Nontrivial Solutions for a System of Second-Order Discrete Boundary Value Problems

Abstract

In this work, we shall study the existence of nontrivial solutions for a system of second-order discrete boundary value problems. Under some conditions concerning the eigenvalues of relevant linear operator, we use the topological degree theory to obtain our main results.

1. Introduction

Nonlinear discrete problems appear in many mathematical models, such as computer science, mechanical engineering, control systems, economics, and fluid mechanics (see [1–4]). Owing to the wide applications, in recent years, there are a large number of researchers paying special attention in this direction (we refer to some results [5–15] and the references therein). For example, in [5], the authors used the Guo–Krasnosel’skii fixed point theorem to study the existence of positive solutions for the following second-order discrete boundary value problem:and the following discrete second-order system:where is a positive integer, is the forward difference operator, i.e., , and .

In [6], the authors used the monotone iterative technique to investigate the existence and uniqueness of positive solutions for the following discrete -Laplacian fractional boundary value problem:where is a real number, is a discrete fractional operator, and is the -Laplacian with .

Coupled systems of discrete problems have also been investigated by many authors; some results can be found in a series of papers [11–15] and the references cited therein (also see some results on differential systems [16–24]). For example, in [11], the authors used the Guo–Krasnosel’skii fixed point theorem to study the following systems of three-point discrete boundary value problems:where , . They offered some values for the parameters to yield a positive solution for the above system.

In [12], the authors used the fixed point index to study the positive solutions for the following system of first-order discrete fractional boundary value problems:

By discrete Jensen’s inequality, the authors adopted some appropriate nonnegative concave and convex functions to characterize the coupling behavior of the nonlinearities .

Motivated by the aforementioned works, in this paper, by means of the topological degree theory, we study the existence of nontrivial solutions for the following system of second-order discrete boundary value problems:where is a fixed positive integer number, are continuous and satisfy the following conditions: (H1) There exist three nonnegative functions and on such that where . (H2) , . (H3) , uniformly on , where . (H4) , uniformly on .

Now, we state our main result here.

Theorem 1. Suppose that (H1)–(H4) hold. Then, (6) has at least one nontrivial solution.

2. Preliminaries

Let be the Banach space of real valued functions defined on the discrete interval with the norm , where . Define the following sets:and for . Then, are cones on , and is an open ball in .

Lemma 1 (see [11, 15]). Let . Then, the discrete boundary value problemhas a solution with the formwhere

Furthermore, has the following properties (see [13, 15]):(i) and for .(ii), for .

By Lemma 1, system (6) is equivalent to

Then, we can define operators byand operator by

Note that are completely continuous operators (see [11]), and solves (6) if and only if is a fixed point of the operator .

Lemma 2 (see [7, 15]). Let . Then, .
Define a linear operator as follows:

Then, we haveand we have the following lemma.

Lemma 3. If , then .

This is a direct result by Lemma 1 (ii), so we omit the proof.

Remark 1. in Lemma 2.

Lemma 4 (see [25, Theorem A.3.3]). Let be a bounded open set in a Banach space and be a continuous compact operator. If there exists such thatthen the topological degree .

Lemma 5 (see [25, Lemma 2.5.1]). Let be a bounded open set in a Banach space with and be a continuous compact operator. Ifthen the topological degree .

3. Main Results

In order to obtain the Proof of Theorem 1, we first provide a lemma.

Lemma 6. There exists a sufficiently large such that

Proof. By (H3), there exist and such thatNote that when , the functions and are bounded, so we can choose some appropriate positive numbers such thatwhereFrom (H2), for any given with , there is such thatLet and Then,Thus, we haveNote that can be chosen arbitrarily small, so we can letwhereNow, we provewhere . We argue this claim by indirection. Suppose that there exist such thatThen by Lemma 3, , and we also haveUsing (24) and (25), we haveSo, from Lemma 3 and Remark 1, we haveNote that , and using (24), , we haveIt is noted that , and . Therefore, we getUsing , we haveand implies thatConsequently, we obtainAs a result, we getIn view of (31) and (32), we seeDefine . Then, , and . From , we obtainHence,which contradicts the definition of . Therefore, (29) holds, and from Lemma 4, we obtainThis completes the proof.

Proof of Theorem 1. From (H4), there exist and such thatThis implies thatConsequently, we haveNow, we prove thatfor all and . We argue by contradiction. Suppose that there exist and such thatTherefore,Hence, we haveFrom Lemma 1 (i) and Lemma 2, we haveMultiplying both sides of (53) by , then summing from 1 to , and using (54), we obtainThis implies thatBecause for , we have . This contradicts . Therefore, (50) holds, and Lemma 5 implies thatCombining this with Lemma 6, we haveTherefore, the operator has at least one fixed point in , and (6) has at least one nontrivial solution. This completes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

This study was carried out in collaboration among all authors. All authors read and approved the final manuscript.

Acknowledgments

This study was supported by the project of National Social Science Fund of China (NSSFC) (18BTY015) and Shandong Province Higher Educational Science and Technology Program (J16LI01).