- Research Article
- Open Access

- Alberto Cabada
^{1}Email author, - Chengyue Li
^{2}and - Stepan Tersian
^{3}

**2010**:195376

https://doi.org/10.1155/2010/195376

© Alberto Cabada et al. 2010

**Received:**5 July 2010**Accepted:**27 October 2010**Published:**31 October 2010

## Abstract

## Keywords

- Banach Space
- Hamiltonian System
- Difference Equation
- Laplacian Equation
- Variational Approach

## 1. Introduction

is referred to as the -Laplacian difference operator, and functions and are -periodic in and satisfy suitable conditions.

In the theory of differential equations, a trajectory , which is asymptotic to a constant as is called doubly asymptotic or homoclinic orbit. The notion of homoclinic orbit is introduced by Poincaré [1] for continuous Hamiltonian systems.

Recently, there is a large literature on the use of variational methods to the existence of homoclinic or heteroclinic orbits of Hamiltonian systems; see [2–7] and the references therein.

In the recent paper of Li [8] a unified approach to the existence of homoclinic orbits for some classes of ODE's with periodic potentials is presented. It is based on the Brezis and Nirenberg's mountain-pass theorem [9]. In this paper we extend this approach to homoclinic orbits for discrete -Laplacian type equations.

Discrete boundary value problems have been intensively studied in the last decade. The studies of such kind of problems can be placed at the interface of certain mathematical fields, such as nonlinear differential equations and numerical analysis. On the other hand, they are strongly motivated by their applicability to mathematical physics and biology.

The variational approach to the study of various problems for difference equations has been recently applied in, among others, the papers of Agarwal et al. [10], Cabada et al. [11], Chen and Fang [12], Fang and Zhao [13], Jiang and Zhou [14], Ma and Guo [15], Mihăilescu et al. [16], Kristály et al. [17].

Let us consider functions satisfying the following assumptions.

(*F*_{1}) The function
is continuous in
and
-periodic in
.

(*F*_{2}) The potential function
of

satisfies the Rabinowitz's type condition:

where and we are looking for its homoclinic solutions, that is, solutions of (1.10) such that as .

In order to obtain homoclinic solutions of (1.10), we will use variational approach and Brezis-Nirenberg mountain pass theorem [9].

Our main result is the following.

Theorem 1.1.

Suppose that the function is positive and -periodic and the functions satisfy assumptions − . Then, for each , (1.10) has a nonzero homoclinic solution , which is a critical point of the functional .

Moreover, given a nontrivial solution of problem (1.10), there exist two integer numbers such that for all and , the sequence is strictly monotone.

The paper is organized as follows. In Section 2, we present the proof of the main result and discuss the optimality of the condition . In Section 3, we give some examples of equations modeled by this kind of problems and present some additional remarks.

## 2. Proof of the Main Result

It is well known that if , then . Indeed, if , there exists a positive integer number , such that for all satisfying it is verified that and, as consequence, and the series is convergent too.

with given in (1.7) and defined in (1.8).

We have the following result.

Lemma 2.1.

The functional is well defined, -differentiable, and its critical points are solutions of (1.10).

Proof.

Then, the series is convergent and the functional is well defined on .

are continuous functions.

Moreover the functional is continuously Fréchet-differentiable in . It is clear, by (2.7), that the critical points of are solutions of (1.10).

*condition*if every sequence of such that

has a convergent subsequence. A sequence
such that (2.8) holds is referred to as
*-sequence*.

Theorem 2.2 (mountain-pass theorem, Brezis and Nirenberg [9]).

Then, there exists a sequence for . Moreover, if satisfies the condition, then is a critical value of , that is, there exists such that and .

Lemma 2.3.

Proof.

By , there exist , such that for all and .

Then, we can take large enough, such that for , and (2.14) holds.

Lemma 2.4.

Suppose that the assumptions of Lemma 2.3 hold. Then, there exists and a -bounded sequence for .

Proof.

and is defined in the proof of Lemma 2.3.

which implies that the sequence is bounded in .

Now we are in a position to prove Theorem 1.1.

Proof of Theorem 1.1.

From the density of in , we deduce that the previous equality is fulfilled for all and, in consequence, is a critical point of the functional , that is, is a solution of (1.10).

Since is bounded in , and is arbitrary, by (2.28) we obtain a contradiction with . The proof of the first part is complete.

Now, let be a nonzero homoclinic solution of problem (1.10). Assume that it attains positive local maximums and/or negative local minimums at infinitely many points . In particular we can assume that . In consequence and .

Suppose now that function vanishes at infinitely many points . From condition we conclude that and, in consequence, . Therefore it has an unbounded sequence of positive local maximums and negative local minimums, in contradiction with the previous assertion.

As a direct consequence of the two previous properties, we deduce that, for large enough, function has constant sign and it is strictly monotone.

To illustrate the optimality of the obtained results, we present in the sequel an example in which it is pointed out that condition cannot be removed to deduce the existence result proved in Theorem 1.1.

Example 2.5.

It is clear that for all and that for all if and only if .

As consequence, the inequality for all is satisfied if and only if , that is, condition does not hold.

Let us see that this problem has only the trivial solution for small values of the parameter .

Analogously it can be verified that the solution has no negative values on .

## 3. Remarks and Examples

- (A)

arising in mathematical physics and biology.

(A) *Second-Order Discrete*
*-Laplacian Equations.*

where is a constant depending on , which implies that and as . It implies that for a given , there exists such that for any , the problem has a solution for every .

We extend this phenomenon, looking for homoclinic solutions of (3.1). Applying Theorem 1.1 with and , we obtain the following.

Corollary 3.1.

Suppose that the function is positive and -periodic and . Then, for each , (3.1) has a nonzero homoclinic solution.

Moreover, given a nontrivial solution of problem (3.1), there exist two integer numbers such that for all and , the sequence is strictly monotone.

(B) *Higher Even-Order Difference Equations.*

where for each , satisfy the assumptions − .

Now following the steps of the proof of Theorem 1.1 one can prove the following.

Theorem 3.2.

Suppose that , the function is positive and -periodic and the functions satisfy assumptions − and , . Then, for each , (3.8) has a nonzero homoclinic solution , which is a critical point of the functional .

A typical example of (3.8) is (3.2), which is a discretization of a fourth-order extended Fisher-Kolmogorov equation. Homoclinic solutions for fourth-order ODEs are studied in [7] using variational approach and concentration-compactness arguments. As a consequence of Theorem 3.2 we obtain the following corollary.

Corollary 3.3.

Suppose that , the function is positive and -periodic and . Then, for each , (3.2) has a nonzero homoclinic solution .

(C) *Second-Order Difference Equations with Cubic and Quintic Nonlinearities.*

Our next example is (3.3), known as stationary Ginzubrg-Landau equation with cubic-quintic nonlinearity. We refer to [18, 19] and references therein. From physical point of view it is interesting the case , , . Theorem 1.1 can be applied for with , -periodic, and positive. Then satisfies assumptions − with and as a consequence we have the following corollary.

Corollary 3.4.

Suppose that the functions , and are -periodic and and are positive. Then, for each , (3.3) has a nonzero homoclinic solution .

Moreover, given a nontrivial solution of problem (1.10), there exist two integer numbers such that for all and , the sequence is strictly monotone.

Moreover, we can prove that if in addition to conditions − the following condition holds:

the homoclinic solution of (1.10) is positive.

in contradiction with . Then for every .

If for some , we know that and, in consequence, , and we arrive at a contradiction as in the previous case.

We summarize above observations in the following.

Theorem 3.5.

Suppose that the function is positive and -periodic and the functions satisfy assumptions , , and . Then, for each , (1.10) has a nonzero homoclinic solution . If moreover holds, is a positive solution on that is strictly monotone for large enough.

In the case we can estimate the maximum of the solution , provided the additional assumption

(*F*_{5}) Assume that for all
and
function
has the form
, where
is
-periodic in
,
and for each
,
is increasing in
for
.

We summarize above observation in the following.

Corollary 3.6.

satisfies the estimate (3.15).

Clearly, the positive solution of (3.17) is a positive solution of (3.4) too.

and (3.22) shows that blows up, that is, tends to as .

We summarize above facts in the following.

Corollary 3.7.

Let and and be -periodic sequences.

Let be such that . Since is an infinite sequence of integers, by Dirichlet principle, there exists a fixed and a subsequence of , still denoted by , such that and . Note that if , then or .

## Dedication

This work is dedicated to Professor Gheorghe Moroşanu on the occasion of his 60-th birthday.

## Declarations

### Acknowledgment

S. Tersian is thankful to Department of Mathematical Analysis at University of Santiago de Compostela, Spain, where a part of this work was prepared during his visit. A. Cabada partially supported by Ministerio de Educación y Ciencia, Spain, project MTM2007-61724.

## Authors’ Affiliations

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