Abstract and Applied Analysis

Volume 2013 (2013), Article ID 804152, 9 pages

http://dx.doi.org/10.1155/2013/804152

Research Article

Some Difference Inequalities for Iterated Sums with Applications

Wu-Sheng Wang¹ and Shanhe Wu²

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

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

Received 4 September 2013; Revised 13 November 2013; Accepted 28 November 2013

Academic Editor: Jaume Giné

Copyright © 2013 Wu-Sheng Wang and Shanhe Wu. 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.

The main objective of this paper is to establish two new nonlinear sum-difference inequalities with multiple iterated sums. Under several practical assumptions, the inequalities are solved through rigorous analysis, and explicit bounds for the unknown functions are given clearly. These new inequalities can be used as handy tools in the study of the estimation of solutions of difference equations.

1. Introduction

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

It has become one of the very few classic and most influential results in the theory and applications of inequalities. Because of its fundamental importance, over the years, many generalizations and analogous results of (2) have been established, such as [3–14].

Among these references, Baĭnov and Simeonov [4, P. 107] considered the following interesting Gronwall-type inequality:

Kim [8] considered analogous Gronwall-type integral inequalities involving iterated integrals,

In 2011, Abdeldaim and Yakout [12] studied some new integral inequalities of Gronwall-Bellman-Pachpatte type such as

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; for detailed information, please refer to the literatures [15–35]. For instance, Pachpatte [19] considered the following discrete inequality:

In 2006, Cheung and Ren [24] studied

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

In 2012, Zhou et al. [33] studied the following inequalities:

However, the above results are not applicable to some certain inequalities with multiple iterated sums. Hence, it is desirable to consider more general difference inequalities of these extended types. They can be used in the study of certain classes of difference equations or applied in many practical engineering problems.

Motivated by the results given in [7, 8, 12, 19, 24, 25, 29, 33], in this paper we discuss the following two types of inequalities:

All the assumptions on (10) and (11) are given in the next sections. The inequalities (10) and (11) consist of multiple iterated sums. Under several practical assumptions, the inequalities are solved through rigorous analysis, and explicit bounds for the unknown functions are given clearly. Further, the derived results are applied to study the estimation of solutions of difference equations.

2. Main Results

Throughout this paper, let and . 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 .

Lemma 1. Let , and be real-valued nonnegative functions defined on and satisfy the inequality

where is a nonnegative constant. Then,

Proof. From (12), we have

Multiplying by on both sides of the above inequality (14) and summing up both sides from to , we obtain

From (15), we obtain the desired estimate (13).

Theorem 2. Let and be nonnegative functions defined on with nondecreasing on . Moreover, let , , be nonnegative functions for    and nondecreasing in for fixed . If and is not equal to 1, then the discrete inequality (10) gives

where can be successively determined from the formulas

for , ,

where is the inverse functions of , , is chosen arbitrarily, and is the largest natural number such that

Remark 3. Firstly, from (17) and (18), we obtain ; then let , and we get since is chosen arbitrarily.

Remark 4. We can obtain using MATLAB program: firstly let , when ; let , when ; let , and so on until . If , for all , then .

Proof. Fix , where is chosen arbitrarily and is defined by (20). For , from (10) we have

Now we introduce the functions

For and , then , , are all positive and nondecreasing functions on with , , and the inequalities (22) and (23) imply that

From (22), we observe that

We claim that

for , .

Now we prove (26) and (27) by induction. Obviously, (26) is true for by (26). We make the inductive assumption that (26) is true for . By the inductive assumption and (24), from (23) we obtain

It actually proves (26) by induction. From (23) and (26), we have

It proves (27). From (27), we have

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

By setting in (31) and substituting , successively, we obtain

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

Then, (32) is equivalent to

By the definition of , we obtain

From (34) and (35), we get

Once again, performing the similar procedure from (30) to (32), (36) gives

where is defined by (19). By combining (33), (34), and (37), we can obtain that

where is defined by (17). Applying Lemma 1 to (26) for , we have

where is defined by (18). From (24) and (39), we have

Since is arbitrary, from (40), we get the required estimate

where is defined by (20). Theorem 2 is proved.

Theorem 5. Let , , and be nonnegative functions defined on with and nondecreasing on , and let , , be nonnegative functions for , , which are nondecreasing in for fixed . Suppose that is a nondecreasing continuous function on with for . The inequality (11) implies that

where is the inverse function of ,

and is the largest natural number such that

Remark 6. We can obtain using Matlab program similar to Remark 4.

Proof. Let the function be positive. Fix , where is chosen arbitrarily and is defined by (45). For , from (11) we have

We denote the right-hand side of (46) by for . Then , the function is positive and nondecreasing in , , and

Define a function by

for all . From (47) and (48), we have

From (48), we have

From (50), we get

where

for all .

Continuing in this way, we obtain

where

From (54) and the inequality , we have

We define the functions , which are nondecreasing and continuously differentiable on with , on .

On the other hand, by the formula of partial integration, we have

By the monotonicity of , , from (56) we have

By the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that

From (55), (57), and (58), we have

Next, from the inequalities (53) and (59), we have

Once again, applying the same procedure from (56) to (59) to the inequality (60), we get

Proceeding in this way, we obtain

Using the inequalities (49) and (62), we have

where is defined by (44).

Once again, using the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that

where is defined by (43). Using (63), (64), and , we obtain

In (65), let ; we have

Due to the randomness of , (42) is achieved immediately from (66).

3. Application

In this section, we apply Theorem 5 to the following difference equation:

Corollary 7. Assume that is defined on , and there exist nonnegative functions , , such that

where the function is as in Theorem 5. If is any solution of the problem (67), then

where the functions , are as in Theorem 5,

and is the largest natural number such that

Proof. The difference equation (67) is equivalent to

Using (68), from (72), we have

Notice that, by the assumption, all functions in (73) satisfy the conditions of Theorem 5. Applying Theorem 5 to the inequality (73), (69) is immediately derived. This completes the proof of Corollary 7.

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgments

This research was supported, in part, by the National Natural Science Foundation of China under Grant 11161018 by the Guangxi Natural Science Foundation of China under Grant 2012GXNSFAA053009, and by the Natural Science Foundation of Fujian province of China under Grant 2012J01014. The authors are grateful to the anonymous referees for their careful comments and valuable suggestions on this paper.

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