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Multiple homoclinic orbits for second order discrete Hamiltonian systems without symmetric condition
Advances in Difference Equations volume 2015, Article number: 199 (2015)
Abstract
In this paper, we investigate the second order self-adjoint discrete Hamiltonian system \(\Delta[p(n)\Delta u(n-1)]-L(n)u(n)+\lambda a(n)\nabla G(u(n))+\mu b(n)\nabla F(u(n))=0\), where \(p,L:\mathbb{Z}\rightarrow\mathbb{R}^{N\times N}\) are both positive definite for all \(n\in\mathbb{Z}\), and no symmetric condition on G and F is needed. We establish two new criteria to guarantee that the above system has at least two nontrivial homoclinic solutions or infinitely many homoclinic solutions via critical point theory.
1 Introduction
In this paper, we consider the following second order self-adjoint discrete Hamiltonian system:
where \(u\in\mathbb{R}^{N}\), \(\Delta u(n)=u(n+1)-u(n)\) is the forward difference, \(p,L:\mathbb{Z}\rightarrow\mathbb{R}^{N\times N}\). As usual, we say that a solution \(u(n)\) of system (1.1) is homoclinic (to 0) if \(u(n)\rightarrow0\) as \(|n|\rightarrow\infty\). In addition, if \(u(n)\not\equiv0\) then \(u(n)\) is called a nontrivial homoclinic solution.
The discrete Hamiltonian system has found a great deal of interest last years because not only it is important in applications but it provides a good model for developing mathematical methods. In general, system (1.1) may be regarded as a discrete analog of the following second order Hamiltonian system:
With the aid of variational methods, the existence and multiplicity of homoclinic orbits for system (1.2) or its special form has been extensively investigated in many recent papers; see [1–16]. System (1.1) is the best approximation of (1.2) when one lets the step size not be equal to 1 but the variable’s step size go to 0, so solutions of system (1.1) can give some desirable numerical features for system (1.2). Moreover, system (1.1) does have applications as is shown in the monographs [17, 18].
In the past ten years, many authors have studied the existence and multiplicity of homoclinic solutions for system (1.1) or its special form (with \(F=0\)) via variational methods; see [19–31] and the references therein. In particular, see [20, 22, 24, 25, 28, 29, 31]. The authors have studied the existence of multiple homoclinic solutions for system (1.1) under the assumption that the nonlinear term is symmetric.
When no symmetric condition on the nonlinear term is assumed, as far as the authors are aware, there is no research about the existence of multiple homoclinic solutions for system (1.1). Motivated by the above facts, in this paper, our aim is to study the existence of multiple homoclinic solutions for system (1.1) under the condition that the nonlinear term possesses no symmetric condition.
We will use the following conditions:
- (F1):
-
\(p(n)\) is a real symmetric positive definite matrix for all \(n\in\mathbb{Z}\).
- (F2):
-
\(L(n)\) is a real symmetric positive definite matrix for all \(n\in\mathbb{Z}\) and
$$l(n)=\inf_{|u|=1}\bigl(L(n)u,u\bigr)\rightarrow+\infty $$as \(|n|\rightarrow\infty\).
- (F3):
-
\(G\in C^{1}(\mathbb{R}^{N},\mathbb{R})\) and \(G(0)=0\).
- (F4):
-
\(a\in l^{\frac{2}{1-\gamma}}(\mathbb{Z},\mathbb {R})\) is a non-negative, non-zero function (for some \(\gamma\in(0,1)\)).
- (F5):
-
\(\lim_{u\rightarrow0}\frac{|\nabla G(u)|}{|u|}=0\).
- (F6):
-
\(\lim_{|u|\rightarrow\infty}\frac{|\nabla G(u)|}{|u|}=0\).
- (F7):
-
There exists \(\zeta=(\zeta_{1},\zeta_{2},\ldots,\zeta _{N})\in\mathbb{R}^{N}\) such that \(G(\zeta)>0\).
Denote by Λ the set of functions \(F\in C^{1}(\mathbb {R}^{N},\mathbb{R})\), such that \(F(0)=0\) and satisfying the property:
- (F8):
-
There exist \(d>0\) and \(\alpha>1\) such that
$$\bigl\vert \nabla F(u)\bigr\vert \leq d\bigl(|u|+|u|^{\alpha}\bigr), \quad \forall u\in\mathbb{R}^{N}. $$ - (F9):
-
\(a\in l^{1}(\mathbb{Z},\mathbb{R})\) is a non-negative, non-zero function.
- (F10):
-
\(\frac{\|a\|_{1}}{\sqrt{(\varrho_{2}+4\varrho_{1})\varrho_{2}}}\liminf_{\eta\rightarrow\infty} \frac{\max_{|u|\leq\eta}G(u)}{\eta^{2}} <\frac{a(n_{0})}{2l_{1}+l_{2}}\limsup_{|u|\rightarrow\infty} \frac{G(u)}{|u|^{2}}\), where \(n_{0}\in\mathbb{Z}\) with \(a(n_{0})=\max\{a(n):n\in\mathbb{Z}\}\), \(\varrho_{1}=\inf\{(p(n)x,x):n\in \mathbb{Z},x\in\mathbb{R}^{N},|x|=1\}\), \(\varrho_{2}=\inf\{l(n):n\in \mathbb{Z}\}\), \(l_{1}=\sup\{(p(n)x,x):n=n_{0},n_{0}+1,x\in\mathbb {R}^{N},|x|=1\}\) and \(l_{2}=\sup\{(L(n_{0})x,x):x\in\mathbb{R}^{N},|x|=1\}\).
- (F11):
-
There exist \(M>0\) and \(\nu>2\) such that
$$\bigl\vert \nabla G(u)\bigr\vert \leq M\bigl(|u|+|u|^{\nu-1}\bigr), \quad \forall u\in\mathbb{R}^{N}. $$
Now, we state our main results.
Theorem 1.1
Assume that (F1)-(F7) hold. Then there exists \(\lambda_{1}>0\) such that for each \(\lambda>\lambda_{1}\), and for every function \(b\in l^{\infty}(\mathbb{Z},\mathbb{R})\) and every function \(F\in\Lambda\), there is \(\sigma>0\) with property that for each \(\mu \in[0,\sigma]\), system (1.1) has at least two nontrivial homoclinic solutions.
Example 1.1
\(a(n)=(\frac{1}{1+n^{2}})^{1/4}\), \(G(u)=\min\{|u|^{\sigma _{1}},|u|^{\sigma_{2}}\}\) with \(1<\sigma_{1}<2<\sigma_{2}\).
Example 1.2
\(a(n)=(\frac{1}{1+n^{2}})^{1/4}\), \(G(u)=((1,1,\ldots,1),u)\ln (1+|u|^{2})\). It is clear that \(G(u)\) is not even.
Remark 1.1
In Example 1.1 and Example 1.2, one cannot obtain the existence of homoclinic solutions in [20, 22, 24, 25, 28, 29, 31]. But in this paper, we obtain the existence of two homoclinic solutions.
When \(F=0\), a new result is obtained.
Theorem 1.2
Assume that (F1)-(F3) and (F9)-(F11) hold. Then, for every
system (1.1) (with \(F=0\)) possesses an unbounded sequence of homoclinic solutions.
Remark 1.2
In Theorem 1.2, we can substitute \(\eta\rightarrow \infty\) and \(|u|\rightarrow\infty\) with \(\eta\rightarrow0^{+}\) and \(|u|\rightarrow0^{+}\) by applying part (c) of Theorem 2.2 instead of part (b) of Theorem 2.2 in the proof, and obtaining a sequence of pairwise distinct homoclinic solutions.
The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of Theorems 1.1 and 1.2.
2 Preliminaries
In this section, the following Ricceri-type three critical points theorem will be needed in our argument. If X is a real Banach space, denote by \(\Gamma_{X}\) the class of all functionals \(\psi:X\rightarrow \mathbb{R}\) possessing the following property: if \(\{u_{n}\}\) is a sequence in X converging weakly to \(u\in X\) and \(\liminf_{n\rightarrow\infty} \psi(u_{n})\leq\psi(u)\), then \(\{u_{n}\} \) has a subsequence converging strongly to u.
For example, if X is uniformly convex and \(g:[0,+\infty)\rightarrow \mathbb{R}\) is a continuous, strictly increasing function, then, by a classical result, the functional \(u\rightarrow g(\|u\|)\) belongs to the class \(\Gamma_{X}\).
Theorem 2.1
([32], Theorem 2)
Let X be a separable and reflexive real Banach space; let \(\Phi :X\rightarrow\mathbb{R}\) be a coercive, sequentially weakly lower semicontinuous \(C^{1}\) functional, belonging to \(\Gamma_{X}\), bounded on each bounded subset of X and whose derivative admits a continuous inverse on \(X^{\ast}\); \(J:X\rightarrow\mathbb{R}\) a \(C^{1}\) functional with compact derivative. Assume that Φ has a strict local minimum \(v_{0}\) with \(\Phi(v_{0})=J(v_{0})=0\). Finally, setting
assume that \(\alpha_{1}<\alpha_{2}\).
Then, for each compact interval \([b_{1},b_{2}]\subset(\frac{1}{\alpha _{2}},\frac{1}{\alpha_{1}})\) (with the conventions \(\frac{1}{0}=+\infty\), \(\frac{1}{+\infty}=0\)), there exists \(N_{1}>0\) with the following property: for every \(\lambda\in[b_{1},b_{2}]\) and every \(C^{1}\) functional \(\Psi:X\rightarrow\mathbb{R}\) with compact derivative, there exists \(\sigma>0\) such that, for each \(\mu\in[0,\sigma]\), the equation
has at least three solutions in X whose norms are less than \(N_{1}\).
In the next section we shall prove our results applying the classical Ricceri variational principle ([33], Theorem 2.5).
Let X be a non-empty set and let \(\Phi,J:X\rightarrow\mathbb{R}\) be two functionals.
For all \(r>\inf_{X}\Phi\), we put
and
Theorem 2.2
([33], Theorem 2.5)
Let X be a reflexive real Banach space, and let \(\Phi,J:X\rightarrow \mathbb{R}\) be two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, (strongly) continuous, and coercive and J is sequentially weakly continuous. One has:
-
(a)
For every \(r>\inf_{X}\Phi\) and every \(\lambda\in (0,\frac{1}{\varphi_{1}(r)})\), the restriction of the functional \(\Phi -\lambda J\) to \(\Phi^{-1}((-\infty,r))\) admits a global minimum, which is a critical point (local minimum) of \(\Phi -\lambda J\) in X.
-
(b)
If \(\gamma_{1}<+\infty\) then, for each \(\lambda\in (0,\frac{1}{\gamma_{1}})\), the following alternative holds: either
- (b1):
-
\(\Phi-\lambda\Psi\) possesses a global minimum, or
- (b2):
-
there is a sequence \(\{u_{n}\}\) of critical points (local minima) of \(\Phi-\lambda J\) such that \(\lim_{n\rightarrow+\infty}\Phi(u_{n})=+\infty\).
-
(c)
If \(\gamma_{2}<+\infty\) then, for each \(\lambda\in (0,\frac{1}{\gamma_{2}})\), the following alternative holds: either
- (c1):
-
there is a global minimum of Φ which is a local minimum of \(\Phi-\lambda J\), or
- (c2):
-
there is a sequence of pairwise distinct critical points (local minima) of \(\Phi-\lambda J\) which weakly converges to a global minimum of Φ.
In what follows, we always assume that \(p(n)\) and \(L(n)\) are real symmetric positive definite matrices for all \(n\in\mathbb{Z}\). Let
and for \(u,v\in X\), let
and the corresponding norm is
Then X is a Hilbert space with the above inner product. As usual, for \(1\leq q<+\infty\), \(j=1\mbox{ or }N\), let
and
and their norms are defined by
respectively.
Lemma 2.1
(see [28])
For \(u\in X\),
where \(\varrho_{1}=\inf\{(p(n)x,x):n\in\mathbb{Z},x\in\mathbb {R}^{N},|x|=1\}\) and \(\varrho_{2}=\inf\{l(n):n\in\mathbb{Z}\}\) and, for \(u\in l^{2}(\mathbb{Z},\mathbb{R}^{N})\),
Lemma 2.2
(see [29])
Assume that L satisfies (F2). Then X is compactly embedded in \(l^{q}(\mathbb{Z},\mathbb {R}^{N})\) for any \(2\leq q<\infty\), and
For any \(u\in X\), put
Lemma 2.3
Assume that (F1)-(F8) hold. Then the functional \(I:X\rightarrow\mathbb{R}\) defined by
is well defined and of class \(C^{1}(X,\mathbb{R})\) and
Furthermore, \(J^{\prime},\Psi^{\prime}:X\rightarrow X^{\ast}\) are compact and the critical points of I in X are solutions of system (1.1) with \(u(\pm\infty)=0\).
Proof
We first show that \(J:X\rightarrow\mathbb{R}\). It follows from (F5) and (F6) that for any \(\epsilon>0\), there exists \(D_{\epsilon}\) such that
By (2.8), (F4), and (2.3), we have
for some \(D>0\). Analogously, we see that Ψ is well defined on X. Thus, I is well defined on X.
Now, we show that J is Gâteaux differentiable on X. By virtue of (2.8) and (F4), for any \(\xi\in[0,1]\), it is easy to check that
where C is a constant independent of ξ. Therefore, for any \(u,v\in X\), by the mean value theorem and the Lebesgue dominated convergence theorem, we have
where \(\theta_{n}\in[0,1]\) depends on u, v, h. It is easy to see that \(W(u,v)\) is linear. Next we show that \(W(u,v)\) is bounded. In fact, for any \(u\in X\), by (F3), (F4), (2.3), and Hölder’s inequality, we have
for some \(C_{1}>0\). Therefore, \(DJ(u)=W(u,\cdot)\) is the Gâteaux derivative of J at u.
Next, we prove that \(DJ(u)\) is weakly continuous in u. To this end, we first claim that if \(u_{k}\rightharpoonup u\) in X, then \(f_{k}\rightarrow f\) in \(l^{2}(\mathbb{Z},\mathbb{R}^{N})\), where \(f_{k}=\{ \nabla G(u_{k}(n))\}_{n\in\mathbb{Z}}\) and \(f=\{\nabla G(u(n))\}_{n\in \mathbb{Z}}\). Arguing indirectly, by Lemma 2.2, we may assume that there exists a subsequence \(\{u_{k_{i}}\}\) such that
and
for some \(\varepsilon>0\). By (2.12), passing to a subsequence if necessary, we can assume that
and
Let
and
then \(\{e_{1}(n)\}_{n\in\mathbb{Z}}\in l^{2}(\mathbb{Z},\mathbb{R})\) and \(\{e_{2}(n)\}_{n\in\mathbb{Z}}\in l^{4}(\mathbb{Z},\mathbb{R})\). It follows from (2.8) that
for all \(i,n\in\mathbb{Z}\). Combining (2.12) and (2.14), by Lebesgue dominated convergence, we have
which contradicts (2.13). Hence the claim above is true. We assume that \(u_{k}\rightharpoonup u\) in X, then \(f_{k}\rightarrow f\) in \(l^{2}(\mathbb{Z},\mathbb{R}^{N})\), where \(f_{k}=\{\nabla G(u_{k}(n))\} _{n\in\mathbb{Z}}\) and \(f=\{\nabla G(u(n))\}_{n\in\mathbb{Z}}\). By (2.3), (F4), and Hölder’s inequality, we obtain
as \(k\rightarrow\infty\). This implies that \(DJ(u)\) is weakly continuous in u. Hence, \(J^{\prime}(u)=DJ(u)\), i.e., \(J\in C^{1}(X,\mathbb{R})\). Furthermore, \(J^{\prime}\) is compact by the weakly continuity of \(J^{\prime}\) since X is a Hilbert space. Similarly, we can prove that \(\Psi\in C^{1}(X,\mathbb{R})\) and \(\Psi^{\prime}\) is compact. Due to the form of I in (2.6), (2.7) is verified and hence \(I\in C^{1}(X,\mathbb{R})\).
Finally, we show that the critical points of I in X are solutions of system (1.1) with \(u(\pm\infty)=0\). Observe that, for any \(u,v\in X\),
It follows from (2.7) and the above equations that \(\langle I^{\prime}(u),v\rangle=0\) for all \(v\in X\) if and only if
So, the critical points of I in X are the solutions of system (1.1). On the other hand, it follows from (2.3) that \(u(\pm\infty)=0\). This completes the proof. □
Lemma 2.4
Φ is coercive, sequentially weakly lower semicontinuous, bounded on each bounded subset of X and its derivative admits a continuous inverse on \(X^{\ast}\).
Proof
It is easy to verify that Φ is coercive. Let \(u_{k}\rightharpoonup u\) in X, we see that \(\liminf_{n\rightarrow \infty}\|u_{k}\|\geq\|u\|\). Then we have
So Φ is sequentially weakly lower semicontinuous. Moreover, it is easy to see that Φ is bounded on each bounded subset of X.
Next we will show that \(\Phi^{\prime}\) admits a continuous inverse on \(X^{\ast}\). For each \(u\in X\backslash\{0\}\), by (2.7), we have
So \(\lim_{\|u\|\rightarrow+\infty}\langle\Phi^{\prime }(u),u\rangle/\|u\|=+\infty\), that is, \(\Phi^{\prime}\) is coercive. For any \(u, v\in X\), in view of (2.7), we have
So \(\Phi^{\prime}\) is uniformly monotone. By ([34], Theorem 26.A(d)), we see that \(\Phi^{\prime}\) admits a continuous inverse on \(X^{\ast}\). □
3 Proof of Theorems 1.1 and 1.2
We will prove Theorem 1.1 by using Theorem 2.1. First, we give the following four useful lemmas.
Lemma 3.1
\(\limsup_{u\rightarrow0}\frac {J(u)}{\Phi(u)}\leq0\).
Proof
It follows from (F3), (F5), and (F6) that, for any \(\epsilon>0\), there exists \(T_{\epsilon}\) such that
Thus, for each \(u\in X\backslash\{0\}\),
Taking the ‘lim sup’ of the above estimation when \(u\rightarrow0\), the arbitrariness of ϵ gives the required inequality. □
Lemma 3.2
\(\limsup_{\|u\|\rightarrow\infty }\frac{J(u)}{\Phi(u)}\leq0\).
Proof
By (F5) and (F6), for every \(\epsilon>0\). there exists \(\eta_{\epsilon}\in(0,1)\) such that
Since \(G\in C^{1}(\mathbb{R}^{N},\mathbb{R})\), there exists a constant \(C_{\epsilon}>0\) such that
where γ is given in (F4). By virtue of (3.2), (3.3), and (F3), we have
By (F4), (3.4), (2.3), and Hölder’s inequality, we obtain
Thus, for each \(u\in X\backslash\{0\}\),
Taking the ‘lim sup’ of the above estimation when \(\|u\|\rightarrow \infty\), the arbitrariness of ϵ gives the required inequality. □
Lemma 3.3
\(\sup_{\Phi(u)>0}\frac{J(u)}{\Phi(u)}>0\).
Proof
Since a is a non-negative, non-zero function, there exists \(n_{1}\in\mathbb{Z}\) such that \(a(n_{1})>0\). Define
Obviously, \(\Phi(v)=\frac{1}{2}\|v\|>0\). It follows from (3.5), (F3), and (F7) that
Hence we have
The proof is complete. □
Lemma 3.4
\(\lambda_{1}=\frac{1}{\alpha_{2}}\geq \frac{1}{2D\varrho_{2}^{-1}\|a\|_{\infty}}\), where \(\alpha_{2}\) is defined in Theorem 2.1 and \(D=\max_{u\neq0}\frac {G(u)}{|u|^{2}}\).
Proof
It follows from (F4) and (2.3) that
By (3.7), we have
which ends the proof. □
Now we give the proof of Theorem 1.1.
Proof
Obviously, X is a separable, reflexive and uniformly convex Banach space. It follows from Lemmas 2.3, 2.4, and Lemmas 3.1-3.4 that Φ, J, and Ψ satisfy all conditions of Theorem 2.1. Thus, for each \(\lambda>\lambda_{1}\geq \frac{1}{2D\varrho_{2}^{-1}\|a\|_{\infty}}\), there is \(\sigma>0\) with the property that for each \(\mu\in[0,\sigma]\), I has at least three solutions in X whose norms are less than \(N_{1}\). It is easy to see that 0 is a solution of (1.1). Hence, system (1.1) has at least two nontrivial homoclinic solutions. □
Now we give the proof of Theorem 1.2.
Proof
Obviously, X be a reflexive real Banach space. With similar arguments to those used in the proofs of Lemmas 2.3 and 2.4, we can see that \(\Phi ,J:X\rightarrow R\) are two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous, continuous, and coercive and J is sequentially weakly continuous. Our aim is to apply part (b) of Theorem 2.2.
Next, we show that \(\gamma_{1}<+\infty\). Let \(\{\eta_{k}\}\) be a sequence of positive numbers such that \(\lim_{k\rightarrow\infty }\eta_{k}=+\infty\) and
Take \(r_{k}=\frac{\eta_{k}^{2}\sqrt{(\varrho_{2}+4\varrho_{1})\varrho _{2}}}{2}\) for all \(k\in\mathbb{N}\). By (2.1), one has
for any \(v\in X\) with \(\|v\|^{2}<2r_{k}\). In view of (3.8), (3.9), and \(\Phi(0)=J(0)=0\), we have
Thus, by (F10), we have
In view of (F10), we can consider the interval Ω. A simple reasoning related to (3.10) and (F10) shows that
Now, we will verify that I is unbounded from below. It follows from the choice of λ that there exists a sequence \(\{\rho_{k}\}\) in \(\mathbb{R}^{N}\) with \(|\rho_{k}|\rightarrow\infty\) such that
We define the function
It is easy to see that \(v_{k}\in X\). By (F10), (F11), (3.11), and (3.12), we have
Combining (3.11) and (3.13), we find that I is unbounded from below, that is, I has no global minimum.
By part (b) of Theorem 2.2, there is a sequence \(\{u_{k}\}\) of critical points (local minima) of \(\Phi-\lambda J\) such that \(\lim_{k\rightarrow+\infty}\Phi(u_{k})=+\infty\). Hence, system (1.1) possesses an unbounded sequence of homoclinic solutions. □
4 Example
In order to illustrate Theorem 1.2, we give an example.
Example 4.1
Consider the following problem for the case of \(N=1\):
where \(p=L=1\), \(a(n)=\frac{1}{1+n^{2}}\), and
with \(d_{k}=\frac{k!(k+2)!-1}{4(k+1)!}\) and \(e_{k}=\frac {k!(k+2)!+1}{4(k+1)!}\) for every \(k\in\mathbb{N}\).
Define
for all \(u\in\mathbb{R}\). It is easy to verify that
moreover, for any \(k\in\mathbb{N}\),
By (4.2) and (4.3), we see that \(G\in C^{1}(\mathbb{R})\) is nondecreasing. For any \(k\in\mathbb{N}\) one has
and
Thus,
and in fact, since \(\lim_{u\rightarrow-\infty}\frac{G(u)}{u^{2}}=0\), it is a simple computation to verify that
Applying Theorem 1.2, then, for \(\lambda>\frac{3}{32}\), problem (4.1) has an unbounded sequence of homoclinic solutions.
Remark 4.1
In Example 4.1, one cannot obtain the existence of homoclinic solutions in [20, 22, 24, 25, 28, 29, 31]. But in this paper, we obtain the existence of infinitely many homoclinic solutions.
References
Chen, HW, He, ZM: Infinitely many homoclinic solutions for a class of second-order Hamiltonian systems. Adv. Differ. Equ. 2014, 161 (2014)
Caldiroli, P, Montecchiari, P: Homoclinic orbits for second order Hamiltonian systems with potential changing sign. Commun. Appl. Nonlinear Anal. 1(2), 97-129 (1994)
Carrião, PC, Miyagaki, OH: Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems. J. Math. Anal. Appl. 230, 157-172 (1999)
Coti Zelati, V, Ekeland, I, Séré, E: A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann. 288(1), 133-160 (1990)
Coti Zelati, V, Rabinowitz, PH: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Am. Math. Soc. 4, 693-727 (1991)
Ding, YH: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25, 1095-1113 (1995)
Izydorek, M, Janczewska, J: Homoclinic solutions for a class of the second order Hamiltonian systems. J. Differ. Equ. 219, 375-389 (2005)
Omana, W, Willem, M: Homoclinic orbits for a class of Hamiltonian systems. Differ. Integral Equ. 5, 1115-1120 (1992)
Paturel, E: Multiple homoclinic orbits for a class of Hamiltonian systems. Calc. Var. Partial Differ. Equ. 12, 117-143 (2001)
Rabinowitz, PH: Homoclinic orbits for a class of Hamiltonian systems. Proc. R. Soc. Edinb., Sect. A 114, 33-38 (1990)
Rabinowitz, PH, Tanaka, K: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206, 473-499 (1991)
Salvatore, A: Homoclinic orbits for a special class of nonautonomous Hamiltonian systems. Nonlinear Anal. 30, 4849-4857 (1997)
Sun, J, Chen, H, Nieto, JJ: Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 373, 20-29 (2011)
Zhang, Q, Liu, C: Infinitely many homoclinic solutions for second order Hamiltonian systems. Nonlinear Anal. 72, 894-903 (2010)
Zhang, ZH, Yuan, R: Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems. Nonlinear Anal. 71, 4125-4130 (2009)
Zou, WM, Li, SJ: Infinitely many homoclinic orbits for the second-order Hamiltonian systems. Appl. Math. Lett. 16, 1283-1287 (2003)
Agarwal, RP: Difference Equations and Inequalities: Theory, Methods, and Applications, 2nd edn. Dekker, New York (2000)
Ahlbrandt, CD, Peterson, AC: Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations. Kluwer Academic, Dordrecht (1996)
Chen, HW, He, ZM: Homoclinic solutions for second order discrete Hamiltonian systems with superquadratic potentials. J. Differ. Equ. Appl. 19, 1147-1160 (2013)
Chen, HW, He, ZM: Infinitely many homoclinic solutions for second-order discrete Hamiltonian systems. J. Differ. Equ. Appl. 19, 1940-1951 (2013)
Chen, HW, He, ZM: Homoclinic solutions for second-order discrete Hamiltonian systems with asymptotically quadratic potentials. Math. Methods Appl. Sci. 37, 2451-2462 (2014)
Chen, P: Existence of homoclinic orbits in discrete Hamiltonian systems without Palais-Smale condition. J. Differ. Equ. Appl. 19, 1781-1794 (2013)
Deng, XQ, Chen, G: Homoclinic orbits for second order discrete Hamiltonian systems with potentials changing sign. Acta Appl. Math. 103, 301-314 (2008)
Lin, XY, Tang, XH: Existence of infinitely homoclinic orbits in discrete Hamiltonian systems. J. Math. Anal. Appl. 373, 59-72 (2011)
Ma, MJ, Guo, ZM: Homoclinic orbits for second order self-adjoint difference equations. J. Math. Anal. Appl. 323(1), 513-521 (2006)
Ma, MJ, Guo, ZM: Homoclinic orbits and subharmonics for nonlinear second order difference equations. Nonlinear Anal. 67, 1737-1745 (2007)
Tang, XH, Chen, J: Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems. Adv. Differ. Equ. 2013, 242 (2013)
Tang, XH, Lin, XY: Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential. J. Differ. Equ. Appl. 17, 1617-1634 (2011)
Tang, XH, Lin, XY: Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential. J. Differ. Equ. Appl. 19, 796-813 (2013)
Tang, XH, Lin, XY, Xiao, L: Homoclinic solutions for a class of second order discrete Hamiltonian systems. J. Differ. Equ. Appl. 16, 1257-1273 (2010)
Zhang, X, Shi, YM: Homoclinic orbits of a class of second-order difference equations. J. Math. Anal. Appl. 396, 810-828 (2012)
Ricceri, B: A further three critical points theorem. Nonlinear Anal. 71, 4151-4157 (2009)
Ricceri, B: A general variational principle and some of its applications. J. Comput. Appl. Math. 113, 401-410 (2000)
Zeidler, E: Nonlinear Functional Analysis and Its Applications, vol. 2. Springer, Berlin (1990)
Acknowledgements
The second author was supported by the Doctor Priming Fund Project of University of South China (2014XQD13). The authors would like to thank the referees for their valuable suggestions and comments, which led to improvement of the manuscript.
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He, Z., Chen, H. Multiple homoclinic orbits for second order discrete Hamiltonian systems without symmetric condition. Adv Differ Equ 2015, 199 (2015). https://doi.org/10.1186/s13662-015-0545-0
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DOI: https://doi.org/10.1186/s13662-015-0545-0
MSC
- 37J45
- 39A12
- 58E05
- 70H05
Keywords
- homoclinic solutions
- discrete Hamiltonian systems
- critical point theory
- variational methods