Advances in Difference Equations 
Volume 2008 (2008), Article ID 247071, 11 pages
doi:10.1155/2008/247071
Research Article

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

Xiaochun Cai1 and Jianshe Yu2

1College of Statistics, Hunan University, Changsha, Hunan 410079, China
2College of Mathematics and Information Science, Guangzhou University, Guangzhou 510405, China

Received 14 August 2007; Accepted 14 November 2007

Recommended by Patricia J. Y. Wong

Abstract

The authors consider the second-order nonlinear difference equation of the type Δ(pn(Δxn−1)δ)+qnxnδ=f(n,xn),n∈ℤ, 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.

Acknowledgment

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

References

  1. C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems. Difference Equations, Continued Fractions, and Riccati Equations, vol. 16 of Kluwer Texts in the Mathematical Sciences, Kluwer Academic, Dordrecht, The Netherlands, 1996.
  2. S. S. Cheng, H. J. Li, and W. T. Patula, “Bounded and zero convergent solutions of second-order difference equations,” Journal of Mathematical Analysis and Applications, vol. 141, no. 2, pp. 463–483, 1989.
  3. T. Peil and A. Peterson, “Criteria for C-disfocality of a selfadjoint vector difference equation,” Journal of Mathematical Analysis and Applications, vol. 179, no. 2, pp. 512–524, 1993.
  4. F. Dannan, S. Elaydi, and P. Liu, “Periodic solutions of difference equations,” Journal of Difference Equations and Applications, vol. 6, no. 2, pp. 203–232, 2000.
  5. S. Elaydi and S. Zhang, “Stability and periodicity of difference equations with finite delay,” Funkcialaj Ekvacioj, vol. 37, no. 3, pp. 401–413, 1994.
  6. Z. M. Guo and J. S. Yu, “The existence of periodic and subharmonic solutions for second-order superlinear difference equations,” Science in China Series A, vol. 3, pp. 226–235, 2003.
  7. R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000.
  8. R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, vol. 404 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1997.
  9. M. Cecchi, Z. Došlá, and M. Marini, “Positive decreasing solutions of quasi-linear difference equations,” Computers & Mathematics with Applications, vol. 42, no. 10-11, pp. 1401–1410, 2001.
  10. P. J. Y. Wong and R. P. Agarwal, “Oscillations and nonoscillations of half-linear difference equations generated by deviating arguments,” Computers & Mathematics with Applications, vol. 36, no. 10–12, pp. 11–26, 1998, Advances in difference equations, I.
  11. P. J. Y. Wong and R. P. Agarwal, “Oscillation and monotone solutions of second order quasilinear difference equations,” Funkcialaj Ekvacioj, vol. 39, no. 3, pp. 491–517, 1996.
  12. M. Cecchi, M. Marini, and G. Villari, “On the monotonicity property for a certain class of second order differential equations,” Journal of Differential Equations, vol. 82, no. 1, pp. 15–27, 1989.
  13. M. Marini, “On nonoscillatory solutions of a second-order nonlinear differential equation,” Bollettino della Unione Matematica Italiana, vol. 3, no. 1, pp. 189–202, 1984.
  14. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986.
  15. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
  16. K. C. Chang and Y. Q. Lin, Functional Analysis, Peking University Press, Beijing, China, 1986.