue of corresponding theory of differential equations. 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 (such as [24–32]). Sugiyama [26] established the most precise and complete discrete analogue of the Gronwall inequality in the following form:

For instance, Pachpatte [27] considered the following discrete inequality:

In 2006, Cheung and Ren [29] studied

Later, Zheng et al. [31] discussed the following discrete inequality:

Motivated by the results given in [2, 20, 21, 32], in this paper, we discuss a new linear singular integral inequality

where , , , and .

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

Lemma 1 (discrete Jensen inequality [28]). Let be nonnegative real numbers, is a real number, and is a natural number. Then

Lemma 2 (discrete Hölder inequality [30]). Let    be nonnegative real numbers, and let be positive numbers such that ; then

Lemma 3. Let , , , and . If , then

where , is the well-known -function.

Proof. By the definition of integration and the conditions in Lemma 3, we have

Using a change of variables and , we have the estimation

Since , , , and , from (14) and (15), we have the relation (13).

2. Main Result

In this section, we give the estimation of unknown function in (10). Let . For function , its 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 .

Theorem 4. Suppose that is a constant, , are nonnegative and nondecreasing functions defined on , are nonnegative, nondecreasing, and continuous functions defined on , , , , and . If satisfies (10), then

where

and , , , is the largest integer number such that

Proof. From (10), we have

Applying Lemma 2 with , to (24), we obtain that

where is used. Applying Lemma 3, we have

By discrete Jensen inequality (11) with , from (26) we obtain that

Again using discrete Jensen inequality (11) with , , from (27) we obtain that

For in (28), applying Lemma 2 with , , we obtain that

here Lemma 3 is used. Substituting (29) into (28), we have

where and are defined by (21) and (22), respectively. Let ; from (30) we have

Since , are nondecreasing functions, from (31) we have

where is chosen arbitrarily.

Let denote the function on the right-hand side of (32), which is a positive and nondecreasing function on . From (32), we have

Using and (33), we obtain

for all .

Let

Then

From (35), we have

It implies that, for all ,

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

for all , where is defined by (18). From (38) and (39), we have

Taking in (40) and summing up over from to , from (40) we obtain

Let denote the function on the right-hand side of (41), which is a positive and nondecreasing function on . From (41), we have

Using and (42), we obtain

From (43), we have

for all . Again by the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that

where is defined by (19). From (44) and (45), we have

Taking in (46) and summing up over from to , from (46) we obtain

for all . Let

Then

From (48) and (49), we have

Using the mean-value theorem for integrals, from (50) we have

where is defined by (20). From (36), (42), (49), and (51), we have

Using and (33), from (52) we obtain that

Since is chosen arbitrarily, from (53) we have

This is our required estimation (16) of unknown function in (10).

3. Application

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

Example 5. Suppose that satisfies the difference equation

where , , , and . Then we have

for all . Let , . From (18) to (20) we obtain that

Using Theorem 4, we get

which is an upper bound of in (55).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by National Natural Science Foundation of China (Project no. 11161018), the NSF of Guangxi Zhuang Autonomous Region (no. 2012GXNSFAA053009), the high school specialty and curriculum integration project of Guangxi Zhuang Autonomous Region (no. GXTSZY2220), the Science Innovation Project of Department of Education of Guangdong province (Grant 2013KJCX0125), and the NSF of Guangdong Province (no. s2013010013385). The authors are very grateful to the editor and the referees for their careful comments and valuable suggestions on this paper.

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