Nontrivial Periodic Solutions to Some Semilinear Sixth-Order Difference Equations

Abstract

We establish some new criteria to guarantee nonexistence, existence, and multiplicity of nontrivial periodic solutions of some semilinear sixth-order difference equations by using minmax method, index theory, and variational technique. Our results only make some assumptions on the period , which are very easy to verify and rather relaxed.

1. Introduction

In the present paper, we deal with the following sixth-order difference equation: with , where is an integer and denotes the discrete interval . is the forward difference operator defined by and . and are positive constants satisfying .

By using index theory in combination with variational technique, we will prove nonexistence, existence, and multiplicity of nontrivial periodic solutions to (1) under convenient assumptions on . All our results only depend on and and are easy to satisfy.

Periodic solution problems for difference equations have been extensively studied (see the monographs of Lakshmikantham and Trigiante [1] and of Agarwal [2]). The classical theory of difference equations employs numerical analysis and features from the linear and nonlinear operator theory, such as fixed point methods; we remark that, usually, the applications of fixed point methods yield existence results only. Recently, although many new results have been established by applying variational methods, we recall here the works of Cai and Yu [3], Guo and Yu [4], and Deng et al. [5]. The variational approach represents an important advance as it allows proving multiplicity results as well.

In general, (1) may be regarded as a discrete analogue of the following sixth-order differential equation: As to (2), it is a model for describing the behavior of phase fronts in materials that are undergoing a transition between the liquid and solid state and be widely studied; one can refer to [6–8] and references therein.

Difference equations, the discrete analogs of differential equations, represent the discrete counterpart of ordinary differential equations and are usually studied in connection with numerical analysis. They occur widely in numerous settings and forms, both in mathematics itself and in its applications to computing, statistics, electrical circuit analysis, biology, dynamical systems, economics, and other fields. For the general background of difference equations, one can refer to monographs [2, 9–13] for details.

Since 2003, critical point theory has been a powerful tool to establish sufficient conditions on the existence of periodic solutions of difference equations and many significant results have been obtained; see, for example, [4, 14, 15]. Compared to first-order or second-order difference equations, the study of higher order difference equations has received considerably less attention. For example, [16] studied in the context of discrete calculus of variational functional, and Peil and Peterson [17] studied the asymptotic behaviour of solutions of (3) with for . In 2007, based on Linking Theorem, [3] gave some criteria for the existence of periodic solutions of for the case where grows superlinear at both 0 and . Results in [3] made many assumptions on and they are not easy to verify. The aim of this paper is to apply critical point theory to deal with the periodic solution problems of (1) when it is semilinear under concise and explicit assumptions on the period . The main results of this paper are the following three theorems.

Theorem 1. Let , , and integer satisfy where is defined as then (1) has only the trivial solution.

Theorem 2. Let , , and integer satisfy where is defined as there exists a nontrivial solution of (1).

Theorem 3. Let , , and for some ; there exist at least geometrically distinct nontrivial solutions of (1).

The remaining of the paper is organized as follows. In Section 2 we establish the variational framework associated with (1) and transfer the problem on the existence of periodic solutions of (1) into the existence of critical points of the corresponding functional. We also state some fundamental lemmas for later use. Then we present the detailed proofs of main results in Section 3. Finally, we exhibit a simple example to illustrate our conclusions.

2. Preliminaries

In order to study the periodic solutions of (1), we will state some basic notations and lemmas, which will be used in the proofs of our main results. Let For a given integer , is defined as a subspace of by and for , let Then is a finite dimensional Hilbert space with above inner product, and the induced norm is As usual, for , let and its norm is defined by

Define the functional as follows: Clearly, and for any one can easily check that For any , by using for any , We can compute the partial derivative as Then, is a critical point of on ; that is, if and only if By the periodicity of , we have reduced the existence of periodic solutions of (1) to the existence of critical points of on . For convenience, we identity with , so we draw a conclusion as follows.

Lemma 4. Suppose that is a critical point of the functional ; then is a -periodic solution of (1).

We provide some lemmas which will be needed in proofs of our main results.

Lemma 5. For any , , , and , where , , and .

Lemma 6. Let be a critical point of ; for every , there hold

Proof. Let be a critical point of , according to the definition of and the periodicity of and ; then we have Similarly, we get
Let be a real Banach space. is the subset of , which is closed and symmetric with respect to 0; that is, For any , the geometric index, also called genus, of is defined by when there exists no such finite , set . Finally set .

The following lemma is trivial.

Lemma 7 (Chang [18] and Rabinowitz [19]). Let be a subset of with dimension and the unit sphere of and then let .

Next, let us recall the definition of Palais-Smale condition.

Let be a real Banach space, . is a continuously Frechet differentiable functional defined on . is said to be satisfied Palais-Smale condition (P.S. condition for short) if any sequence for which is bounded and possesses a convergent subsequence in .

Lemma 8 (see [20]). Let be a real Banach space and let be a functional and satisfy P.S. condition. If is bound from below, then is a critical value of .

Lemma 9 (see [19]). Let be an even functional defined on and satisfy P.S. condition. For positive integer , define Then and(1)assume ; then is a critical value of ;(2)if , then , where .

3. Proofs of Main Results

With the above preparations, we will prove our main results in this section. In order to give proofs of theorems, we need the following lemmas.

Lemma 10. For any , if is a critical of , we have

Proof. From Agarwal [2], we have inequality (30) and then that is inequality (31).
From Lemmas 5 and 6, there holds and it follows Using inequality (31), we have inequality (32) is true.

Lemma 11. Suppose and , is a critical point of , and then

Proof. From Lemmas 5 and 6, we have so inequality (37) holds.
To prove inequality (38), let Similar to the proof of inequality (37), we get inequality (38).

Now we will give the proof of Theorem 1.

Proof of Theorem 1. Suppose all conditions of Theorem 1 hold. Let be a nontrivial solution of (1); then
By (37), (30), (5), and (6), we have which is a contradiction. Moreover, by (38), (31), (5), and (6), we have which is a contradiction. Thus should be the trivial solution of (1); in other words, (1) has no nontrivial solution when (5) and (6) hold.

To apply Lemmas 8 and 9 to look for nontrivial solutions for (1), next we prove that satisfies P.S. condition.

Lemma 12. Let and , and then the functional is bounded from below on and satisfies P.S. condition.

Proof. Denote , and then . From (38) and the elementary inequality there hold which means that the functional is bounded from below on .
Next we will show that satisfies P.S. condition. Suppose satisfy that is a bounded sequence from above; that is, there exists a positive constant such that By (38), we have and it follows And together with (32), there holds From (37), it follows and then thus Therefore, by (48)–(52), we get that there exists a positive constant such that that is, As a consequence, possesses a convergence subsequence in the finite dimensional Hilbert space and satisfies P.S. condition. This completes the proof of Lemma 12.

From Lemmas 8 and 12, there exists is a critical value of , which means that there exists a critical point of on . Next, we devote ourselves to verifying the critical point is nontrivial.

Set In Theorem 1, we have shown that if . To complete the proof of Theorem 2, we will show this does not hold for large . We prove that for , where is an appropriate number defined as (8), it holds and the corresponding critical point is nontrivial.

Proof of Theorem 2. By Lemma 12, is nontrivial. So if we can find some critical point such that , then is a nontrivial solution of (1). Let us take where will be chosen later. By direct computation, it follows Since if for sufficiently small , we have . We show that there exists , when , such that (58) is true. Denote , inequality (58) is equivalent to The roots of the polynomial are where and . Therefore, (59) holds for and (58) holds for ; that is, . Then (1) has a nontrivial solution for , where is defined as (8).

We can prove the multiplicity result in Theorem 3 using Lemma 9.

Denote the open ball in with radius about 0; is the boundary of .

Proof of Theorem 3. For integer , let us consider the subset , where is an appropriate positive number. Give an odd continuous mapping As defined in (26), by the monotonicity and superinvariance of genus and Lemma 7, for any , we have is also a subset of the -dimensional space equipped with the norm Then for every there exist positive constants and such that In particular, Since for , we have For , . Using (67) and (69), there holds for sufficiently small . It follows where is defined in Lemma 9.
According to Lemma 9, has at least geometrically distinct critical points. Furthermore, ; similar to the proof of Theorem 2, we draw a conclusion that all distinct critical points we have obtained are all nontrivial. And the proof of Theorem 3 is completed.

Finally, we exhibit a simple example to illustrate our conclusions.

Example. Consider system (1) with and .

Solution. Here , , , and and it follows we can choose . When , (1) has a nontrivial solution. If one let , where is a integer, then (1) has distinct nontrivial solution.

Remark. From the given example, one can find our results only depend on coefficients and which are very easy to verify and rather relaxed.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author thanks the anonymous referee for his/her valuable suggestions. This work was supported by National Natural Science Foundation of China (no. 11101098), the Foundation of Guangzhou Education Bureau (no. 2012A019 ), and PCSIR (no. IRT1226).