- Research Article
- Open Access

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

- Xiaochun Cai
^{1}Email author and - Jianshe Yu
^{2}

**2008**:247071

https://doi.org/10.1155/2008/247071

© X. Cai and J. Yu. 2008

**Received:**14 August 2007**Accepted:**14 November 2007**Published:**29 November 2007

## 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.

## Keywords

- Periodic Solution
- Difference Equation
- Point Theorem
- Discrete Analogue
- Mountain Pass

## 1. Introduction

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

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 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:

(I_{1}) there exist constants
and
such that
for all
, where

(I_{2})
and there exists
such that

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

(I_{3}) there exist constants
,
such that

(I_{4}) there is
and a constant
such that
.

where

## 2. Preliminaries

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

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:

satisfies P-S condition.

Proof.

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:

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

Proof.

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:

possesses at least one -periodic solution.

First, we proved the following lemma.

Lemma 3.4.

satisfies P-S condition on .

Proof.

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.

where

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:

(G_{3})

(G_{4})

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

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.

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.

## Declarations

### Acknowledgment

This project is supported by specialized research fund for the doctoral program of higher education, Grant no. 20020532014.

## Authors’ Affiliations

## References

- Ahlbrandt CD, Peterson AC:
*Discrete Hamiltonian Systems. Difference Equations, Continued Fractions, and Riccati Equations, Kluwer Texts in the Mathematical Sciences*.*Volume 16*. Kluwer Academic, Dordrecht, The Netherlands; 1996:xiv+374.Google Scholar - Cheng SS, Li HJ, Patula WT: Bounded and zero convergent solutions of second-order difference equations.
*Journal of Mathematical Analysis and Applications*1989, 141(2):463-483. 10.1016/0022-247X(89)90191-1MATHMathSciNetView ArticleGoogle Scholar - Peil T, Peterson A:Criteria for
-disfocality of a selfadjoint vector difference equation.
*Journal of Mathematical Analysis and Applications*1993, 179(2):512-524. 10.1006/jmaa.1993.1366MATHMathSciNetView ArticleGoogle Scholar - Dannan F, Elaydi S, Liu P: Periodic solutions of difference equations.
*Journal of Difference Equations and Applications*2000, 6(2):203-232. 10.1080/10236190008808222MATHMathSciNetView ArticleGoogle Scholar - Elaydi S, Zhang S: Stability and periodicity of difference equations with finite delay.
*Funkcialaj Ekvacioj*1994, 37(3):401-413.MATHMathSciNetGoogle Scholar - Guo ZM, Yu JS: The existence of periodic and subharmonic solutions for second-order superlinear difference equations.
*Science in China Series A*2003, 3: 226-235.Google Scholar - Agarwal RP:
*Difference Equations and Inequalities: Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 228*. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.Google Scholar - Agarwal RP, Wong PJY:
*Advanced Topics in Difference Equations, Mathematics and Its Applications*.*Volume 404*. Kluwer Academic, Dordrecht, The Netherlands; 1997:viii+507.View ArticleGoogle Scholar - Cecchi M, Došlá Z, Marini M: Positive decreasing solutions of quasi-linear difference equations.
*Computers & Mathematics with Applications*2001, 42(10-11):1401-1410. 10.1016/S0898-1221(01)00249-8MATHMathSciNetView ArticleGoogle Scholar - Wong PJY, Agarwal RP: Oscillations and nonoscillations of half-linear difference equations generated by deviating arguments.
*Computers & Mathematics with Applications*1998, 36(10–12):11-26. Advances in difference equations, IIMATHMathSciNetView ArticleGoogle Scholar - Wong PJY, Agarwal RP: Oscillation and monotone solutions of second order quasilinear difference equations.
*Funkcialaj Ekvacioj*1996, 39(3):491-517.MATHMathSciNetGoogle Scholar - Cecchi M, Marini M, Villari G: On the monotonicity property for a certain class of second order differential equations.
*Journal of Differential Equations*1989, 82(1):15-27. 10.1016/0022-0396(89)90165-4MATHMathSciNetView ArticleGoogle Scholar - Marini M: On nonoscillatory solutions of a second-order nonlinear differential equation.
*Bollettino della Unione Matematica Italiana*1984, 3(1):189-202.MATHMathSciNetGoogle Scholar - Rabinowitz PH:
*Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics*.*Volume 65*. American Mathematical Society, Providence, RI, USA; 1986:viii+100.Google Scholar - Mawhin J, Willem M:
*Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences*.*Volume 74*. Springer, New York, NY, USA; 1989:xiv+277.View ArticleGoogle Scholar - Chang KC, Lin YQ:
*Functional Analysis*. Peking University Press, Beijing, China; 1986.Google Scholar

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