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

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 [1–15]. Discrete inequalities, especially the discrete Volterra-Fredholm-type inequalities, have been applied to study the discrete equations widely. For example, see [1–3, 9–11] 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 byThenorBy Lemma 3, we havewhereand are defined in (54)–(56).
Similar to the process of (17)–(23), we getwhere are defined in (52) and (57).
Combining (61) and (64), we get the desired result.

Theorem 9. Let , , , , , , be defined as in Theorem 4. Assume that is nondecreasing in the first variable. are defined as in Theorem 7. Ifthen, for , we haveprovided that , where

The proof for Theorem 9 is similar to the combination of Theorems 7 and 8, and we omit the details here.

3. Applications

In this section, we will present some applications for the established results to study boundedness, uniqueness, and continuous dependence of solutions of certain difference equations.

Consider the following Volterra-Fredholm sum-difference equations:where is an odd number, .

Theorem 10. Assume that functions in equation (68) satisfy the following conditions:for , where are nonnegative constants satisfying which are nondecreasing in the last two variables; then one hasprovided that , where

Proof. Using conditions (69) to (68), we haveThen a suitable application of Theorem 4 (with ) to (72) yields the desired result.
The following theorem deals with the uniqueness of the solutions of (68).

Theorem 11. Supposing thathold for , where are nondecreasing in the last two variables, then (68) has at most one solution.

Proof. Assume that are two solutions of (68). Then Treat as one variable, and a suitable application of Corollary 6 yields , which implies that . Since is an odd number, then we have , and the proof is complete.

Finally we study the continuous dependence of the solutions of (68) on functions . For this, we consider the following variation of (68):where and is an odd number.

Theorem 12. Consider (68) and (76). If hold for , where , and are nondecreasing in the last two variables, furthermore, for all solution of (76), the following conditions hold for : where is an arbitrary constant. Thenwhere , and for . That is, depends continuously on the functions .

Proof. Let and be solutions of (68) and (76), respectively. Then satisfies (68) and satisfies (76). Hence Treat as one variable, and a suitable application of Corollary 6 (with ) yields the desired result (79). Hence depends continuously on .

4. Conclusions

The author carried out some new Volterra-Fredholm-type discrete inequalities involving four iterated infinite sums and their corresponding applications. The results are more effective to qualitative analysis of solutions for sum-difference equations, such as the boundedness, uniqueness, and continuous dependence on solutions.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

Run Xu received a grant from National Science Foundation of China (11671227).