Existence Theorems of Periodic Solutions for Second-Order Nonlinear Difference Equations

Abstract

The authors consider the second-order nonlinear difference equation of the type using critical point theory, and they obtain some new results on the existence of periodic solutions.

1. Introduction

We denote by the set of all natural numbers, integers, and real numbers, respectively. For , define when .

Consider the nonlinear second-order difference equation

(1.1)

where the forward difference operator is defined by the equation and

(1.2)

In (1.1), the given real sequences satisfy for any , is continuous in the second variable, and for a given positive integer and for all . and is the ratio of odd positive integers. By a solution of (1.1), we mean a real sequence , satisfying (1.1).

In [1, 2], the qualitative behavior of linear difference equations of type

(1.3)

has been investigated. In [3], the nonlinear difference equation

(1.4)

has been considered. However, results on periodic solutions of nonlinear difference equations are very scarce in the literature, see [4, 5]. In particular, in [6], by critical point method, the existence of periodic and subharmonic solutions of equation

(1.5)

has been studied. Other interesting contributions can be found in some recent papers [7–11] and in references contained therein. It is interesting to study second-order nonlinear difference equations (1.1) because they are discrete analogues of differential equation

(1.6)

In addition, they do have physical applications in the study of nuclear physics, gas aerodynamics, infiltrating medium theory, and plasma physics as evidenced in [12, 13].

The main purpose here is to develop a new approach to the above problem by using critical point method and to obtain some sufficient conditions for the existence of periodic solutions of (1.1).

Let be a real Hilbert space, , , which implies that is continuously Fréchet differentiable functional defined on . is said to be satisfying Palais-Smale condition (P-S condition) if any sequence is bounded, and as possesses a convergent subsequence in . Let be the open ball in with radius and centered at , and let denote its boundary.

Lemma 1.1 (mountain pass lemma, see [14]).

Let be a real Hilbert space, and assume that satisfies the P-S condition and the following conditions:

(I₁) there exist constants and such that for all , where

(I₂) and there exists such that

Then is a positive critical value of , where

(1.7)

Lemma 1.2 (saddle point theorem, see [14, 15]).

Let be a real Banach space, where and is finite dimensional. Suppose satisfies the P-S condition and

(I₃) there exist constants , such that

(I₄) there is and a constant such that .

Then possesses a critical value and

(1.8)

where

2. Preliminaries

In this section, we are going to establish the corresponding variational framework for (1.1).

Let be the set of sequences

(2.1)

that is,

(2.2)

For any , is defined by

(2.3)

Then is a vector space. For given positive integer is defined as a subspace of by

(2.4)

Clearly, is isomorphic to , and can be equipped with inner product

(2.5)

by which the norm can be induced by

(2.6)

It is obvious that with the inner product defined by (2.5) is a finite-dimensional Hilbert space and linearly homeomorphic to . Define the functional on as follows:

(2.7)

where . Clearly, , and for any , by using , we can compute the partial derivative as

(2.8)

Thus is a critical point of on if and only if

(2.9)

By the periodicity of and in the first variable , we have reduced the existence of periodic solutions of (1.1) to that of critical points of on . In other words, the functional is just the variational framework of (1.1). For convenience, we identify with . Denote and such that . Denote other norm on as follows (see, e.g., [16]): , for all and . Clearly, . Due to and being equivalent when there exist constants , , , and such that , , and

(2.10)

(2.11)

for all , and .

3. Main Results

In this section, we will prove our main results by using critical point theorem. First, we prove two lemmas which are useful in the proof of theorems.

Lemma 3.1.

Assume that the following conditions are satisfied:

(F₁) there exist constants , , and such that

(3.1)

(F₂)

(3.2)

Then the functional

(3.3)

satisfies P-S condition.

Proof.

For any sequence with being bounded and as there exists a positive constant such that Thus, by ,

(3.4)

Set

(3.5)

Then . Also, by the above inequality, we have

(3.6)

In view of

(3.7)

we have

(3.8)

Then we get

(3.9)

Therefore, for any ,

(3.10)

Since the above inequality implies that is a bounded sequence in Thus possesses convergent subsequences, and the proof is complete.

Theorem 3.2.

Suppose that and following conditions hold:

for each ,

(3.11)

(3.12)

Then there exist at least two nontrivial -periodic solutions for (1.1).

Proof.

We will use Lemma 1.1 to prove Theorem 3.2. First, by Lemma 3.1, satisfies P-S condition. Next, we will prove that conditions and hold. In fact, by , there exists such that for any and

(3.13)

where Thus for any for all we have

(3.14)

Taking we have

(3.15)

and the assumption is verified. Clearly, For any given with and a constant

(3.16)

Thus we can easily choose a sufficiently large such that and for Therefore, by Lemma 1.1, there exists at least one critical value We suppose that is a critical point corresponding to , that is, and By a similar argument to the proof of Lemma 3.1, for any there exists such that . Clearly, If and the proof is complete; otherwise, and By Lemma 1.1,

(3.17)

where Then for any By the continuity of in , and show that there exists some such that If we choose such that the intersection is empty, then there exist such that Thus we obtain two different critical points , of in . In this case, in fact, we may obtain at least two nontrivial critical points which correspond to the critical value The proof of Theorem 3.2 is complete. When , we have the following results.

Theorem 3.3.

Assume that the following conditions hold:

(G₁)

(3.18)

(G₂)

(3.19)

where is a constant in (2.10), and is the minimal positive eigenvalue of the matrix

(3.20)

Then equation

(3.21)

possesses at least one -periodic solution.

First, we proved the following lemma.

Lemma 3.4.

Assume that holds, then the functional

(3.22)

satisfies P-S condition on .

Proof.

For any sequence with being bounded and as there exists a positive constant such that In view of and

(3.23)

we have

(3.24)

By , the above inequality implies that is a bounded sequence in . Thus possesses a convergent subsequence, and the proof of Lemma 3.4 is complete. Now we prove Theorem 3.3 by the saddle point theorem.

Proof of Theorem 3.3.

For any we have

(3.25)

Take then

(3.26)

Set

(3.27)

then we have

(3.28)

On the other hand, for any we have

(3.29)

where

Clearly, is an eigenvalue of the matrix and is an eigenvector of corresponding to , where . Let be the other eigenvalues of . By matrix theory, we have for all . Without loss of generality, we may assume that then for any

(3.30)

as one finds by minimizing with respect to That is

(3.31)

Set

(3.32)

then by , we have

(3.33)

This implies that the assumption of saddle point theorem is satisfied. Thus there exists at least one critical point of on , and the proof is complete. When we have the following result.

Theorem 3.5.

Assume that the following conditions are satisfied:

(G₃)

(G₄)

where

Then (3.21) possesses at least one -periodic solution.

Before proving Theorem 3.5, first, we prove the following result.

Lemma 3.6.

Assume that holds, then defined by (3.22) satisfies P-S condition.

Proof.

For any sequence with being bounded and as there exists a positive constant such that

Thus

(3.34)

That is,

(3.35)

By , the above inequality implies that is a bounded sequence in Thus possesses convergent subsequences, and the proof is complete.

Proof of Theorem 3.5.

For any we have

(3.36)

Take then

(3.37)

Set

(3.38)

then for all On the other hand, for any we have

(3.39)

Set then Thus satisfies the assumption of saddle point theorem, that is, there exists at least one critical point of on This completes the proof of Theorem 3.5.