# Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations

- Peng Chen
^{1}and - XH Tang
^{1}Email author

**2010**:470375

**DOI: **10.1155/2010/470375

© Peng Chen and X. H. Tang. 2010

**Received: **5 May 2010

**Accepted: **2 August 2010

**Published: **17 August 2010

## Abstract

By using the critical point theory, we establish some existence criteria to guarantee that the nonlinear difference equation has at least one homoclinic solution, where , and is non periodic in . Our conditions on the nonlinear term are rather relaxed, and we generalize some existing results in the literature.

## 1. Introduction

where is the forward difference operator defined by , , is the ratio of odd positive integers, and are real sequences, . . As usual, we say that a solution of (1.1) is homoclinic (to 0) if as . In addition, if , then is called a nontrivial homoclinic solution.

Difference equations have attracted the interest of many researchers in the past twenty years since they provided a natural description of several discrete models. Such discrete models are often investigated in various fields of science and technology such as computer science, economics, neural network, ecology, cybernetics, biological systems, optimal control, and population dynamics. These studies cover many of the branches of difference equation, such as stability, attractiveness, periodicity, oscillation, and boundary value problem. Recently, there are some new results on periodic solutions of nonlinear difference equations by using the critical point theory in the literature; see [1–3].

which has arose in the study of fluid dynamics, combustion theory, gas diffusion through porous media, thermal self-ignition of a chemically active mixture of gases in a vessel, catalysis theory, chemically reacting systems, and adiabatic reactor (see, e.g., [4–6] and their references). In the case of , (1.2) has been discussed extensively in the literature; we refer the reader to the monographs [7–10].

It is well known that the existence of homoclinic solutions for Hamiltonian systems and their importance in the study of the behavior of dynamical systems have been already recognized from Poincar*é*; homoclinic orbits play an important role in analyzing the chaos of dynamical system. In the past decade, this problem has been intensively studied using critical point theory and variational methods.

where , , , and .

Theorem A (see [13]).

Assume that and satisfy the following conditions:

- (p)
for all ;

- (q)
for all and ;

- (f1)there is a constant such that(1.4)
- (f2)
uniformly with respect to .

Then (1.3) possesses a nontrivial homoclinic solution.

It is worth pointing out that to establish the existence of homoclinic solutions of (1.3), condition (f1) is the special form (with ) of the following so-called global Ambrosetti-Rabinowitz condition on ; see [15].

However, it seems that results on the existence of homoclinic solutions of (1.1) by critical point method have not been considered in the literature. The main purpose of this paper is to develop a new approach to the above problem by using critical point theory.

Motivated by the above papers [13, 14], we will obtain some new criteria for guaranteeing that (1.1) has one nontrivial homoclinic solution without any periodicity and generalize Theorem A. Especially, satisfies a kind of new superquadratic condition which is different from the corresponding condition in the known literature.

In this paper, we always assume that , , . Our main results are the following theorems.

Theorem 1.1.

Assume that and satisfy the following conditions:

- (p)
for all ;

- (q)
for all and ;

uniformly in ;

(F2) there is a constant such that

(F3) and there is a constant such that

Then (1.1) possesses a nontrivial homoclinic solution.

Theorem 1.2.

Assume that , and satisfy , , and the following assumption:

uniformly in . Then (1.1) possesses a nontrivial homoclinic solution.

Remark 1.3.

*Obviously, both conditions*
*and*
*are weaker than*
*. Therefore, both Theorems 1.1 and 1.2 generalize Theorem A by relaxing conditions*
*and*
*.*

When is subquadratic at infinity, as far as the authors are aware, there is no research about the existence of homoclinic solutions of (1.1). Motivated by the paper [16], the intention of this paper is that, under the assumption that is indefinite sign and subquadratic as , we will establish some existence criteria to guarantee that (1.1) has at least one homoclinic solution by using minimization theorem in critical point theory.

Now we present the basic hypothesis on , and in order to announce the results in this paper.

*For every*

*,*

*is continuously differentiable in*

*, and there exist two*constants

*and two functions*

*such that*

(F5) *There exist two functions*
*and*
*such that*

*where*
*as*
*,*
*is a positive constant*.

(F6) *There exist*
*and two constants*
*and*
*such that*

Up to now, we can state our main results.

Theorem 1.4.

Assume that , and satisfy , , , and . Then (1.1) possesses at least one nontrivial homoclinic solution.

By Theorem 1.4, we have the following corollary.

Corollary 1.5.

Assume that , , and satisfy , and the following conditions:

(F7) , where and , is a constant such that for some .

(F9) as is a positive constant.

Then (1.1) possesses at least one nontrivial homoclinic solution.

## 2. Preliminaries

Then is a uniform convex Banach space with this norm.

respectively.

Furthermore, the critical points of in are classical solutions of (1.1) with .

We will obtain the critical points of *I* by using the Mountain Pass Theorem. We recall it and a minimization theorem as follows.

- (i)
;

- (ii)
there exist constants such that ;

- (iii)
there exists such that .

where is an open ball in of radius centered at and

Lemma 2.2.

where .

Proof.

The proof is completed.

Lemma 2.3.

Assume that and hold. Then for every ,

- (i)
is nondecreasing on ;

- (ii)
is nonincreasing on .

The proof of Lemma 2.3 is routine and so we omit it.

Lemma 2.4 . (see [18]).

Let be a real Banach space and satisfy the (PS)-condition. If is bounded from below, then is a critical value of .

## 3. Proofs of Theorems

Proof of Theorem 1.1..

Since , it follows from (3.11) and (3.12) that in . Hence, satisfies the (PS)-condition.

Then function is a desired classical solution of (1.1). Since , is a nontrivial homoclinic solution. The proof is complete.

Proof of Theorem 1.2.

Equation (3.29) shows that implies that , that is, assumption (ii) of Lemma 2.1 holds. The proof of Theorem 1.2 is completed.

Proof of Theorem 1.4.

Since , (3.30) implies that as . Consequently, is bounded from below.

Since , it follows from (3.38) and (3.39) that in . Hence, satisfies (PS)-condition.

By Lemma 2.4, is a critical value of , that is, there exists a critical point such that .

Since , it follows from (3.40) that for small enough. Hence , therefore is nontrivial critical point of , and so is a nontrivial homoclinic solution of (1.1). The proof is complete.

Proof of Corollary 1.5..

This shows that (F6) holds also. Hence, by Theorem 1.4, the conclusion of Corollary 1.5 is true. The proof is complete.

## 4. Examples

In this section, we give some examples to illustrate our results.

Example 4.1.

Then it is easy to verify that all conditions of Theorem 1.1 are satisfied. By Theorem 1.1, (1.1) has at least a nontrivial homoclinic solution.

Example 4.2.

Then it is easy to verify that all conditions of Theorem 1.2 are satisfied. By Theorem 1.2, (1.1) has at least a nontrivial homoclinic solution.

Example 4.3.

By Theorem 1.4, (1.1) has at least a nontrivial homoclinic solution.

## Declarations

### Acknowledgments

The authors would like to express their thanks to the referees for their helpful suggestions. This paper is partially supported by the NNSF (no: 10771215) of China and supported by the Outstanding Doctor degree thesis Implantation Foundation of Central South University (no: 2010ybfz073).

## Authors’ Affiliations

## References

- Guo Z, Yu J:
**Existence of periodic and subharmonic solutions for second-order superlinear difference equations.***Science in China A*2003,**46**(4):506-515.MathSciNetView ArticleMATHGoogle Scholar - Guo Z, Yu J:
**Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems.***Nonlinear Analysis: Theory, Methods & Applications*2003,**55**(7-8):969-983. 10.1016/j.na.2003.07.019MathSciNetView ArticleMATHGoogle Scholar - Guo Z, Yu J:
**The existence of periodic and subharmonic solutions of subquadratic second order difference equations.***Journal of the London Mathematical Society. Second Series*2003,**68**(2):419-430. 10.1112/S0024610703004563MathSciNetView ArticleMATHGoogle Scholar - Castro A, Shivaji R:
**Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric.***Communications in Partial Differential Equations*1989,**14**(8-9):1091-1100. 10.1080/03605308908820645MathSciNetView ArticleMATHGoogle Scholar - Esteban JR, Vázquez JL:
**On the equation of turbulent filtration in one-dimensional porous media.***Nonlinear Analysis: Theory, Methods & Applications*1986,**10**(11):1303-1325. 10.1016/0362-546X(86)90068-4MathSciNetView ArticleMATHGoogle Scholar - Kaper HG, Knaap M, Kwong MK:
**Existence theorems for second order boundary value problems.***Differential and Integral Equations*1991,**4**(3):543-554.MathSciNetMATHGoogle Scholar - Agarwal RP, Stanek S:
**Existence of positive solutions to singular semi-positone boundary value problems.***Nonlinear Analysis: Theory, Methods & Applications*2002,**51**(5):821-842. 10.1016/S0362-546X(01)00864-1MathSciNetView ArticleMATHGoogle Scholar - Cecchi M, Marini M, Villari G:
**On the monotonicity property for a certain class of second order differential equations.***Journal of Differential Equations*1989,**82**(1):15-27. 10.1016/0022-0396(89)90165-4MathSciNetView ArticleMATHGoogle Scholar - Li W-T:
**Oscillation of certain second-order nonlinear differential equations.***Journal of Mathematical Analysis and Applications*1998,**217**(1):1-14. 10.1006/jmaa.1997.5680MathSciNetView ArticleMATHGoogle Scholar - Marini M:
**On nonoscillatory solutions of a second-order nonlinear differential equation.***Unione Matematica Italiana. Bollettino. C. Serie VI*1984,**3**(1):189-202.MathSciNetMATHGoogle Scholar - Cai X, Yu J:
**Existence theorems for second-order discrete boundary value problems.***Journal of Mathematical Analysis and Applications*2006,**320**(2):649-661. 10.1016/j.jmaa.2005.07.029MathSciNetView ArticleMATHGoogle Scholar - Ma M, Guo Z:
**Homoclinic orbits and subharmonics for nonlinear second order difference equations.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(6):1737-1745. 10.1016/j.na.2006.08.014MathSciNetView ArticleMATHGoogle Scholar - Ma M, Guo Z:
**Homoclinic orbits for second order self-adjoint difference equations.***Journal of Mathematical Analysis and Applications*2006,**323**(1):513-521. 10.1016/j.jmaa.2005.10.049MathSciNetView ArticleMATHGoogle Scholar - Lin X, Tang XH:
**Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems.***Journal of Mathematical Analysis and Applications*2011,**373**(1):59-72. 10.1016/j.jmaa.2010.06.008MathSciNetView ArticleMATHGoogle Scholar - Rabinowitz PH:
*Minimax Metods in Critical Point Theory with Applications in Differential Equations, CBMS Regional Conference Series, no. 65*. American Mathematical Society, Providence, RI, USA; 1986.View ArticleGoogle Scholar - Zhang Z, Yuan R:
**Homoclinic solutions of some second order non-autonomous systems.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(11):5790-5798. 10.1016/j.na.2009.05.003MathSciNetView ArticleMATHGoogle Scholar - Izydorek M, Janczewska J:
**Homoclinic solutions for a class of the second order Hamiltonian systems.***Journal of Differential Equations*2005,**219**(2):375-389. 10.1016/j.jde.2005.06.029MathSciNetView ArticleMATHGoogle Scholar - Mawhin J, Willem M:
*Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences*.*Volume 74*. Springer, New York, NY, USA; 1989:xiv+277.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.