Journal of Applied Mathematics

Volume 2014 (2014), Article ID 763761, 9 pages

http://dx.doi.org/10.1155/2014/763761

Research Article

Estimation of Unknown Functions of Iterative Difference Inequalities and Their Applications

Ricai Luo,¹ Wu-Sheng Wang,² and Honglei Xu³

¹School of Computer and Information Engineering, Hechi University, Guangxi, Yizhou 546300, China

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

³Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia

Received 28 March 2014; Accepted 10 July 2014; Published 22 July 2014

Academic Editor: Youjun Deng

Copyright © 2014 Ricai Luo 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 establish some new nonlinear retarded finite difference inequalities. The results that we propose here can be used as tools in the theory of certain new classes of finite difference equations in various difference situations. We also give applications of the inequalities to show the usefulness of our results.

1. Introduction

An integral inequality that provides an explicit bound to the unknown function furnishes a handy tool to investigate qualitative properties of solutions of differential and integral equations. One of the best known and widely used inequalities in the study of nonlinear differential equations is Gronwall-Bellman inequality [1, 2], which can be stated as follows. If and are nonnegative continuous functions on an interval satisfying

for some constant , then

Being an important tool in the study of 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 [6–12] and the references therein. Along with the development of the theory of integral inequalities and the theory of difference equations, more and more attentions are paid to discrete versions of Gronwall-type inequalities; see [13–36] and the references cited therein.

Sugiyama [13] established the most precise and complete discrete analogue of the Gronwall inequality [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 [15] established some 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

Lemma 1 (see [16]). Suppose that is a nonnegative constant and , , , , and are nonnegative functions defined on , for all . If satisfies the inequality

then

where

in which

and for all .

Lemma 2 (see [14, 18]). 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 [18, 19] also established some difference inequalities of product form as follows. Let 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 (7), and is a solution of

Let 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 (7), and is a solution of the difference equation

Motivated by the results given in [16, 18, 19], in this paper, we discuss new nonlinear finite difference inequalities:

Our inequalities can be used as tools in the study of certain classes of finite difference equations. We also present some immediate applications to show the importance of our results to study the various problems in the theory of finite difference equations.

2. Main Results

Throughout this paper, let , . 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 constant, positive functions defined on , a monotone increasing function, and a monotone decreasing function. Let be a nonnegative function defined on such that

(i)Suppose . If , then

(ii)Suppose . Then

Proof. (i) We apply mean value theorem for differentiation to the function

and then there exists between and such that

Because is monotone increasing and is monotone decreasing and , we see that and . So for all values of between and we have

From (22) and (27), we have

Since , from (26) and (28) we have

Taking in (29) and summing up over from to , we obtain

From (30), we obtain our required estimation (23).

(ii) Now by following the same steps as in the proof of (i) before (29) we have

because . Taking in (31) and summing up over from to , we obtain

From (32), we obtain our required estimation (24).

Theorem 4. Let be a positive and monotone nondecreasing function defined on and nonnegative functions defined on . If satisfies

then

where

in which

and for all .

Proof. Fix , where is chosen arbitrarily, since is a nonnegative and monotone nondecreasing function, from (33), we have

Define a function by the right-hand side of (37). Then is a positive and monotone nondecreasing function defined on . We have

Using the definitions of the operator and , we obtain

Let

Then

It follows that

Adding to both sides of the above inequality we have

Put

and then , and

We see that the inequality

Define a function

Multiplying by to both sides of (46) we obtain

Let , , , and . Because is monotone increasing, is monotone decreasing and ; applying Theorem 3 to (48) we obtain

where , are used. Define a function of the right-hand side of (49). Substituting (49) in (43) we obtain

Performing the same derivation as in (46)–(49), we obtain from (50) that

Define a function of the right-hand side of (51). Substituting (51) in (39) we obtain

Using (38), from (52) it follows that

Since is arbitrary, from (53), we get the required estimate (35).

Theorem 5. Let be nonnegative functions defined for and a nonnegative constant. Let be a real-valued function defined for , , and monotone nondecreasing with respect to for any fixed . If satisfies (21), then

where

in which

and is a solution of the difference equation

where

in which

and for all .

Proof. We first assume that and define a function by the right-hand side of (21). Then is a positive and monotone nondecreasing function defined on . We have

Using the definitions of the operator and , we obtain

From (61) it follows that the inequality

holds for all . Setting in (62) and substituting , successively, we get

Define a function by

Then and

Using (64), the inequality (63) can be written as

Since is positive and monotone nondecreasing for , satisfy the conditions in Theorem 4. Now an application of Theorem 4 to (66) yields

where

in which

Since is monotone nondecreasing with respect to for any fixed , from (65) and (67), we have

Now as a suitable application of Lemma 2, we obtain

where is a solution of (57). Using (60), (67), and (71), we obtain our required estimation (54).

If is nonnegative, we can carry out the above procedure with instead of where is an arbitrary small number. Letting , we obtain (54).

3. Application to Finite Difference Equations

In this section, we consider the following difference equation:

where , , are real-valued functions defined, respectively, on , , , is as defined in Theorem 5, and is a constant. We assume that

where , , are as defined in Theorem 5. Using the definitions of the operator , from (72), we see that the inequality

holds for all . It follows that

From (76), we have

Using the conditions (74), we obtain

Now an application of Theorem 5 to (78) yields the estimation of the difference equation (72), that is,

where

in which

and is a solution of the difference equation

where

in which

and for all .

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. 11161018,11171079), the NSF of Guangxi Zhuang Autonomous Region (no. 2012GXNSFAA053009), and the SRF of the Education Department of Guangxi Zhuang Autonomous Region (nos. 201106LX599 and 201106LX591).

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