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Homoclinic solutions for a class of nonlinear difference systems with classical \((\phi_{1}, \phi_{2})\)-Laplacian
Advances in Difference Equations volume 2015, Article number: 149 (2015)
Abstract
In this paper, we consider the existence of homoclinic solutions for a class of nonlinear difference systems involving classical \((\phi_{1}, \phi_{2})\)-Laplacian. First, we improve some inequalities in known literature. Then, by using the variational method, some new existence results are obtained. Finally, some examples are given to verify our results.
1 Introduction and main results
Let \(\mathbb{R}\) denote the real numbers and \(\mathbb{Z}\) the integers. Given \(a< b \) in \(\mathbb{Z}\). Let \(\mathbb{Z}[a,b]=\{a,a+1,\ldots,b\}\). Let \(T>1\) and N be fixed positive integers.
In this paper, we investigate the existence of homoclinic solutions for the following nonlinear difference systems involving classical \((\phi _{1},\phi_{2})\)-Laplacian:
where \(t\in\mathbb{Z}\), \(u_{m}(t)\in\mathbb{R}^{N}\), \(m=1,2\), \(V(t,x_{1},x_{2})=-K(t,x_{1},x_{2})+W(t,x_{1},x_{2})\), \(K,W:\mathbb{Z}\times \mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}\) and \(\phi_{m}\), \(m=1,2\), satisfy the following condition:
- (\(\mathcal{A}0\)):
-
\(\phi_{m}\) is a homeomorphism from \(\mathbb{R}^{N}\) onto \(\mathbb{R}^{N}\) such that \(\phi_{m}(0)=0\), \(\phi_{m}=\nabla\Phi_{m}\), with \(\Phi_{m}\in C^{1}(\mathbb{R}^{N},[0,+\infty])\) strictly convex and \(\Phi_{m}(0)=0\), \(m=1,2\).
Remark 1.1
Assumption (\(\mathcal{A}0\)) is given in [1], which is used to characterize the classical homeomorphism. If, furthermore, \(\Phi_{m}:\mathbb{R}^{N}\rightarrow\mathbb{R}\) is coercive (i.e., \(\Phi_{m}(x)\rightarrow+\infty\) as \(|x|\rightarrow\infty\)), there exists \(\delta_{m}>0\) such that
where \(\delta_{m}=\min_{|x|=1}\Phi_{m}(x)\), \(m=1,2\) (see [1]).
We call \(u=(u_{1},u_{2})\) a nontrivial homoclinic solution of system (1.1) if u satisfies system (1.1), \(u\neq0\) and \(u(t)\to0\) as \(t\to \infty\).
It is well known that the variational method has become an important tool to study the existence and multiplicity of solutions for various difference systems. Lots of contributions have been obtained (for example, see [1–20]). It is remarkable that, to the best of our knowledge, few people investigated system (1.1). Recently, in [1] and [2], by using the variational approach, J Mawhin investigated the following second order nonlinear difference systems with ϕ-Laplacian:
where \(\phi=\nabla\Phi\), Φ strictly convex, is a homeomorphism of \(\mathbb{R}^{N}\) onto the ball \(B_{a}\subset\mathbb{R}^{N}\) or of \(B_{a}\) onto \(\mathbb{R}^{N}\). By using the variational approach, under different conditions, the author obtained that system (1.3) has at least one or \(N+1\) geometrically distinct T-periodic solutions. It is interesting that J Mawhin considered three kinds of ϕ: (1) \(\phi:\mathbb{R}^{N}\to\mathbb{R}^{N}\) is a classical homeomorphism, for example, \(\phi(x)=|x|^{p-1}x\) for some \(p>1\) and all \(x\in\mathbb {R}^{N}\); (2) \(\phi:\mathbb{R}^{N}\to B_{a}\) (\(a<+\infty\)) is a bounded homeomorphism, for example, \(\phi(x)=\frac{x}{\sqrt{1+|x|^{2}}}\in B_{1}\) for all \(x\in\mathbb{R}^{N}\); (3) \(\phi: B_{a}\subset\mathbb {R}^{N}\to\mathbb{R}^{N}\) is a singular homeomorphism, for example, \(\phi(x)=\frac{x}{\sqrt{1-|x|^{2}}}\) for all \(x\in B_{1}\). Recently, in [17], we generalized some results in [2] for classical homeomorphism and bounded homeomorphism to system (1.1), which seem to be the first results for system (1.1).
In 2011, He and Chen [16] investigated the existence of homoclinic solutions for the following discrete p-Laplacian systems:
where \(p>1\). They obtained homoclinic orbits as the limit of the subharmonics for system (1.4).
In this paper, motivated by [1, 2, 15, 16] and [17], we first improve some inequalities in [16] and then investigate the existence of homoclinic solutions for system (1.1) with classical homeomorphism. Next we make the following assumption:
- (\(\mathcal{A}1\)):
-
Let \(p>1\). Assume that there exist positive constants \(d_{1}\), \(d_{2}\), \(d_{3}\), \(d_{4}\) such that
$$ d_{1}|x|^{p}\le\Phi_{1}(x)\leq d_{3}|x|^{p},\qquad d_{2}|y|^{p}\le \Phi_{2}(y)\leq d_{4}|y|^{p}, \quad \forall x,y\in \mathbb{R}^{N} $$and
$$ \bigl(\phi_{1}(x),x\bigr)\le p\Phi_{1}(x),\qquad \bigl( \phi_{2}(y),y\bigr)\le p\Phi_{2}(y),\quad \forall x,y\in \mathbb{R}^{N}. $$
For every \(s\in\mathbb{N}\), define
with the norm
Let \(p'>1\) be such that \(\frac{1}{p}+\frac{1}{p'}=1\) and
Next, we present our main results.
Theorem 1.1
Assume that (\(\mathcal{A}1\)) holds, \(f_{i}\neq0\), \(i=1,2\), W and K satisfy the following conditions:
- (\(\mathcal{V}\)):
-
\(V(t,x_{1},x_{2})=-K(t,x_{1},x_{2})+W(t,x_{1},x_{2})\), where \(K,W:\mathbb{Z}\times\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb {R}\), \(K(t,x_{1},x_{2})\) and \(W(t,x_{1},x_{2})\) are T-periodic and for every \(t\in\mathbb{Z}\), \(K,W\in C^{1}(\mathbb{Z}\times\mathbb{R}^{N}\times \mathbb{R}^{N},\mathbb{R})\);
- (H1):
-
there exist \(\gamma\in(1,p)\) and \(a_{1},a_{2}>0\) such that
$$K(t,x_{1},x_{2})\ge a_{1}|x_{1}|^{\gamma}+a_{2}|x_{2}|^{\gamma} \quad \textit{for all } (t,x_{1},x_{2})\in \mathbb{Z}[0,T-1] \times\mathbb{R}^{N}\times\mathbb{R}^{N}; $$ - (H2):
-
\(K(t,0,0)\equiv0\) and
$$\begin{aligned}& \bigl(x_{1},\nabla_{x_{1}} K(t,x_{1},x_{2}) \bigr)+\bigl(x_{2},\nabla_{x_{2}} K(t,x_{1},x_{2}) \bigr) \\& \quad \le p K(t,x_{1},x_{2})\quad \textit{for all } (t,x_{1},x_{2})\in \mathbb{Z}[0,T-1]\times \mathbb{R}^{N}\times\mathbb{R}^{N}; \end{aligned}$$ - (H3):
-
-
(i)
there exist \(r\in(0,1]\), \(0< b_{1}<a_{1}C_{*}^{\gamma-p}\), and \(0< b_{2}<a_{2}C_{*}^{\gamma-p}\) such that
$$\begin{aligned}& W(t,x_{1},x_{2}) \le b_{1}|x_{1}|^{p}+b_{2}|x_{2}|^{p}, \\& \quad \forall t\in\mathbb {Z}[0,T-1], |x_{1}|\le rC_{*},|x_{2}| \le rC_{*}; \end{aligned}$$(1.5) -
(ii)
there exist \(r>1\), \(0< b_{1}<a_{1}(C_{*}r)^{\gamma-p}\), and \(0< b_{2}<a_{2}(C_{*}r)^{\gamma-p}\) such that (1.5) holds;
-
(i)
- (H4):
-
$$\lim_{|x_{1}|+|x_{2}|\to+\infty}\frac {W(t,x_{1},x_{2})}{|x_{1}|^{p}+|x_{2}|^{p}}>d_{3}+d_{4}+2^{p-1}A_{0} \quad \textit{for all } t\in\mathbb{Z}[0,T-1], $$
where
$$A_{0}=\max_{|x_{1}|\le1,|x_{2}|\le1,t\in\mathbb{Z}[0,T-1]}K(t,x_{1},x_{2}); $$ - (H5):
-
there exist positive constants ξ, \(\eta_{1}\), \(\eta_{2}\) and \(\nu \in[0,\gamma-1)\) such that
$$\begin{aligned} 0 \le& \biggl( p+\frac{1}{\xi+\eta_{1}|x_{1}|^{\nu}+\eta_{2}|x_{2}|^{\nu }} \biggr)W(t,x_{1},x_{2}) \\ \le& \bigl(\nabla_{x_{1}} W(t,x_{1},x_{2}),x_{1} \bigr)+\bigl(\nabla _{x_{2}} W(t,x_{1},x_{2}),x_{2} \bigr) \end{aligned}$$for all \((t,x_{1},x_{2})\in\mathbb{Z}[0,T-1]\times\mathbb{R}^{N}\times \mathbb{R}^{N}\);
- (H6):
-
\(f_{1},f_{2}\in l^{p'}\cap l^{\frac{p-\nu}{p-\nu-1}}\) and
-
(i)
when \(r\in (0,1]\),
$$\begin{aligned}& \max\bigl\{ \|f_{1}\|_{l^{p'}},\|f_{2}\|_{l^{p'}}\bigr\} \\& \quad < \frac{1}{2^{p-1}} \min \bigl\{ d_{1},d_{2}, a_{1}C_{*}^{\gamma-p}-b_{1},a_{2}C_{*}^{\gamma -p}-b_{2} \bigr\} r^{p-1}; \end{aligned}$$ -
(ii)
when \(r\in (1,+\infty)\),
$$\begin{aligned}& \max\bigl\{ \|f_{1}\|_{l^{p'}},\|f_{2}\|_{l^{p'}}\bigr\} \\& \quad < \frac{1}{2^{p-1}} \min \bigl\{ d_{1},d_{2}, a_{1} (C_{*}r )^{\gamma-p}-b_{1},a_{2} (C_{*}r )^{\gamma-p}-b_{2} \bigr\} r^{p-1}. \end{aligned}$$
-
(i)
Then system (1.1) possesses a nontrivial homoclinic solution.
Theorem 1.2
Assume that (\(\mathcal{A}1\)) holds, \(f_{i}\neq0\), \(i=1,2\), W and K satisfy (\(\mathcal{V}\)), (H1)-(H5) and the following conditions:
- (H6)′:
-
\(f_{1},f_{2}\in l^{1}\) and
-
(i)
when \(r\in (0,1]\),
$$\begin{aligned}& \max\bigl\{ \|f_{1}\|_{l^{1}},\|f_{2}\|_{l^{1}}\bigr\} \\& \quad < \frac{1}{2^{p-1}C_{*}} \min \bigl\{ d_{1},d_{2}, a_{1}C_{*}^{\gamma-p}-b_{1},a_{2}C_{*}^{\gamma -p}-b_{2} \bigr\} r^{p-1}; \end{aligned}$$ -
(ii)
when \(r\in (1,+\infty)\),
$$\begin{aligned}& \max\bigl\{ \|f_{1}\|_{l^{1}},\|f_{2}\|_{l^{1}}\bigr\} \\& \quad < \frac{1}{2^{p-1}C_{*}} \min \bigl\{ d_{1},d_{2}, a_{1} (C_{*}r )^{\gamma-p}-b_{1},a_{2} (C_{*}r )^{\gamma-p}-b_{2} \bigr\} r^{p-1}. \end{aligned}$$
-
(i)
Then system (1.1) possesses a nontrivial homoclinic solution.
Theorem 1.3
Assume that (\(\mathcal{A}1\)) holds, \(f_{i}\neq0\), \(i=1,2\), W and K satisfy (\(\mathcal{V}\)), (H2), (H4), (H5) and the following conditions:
- (H1)′:
-
there exist \(a_{1}, a_{2}>0\) such that
$$K(t,x_{1},x_{2})\ge a_{1}|x_{1}|^{p}+a_{2}|x_{2}|^{p} \quad \textit{for all } (t,x_{1},x_{2})\in \mathbb{Z}[0,T-1] \times\mathbb{R}^{N}\times\mathbb{R}^{N}; $$ - (H3)′:
-
there exist \(r>0\) and \(0< b_{1}<a_{1}\), \(0< b_{2}<a_{2}\) such that
$$W(t,x_{1},x_{2})\le b_{1}|x_{1}|^{p}+b_{2}|x_{2}|^{p}, \quad \forall |x_{1}|\le rC_{*},|x_{2}|\le rC_{*} ; $$ - (H6)″:
-
\(f_{1},f_{2}\in l^{p'}\cap l^{\frac{p-\nu}{p-\nu-1}}\) and
$$\max\bigl\{ \|f_{1}\|_{l^{p'}},\|f_{2}\|_{l^{p'}}\bigr\} < \frac{1}{2^{p-1}} \min \{d_{1},d_{2}, a_{1}-b_{1},a_{2}-b_{2} \}r^{p-1}. $$
Then system (1.1) possesses a nontrivial homoclinic solution.
Theorem 1.4
Assume that (\(\mathcal{A}1\)) holds, \(f_{i}\neq0\), \(i=1,2\), W and K satisfy (\(\mathcal{V}\)), (H1)′, (H2), (H3)′, (H4), (H5) and the following condition:
- (H6)‴:
-
\(f_{1},f_{2}\in l^{1}\) and
$$\max\bigl\{ \|f_{1}\|_{l^{1}},\|f_{2}\|_{l^{1}}\bigr\} < \frac{1}{2^{p-1}C_{*}} \min \{d_{1},d_{2}, a_{1}-b_{1},a_{2}-b_{2} \}r^{p-1}. $$
Then system (1.1) possesses a nontrivial homoclinic solution.
Remark 1.2
Theorem 1.3 and Theorem 1.4 show that \(f_{1}\), \(f_{2}\) can be large when r is large.
2 Preliminaries
Similar to [15] and [16], we will obtain the homoclinic orbit of system (1.1) as a limit of solutions of a sequence of difference systems:
where \(f_{m,k}:\mathbb{Z}\to\mathbb{R}^{N}\) is a \(2kT\)-periodic extension of restriction of \(f_{m}\) to the interval \(\mathbb{Z}[-kT,kT-1]\), \(k\in\mathbb{N}\), \(m=1,2\).
Next, we present some basic notations. We use \(|\cdot|\) to denote the usual Euclidean norm in \(\mathbb{R}^{N}\). Define
\(\mathcal{H}\) is defined as a subspace of \(\mathcal{V}\) by
Define
Then \(\mathcal{H}_{k}=\mathcal{H}_{1,k}\times\mathcal{H}_{2,k}\). For \(u_{m}\in\mathcal{H}_{m,k}\), set
Moreover, \(l_{2kT}^{\infty}\) denote the space of all bounded real functions on \(\mathbb{Z}[-kT,kT-1]\) endowed with the norm
For \(1< p<+\infty\), on \(\mathcal{H}_{m,k}\), we define
For \(u=(u_{1},u_{2})^{\tau}\in\mathcal{H}_{k}\), define
Then \((\mathcal{H}_{k}, \|u\|_{\mathcal{H}_{k}} )\), \((\mathcal{H}_{1,k}, \|u\|_{\mathcal{H}_{1,k}} )\) and \((\mathcal{H}_{2,k}, \|u\|_{\mathcal{H}_{2,k}} )\) are reflexive Banach spaces.
Lemma 2.1
Let \(a,b\in\mathbb{Z}\), \(a\ge1\), \(b\ge0\), \(\varrho>1\), \(u_{m}\in\mathcal{H}_{m,k}\), \(m=1,2\). Then, for every \(t\in\mathbb{Z}\),
where \(m=1,2\).
Proof
Fix \(t\in\mathbb{Z}\). For every \(\tau\in\mathbb {Z}[t-a,t-1]\), we have
and for every \(\tau\in\mathbb{Z}[t, t+b]\),
Summing (2.3) over \(\mathbb{Z}[t-a,t-1]\) and (2.4) over \(\mathbb{Z}[t, t+b]\), we have
and
Set
Combining (2.5) with (2.6) and using Hölder’s inequality, we obtain
Since
and
Equation (2.7) implies that
which implies that (2.2) holds. Thus the proof is complete. □
Corollary 2.1
Let \(u_{m}\in\mathcal{H}_{m,k}\), \(m=1,2\). Then
where \(m=1,2\).
Proof
Obviously, there exists \(t^{*}\in\mathbb{Z}[-kT,kT-1]\) such that
In Lemma 2.1, let \(a=T\) and \(b=T-1\),
The proof is complete. □
Corollary 2.2
Let \(u_{m}\in\mathcal{H}_{m,k}\), \(m=1,2\). Then
where \(m=1,2\).
Proof
In Corollary 2.1, let \(\varrho=p\) and then use Hölder’s inequality. Then the proof is completed easily. □
Remark 2.1
As \(a\ge1\), Lemma 2.1, Corollary 2.1, and Corollary 2.2 improve Lemma 3.1, Corollary 3.1, and Corollary 3.2 in [16], respectively.
By Lemma 2.3 and Lemma 2.4 in [17], we have the following two lemmas.
Lemma 2.2
(see [17])
For any \(u=(u_{1},u_{2}),v=(v_{1},v_{2})\in\mathcal{H}_{k} \), the following two equalities hold:
Lemma 2.3
(see [17])
Let \(L:\mathbb{Z}[-kT,kT-1]\times\mathbb{R}^{N}\times\mathbb {R}^{N}\times\mathbb{R}^{N}\times\mathbb{R}^{N}\rightarrow \mathbb{R}\), \((t,x_{1},x_{2},y_{1},y_{2})\rightarrow L(t,x_{1},x_{2},y_{1},y_{2})\) and assume that L is continuously differential in \((x_{1},x_{2},y_{1},y_{2})\) for all \(t\in\mathbb{Z}[-kT,kT-1]\). Then the function \(\varphi_{k}:\mathcal{H}_{k}\to\mathbb{R} \) defined by
is continuously differentiable on \(\mathcal{H}_{k}\) and
where \(u,v\in\mathcal{H}_{k}\).
Let
and define \(\eta_{k}:\mathcal{H}_{k}\to[0,+\infty)\) by
Then
It follows from (\(\mathcal{A}0\)), (\(\mathcal{V}\)) and Lemma 2.3 that
By Lemma 2.2, it is easy to see that critical points of \(\varphi_{k}\) in \(\mathcal{H}_{k}\) are \(2kT\)-periodic solutions of system (2.1).
We shall use one linking method in [21] to obtain the critical points of φ (the details can be seen in [21]). Let \((E,\|\cdot\| )\) be a Banach space. Define a continuous map \(\Gamma:[0,1]\times E\to E\) by \(\Gamma(t,x)=\Gamma (t)x\), where \(\Gamma(t)\) satisfies the following conditions:
-
(1)
\(\Gamma(0)=I\), the identity map.
-
(2)
For each \(t\in[0,1)\), \(\Gamma(t)\) is a homeomorphism of E onto E and \(\Gamma^{-1}(t)\in C(E\times[0,1),E)\).
-
(3)
\(\Gamma(1)E\) is a single point in E and \(\Gamma(t)A\) converges uniformly to \(\Gamma(1)E\) as \(t\to1\) for each bounded set \(A\subset E\).
-
(4)
For each \(t_{0}\in[0,1)\) and each bounded set \(A\subset E\),
$$\sup_{\substack{0\le t\le t_{0}\\ u\in A}}\bigl\{ \bigl\Vert \Gamma(t)u\bigr\Vert +\bigl\Vert \Gamma^{-1}(t)u\bigr\Vert \bigr\} < \infty. $$
Let Φ be the set of all continuous maps Γ as defined above.
Definition 2.1
(see [21], Definition 3.2)
We say that A links \(B[\mathrm{hm}]\) if A and B are subsets of E such that \(A\cap B=\emptyset\), and for each \(\Gamma\in\Phi\), there is \(t'\in(0,1]\) such that \(\Gamma(t')A\cap B\neq\emptyset\).
Example 1
(see [21], p.21)
Let B be an open set in E, and let A consist of two points \(e_{1}\), \(e_{2}\) with \(e_{1}\in B\) and \(e_{2}\notin\bar{B}\). Then A links \(\partial B[\mathrm{hm}]\).
We use the following theorem to prove our main results.
Theorem 2.1
(see [21], Theorem 3.4 and Theorem 2.12)
Let E be a Banach space, \(\varphi\in C^{1}(E, \mathbb{R})\) and A and B be two subsets of E such that A links \(B[\mathrm{hm}]\). Assume that
and
Let \(\psi(t)\) be a positive, nonincreasing, locally Lipschitz continuous function on \([0,\infty)\) satisfying \(\int_{0}^{\infty}\psi (r)\, dr=\infty\). Then there exists a sequence \(\{u_{n}\}\subset E\) such that \(\varphi(u_{n})\to c \) and \(\varphi'(u_{n})/\psi(\|u_{n}\|)\to 0\), as \(n\to\infty\). Moreover, if \(c=\sup_{A} \varphi\), then there is a sequence \(\{u_{n}\}\subset E\) satisfying \(\varphi(u_{n})\to c\), \(\varphi'(u_{n})\to0\), and \(d(u_{n},B)\to0\), as \(n\to\infty\).
Remark 2.2
Since A links B, by Definition 2.1, it is easy to know that \(c\ge\inf_{B}\varphi\). By [21], if we let \(\psi(r)=\frac{1}{1+r}\), the sequence \(\{u_{n}\} \) is the Cerami sequence that is \(\{u_{n}\}\) satisfying
3 Proofs
Lemma 3.1
Suppose that (H2) holds. Then
Proof
Define the function \(\xi\in(0,+\infty)\to K(t,\xi^{-1}x_{1},\xi^{-1}x_{2})(\xi^{p}+\xi^{p})\). Then we have
Hence the function \(\xi\in(0,+\infty)\to K(t,\xi^{-1}x_{1},\xi^{-1}x_{2})(\xi^{p}+\xi^{p})\) is nondecreasing. Moreover, note that
Then the proof can be completed easily. □
Lemma 3.2
Suppose that (H1) holds. Then, for any \(u\in \mathcal{H}_{k}\),
Proof
It follows from (\(\mathcal{A}1\)), (H1), \(\gamma\in (1,p)\) and Corollary 2.2 that
□
Proof of Theorem 1.1
We divide the proof into the following Lemmas 3.3-3.5.
Lemma 3.3
Under the assumptions of Theorem 1.1, for every \(k\in\mathbb{N}\), system (2.1) has a nontrivial solution \(u_{k}\) in \(\mathcal{H}_{k}\).
Proof
We first construct A and B which satisfy the assumptions in Theorem 2.1.
(i) When \(r\in(0,1]\), by Corollary 2.2, (H1), (H3)(i), Hölder’s inequality and \(\gamma< p\), for \(u\in\mathcal{H}_{k}\) with \(\|u\|_{\mathcal {H}_{k}}= r\), we have \(\|u_{1}\|_{l_{2kT}^{\infty}}\le C_{*}\|u_{1}\|_{\mathcal{H}_{1,k}}\le rC_{*} \) and \(\|u_{2}\|_{l_{2kT}^{\infty}}\le C_{*}\|u_{2}\|_{\mathcal{H}_{2,k}}\le rC_{*}\),
(H6)(i) implies that there exists \(\alpha>0\) such that
(ii) When \(r\in(1,+\infty)\), by Corollary 2.2, (H1), (H3)(ii), Hölder’s inequality and \(\gamma< p\), for \(u\in \mathcal{H}_{k}\) with \(\|u\|_{\mathcal{H}_{k}}= r\), we have
(H6)(ii) implies that there exists \(\alpha>0\) such that
By Lemma 3.1 and the T-periodicity of K, there exists a constant \(B_{0}>0\) such that
where
By (H4), we know that there exist \(\varepsilon_{0}>0\) and \(L>0\) such that
By (3.4) and the T-periodicity of W, there exists a constant \(B_{1}>0\) such that
for all \((t,x_{1},x_{2})\in\mathbb{Z}[0,T-1]\times\mathbb{R}^{N}\times \mathbb{R}^{N}\). For any \(k\in\mathbb{N}\), define \({w}^{(k)}\in \mathcal{H}_{k}\) by
where
Since \(K(t,0,0)\equiv0\) and \(W(t,0,0)\equiv0\), which are implied by (H2) and (H5), then by (3.3) and (3.5) we have
So there exists \(\xi_{0}\in\mathbb{R}\) such that \(\|\xi_{0} {w}^{(k)}\| >r\) and \(\varphi_{k}(\xi_{0} {w}^{(k)})<0\). Moreover, it is clear that \(\varphi _{k}(0)=0\). Let \(e_{1}=\xi_{0} {w}^{(k)}\) and
Then \(0\in B\) and \(e_{1}\notin\bar{B}\). So by Example 1 in Section 2, we know that A links \(\partial B[\mathrm{hm}]\). So by Theorem 2.1 and Remark 2.2, we have
and there exists a sequence \(\{u_{n}=(u_{1}^{(n)},u_{2}^{(n)})\}_{n=1}^{\infty}\subset \mathcal{H}_{k}\) such that
Then there exists a constant \(C_{1k}>0\) such that
It follows from (H5) and the T-periodicity and continuity of W, \(\nabla_{x_{1}} W\) and \(\nabla_{x_{2}} W\) that
So by (3.5) and \(p-\nu>1\), there exists \(C_{2}>0\) such that
Hence, it follows from (H2), (3.8) and (3.10) that
The fact \(p-\nu>1\) and the above inequality show that \(\sum_{t=-kT}^{kT-1} |u_{1}^{(n)}(t)|^{p-\nu}\) and \(\sum_{t=-kT}^{kT-1} |u_{2}^{(n)}(t)|^{p-\nu}\) are bounded. By (\(\mathcal{A}1\)), (H1), (H6), (3.8), (3.9), (3.11), Hölder’s inequality and Corollary 2.2, we have
Since \(\nu< p-1\), (3.12) and the boundedness of \(\sum_{t=-kT}^{kT-1} |u_{1}^{(n)}(t)|^{p-\nu}\) and \(\sum_{t=-kT}^{kT-1} |u_{2}^{(n)}(t)|^{p-\nu}\) imply that \(\|u_{1}^{(n)}\|_{\mathcal {H}_{1,k}}\) and \(\|u_{2}^{(n)}\|_{\mathcal{H}_{2,k}}\) are bounded. Since \(\mathcal{H}\) is a finite-dimensional space, \(\{ u^{(n)}=(u_{1}^{(n)},u_{2}^{(n)})\}\) has a convergence subsequence, still denoted by \(\{u^{(n)}=(u_{1}^{(n)},u_{2}^{(n)})\}\), such that \(u^{(n)}=(u_{1}^{(n)},u_{2}^{(n)})\to u_{k}=(u_{1k},u_{2k})\) as \(n\to\infty \). Moreover, by the continuity of \(\varphi_{k}\) and \(\varphi_{k}'\), we obtain \(\varphi_{k}'(u_{k})=0\) and \(\varphi_{k}(u_{k})=c_{k}>0\). It is clear that \(u_{k}\neq0\) and so \(u_{k}\) is a desired nontrivial solution of system (2.1). The proof is complete. □
Lemma 3.4
Let \(\{u_{k}=(u_{1k}, u_{2k})\}_{k\in\mathbb {N}}\) be the solutions of system (2.1). Then there exists \(M_{1}>0\) such that \(\|u_{1k}\|_{l_{2kT}^{\infty}}\le M_{1}\) and \(\|u_{2k}\| _{l_{2kT}^{\infty}}\le M_{1}\).
Proof
First, we prove that the sequence \(\{c_{k}\}_{k\in \mathbb{N}}\) is bounded. For every \(k\in\mathbb{N}\), define \(\Gamma _{k}:[0,1]\times\mathcal{H}_{k}\to\mathcal{H}_{k}\) by
Then \(\Gamma\in\Phi\). Note that the set \(A=\{0,e_{1}\}\). So (3.7) and the argument of (3.6) imply that
where \(M_{2}\) is independent of \(k\in\mathbb{N}\), which implies that the sequence \(\{c_{k}\}_{k\in\mathbb{N}}\) is bounded. Moreover, \(\varphi_{k}'(u_{k})=0\). Then it follows from (\(\mathcal{A}1\)), (H2), and (3.10) that
So
Then
Thus (3.14) and Lemma 3.2 imply that
Note that \(p>\gamma>\nu+1\). So (H6) implies there exists \(M_{3}>0\) (independent of k) such that
By Corollary 2.1,
Let \(M_{1}=\max\{ C_{1*}M_{3}, C_{2*}M_{3}\}\). Thus the proof is complete. □
Lemma 3.5
Let \(\{u_{k}\}\) be determined by Lemma 3.4. Then there exists a subsequence \(\{u_{k_{j}}=(u_{1k_{j}},u_{2k_{j}})\}\) of \(\{u_{k}\}_{k\in\mathbb {N}}\) convergent to a certain function \(u_{\infty}=(u_{1\infty },u_{2\infty})\) and when \(f_{1}\neq0\) and \(f_{2}\neq0\), \(u_{\infty}\) is a nontrivial solution of system (1.1) such that \(u_{\infty}(t)\to0\) and \(\Delta{u}_{\infty}(t-1)\to0\) as \(t\to\pm\infty\).
Proof
Note that
Then, similar to the argument in [15] or [16], one can prove that \(\{u_{mk}\}_{k\in\mathbb{N}}\) has a convergent subsequence \(\{u_{mk_{j}}\}\) such that \(u_{mk_{j}}\to u_{m\infty}\) and \(u_{m\infty}(t)\to0\) and \(\Delta{u}_{m\infty}(t-1)\to0\) as \(t\to \pm\infty\), where \(m=1,2\). Let \(u_{\infty}=(u_{1\infty},u_{2\infty})\). By (3.13) and the continuity of \(\Phi_{m}\), \(K(t,\cdot,\cdot)\), \(W(t,\cdot,\cdot)\) and \(\varphi_{k}'\), similar to the argument in [15] or [16], the proof is easy to be completed. □
□
Proof of Theorem 1.2
The proof is easy to be completed by replacing
with
in the proofs of Lemma 3.3 and Lemma 3.4. □
Proofs of Theorem 1.3 and Theorem 1.4
We only note that in the proof of Lemma 3.3, when \(\gamma=p\), we do not need to consider the case that \(r\in(0,1]\) alone and it is sufficient that \(r>0\). Other proofs are the same as those of Theorem 1.1 and Theorem 1.2, respectively. □
4 Examples
We first give two examples about Φ which satisfy assumption (\(\mathcal{A}1\)).
(I) An example with \(N=1\). Define \(\Phi_{m}:\mathbb{R}\to\mathbb {R}^{N}\), \(m=1,2\), by
where \(\alpha_{1},\alpha_{2}\in[d_{1},d_{3}]\), \(\beta_{1},\beta_{2}\in [d_{2},d_{4}]\). Then it is easy to verify that \(\Phi_{m}\), \(m=1,2\), satisfies (\(\mathcal{A}1\)).
(II) As described in [1], the following classical case with p-Laplacian also satisfies the assumption (\(\mathcal{A}1\)). Define \(\Phi_{m}:\mathbb{R}^{N}\to\mathbb{R}^{N}\), \(m=1,2\), by
where \(\alpha\in[d_{1},d_{3}]\), \(\beta\in[d_{2},d_{4}]\).
Next, we present some examples of K and W which satisfy those assumptions in Theorem 1.1. There are lots of examples of K. For example, let
where \(\gamma\in(1,p)\), \(a_{i}, i=1,2:\mathbb{Z}\to\mathbb{R}^{+}\) are T-periodic. Let \(a_{i}=\min_{t\in\mathbb{Z}[0,T-1]} a_{i}(t)\). Then it is easy to see that K satisfies (H1) and (H2).
For W, we assume that
where \(b:\mathbb{Z}\to\mathbb{R}^{+}\) is T-periodic. Let \(b^{+}=\max_{t\in\mathbb{Z}[0,T-1]}\{b(t)\}\). Then
Let \(b_{1}=b_{2}=b^{+}\ln(|rC_{*}|^{p}+|rC_{*}|^{p}+1)\). If r is sufficiently small, then (H3)(i) holds. It is easy to see that
So (H4) holds. Let \(\nu\in(0,\gamma-1)\). Note that
for all \((x_{1},x_{2})\in \mathbb{R}^{N}\times\mathbb{R}^{N}\), when we choose sufficiently large ξ, \(\eta_{1}\) and \(\eta_{2}\). Hence
for all \((t,x_{1},x_{2})\in\mathbb{Z}[0,T-1]\times\mathbb{R}^{N}\times \mathbb{R}^{N}\), which implies (H5) holds.
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Acknowledgements
This project is supported by the National Natural Science Foundation of China (No. 11301235), Tianyuan Fund for Mathematics of the National Natural Science Foundation of China (No. 11226135) and the Fund for Fostering Talents in Kunming University of Science and Technology (No. KKSY201207032).
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Zhang, X., Wang, Y. Homoclinic solutions for a class of nonlinear difference systems with classical \((\phi_{1}, \phi_{2})\)-Laplacian. Adv Differ Equ 2015, 149 (2015). https://doi.org/10.1186/s13662-015-0467-x
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DOI: https://doi.org/10.1186/s13662-015-0467-x