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Boundary value problems for a coupled system of secondorder nonlinear difference equations
Advances in Difference Equations volume 2017, Article number: 210 (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.
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 \(a\leq b\).
In this paper, we study the following boundary value problems for a coupled system of secondorder nonlinear difference equations:
for all \(k\in\mathbb{Z}(1,N)\), where \(N\in\mathbb{N^{+}}\), \(\omega _{1}\), \(\omega_{2} \in\mathbb{R}\), \(a_{i}\in\mathbb{R}\setminus\{0\} , i=1,2,3\), and \(\{b_{jk}\}\) is a real number sequence, \(j=1,2\). Δ is the forward difference operator defined by \(\Delta\phi (k)=\phi(k+1)\phi(k)\), \(\Delta^{2}\phi(k)=\Delta(\Delta\phi(k))\).
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).
System (1.1) arises from the discretization of the twocomponent system of timedependent nonlinear GrossPitaevskii system (see [16] for more detail) as
where \(k\in\mathbb{N^{+}}\). For the general background on a coupled discrete Schrödinger system, we refer to [17].
Huang and Zhou [18, 19] considered system (1.2), with a few differences in notation, and they studied the solutions to system (1.2) of the form
where \(k\in\mathbb{N^{+}}\).
By substituting (1.3) into (1.2), a routine calculation gives
where \(k\in\mathbb{N^{+}}\). Using the Nehari manifold approach, Huang and Zhou have established sufficient conditions on the existence of nontrivial homoclinic solutions that have both of the components not identically zero to system (1.4), in [18] and [19]. However, we find that no similar results were obtained in the literature for system (1.1). We noticed that most works on the existence of solutions to the boundary value problems are for difference equations, and less is known for coupled difference systems. If we let \(\phi(k)\equiv0\) or \(\psi(k)\equiv0\) in system (1.1), we get a boundary value problem for a single difference equation. Therefore, it is important that we get solutions of (1.1) where both of the components are not zero. Throughout this paper, we will show that the critical point theory is very useful to demonstrate the existence of solutions of 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.
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.
Let S be the Ndimensional Banach space as follows:
which is equipped with the norm
Define \(\Vert \cdot \Vert \), \(\Vert \cdot \Vert _{\infty}\) and \(\Vert (\cdot,\cdot) \Vert _{\infty}\) as
and
respectively.
Define a linear map \(L: S\rightarrow\mathbb{R}^{N}\) by
where \(\cdot^{T}\) denotes the transpose of ⋅.
Clearly,
where
The eigenvalues of matrix C (see, for instance, [15], Section 2, and [20], Section 2) are
one has
and
From [15], Proposition 2.1, we know that
where
Let
for all \(x,y\in\mathbb{R}\) and \(k\in\mathbb{Z}(1,N)\), obviously,
The functional \(I(\phi,\psi)\) is defined by
where
for all \((\phi,\psi)\in S\times S\).
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
Put \(r>0\) such that
Then the functional \(I(\phi,\psi)=A(\phi,\psi)B(\phi,\psi)\) has at least a local minimum \((\phi^{*},\psi^{*})\in S\times S\) such that \(A(\phi^{*},\psi^{*})< r\), \(I(\phi^{*},\psi^{*})\leq I(\phi,\psi)\) for all \((\phi,\psi)\in A^{1}([0,r])\) and \(I^{\prime}(\phi ^{*},\psi^{*})=0\).
Proof
First, we deal with the case in which \(\sup_{A^{1}([0,r])}B(\phi ,\psi)=0\). We find \(0\leq A(\phi,\psi)= A(\phi,\psi)\sup_{A^{1}([0,r])}B(u,v)\leq A(\phi,\psi)B(\phi,\psi)=I(\phi,\psi )\) for all \((\phi,\psi)\in A^{1}([0,r])\). It is easy to show that \((\phi^{*},\psi^{*})=(0,0)\) satisfies our conclusion. Now, suppose \(\sup_{A^{1}([0,r])}B(\phi, \psi)>0\). Remind that \(A(\phi,\psi)\) is continuous and coercive, therefore the set \(A^{1}([0,r])\) is closed and bounded. Hence, there exists \((\phi^{*},\psi^{*})\in A^{1}([0,r])\) such that
Lemma 2.4 will be proved if we can show that \((\phi^{*},\psi^{*})\in A^{1}([0,r[)\). For the sake of contradiction, suppose \(A(\phi^{*},\psi^{*})=r\). Using \(\frac{\sup_{A^{1}([0,r])}B(\phi,\psi)}{r}<1\), we obtain \(\frac{B(\phi ^{*},\psi^{*})}{r}<1\), that is, \(\frac{B(\phi^{*},\psi^{*})}{A(\phi ^{*},\psi^{*})}<1\), so \(I(0,0)=0< I(\phi^{*},\psi^{*})\) and this leads to a contradiction. Hence, \((\phi^{*},\psi^{*})\in A^{1}([0,r[)\). This completes the proof of Lemma 2.4. □
Lemma 2.5
Suppose that there exist \(r>0\) and \((w,w)\in S\times S\), with \(0< A(w,w)< r\), such that
Then the functional \(I(\phi,\psi)=A(\phi,\psi)B(\phi,\psi)\) has at least a local minimum \((\phi^{*},\psi^{*})\in S\times S\) such that \((\phi^{*},\psi^{*})\neq(0,0)\), \(A(\phi^{*},\psi^{*})< r\), \(I(\phi ^{*},\psi^{*})\leq I(\phi,\psi)\) for all \((\phi,\psi)\in A^{1}([0,r])\) and \(I^{\prime}(\phi^{*},\psi^{*})=0\).
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. □
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
Assume that the following condition holds.
 \((J_{1})\) :

\(a_{1}>0\), \(a_{2}>0\), \(a_{3}>0\) or \(a_{3}^{2}< a_{1}a_{2}\) when \(a_{3}<0\).
Proof
Taking into consideration that \((J_{1})\) holds, we have
for all \(k\in\mathbb{Z}(1,N)\). There exist \(\alpha>\frac{1}{2}\lambda _{N}\) and \(\beta>0\) such that \(F_{k}(x,y)\geq\alpha (x^{2}+y^{2})\beta\) for all \(k\in\mathbb{Z}(1,N)\). Let \(\{(\phi _{n},\psi_{n})\}\) be a (PS) sequence such that \(\vert I(\phi _{n},\psi_{n}) \vert \leq M\), where \(M>0\).
First, we need to verify that \(\{(\phi_{n},\psi_{n})\}\) is bounded. We have
We obtain \(\frac{2\alpha\lambda_{N}}{2K_{2}^{2}\lambda_{N}}\Vert (\phi_{n},\psi_{n}) \Vert _{\infty}^{2}\leq M+N\beta\). Notice that \(2\alpha\lambda_{N}>0\), hence, \(\{(\phi_{n},\psi_{n})\}\) is bounded and admits a convergent subsequence.
The next thing to do in the proof is to verify that \(I(\phi,\psi)\) is unbounded from below. From the above discussion, one has \(I(\phi,\psi )\leq\frac{\lambda_{N}2\alpha}{2K_{2}^{2}\lambda_{N}}\Vert (\phi ,\psi) \Vert _{\infty}^{2}+N\beta\). Since \(\lambda_{N}2\alpha <0\), one has
This completes the proof of Lemma 3.1. □
Lemma 3.2
Assume that the following condition holds.
 \((J_{2})\) :

\(a_{1}<0\), \(a_{2}<0\), \(a_{3}<0\) or \(a_{3}^{2}< a_{1}a_{2}\) when \(a_{3}>0\).
Proof
Notice that \((J_{2})\) holds, one has
for all \(k\in\mathbb{Z}(1,N)\). There exist \(\alpha<0\) and \(\beta>0\) such that \(F_{k}(x,y)\leq\alpha(x^{2}+y^{2})+\beta\) for all \(k\in \mathbb{Z}(1,N)\). Then
Since \(\lambda_{1}2\alpha>0\), we have
Therefore, \(I(\phi,\psi)\) is coercive. □
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
Assume that \((J_{1})\) holds. Moreover, there exists \(r>0\) such that
Then \(I(\phi,\psi)\) has at least two distinct critical points.
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
Assume that \((J_{2})\) holds. Furthermore, there exist \(r>0\) and \((w,w)\in S\times S\), with \(r< A(w,w)\), such that
Then the functional \(I(\phi,\psi)=A(\phi,\psi)B(\phi,\psi)\) has at least three distinct critical points.
Proof
Taking into account that \((J_{2})\) holds, we know that \(I(\phi,\psi)\) is coercive from Lemma 3.2. Since
from Lemma 2.4, \(I(\phi,\psi)\) admits a local minimum \((\phi_{1},\psi_{1})\) such that \(A(\phi_{1},\psi_{1})< r\).
Let
Apparently, \(I^{r}(\phi,\psi)=A^{r}(\phi,\psi)B(\phi,\psi)\) is continuous. Since \(A^{r}(\phi,\psi)B(\phi,\psi)\geq A(\phi,\psi )B(\phi,\psi)\) for all \((\phi,\psi)\in S\times S\), then \(I^{r}(\phi,\psi)\) is coercive.
It follows from Lemma 2.1 that \(I^{r}(\phi,\psi)\) has a global minimum \((\phi_{2},\psi_{2})\), that is to say,
We assert that \(A(\phi_{2},\psi_{2})>r\). If otherwise, then \(A(\phi _{2},\psi_{2})\leq r\).
It follows from \(\frac{\sup_{A^{1}([0,r])}B(\phi,\psi)}{r}<1<\frac {B(w,w)}{A(w,w)}\) that
then \(r\sup_{A^{1}([0,r])}B(\phi,\psi)>A(w,w)B(w,w)\). Since \(A(w,w)>r\) and \(A(\phi_{2},\psi_{2})\leq r\), we obtain that \(A^{r}(\phi_{2},\psi_{2})B(\phi_{2},\psi_{2})>A^{r}(w,w)B(w,w)\) and this is contrary to (4.1). Our assertion is proved.
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. □
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
Assume that the following condition holds.
 \((J_{3})\) :

\(a_{1}>0\) and there exists a constant \(c>0\) such that
$$ \omega_{1}\min_{k\in\mathbb{Z}(1,N)}b_{1k}+a_{1}c^{2} \leq0. $$
has no nontrivial solution \(\phi^{*}\) such that \(\Vert \phi^{*} \Vert _{\infty}< c\).
Proof
If the conclusion is not true, then suppose that the boundary value problem (5.1) has a nontrivial solution \(\phi^{*}\) such that \(\Vert \phi^{*} \Vert _{\infty}< c\), hence, \(I(\phi,0)\) has a nontrivial critical point \(\phi^{*}\). Since
where ∇ denotes the gradient. One has
where \(\langle\cdot,\cdot\rangle\) denotes the usual scalar product in \(\mathbb{R}^{N}\). By matrix theory, C is positive definite, we have
that is,
Using \((J_{3})\), we find
where \(\phi^{*}(k)\in\,]{}c,c[\) for all \(k\in\mathbb{Z}(1,N)\).
This leads to a contradiction. This completes the proof. □
Lemma 5.2
Assume that the following condition holds.
 \((J_{4})\) :

\(a_{2}>0\) and there exists a constant \(c>0\) such that
$$ \omega_{2}\min_{k\in\mathbb{Z}(1,N)}b_{2k}+a_{2}c^{2} \leq0. $$
has no nontrivial solution \(\psi^{*}\) such that \(\Vert \psi^{*} \Vert _{\infty}< c\).
Proof
The proof of this lemma is analogous to that in Lemma 5.1 and so is omitted. □
Lemma 5.3
Assume that the following condition holds.
 \((J_{5})\) :

\(a_{1}<0\), \(\min_{k\in\mathbb{Z}(1,N)}b_{1k}\omega _{1}\leq0\) 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
It follows from Lemma 5.1 that
Taking \((J_{5})\) into consideration, we have
where \(\Vert \phi^{*} \Vert _{\infty}>\frac{c}{\sqrt{2N\lambda _{N}}K_{2}} \geq\sqrt{\frac{N(\min_{k\in\mathbb {Z}(1,N)}b_{1k}\omega_{1})}{a_{1}}}\).
This leads to a contradiction. We have thus proved the lemma. □
Lemma 5.4
Assume that the following condition holds.
 \((J_{6})\) :

\(a_{2}<0\), \(\min_{k\in\mathbb{Z}(1,N)}b_{2k}\omega _{2}\leq0\) 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
Assume that there exist two constants c, d, with \(0< d< c\), such that
Then system (1.1) has at least one nontrivial solution \((\phi^{*},\psi^{*})\) such that \(\Vert (\phi^{*},\psi ^{*}) \Vert _{\infty}< c\). Furthermore, if \((J_{3})\) and \((J_{4})\) hold (and c as above), then system (1.1) has at least one nontrivial solution \((\phi^{*},\psi^{*})\) with \(\phi ^{*}\neq0\) and \(\psi^{*}\neq0\) such that \(\Vert (\phi^{*},\psi ^{*}) \Vert _{\infty}< c\), there exist other three nontrivial solutions \((\phi^{*},\psi^{*})\), \((\phi^{*},\psi^{*})\) and \((\phi^{*},\psi^{*})\) to system (1.1).
Proof
Let \(r=\frac{c^{2}}{2K_{2}^{2}}\). From \(\Vert \phi \Vert _{\infty }\leq K_{2}\Vert \phi \Vert \) and \(\Vert \psi \Vert _{\infty }\leq K_{2}\Vert \psi \Vert \), one has \(\Vert (\phi,\psi) \Vert _{\infty}=\max\{\Vert \phi \Vert _{\infty}, \Vert \psi \Vert _{\infty}\}\leq\max\{K_{2}\Vert \phi \Vert ,K_{2}\Vert \psi \Vert \}=K_{2}\max\{\Vert \phi \Vert ,\Vert \psi \Vert \}\leq K_{2}\sqrt{2r}=c\) for all \((\phi, \psi )\in S\times S\) such that \(A(\phi,\psi)\leq r\). Hence,
for all \((\phi,\psi)\in S\times S\) such that \(A(\phi,\psi)\leq r\). As a result,
Put \((w,w)\in\mathbb{R}^{N+2}\times\mathbb{R}^{N+2}\) to be such that \(w(k)=d\) for all \(k\in\mathbb{Z}(1,N)\) and \(w(0)=w(N+1)=0\). Obviously, \((w,w)\in S\times S\). Furthermore, we obtain
Consequently,
It follows that
According to \(0< d< c\), we have
so, \(0< d<\frac{1}{2K_{2}}c\). Hence, \(0< A(w,w)< r\). Lemma 2.5 ensures that system (1.1) has at least one nontrivial solution \((\phi^{*},\psi^{*})\) such that \(\Vert (\phi ^{*},\psi^{*}) \Vert _{\infty}< c\).
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
Assume that \((J_{1})\) holds and there exists \(c>0\) such that
Then system (1.1) has at least one nontrivial solution \((\phi^{*},\psi^{*})\) such that \(\Vert (\phi^{*},\psi ^{*}) \Vert _{\infty}< c\). Moreover, if \((J_{3})\) and \((J_{4})\) hold \((and~c~as~above)\), then system (1.1) has at least one nontrivial solution \((\phi^{*},\psi^{*})\) with \(\phi ^{*}\neq0\) and \(\psi^{*}\neq0\) such that \(\Vert (\phi^{*},\psi ^{*}) \Vert _{\infty}< c\). Also, \((\phi^{*},\psi^{*})\), \((\phi ^{*},\psi^{*})\) and \((\phi^{*},\psi^{*})\) are other three nontrivial solutions to system (1.1).
Proof
As we have stated in the proof of Theorem 5.1, it follows from
that
Taking \((J_{1})\) into consideration, Theorem 4.1 ensures that system (1.1) has at least two solutions, that is, system (1.1) has at least one nontrivial solution. Discussing as in the proof of Theorem 5.1, the remaining conclusion is achieved. □
Theorem 5.3
Assume that \((J_{2})\) holds and there exist two constants c, d, with \(0< c<\sqrt{2}d\), such that
Then system (1.1) has at least two nontrivial solutions \((\phi^{*},\psi^{*})\) and \((\phi^{**},\psi^{**})\) such that \(A(\phi^{**},\psi^{**})>r\), where \(r=\frac{c^{2}}{2K_{2}^{2}}\). Furthermore, if \((J_{5})\) and \((J_{6})\) hold \((and~c~as~above)\), then system (1.1) has at least one nontrivial solution \((\phi^{**},\psi^{**})\) with \(\phi^{**}\neq0\) and \(\psi^{**}\neq 0\) such that \(\Vert (\phi^{**},\psi^{**}) \Vert _{\infty}>\frac {c}{\sqrt{2N\lambda_{N}}K_{2}}\). Then there exist other three nontrivial solutions \((\phi^{**},\psi^{**})\), \((\phi^{**},\psi ^{**})\), and \((\phi^{**},\psi^{**})\) to system (1.1).
Proof
Let \(r=\frac{c^{2}}{2K_{2}^{2}}\). Furthermore, put \((w,w)\in\mathbb {R}\times\mathbb{R}\) to be such that \(w(k)=d\) for all \(k\in\mathbb {Z}(1,N)\) and \(w(0)=w(N+1)=0\). Apparently, \((w,w)\in S\times S\). From the above discussion of Theorem 5.1, we have
It follows from \(0< c<\sqrt{2}d\) and \(\sqrt{2}K_{2}\geq1\) that \(0< c<2K_{2}d\). Hence, \(0< r< A(w,w)\). Taking \((J_{2})\) into consideration, Theorem 4.2 ensures that system (1.1) has at least two nontrivial solutions \((\phi ^{*},\psi^{*})\) and \((\phi^{**},\psi^{**})\) such that \(A(\phi ^{**},\psi^{**})>r\). From \(\Vert \phi \Vert \leq\sqrt{N\lambda _{N}}\Vert \phi \Vert _{\infty}\) and \(\Vert \psi \Vert \leq \sqrt{N\lambda_{N}}\Vert \psi \Vert _{\infty}\), one has \(\Vert (\phi,\psi) \Vert _{\infty}=\sqrt{ \Vert (\phi,\psi) \Vert _{\infty}^{2}}=\sqrt{\max\{\Vert \phi \Vert _{\infty }^{2},\Vert \psi \Vert _{\infty}^{2}\}}\geq \sqrt{\frac{1}{2}(\Vert \phi \Vert _{\infty}^{2}+\Vert \psi \Vert _{\infty}^{2})}\geq\sqrt{\frac{1}{2N\lambda_{N}}(\Vert \phi \Vert ^{2}+\Vert \psi \Vert ^{2})} >\sqrt{\frac{r}{N\lambda_{N}}}=\frac{c}{\sqrt{2N\lambda_{N}}K_{2}}\) for all \(A(\phi,\psi)>r\). Hence, \(\Vert (\phi^{**},\psi^{**}) \Vert _{\infty}>\frac{c}{\sqrt{2N\lambda_{N}}K_{2}}\).
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. □
Application
Example 6.1
Fix \(N=1\), \(\omega_{1}=9\), \(\omega_{2}=11\), \(b_{11}=1\), \(b_{21}=3\), \(a_{1}=2\), \(a_{2}=5\) and \(a_{3}=4\). Theorem 5.2 ensures that the system
where \(k=1\), admits at least four nontrivial solutions where both of the components are not zero. We observe that \(K_{2}=\frac{\sqrt{2}}{2}\) and \(F_{1}(x,y)=5x^{2}7y^{2}+\frac{1}{2}x^{4}+\frac{5}{4}y^{4}+x^{2}y^{2}\). Choosing \(c=\frac{33}{20}\), we have \(2K_{2}^{2}\frac{\max_{(x,y)\in [0,c]\times[0,c]}F_{1}(x,y)}{c^{2}}=2K_{2}^{2}\frac {F_{1}(0,0)}{c^{2}}=0<1\). Furthermore, \(\omega _{1}b_{11}+a_{1}c^{2}<0\), \(\omega_{2}b_{21}+a_{2}c^{2}<0\) show that conditions \((J_{3})\) and \((J_{4})\) hold. The conclusion follows from Theorem 5.2. Indeed, a simple calculation establishes \(((0,0),(\frac{\sqrt{6}}{3},\frac{2\sqrt{6}}{3}),(0,0))\), \(((0,0),(\frac{\sqrt{6}}{3},\frac{2\sqrt{6}}{3}),(0,0))\), \(((0,0),(\frac{\sqrt{6}}{3},\frac{2\sqrt{6}}{3}),(0,0))\) and \(((0,0),(\frac{\sqrt{6}}{3},\frac{2\sqrt{6}}{3}),(0,0))\) as the four nontrivial solutions of system (6.1).
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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).
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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).
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Tan, J., Zhou, Z. Boundary value problems for a coupled system of secondorder nonlinear difference equations. Adv Differ Equ 2017, 210 (2017). https://doi.org/10.1186/s1366201712574
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Keywords
 boundary value problem
 coupled system
 nonlinear difference equation
 critical point theory