Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations

Consider the nonlinear difference equation of the form

(1.1)

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].

In general, (1.1) may be regarded as a discrete analogue of a special case of the following second-order differential equation:

(1.2)

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.

In some recent papers [1–3, 11–14], the authors studied the existence of periodic solutions, subharmonic solutions, and homoclinic solutions of some special forms of (1.1) by using the critical point theory. These papers show that the critical point method is an effective approach to the study of periodic solutions for difference equations. Along this direction, Ma and Guo [13] applied the critical point theory to prove the existence of homoclinic solutions of the following special form of (1.1):

(1.3)

where , , , and .

Theorem A (see [13]).

Assume that and satisfy the following conditions:

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].

(AR) For every , is continuously differentiable in , and there is a constant  such that

(1.5)

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:

(F1) , for every , and are continuously differentiable in , and there is a bounded set such that

(1.6)

uniformly in ;

(F2) there is a constant such that

(1.7)

(F3) and there is a constant such that

(1.8)

Then (1.1) possesses a nontrivial homoclinic solution.

Theorem 1.2.

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

(F1') , for every , and are continuously differentiable in and

(1.9)

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.

(F4) For every , is continuously differentiable in , and there exist two constants and two functions such that

(1.10)

(F5) There exist two functions and such that

(1.11)

where as , is a positive constant.

(F6) There exist and two constants and such that

(1.12)

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 .

(F8) There exist constants , and such that

(1.13)

(F9) as is a positive constant.

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

Let

(2.1)

and for , let

(2.2)

Then is a uniform convex Banach space with this norm.

As usual, for , let

(2.3)

and their norms are defined by

(2.4)

respectively.

For any with , we let , and for function and , we set

(2.5)

Let be defined by

(2.6)

If , and , or holds, then and one can easily check that

(2.7)

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.

Lemma 2.1 .(see [15, 17]).

Let be a real Banach space and satisfy (PS)-condition. Suppose that satisfies the following conditions:

1. (i)

   ;

    
2. (ii)

   there exist constants such that ;

    
3. (iii)

   there exists such that .

    

Then possesses a critical value given by

(2.8)

where is an open ball in of radius centered at and

Lemma 2.2.

For

(2.9)

where .

Proof.

Since , it follows that . Hence, there exists such that

(2.10)

So, we have

(2.11)

The proof is completed.

Lemma 2.3.

Assume that and hold. Then for every ,

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 .

Proof of Theorem 1.1..

In our case, it is clear that . We show that satisfies the (PS)-condition. Assume that is a sequence such that is bounded and as . Then there exists a constant such that

(3.1)

From (2.6), (2.7), (3.1), (F2), and (F3), we obtain

(3.2)

It follows that there exists a constant such that

(3.3)

Then, is bounded in . Going if necessary to a subsequence, we can assume that in . For any given number , by (F1), we can choose such that

(3.4)

Since , we can also choose an integer such that

(3.5)

By (3.3) and (3.5), we have

(3.6)

Since in , it is easy to verify that converges to pointwise for all , that is,

(3.7)

Hence, we have by (3.6) and (3.7)

(3.8)

It follows from (3.7) and the continuity of on that there exists such that

(3.9)

On the other hand, it follows from (3.3), (3.4), (3.6), and (3.8) that

(3.10)

Since is arbitrary, combining (3.9) with (3.10), we get

(3.11)

It follows from (2.7) and the H lder's inequality that

(3.12)

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

We now show that there exist constants such that satisfies assumption (ii) of Lemma 2.1. By (F1), there exists such that

(3.13)

It follows from that

(3.14)

Set

(3.15)

(3.16)

If , then by Lemma 2.2, for , we have by (q), (3.15), and Lemma 2.3(i) that

(3.17)

Set . Hence, from (2.6), (3.14), (3.17), (q), and (F1), we have

(3.18)

Equation (3.18) shows that implies that , that is, satisfies assumption (ii) of Lemma 2.1. Finally, it remains to show that satisfies assumption (iii) of Lemma 2.1. For any , it follows from (2.9) and Lemma 2.3(ii) that

(3.19)

where

(3.20)

Take such that

(3.21)

and for . For , by Lemma 2.3(i) and (3.21), we have

(3.22)

where . By (2.6), (3.19), (3.21), and (3.22), we have for

(3.23)

Since and , (3.23) implies that there exists such that and . Set . Then , and . By Lemma 2.1, possesses a critical value given by

(3.24)

where

(3.25)

Hence, there exists such that

(3.26)

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.

In the proof of Theorem 1.1, the condition that for , in (F1), is only used in the the proofs of assumption (ii) of Lemma 2.1. Therefore, we only proves assumption (ii) of Lemma 2.1 still hold that using (F1') instead of (F1). By (F1'), there exists such that

(3.27)

Since , it follows that

(3.28)

If , then by Lemma 2.2, for . Set . Hence, from (2.6) and (3.28), we have

(3.29)

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.

In view of Lemma 2.4, . We first show that is bounded from below. By (F4), (2.6), and H lder inequality, we have

(3.30)

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

Next, we prove that satisfies the (PS)-condition. Assume that is a sequence such that is bounded and as . Then by (2.6), (2.9), and (3.30), there exists a constant such that

(3.31)

So passing to a subsequence if necessary, it can be assumed that in . It is easy to verify that converges to pointwise for all , that is,

(3.32)

Hence, we have, by (3.31) and (3.32),

(3.33)

By (F5), there exists such that

(3.34)

For any given number , by (F5), we can choose an integer such that

(3.35)

It follows from (3.32) and the continuity of on that there exists such that

(3.36)

On the other hand, it follows from (3.31), (3.33), (3.34), (3.35), and (F5) that

(3.37)

Since is arbitrary, combining (3.36) with (3.37), we get

(3.38)

Similar to the proof of Theorem 1.1, it follows from (3.12) that

(3.39)

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 .

Finally, we show that . Let and for . Then by (F4) and (F6), we have

(3.40)

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..

Obviously, (F7) and (F8) imply that (F4) holds, and (F7) and (F9) imply that (F5) holds with . In addition, by (F7) and (F8), we have

(3.41)

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.

Example 4.1.

In (1.1), let and

(4.1)

where such that as , , . Let , and

(4.2)

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.

In (1.1), let , for all and , and let

(4.3)

where , . Let , and

(4.4)

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.

In (1.1), let such that as and

(4.5)

Then

(4.6)

We can choose such that

(4.7)

Let

(4.8)

Then

(4.9)

These show that all conditions of Theorem 1.4 are satisfied, where

(4.10)

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