Abstract and Applied Analysis

Volume 2014 (2014), Article ID 158375, 6 pages

http://dx.doi.org/10.1155/2014/158375

Research Article

Some Retarded Difference Inequalities of Product Form and Their Application

Zongyi Hou,¹ Shanhe Wu,² and Wu-Sheng Wang¹

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

²Department of Mathematics and Computer Science, Longyan University, Longyan, Fujian 364012, China

Received 18 February 2014; Accepted 8 July 2014; Published 22 July 2014

Academic Editor: Weinian Zhang

Copyright © 2014 Zongyi Hou 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.

A class of new nonlinear retarded difference inequalities is established. An application of the obtained inequalities to the estimation of finite difference equations is given.

1. Introduction

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. Various investigators have discovered many useful and new difference inequalities, mainly inspired by their applications in various branches of difference equations; see [1–25] and the references cited therein.

Sugiyama [2] established the most precise and complete discrete analogue of the Gronwall inequality (see [1]) in the following form.

Let and be nonnegative functions defined for , and suppose that for every . If

where is the set of points , is a given integer, and is a nonnegative constant, then

Pachpatte [4] established a generalized discrete analogue of the Gronwall inequality in the following form.

Let be a positive and monotone nondecreasing function on , and let , be nonnegative functions on . If satisfies

then

where

Besides the results mentioned above, the following results are closely related to the investigation of the present paper, and, particularly, they will be used as lemmas for the proofs of our main results in Theorems 3 and 4.

Lemma 1 (see [5]). Let , , , , and be nonnegative functions defined on , for which the inequality

holds, where is a nonnegative constant and . If and for all , then

Lemma 2 (see [3, 7]). Let be a real-valued function defined for and monotone nondecreasing with respect to for any fixed . Let be a real-valued function defined for such that

Let be a solution of

such that . Then

Pachpatte [7, 8] also established some difference inequalities of product form as follows.

Let , and be nonnegative functions defined on and let be a nonnegative constant. Let be a nonnegative function defined for , and monotone nondecreasing with respect to for any fixed . If satisfies

then

where is defined by (5) and is a solution of

Let , and be nonnegative functions defined for and let be a nonnegative constant. Let be a nonnegative function defined for , and monotone nondecreasing with respect to for any fixed . If satisfies

then

where is defined by (5) and is a solution of the difference equation

Motivated by the results given in [5, 7, 8], in this paper, we discuss new nonlinear difference inequalities:

It is important to note that the inequality given above can be used as tools in the study of certain classes of finite difference equations. In Section 3 we provide an application of our results to the estimation of finite difference equations.

2. Main Results

Throughout this paper, let and , . For function , , we define the operator by . Obviously, the linear difference equation with the initial condition has the solution . For convenience, in the sequel we complementarily define that and

Theorem 3. Let be a nonnegative and monotone nondecreasing function defined on , and let , , and be nonnegative functions defined on . Let be a constant with . If satisfies

then

where

Proof. Fix , where is chosen arbitrarily; since   is a nonnegative and monotone nondecreasing function, from (18), we have

Now an application of Lemma 1 to (21) yields

Since is arbitrary, from (22), we get the required estimate (19).

Theorem 4. Let ,and be nonnegative functions defined for and let be a nonnegative constant. Let be a real-valued function defined for , and monotone nondecreasing with respect to for any fixed . Let be a constant with . If satisfies (17), then

where is defined by (20) in Theorem 3 and is a solution of the difference equation

Proof. We first assume that and define a function by the right-hand side of (17). Then is a nonnegative and monotone nondecreasing function defined on . We have

Using the definitions of the operator and , we obtain

From (26), we have

Setting in (27) and substituting , successively, we get

Define a function by

Then and

Using (29), inequality (28) can be written as

since is positive and monotone nondecreasing for and , and satisfy the conditions in Theorem 3. Now an application of Theorem 3 to (31) yields

where is defined by (20) in Theorem 3. Since is monotone nondecreasing with respect to for any fixed , from (30) and (32), we have

Now with a suitable application of Lemma 2, we obtain

where is a solution of (24). Using (25), (32), and (34), we obtain our required estimation (23).

If is nonnegative, we can carry out the above procedure with instead of , where is an arbitrary small number. Letting , we obtain (23).

3. Application to Finite Difference Equations

In this section, we apply our result to the following difference equation:

where , and are real-valued functions defined, respectively, on , , and and is as defined in Theorem 4. We assume that

where , and are as defined in Theorem 4. From (35), we have

From (37), we have

Using conditions (36), we obtain

Now an application of Theorem 4 to (39) yields the estimation of the difference equation (35) as follows:

where is a solution of the difference equation

Conflict of Interests

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

Acknowledgments

The present investigation was supported by the National Natural Science Foundation of China (no. 11161018), the Foundation of Scientific Research Project of Fujian Province Education Department (no. JK2012049), the Guangxi Natural Science Foundation (no. 2012GXNSFAA053009), and the Scientific Research Foundation of the Education Department of Guangxi Zhuang Autonomous Region (no. 201106LX599 and no. 201204LX391). The authors are grateful to the anonymous referees for their careful comments and valuable suggestions on this paper.

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