#### 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).