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# Homoclinic solutions of some second-order non-periodic discrete systems

- Yuhua Long
^{1}Email author

**2011**:64

https://doi.org/10.1186/1687-1847-2011-64

© Long; licensee Springer. 2011

**Received:**15 July 2011**Accepted:**20 December 2011**Published:**20 December 2011

## Abstract

In this article, we discuss how to use a standard minimizing argument in critical point theory to study the existence of non-trivial homoclinic solutions of the following second-order non-autonomous discrete systems

without any periodicity assumptions. Adopting some reasonable assumptions for *A* and *L*, we establish that two new criterions for guaranteeing above systems have one non-trivial homoclinic solution. Besides that, in some particular case, for the first time the uniqueness of homoclinic solutions is also obtained.

**MSC:** 39A11.

## Keywords

- homoclinic solution
- variational functional
- critical point
- subquadratic second-order discrete system

## 1. Introduction

The theory of nonlinear discrete systems has widely been used to study discrete models appearing in many fields such as electrical circuit analysis, matrix theory, control theory, discrete variational theory, etc., see for example [1, 2]. Since the last decade, there have been many literatures on qualitative properties of difference equations, those studies cover many branches of difference equations, see [3–7] and references therein. In the theory of differential equations, homoclinic solutions, namely doubly asymptotic solutions, play an important role in the study of various models of continuous dynamical systems and frequently have tremendous effects on the dynamics of nonlinear systems. So, homoclinic solutions have extensively been studied since the time of Poincaré, see [8–13]. Similarly, we give the following definition: if *x*
_{
n
} is a solution of a discrete system, *x*
_{
n
} will be called a homoclinic solution emanating from 0 if *x*
_{
n
} → 0 as |*n*| → +∞. If *x*
_{
n
} ≠ 0, *x*
_{
n
} is called a non-trivial homoclinic solution.

For our convenience, let **N**, **Z**, and **R** be the set of all natural numbers, integers, and real numbers, respectively. Throughout this article, | · | denotes the usual norm in **R**
^{
N
} with *N* ∈ **N**, (·,·) stands for the inner product. For *a*, *b* ∈ **Z**, define **Z**(*a*) = {*a*, *a* + 1, *...*}, **Z**(*a*, *b*) = {*a*, *a* + 1, ..., } when *a* ≤ *b*.

without any periodicity assumptions, where *A* is an antisymmetric constant matrix, *L*(*n*) ∈ *C*
^{1}(**R**, **R**
^{
N×N
}) is a symmetric and positive definite matrix for all *n* ∈ **Z**, *W*(*n*, *x*
_{
n
}) = *a*(*n*)*V*(*x*
_{
n
}), and *a:* **R** → **R**
^{+} is continuous and *V* ∈ C^{1}(**R**
^{
N
}, **R**). The forward difference operator Δ is defined by Δ*x*
_{
n
} = *x*
_{
n+1
} - *x*
_{
n
} and Δ^{2}
*x*
_{
n
} *=* Δ(Δ*x*
_{
n
}).

*A*= 0, (1.1) becomes the second-order self-adjoint discrete system

which is in some way a type of the best expressive way of the structure of the solution space for recurrence relations occurring in the study of second-order linear differential equations. So, (1.3) arises with high frequency in various fields such as optimal control, filtering theory, and discrete variational theory and many authors have extensively studied its disconjugacy, disfocality, boundary value problem oscillation, and asymptotic behavior. Recently, Bin [16] studied the existence of non-trivial periodic solutions for asymptotically superquadratic and subquadratic system (1.1) when *A* = 0. Ma and Guo [17, 18] gave results on existence of homoclinic solutions for similar system (1.3). In this article, we establish that two new criterions for guaranteeing the above system have one non-trivial homoclinic solution by adopting some reasonable assumptions for *A* and *L*. Besides that, in some particular case, we obtained the uniqueness of homoclinic solution for the first time.

Now we present some basic hypotheses on *L* and *W* in order to announce our first result in this article.

(*H*
_{1}) *L*(*n*) ∈ *C*
^{1}(**Z**, **R**
^{
N×N
}) is a symmetric and positive definite matrix and there exists a function *α*: **Z** → **R**
^{+} such that (*L*(*n*)*x*, *x*) ≥ *α*(*n*)|*x|*
^{2} and *α*(*n*) → + ∞ as |*n*| → +∞;

(*H*
_{2}) *W*(*n*, *x*) = *a*(*n*) |*x|*
^{
γ
}, i.e., *V*(*x*) = |*x|*
^{
γ
}, where *a:* **Z** → **R** such that *a*(*n*
_{0}) *>* 0 for some *n*
_{0} ∈ **Z**, 1 *< γ <* 2 is a constant.

**Remark 1.1**From (

*H*

_{1}), there exists a constant

*β >*0 such that

*H*

_{2}), we see

*V*(

*x*) is subquadratic as |

*x|*→ +∞ and

In addition, we need the following estimation on the norm of *A*. Concretely, we suppose that (*H*
_{3}) *A* is an antisymmetric constant matrix such that $\parallel A\parallel <\sqrt{\beta}$, where *β* is defined in (1.4).

**Remark 1.2** In order to guarantee that (*H*
_{3}) holds, it suffices to take *A* such that ||*A*|| is small enough.

Up until now, we can state our first main result.

**Theorem 1.1** If (*H*
_{1})-(*H*
_{3}) are hold, then (1.1) possesses at least one non-trivial homoclinic solution.

Substitute (*H*
_{2})' by (*H*
_{2}) as follows

*H*

_{2})'

*W*(

*n*,

*x*) =

*a*(

*n*)

*V*(

*x*), where

*a:*

**Z**→

**R**such that

*a*(

*n*

_{1})

*>*0 for some

*n*

_{1}∈

**Z**and

*V*∈

*C*

^{1}(

**R**

^{ N },

**R**), and

*V*(0) = 0. Moreover, there exist constants

*M >*0,

*M*

_{1}

*>*0, 1

*< θ <*2 and 0

*< r*≤ 1 such that

**Remark 1.3**By

*V*(0) = 0,

*V*∈

*C*

^{1}(

**R**

^{ N },

**R**) and (1.7), we have

which yields that *V*(*x*) is subquadratic as |*x|* → +∞.

We have the following theorem.

**Theorem 1.2**Assume that (

*H*

_{1}), (

*H*

_{2})' and (

*H*

_{3}) are satisfied, then (1.1) possesses at least one non-trivial homoclinic solution. Moreover, if we suppose that

*V*∈

*C*

^{2}(

**R**

^{ N },

**R**) and there exists constant

*ω*with $0<\omega <\beta -\sqrt{\beta}\parallel A\parallel $ such that

then (1.1) has one and only one non-trivial homoclinic solution.

The remainder of this article is organized as follows. After introducing some notations and preliminary results in Section 2, we establish the proofs of our Theorems 1.1 and 1.2 in Section 3.

## 2. Variational structure and preliminary results

In this section, we are going to establish suitable variational structure of (1.1) and give some lemmas which will be fundamental importance in proving our main results. First, we state some basic notations.

*E*, for all

*x*,

*y*∈

*E*there holds

*E*,

*<*·, ·

*>*) is an inner space with

*E*is a Hilbert space. For later use, given

*β*> 0, define ${l}^{\beta}=\left\{x=\left\{{x}_{n}\right\}\in S:\sum _{n\in Z}\mid {x}_{n}{\mid}^{\beta}<+\infty \right\}$ and the norm

*l*

^{ ∞ }= {

*x*= {

*x*

_{ n }} ∈

*S:*|

*x*

_{ n }

*|*< +∞} and

**Lemma 2.1** Assume that *L* satisfies (*H*
_{1}), {*x*
^{(k)}} ⊂ *E* such that *x*
^{(k)}⇀ *x*. Then *x*
^{(k)}⇀ *x* in *l*
^{2}.

**Proof** Without loss of generality, we assume that *x*
^{(k)}⇀ 0 in *E*. From (*H*
_{1}) we have *α*(*n*) > 0 and *α*(*n*) → +∞ as *n* → ∞, then there exists *D* > 0 such that $\left|\frac{1}{\alpha \left(n\right)}\right|=\frac{1}{\alpha \left(n\right)}\le \epsilon $ holds for any *ε* > 0 as |*n|* > *D*.

*I*= {

*n*: |

*n*| ≤

*D*,

*n*∈

**Z**} and ${E}_{I}=\{x\in E:\sum _{n\in I}[(\Delta {{x}_{n})}^{2}+L\left(n\right){x}_{n}\cdot {x}_{n}]<+\infty \}$, then

*E*

_{ I }is a 2

*DN*-dimensional subspace of

*E*and clearly

*x*

^{(k)}⇀ 0 in

*E*

_{ I }. This together with the uniqueness of the weak limit and the equivalence of strong convergence and weak convergence in

*E*

_{ I }, we have

*x*

^{(k)}→ 0 in

*E*

_{ I }, so there has a constant

*k*

_{0}> 0 such that

*H*

_{1}), there have

*ε*is arbitrary and

*||x*

^{(k)}

*||*is bounded, then

combing with (2.2) and (2.3), *x*
^{(k)}→ 0 in *l*
^{2} is true.

In order to prove our main results, we need following two lemmas.

**Lemma 2.2**For any

*x*(

*j*)

*>*0,

*y*(

*j*)

*>*0,

*j*∈

**Z**there exists

where *q >* 1, *s >* 1, $\frac{1}{q}+\frac{1}{s}=1$.

**Lemma 2.3**[19] Let

*E*be a real Banach space and

*F*∈

*C*

^{1}(

*E*,

**R**) satisfying the PS condition. If

*F*is bounded from below, then

is a critical point of *F*.

## 3. Proofs of main results

*F: E*→

**R**as follows

**Lemma 3.1** Under conditions of Theorem 1.1, we have *F* ∈ *C*
^{1}(*E*, **R**) and any critical point of *F* on *E* is a classical solution of (1.1) with *x*
_{±∞} = 0.

**Proof**We first show that

*F: E*→

**R**. By (1.4), (2.1), (

*H*

_{2}), and Lemma 2.2, we have

Combining (3.1) and (3.2), we show that *F: E* → **R**.

*F*∈

*C*

^{1}(

*E*,

**R**). Write ${F}_{1}\left(x\right)=\frac{1}{2}\parallel x{\parallel}^{2}+\frac{1}{2}\sum _{n\in Z}\left(A{x}_{n},\Delta {x}_{n}\right)$, ${F}_{2}\left(x\right)=\sum _{n\in Z}W\left(n,{x}_{n}\right)$, it is obvious that

*F*(

*x*) =

*F*

_{1}(

*x*) -

*F*

_{2}(

*x*) and

*F*

_{1}(

*x*) ∈

*C*

^{1}(

*E*,

**R**). And by use of the antisymmetric property of

*A*, it is easy to check

Therefore, it is sufficient to show that *F*
_{2}(*x*) ∈ *C*
^{1}(*E*, **R**).

*V*(

*x*) = |

*x|*

^{ γ }, i.e.,

*V*∈

*C*

^{1}(

**R**

^{ N },

**R**), let us write

*φ*(

*t*) =

*F*

_{2}(

*x*+

*th*), 0 ≤

*t*≤ 1, for all

*x*,

*h*∈

*E*, there holds

where 0 < *θ*
_{
n
} < 1. It follows that *F*
_{2}(*x*) is Gateaux differentiable on *E*.

*y*∈

*E*, using (2.1), (3.4) and lemma 2.2, it follows

*F*

_{2}(

*x*) at

*x*is ${F}_{2}^{\prime}\left(x\right)\in E$ and

*y*∈

*E*and

*ε*> 0, when

*||y||*≤

*δ*, i.e., $\mid y\mid \le {\alpha}^{-\frac{1}{2}}\delta $ there exists

*δ >*0 such that

*ε*is arbitrary, then ${F}_{2}^{\prime}:E\to {E}^{\prime}$, $x\to {F}_{2}^{\prime}\left(x\right)$ is continuous and

*F*

_{2}(

*x*) ∈

*C*

^{1}(

*E*,

**R**). Hence,

*F*∈

*C*

^{1}(

*E*,

**R**) and for any

*x*,

*h*∈

*E*, we have

this is just (1.1). Then critical points of variational functional (3.1) corresponds to homoclinic solutions of (1.1)

**Lemma 3.2** Suppose that (*H*
_{1}), (*H*
_{2}) in Theorem 1.1 are satisfied. Then, the functional (3.1) satisfies PS condition.

**Proof**Let {

*x*

^{(k)}}

_{ k∈N }⊂

*E*be such that {

*F*(

*x*

^{(k)})}

_{ k∈N }is bounded and {

*F*' (

*x*

^{(k)})} → 0 as

*k*→ +∞. Then there exists a positive constant

*c*

_{1}such that

*x*

^{(k)}}

_{ k∈N }is bounded in

*E*. Combining (3.1), (3.5) and remark 1.1, there holds

*μ <*2, it is not difficult to know that {

*x*

^{(k)}}

_{ k∈N }is a bounded sequence in

*E*. So, passing to a subsequence if necessary, it can be assumed that

*x*

^{(k)}⇀

*x*in

*E*. Moreover, by Lemma 2.1, we know

*x*

^{(k)}⇀

*x*in

*l*

^{2}. So for

*k*→ +∞,

*k*large enough, we have

that is the functional (3.1) satisfies PS condition.

Up until now, we are in the position to give the proof of Theorem 1.1.

**Proof of Theorem 1.1**By (3.1), we have, for every

*m*∈

**R**\ {0} and

*x*∈

*E*\ {0},

*< γ <*2 and $\parallel A\parallel <\sqrt{\beta}$, (3.8) implies that

*F*(

*mx*) → +∞ as |

*m|*→ +∞. Consequently,

*F*(

*x*) is a functional bounded from below. By Lemma 2.3,

*F*(

*x*) possesses a critical value

*c*= inf

_{ x∈E }

*F*(

*x*), i.e., there is a critical point

*x*∈

*E*such that

*H*

_{2}), there exists

*δ*

_{0}

*>*0 such that

*a*(

*n*)

*>*0 for any

*n*∈ [

*n*

_{0}-

*δ*

_{0},

*n*

_{0}+

*δ*

_{0}]. Take

*c*

_{0}∈

**R**

^{ N }\ {0} and let

*y*∈

*E*be given by

which yields that *F*(*my*) *<* 0 for *|m|* small enough since 1 *< γ <* 2, i.e., the critical point *x* ∈ *E* obtained above is non-trivial.

Although the proof of the first part of Theorem 1.2 is very similar to the proof of Theorem 1.1, for readers' convenience, we give its complete proof.

**Lemma 3.3**Under the conditions of Theorem 1.2, it is easy to check that

for all *x*, *y* ∈ *E*. Moreover, *F*(*x*) is a continuously Fréchet differentiable functional defined on *E*, i.e., *F* ∈ *C*
^{1}(*E*, **R**) and any critical point of *F*(*x*) on *E* is a classical solution of (1.1) with *x*
_{±∞
}= 0.

**Proof**By (1.8) and (2.1), we have

*F: E →*

**R**. In the following, according to the proof of Lemma 3.1, it is sufficient to show that for any

*y*∈

*E*,

which implies that $\sum _{n\in Z}\left(\nabla W\left(n,{x}_{n}\right),{y}_{n}\right)$ is bounded for any *x*, *y* ∈ *E*.

Using Lemma 2.1, the remainder is similar to the proof of Lemma 3.1, so we omit the details of its proof.

**Lemma 3.4** Under the conditions of Theorem 1.2, *F*(*x*) satisfies the PS condition.

**Proof** From the proof of Lemma 3.2, we see that it is sufficient to show that for any sequence {*x*
^{(k)}}_{
k∈N
}⊂ *E* such that {*F*(*x*
^{(k)})}_{
k∈N
}is bounded and *F'* (*x*
^{(k)}) → 0 as *k* → +∞, then {*x*
^{(k)}}_{
k∈N
}is bounded in *E*.

*F*(

*x*

^{(k)})}

_{ k∈N }is bounded, there exists a constant

*C*

_{2}

*>*0 such that

which implies that {*x*
^{(k)}}_{
k ÎN
}is bounded in *E*, since $\parallel A\parallel <\sqrt{\beta}$.

Combining Lemma 2.1, the remainder is just the repetition of the proof of Lemma 3.2, we omit the details of its proof.

With the aid of above preparations, now we will give the proof of Theorem 1.2.

**Proof of Theorem 1.2**By(1.8), (2.1), (3.1), and Lemma 2.2, we have, for every

*m*∈

**R**\ {0} and

*x*∈

*E*\ {0},

*F*(

*mx*) → +∞ as |

*m|*→ +∞, since $\parallel A\parallel <\sqrt{\beta}$. Consequently,

*F*(

*x*) is a functional bounded from below. By Lemmas 2.3 and 3.4,

*F*(

*x*) possesses a critical value

*c*= inf

_{ x∈E }

*F*(

*x*), i.e., there is a critical point

*x*∈

*E*such that

*x*obtained above is non-trivial. From (

*H*

_{2})', there exists

*δ*

_{1}> 0 such that

*a*(

*n*)

*>*0 for any

*n*∈ [

*n*

_{1}-

*δ*

_{1},

*n*

_{1}+

*δ*

_{1}]. Take

*c*

_{1}∈

**R**

^{ N }with 0 < |c

_{1}

*|*=

*r*where

*r*is defined in (

*H*

_{2})' and let

*y*∈

*E*be given by

*n*∈

**Z**, |

*y|*≤

*r*≤ 1. By (1.6), (2.1), and (3.1), we obtain that

which yields that *F*(*my*) *<* 0 for |*m|* small enough since 1 *< θ <* 2, i.e., the critical point *x* ∈ *E* obtained above is non-trivial.

*x*and

*y*, by Lemma 3.3, we have

where *z* ∈ *E* and *z* ∈ (*x*, *y*), which implies that ||*x - y*|| = 0, since $0<\omega <\beta -\sqrt{\beta}\parallel A\parallel $, that is, *x* ≡ *y* for all *n* ∈ **Z**.

## Declarations

### Acknowledgements

This study was supported by the Xinmiao Program of Guangzhou University, the Specialized Fund for the Doctoral Program of Higher Eduction (No. 20071078001) and the project of Scientific Research Innovation Academic Group for the Education System of Guangzhou City. The author would like to thank the reviewer for the valuable comments and suggestions.

## Authors’ Affiliations

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