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Journal of Applied Mathematics
Volume 2017, Article ID 1474052, 14 pages
https://doi.org/10.1155/2017/1474052
Research Article

Some New Volterra-Fredholm-Type Nonlinear Discrete Inequalities with Two Variables Involving Iterated Sums and Their Applications

Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China

Correspondence should be addressed to Run Xu; moc.361@5002_nurux

Received 27 April 2017; Accepted 13 August 2017; Published 27 September 2017

Academic Editor: Samir H. Saker

Copyright © 2017 Run Xu. 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

Some generalized discrete Volterra-Fredholm-type inequalities were developed, which can be used as effective tools in the qualitative analysis of the solution to difference equations.

1. Introduction

In recent years, various forms of inequalities played increasingly important roles in the study of quantitative properties of solutions of differential and integral equations [115]. Discrete inequalities, especially the discrete Volterra-Fredholm-type inequalities, have been applied to study the discrete equations widely. For example, see [13, 911] and the references therein. In this paper, some new Volterra-Fredholm-type discrete inequalities involving four iterated infinite sums were established. Furthermore, to illustrate the usefulness of the established results, some examples were provided for the studying of their solutions on the boundedness, uniqueness, and continuous dependence.

We design the needed symbols as follows:(a) denotes the set of nonnegative integers and denotes the set of integers, while denotes the set of real numbers .(b)Let , where , and are two constants.(c) are all constants, and are two constants.(d)If is a lattice, then we denote the set of all valued functions on by and denote the set of all valued functions on by .(e)For a function , we have provided .

We need the following lemmas in the discussions of our main results.

Lemma 1 (see [4]). Let be nondecreasing in the third variable; is a constant. For , if then

Lemma 2 (see [4]). Let . If is nondecreasing in the first variable, then, for , then

Lemma 3 (see [5]). Let , and ; then, for any ,

2. Main Results

Theorem 4. Suppose that , , , , , and are nonnegative constants with , , , and being nondecreasing in the last two variables, and are also nondecreasing. Ifthen, for , we haveprovided that , where

Proof. Given , for , we haveDefine a function byThenorBy using Lemma 3, for any , we havewhereand is defined in (11). Then, using that is nondecreasing in every variable, we getwhere is defined in (10).
Since is nondecreasing and are nondecreasing in the last two variables, then is also nondecreasing in the last two variables, and, by Lemma 1 and (19), we getwhere is defined in (9). Considering the definition of and (20), we havewhere is defined in (12). Then,Combining (20) and (22), we deducewhere are defined in (9) and (12).
Then, combining (16) and (23), we obtain the desired result.

Corollary 5. Let , and be nondecreasing in every variable. are defined as in Theorem 4. Ifthen, for , we haveprovided that , where

The proof of Corollary 5 can be completed by setting , in Theorem 4.

Letting , we get the following corollary.

Corollary 6. Let ,, , be defined as in Theorem 4. Ifthen, for , we haveprovided that , where

Theorem 7. Let , , , , , , , be defined as in Theorem 4. Assume that is nondecreasing in the first variable. Ifthen, for , we haveprovided that , where

Proof. Given , for , we haveDefine function byThen,Clearly is nondecreasing in the first variable. Then, by Lemma 2, we getwhere is defined in (32). Define functionFrom (40), we getThen (42) becomesBy (45) and Lemma 3, from (43), we havewhere and are defined in (34) and (37), respectively.
Similar to the process of (17)–(23), we deduce thatwhere are defined in (35) and (38).
Combining (45) and (48), we get the desired result.

Theorem 8. Let , , , , , , be defined as in Theorem 4. satisfies for . Ifthen, for , we haveprovided that , where

Proof. Given , for , we haveDefine function by