Boundary value problems for a coupled system of secondorder nonlinear difference equations
 Jianpeng Tan^{1} and
 Zhan Zhou^{1, 2}Email author
https://doi.org/10.1186/s1366201712574
© The Author(s) 2017
Received: 19 April 2017
Accepted: 28 June 2017
Published: 28 July 2017
Abstract
We discuss the existence of nontrivial solutions to the boundary value problems for a coupled system of secondorder nonlinear difference equations by using the critical point theory. The nontrivial solutions where neither of the components is identically zero are achieved under some sufficient conditions.
Keywords
boundary value problem coupled system nonlinear difference equation critical point theory1 Introduction
Put \(\mathbb{N^{+}}\), \(\mathbb{Z}\), \(\mathbb{R}\) to be the sets of positive integers, integers and real numbers, respectively. For \(a,b \in\mathbb{Z}\), define \(\mathbb{Z}(a,b)=\{ a,a+1,\ldots,b\}\) when \(a\leq b\).
Boundary value problems for a coupled system of nonlinear differential equations have been the subject of many investigations [1–7]. Among the main methods are the Schauder fixed point theorem, the Banach fixed point theorem, the synchronization manifold approach and the coincidence degree theory. However, the results on boundary value problems for a coupled system of secondorder nonlinear difference equations are relatively rare. The critical point theory is a strong tool to study the periodic solutions [8–10], the homoclinic solutions [11, 12] and the boundary value problems [13, 14] for difference equations. Recently, Bonanno et al. [15] developed a new approach to discuss the boundary value problems for a secondorder difference equation by using critical point theory. Motivated by [15], we try to use the variants of the mountain pass theorem, the local minimum theorem and its variant to study system (1.1).
An outline of this paper is as follows. In Section 2, we establish the variational framework and introduce two variants of the mountain pass theorem, the local minimum theorem and its variant. Some notations will also be given. Then, in Section 3, the coerciveness and compactness of the variational functional are given by different assumptions of the coefficients. A two critical points theorem and a three critical points theorem are stated in Section 4. The existence of nontrivial solutions with both of the components being not zero to system (1.1) are then established in Section 5. Finally, as an application, an example is presented in Section 6.
2 Preliminaries
In this section, we define some notations and give the variational functional of system (1.1). Some related fundamental theorems are presented at the end of this section.
A standard argument gives that the critical points of \(I(\phi,\psi)\) are the solutions of system (1.1).
The following definition and lemma are taken from [21, 22].
Definition 2.1
The Gâteaux differentiable function I satisfies the PalaisSmale condition (PS) if every sequence \(\{x_{j}\}\) such that \(I(x_{j})\) is bounded and \(I'(x_{j})\rightarrow0\) for \(j\rightarrow\infty\) contains a convergent subsequence.
Lemma 2.1
Let E be a finite dimensional Banach space. Suppose that \(I:E\rightarrow\mathbb{R}\) is lower semicontinuous and coercive. Then I admits a global minimum.
We state two variants of the mountain pass theorem (see [23], Theorem 2.2) due to [15], Corollary 3.2, and [24], Theorem 1.1, Chapter II.
Lemma 2.2
Let E be a real finite dimensional Banach space. Suppose that \(I:E\rightarrow\mathbb{R}\) is continuously Gâteaux differentiable, unbounded from below and satisfies (PS). Further suppose that I possesses a local minimum \(x_{1}\). Then I possesses a distinct second critical point.
Lemma 2.3
Let E be a real finite dimensional Banach space. Suppose that \(I:E\rightarrow\mathbb{R}\) is continuously Gâteaux differentiable and coercive. Further suppose that I possesses two distinct local minima \(x_{1}\) and \(x_{2}\). Then I possesses a third critical point \(x_{3}\) which is distinct from \(x_{1}\) and \(x_{2}\).
The local minimum theorem and its variant are presented below.
Lemma 2.4
Local minimum theorem
Proof
Lemma 2.5
Proof
As stated in Lemma 2.4, we obtain that \(I(\phi^{*},\psi ^{*})=\min_{A^{1}([0,r])}I(\phi,\psi)\) holds. If \(A(\phi^{*},\psi ^{*})=0\), then \((\phi^{*},\psi^{*})=(0,0)\). It follows from \(1<\frac {B(w,w)}{A(w,w)}\) that \(I(w,w)<0=I(\phi^{*},\psi^{*})\), which leads to a contradiction. If \(A(\phi^{*},\psi^{*})=r\), from \(\frac{\sup_{A^{1}([0,r])}B(\phi,\psi)}{r}<1\), we have \(\frac{B(\phi^{*},\psi ^{*})}{r}<1\), that is, \(\frac{B(\phi^{*},\psi^{*})}{A(\phi^{*},\psi ^{*})}<1\), so \(I(0,0)=0< I(\phi^{*},\psi^{*})\). Again, this leads to a contradiction. Hence, \((\phi^{*},\psi^{*})\in A^{1}(]0,r[)\). Lemma 2.5 is proved. □
3 Compactness and coerciveness of the variational functional
We point out the following two lemmas which will be used in the next section.
Lemma 3.1
 \((J_{1})\) :

\(a_{1}>0\), \(a_{2}>0\), \(a_{3}>0\) or \(a_{3}^{2}< a_{1}a_{2}\) when \(a_{3}<0\).
Proof
Lemma 3.2
 \((J_{2})\) :

\(a_{1}<0\), \(a_{2}<0\), \(a_{3}<0\) or \(a_{3}^{2}< a_{1}a_{2}\) when \(a_{3}>0\).
Proof
4 Multiple critical points theorems
Two consequences of the local minimum theorem were discussed in [15] (see [15], Section 4). In this section, motivated by [15], we state two consequences of Lemma 2.4 as follows. The first one is a two critical points theorem and the second one is a three critical points theorem.
Theorem 4.1
Proof
Taking into consideration that \((J_{1})\) holds, from Lemma 3.1 we obtain that \(I(\phi,\psi)\) satisfies the (PS) condition and it is unbounded from below. Notice that \(\frac{\sup_{A^{1}([0,r])}B(\phi,\psi)}{r}<1\), it follows from Lemma 2.4 that \(I(\phi,\psi)\) admits a local minimum. \(I(\phi,\psi)\) satisfies the conditions in Lemma 2.2, then \(I(\phi,\psi )\) admits a distinct second critical point. Hence, the proof is completed. □
Theorem 4.2
Proof
From inequality (4.1), one has \(A(\phi_{2},\psi_{2})B(\phi _{2},\psi_{2})\leq A(\phi,\psi)B(\phi,\psi)\) for all \((\phi,\psi )\in A^{1}(]r,+\infty[)\). Since \(A^{1}(]r,+\infty[)\) is an open set, we obtain that \((\phi_{2},\psi_{2})\) is a local minimum of \(I(\phi,\psi)\).
To conclude, \(I(\phi,\psi)\) has a local minimum \((\phi_{1},\psi _{1})\) such that \(A(\phi_{1},\psi_{1})< r\) and a local minimum \((\phi _{2},\psi_{2})\) such that \(A(\phi_{2},\psi_{2})>r\). According to Lemma 2.3, the statement in Theorem 4.2 is proved. □
5 Existence of nontrivial solutions
In this section, we establish our main results. The existence of four nontrivial solutions where both of the components are not zero to system (1.1) is ensured by some sufficient conditions.
To prove the main results, we need the following four lemmas.
Lemma 5.1
 \((J_{3})\) :

\(a_{1}>0\) and there exists a constant \(c>0\) such that$$ \omega_{1}\min_{k\in\mathbb{Z}(1,N)}b_{1k}+a_{1}c^{2} \leq0. $$
Proof
This leads to a contradiction. This completes the proof. □
Lemma 5.2
 \((J_{4})\) :

\(a_{2}>0\) and there exists a constant \(c>0\) such that$$ \omega_{2}\min_{k\in\mathbb{Z}(1,N)}b_{2k}+a_{2}c^{2} \leq0. $$
Proof
The proof of this lemma is analogous to that in Lemma 5.1 and so is omitted. □
Lemma 5.3
 \((J_{5})\) :

\(a_{1}<0\), \(\min_{k\in\mathbb{Z}(1,N)}b_{1k}\omega _{1}\leq0\) and there exists a constant \(c>0\) such that$$ \sqrt{\frac{N(\min_{k\in\mathbb {Z}(1,N)}b_{1k}\omega_{1})}{a_{1}}}\leq\frac{c}{\sqrt{2N\lambda_{N}}K_{2}}. $$
Proof
This leads to a contradiction. We have thus proved the lemma. □
Lemma 5.4
 \((J_{6})\) :

\(a_{2}<0\), \(\min_{k\in\mathbb{Z}(1,N)}b_{2k}\omega _{2}\leq0\) and there exists a constant \(c>0\) such that$$ \sqrt{\frac{N(\min_{k\in\mathbb {Z}(1,N)}b_{2k}\omega_{2})}{a_{2}}}\leq\frac{c}{\sqrt{2N\lambda_{N}}K_{2}}. $$
Proof
The proof of this lemma is quite similar to Lemma 5.3 and so is omitted. □
Theorem 5.1
Proof
Furthermore, if \((J_{3})\) and \((J_{4})\) hold, we claim that system (1.1) has at least one nontrivial solution \((\phi^{*},\psi ^{*})\) with \(\phi^{*}\neq0\) and \(\psi^{*}\neq0\) such that \(\Vert (\phi^{*},\psi^{*}) \Vert _{\infty}< c\). For the sake of contradiction, assume that \((\phi^{*},0)\) is a nontrivial solution of system (1.1), that is to say, \(\phi^{*}\) is a nontrivial solution of the boundary value problem (5.1) such that \(\Vert \phi^{*} \Vert _{\infty}< c\), this is contrary to the conclusion of Lemma 5.1. Similarly, we can show that \((0,\psi^{*})\) is not a nontrivial solution of system (1.1). Our claim is proved.
Apparently, \((\phi^{*},\psi^{*})\), \((\phi^{*},\psi^{*})\) and \((\phi^{*},\psi^{*})\) also satisfy system (1.1). Hence, the statements are proved. □
Theorem 5.2
Proof
Theorem 5.3
Proof
If \((J_{5})\) and \((J_{6})\) hold, according to Lemma 5.3 and Lemma 5.4, we assert that system (1.1) has at least one nontrivial solution \((\phi^{**},\psi^{**})\) with \(\phi^{**}\neq0\) and \(\psi^{**}\neq0\) such that \(\Vert (\phi ^{**},\psi^{**}) \Vert _{\infty}>\frac{c}{\sqrt{2N\lambda _{N}}K_{2}}\). Arguing by contradiction, suppose that \((\phi^{**},0)\) is a nontrivial solution of system (1.1), that is to say, \(\phi^{**}\) is a nontrivial solution of the boundary value problem (5.1) such that \(\Vert \phi^{**} \Vert _{\infty}>\frac{c}{\sqrt{2N\lambda_{N}}K_{2}}\), this is contrary to the conclusion of Lemma 5.3. Similarly, we can show that \((0,\psi^{**})\) is not a nontrivial solution of system (1.1). Our assertion is proved.
It is obvious that \((\phi^{**},\psi^{**})\), \((\phi^{**},\psi ^{**})\) and \((\phi^{**},\psi^{**})\) also satisfy system (1.1). This completes the proof. □
6 Application
Example 6.1
Declarations
Acknowledgements
The authors would like to thank the referees for their comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 11726010, No. 11571084) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT_16R16).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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