Periodic Solutions with Minimal Period for Fourth-Order Nonlinear Difference Equations

Abstract

A fourth-order nonlinear difference equation is considered. By making use of critical point theory, some new criteria are obtained for the existence of periodic solutions with minimal period. The main methods used are a variational technique and the Linking Theorem.

1. Introduction

Let , , and denote the sets of all natural numbers, integers, and real numbers, respectively. denotes the greatest-integer function. For any , define when . denotes the transpose of a vector .

Consider the following fourth-order nonlinear difference equation:where is the forward difference operator ; for , and and are -periodic in for a given positive integer .

Equation (1) can be considered as a discrete analogue of continuous versions of problem likewhich is used to describe the stationary states of the deflection of an elastic beam [1]. Equations similar to (2) arise in the study of the existence of solutions to differential equations; we refer the reader to [2–7] and the references therein.

The theory of nonlinear difference equations has been widely used to study discrete models in many fields such as finance insurance, computing, electrical circuit analysis, dynamical systems, physical field, and biology. Because of their importance, many literature and monographs deal with their existence and uniqueness problems; see [8–27].

Using the critical point theory and monotone operator theory, He and Su [13] studied the following discrete nonlinear fourth-order boundary value problems:with three parameters. Some existence, multiplicity, and nonexistence results of nontrivial solutions are obtained.

Chen and Tang [12] in 2011 were concerned with the existence of infinitely many homoclinic orbits from 0 of the fourth-order difference systemby using the symmetric Mountain Pass Lemma and established some existence criteria to guarantee that (4) has infinitely many homoclinic orbits.

In 2012, Ma and Lu [19] showed the existence and multiplicity of positive solutions of the nonlinear discrete fourth-order boundary value problem forby using Dancer’s global bifurcation theorem.

Liu et al. [18] studied the existence and multiplicity of periodic and subharmonic solutions to the following nonlinear difference equation:using variational technique and the Linking Theorem.

By employing the variational methods, Yu et al. [22] got some new criteria for the existence of subharmonic solutions with prescribed minimal period of second-order nonlinear difference equation

Applying the direct method of the calculus of variations and the mountain pass technique, Leszczyński [17] in 2015 proved the existence of at least one and at least two solutions to the fourth-order discrete anisotropic boundary value problem with both advance and retardation of formNonexistence of nontrivial solutions was also obtained.

Motivated by the recent papers [12, 16], our purpose in this work is to apply Linking Theorem in critical point theory to establish some conditions for the nonlinear function which are able to guarantee the existence of at least two nontrivial periodic solutions with minimal period for the above problem.

Throughout this paper, we suppose that is a given integer and . Let To wit, we get the following.

Theorem 1. Assume that the following hypotheses are satisfied:there exists a function with , , and and it satisfies there exist positive constants , such that there exist positive constants , such that there exist positive constants and such that if is a solution of (1) with minimal period , is a rational number, and also has a minimal period , then must be an integer.let be the least prime factor of , Then (1) has at least two nontrivial periodic solutions with minimal period .

Theorem 2. Assume that , , and the following conditions are satisfied: , ;there exist positive constants and such that for and , Then (1) has at least two nontrivial periodic solutions with minimal period .

2. Variational Structure and Some Lemmas

Define the functional as follows: on the finite-dimensional Hilbert space where

For , define

For , define The argument need not be changed and we omit it.

can be rewritten as where .

Clearly, and for any , by using , , we can compute the partial derivative as Therefore, is a critical point of on if and only if Since and in the first variable are -periodic in , we can reduce the existence of periodic solutions of (1) to the existence of critical points of on .

By matrix theory, the eigenvalues of can be given byIt is easy to see that 0 is an eigenvalue of and , , and

Denote Therefore,

Let be the direct orthogonal complement of to ; that is, .

For , the eigenvectors of corresponding to are

When is even, the eigenvector corresponding to 4 is . Let , , and . Then . For any , where , , and are constants.

When is odd, . For any , where , , , and are constants.

Denote Then and

Let denote the open ball in about 0 of radius and let denote its boundary.

Lemma 3 (Linking Theorem [1]). Let be a real Banach space, , where is finite dimensional. Assume that satisfies the PS condition andthere exist constants and such that ;there exists an and a constant such that , where . Then possesses a critical value , where and , where denotes the identity operator.

Lemma 4. Assume that the hypotheses are satisfied. Then the functional is bounded from above in .

Proof. According to and (21), for any , where . Since we getThe desired results are obtained.

Lemma 5. Assume that the hypotheses are satisfied. Then the functional satisfies the PS condition.

Proof. Let be a bounded sequence from below, that is, there exists a constant such that Due to the proof of Lemma 4, it is obvious that Thus, That is, is a bounded sequence in the finite-dimensional space . Consequently, it has a convergent subsequence. The proof is finished.

Lemma 6. If is a critical point of on , then is a critical point of on .

In a similar fashion to the proof of Lemma 4 and the process in [22], we can prove Lemma 6. The detailed proof is omitted.

Set

Lemma 7. Assume that the hypotheses are satisfied. If is a critical point of on , , then has minimal period .

Proof. If not, there is a positive integer such that has minimal period . By , . For any , and then where . Since we get Then which contradicts . The proof is complete.

3. Proof of Results

Proof of Theorem 1. It comes from Lemma 4 that the functional is bounded from above on .
Take On one hand, there exists a sequence on such that On the other hand, from (36), we get Thus, , which implies that is bounded. Therefore, has a convergent subsequence, denoted by . Set Due to the continuity of , it is obvious that . That is, is a critical point of on .
Assumption implies that, for any , , where . Since then Let . Therefore, the functional Thus, we have proved that . At the same time, we have also proved that there exist constants and such that .
By , and assumption , Then and the critical point of corresponding to the critical value is a nontrivial periodic solution of (1).
Let . Then, for any and , let . Then where . Since then Therefore, there is a constant such that, for any , , where . By the Linking Theorem, possesses a critical value , where and .
Similar to the proof of [18], we can prove that (1) has at least two -periodic nontrivial solutions. For simplicity, its proof is omitted.
By , for , we have Then, for , Take . From , we have where is a constant. Note that ; we get Therefore, we have Since then It comes from Lemma 7 that the desired result is obtained.

Remark 8. Similarly to the above argument, we can also prove Theorem 2. For simplicity, we omit its proof.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (no. 11501194) and Philosophy and Social Sciences Planning Research Project in Guangxi (no. 15FGL008). This work was carried out while visiting Central South University. The author Haiping Shi wishes to thank Professor Xianhua Tang for his invitation.