Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations
© Peng Chen and X. H. Tang. 2010
Received: 5 May 2010
Accepted: 2 August 2010
Published: 17 August 2010
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.
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 Poincaré; 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 ).
Assume that and satisfy the following conditions:
for all ;
for all and ;
- (f1)there is a constant such that(1.4)
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 .
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.
Assume that and satisfy the following conditions:
for all ;
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.
Assume that , and satisfy , , and the following assumption:
uniformly in . Then (1.1) possesses a nontrivial homoclinic solution.
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 , 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.
(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.
Assume that , and satisfy , , , and . Then (1.1) possesses at least one nontrivial homoclinic solution.
By Theorem 1.4, we have the following corollary.
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.
Then is a uniform convex Banach space with this norm.
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.
there exist constants such that ;
there exists such that .
where is an open ball in of radius centered at and
The proof is completed.
Assume that and hold. Then for every ,
is nondecreasing on ;
is nonincreasing on .
The proof of Lemma 2.3 is routine and so we omit it.
Lemma 2.4 . (see ).
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.
In this section, we give some examples to illustrate our results.
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.
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.
By Theorem 1.4, (1.1) has at least a nontrivial homoclinic solution.
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).
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