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Hindawi Publishing Corporation Advances in Difference Equations Volume 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 Cai 1 and Jianshe Yu 2 1 College of Statistics, Hunan University, Changsha, Hunan 410079, China 2 College of Mathematics and Information Science, Guangzhou University, Guangzhou 510405, China Correspondence should be addressed to Xiaochun Cai, cxchn8@hnu.cn Received 14 August 2007; Accepted 14 November 2007 Recommended by Patricia J. Y. Wong The authors consider the second-order nonlinear difference equation of the type Δp n Δx n−1  δ  q n x δ n  fn, x n ,n∈ Z, using critical point theory, and they obtain some new results on the existence of periodic solutions. Copyright q 2008 X. Cai and J. Yu. This is an open access article distributed under the Creative Commons Attribution License, which p ermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction We denote by N, Z, R the set of all natural numbers, integers, and real numbers, respectively. For a, b ∈ Z, define Za{a, a  1, }, Za, b{a, a  1, ,b} when a ≤ b. Consider the nonlinear second-order difference equation Δ  p n  Δx n−1  δ   q n x δ n  f  n, x n  ,n∈ Z, 1.1 where the forward difference operator Δ is defined by the equation Δx n  x n1 − x n and Δ 2 x n−1 ΔΔx n−1 Δx n − Δx n−1 . 1.2 In 1.1, the given real sequences {p n }, {q n } satisfy p nT  p n > 0,q nT  q n for any n ∈ Z, f : Z × R → R is continuous in the second variable, and fn  T, zfn, z for a given positive integer T and for all n, z ∈ Z × R. −1 δ  −1,δ>0, and δ is the ratio of odd positive integers. By a solution of 1.1, we mean a real sequence x  {x n },n∈ Z, satisfying 1.1. In 1, 2, the qualitative behavior of linear difference equations of type Δp n Δx n q n x n  0 1.3 2 Advances in Difference Equations has been investigated. In 3, the nonlinear difference equation Δp n Δx n−1 q n x n  fn, x n 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 Δ 2 x n−1  f  n, x n   0,n∈ Z, 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 ptϕu     ft, u0. 1.6 In addition, they do have physical applications in the study of nuclear physics, gas aerody- namics, 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 X be a real Hilbert space, I ∈ C 1 X, R, which implies that I is continuously Fr ´ echet differentiable functional defined on X. I is said to be satisfying Palais-Smale condition P-S condition if any sequence {Iu n } is bounded, and I  u n  → 0asn →∞possesses a conver- gent subsequence in X.LetB ρ be the open ball in X with radius ρ andcenteredat0,andlet ∂B ρ denote its boundary. Lemma 1.1 mountain pass lemma, see 14. Let X be a real Hilbert space, and assume that I ∈ C 1 X, R satisfies the P-S condition and the following conditions: I 1  there exist constants ρ>0 and a>0 such that Ix ≥ a for all x ∈ ∂B ρ ,whereB ρ  {x ∈ X : x X <ρ}; I 2  I0 ≤ 0 and there exists x 0 ∈ B ρ such that Ix 0  ≤ 0. Then c  inf h∈Γ sup s∈0,1 Ihs is a positive critical value of I,where Γ  h ∈ C  0, 1,X  : h00,h1x 0  . 1.7 Lemma 1.2 saddle point theorem, see 14, 15. Let X be a real Banach space, X  X 1 ⊕ X 2 , where X 1 /  {0} and is finite dimensional. Suppose I ∈ C 1 X, R satisfies the P-S condition and I 3  there exist constants σ, ρ>0 such that I| ∂B ρ  X 1 ≤ σ; I 4  there is e ∈ B p  X 1 and a constant ω>σsuch that I| eX 2 ≥ ω. Then I possesses a critical value c ≥ ω and c  inf h∈Γ max u∈B ρ  X 1 Ihu, 1.8 where Γ{h ∈ C B ρ  X 1, X|h| ∂B ρ  X 1  id}. X. Cai and J. Yu 3 2. Preliminaries In this section, we are going to establish the corresponding variational framework for 1.1. Let Ω be the set of sequences x   x n  n∈Z   ,x −n , ,x −1 ,x 0 ,x 1 , ,x n ,  , 2.1 that is, Ω  x   x n  : x n ∈ R,n∈ Z  . 2.2 For any x, y ∈ Ω,a,b∈ R, ax  by is defined by ax  by :  ax n  by n  ∞ n−∞ . 2.3 Then Ω is a vector space. For given positive integer T, E T is defined as a subspace of Ω by E T   x   x n  ∈ Ω : x nT  x n ,n∈ Z  . 2.4 Clearly, E T is isomorphic to R T , and can be equipped with inner product x, y  T  i1 x i y i , ∀x, y ∈ E T , 2.5 by which the norm · can be induced by x :  T  i1 x 2 i  1/2 , ∀x ∈ E T . 2.6 It is obvious that E T with the inner product defined by 2.5 is a finite-dimensional Hilbert space and linearly homeomorphic to R T . Define the functional J on E T as follows: Jx 1 δ  1 T  n1 p n Δx n−1  δ1 − 1 δ  1 T  n1 q n x δ1 n  T  n1 Fn, x n , ∀x ∈ E T , 2.7 where Ft, z  z 0 ft, sds. Clearly, J ∈ C 1 E T , R, and for any x  {x n } n∈Z ∈ E T , by using x 0  x T ,x 1  x T1 , we can compute the partial derivative as ∂J ∂x n  −Δp n Δx n−1  δ  − q n x δ n  fn, x n ,n∈ Z1,T. 2.8 Thus x  {x n } n∈Z is a critical point of J on E T i.e., J  x0 if and only if Δp n Δx n−1  δ q n x δ n  fn, x n ,n∈ Z1,T. 2.9 By the periodicity of x n and fn, z in the first variable n, we have reduced the existence of periodic solutions of 1.1 to that of critical points of J on E T . In other words, the func- tional J is just the variational framework of 1.1. For convenience, we identify x ∈ E T with x x 1 ,x 2 , ,x T  T .DenoteW  {x 1 ,x 2 , ,x T  T ∈ E T : x i ≡ v, v ∈ R,i∈ Z1,T} and W ⊥  Y such that E T  Y ⊕ W. Denote other norm · r on E T as follows see, e.g., 16: x r   T i1 |x i | r  1/r , for all x ∈ E T and r>1. Clearly, x 2  x.Dueto· r 1 and · r 2 being equivalent when r 1 ,r 2 > 1, there exist constants c 1 , c 2 , c 3 ,andc 4 such that c 2 ≥ c 1 > 0, c 4 ≥ c 3 > 0, and c 1 x≤x δ1 ≤ c 2 x, 2.10 c 3 x≤x β ≤ c 4 x, 2.11 for all x ∈ E T , δ>0andβ>1. 4 Advances in Difference Equations 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: F 1  there exist constants a 1 > 0, a 2 > 0,andβ>δ 1 such that  z 0 fn, sds ≤−a 1 |z| β  a 2 , ∀z ∈ R; 3.1 F 2  q n ≤ 0, ∀n ∈ Z. 3.2 Then the functional Jx 1 δ  1 T  n1 p n Δx n−1  δ1 − 1 δ  1 T  n1 q n x δ1 n  T  n1 Fn, x n 3.3 satisfies P-S condition. Proof. For any sequence {x l }⊂E T , with Jx l  being bounded and J  x l  → 0as l → ∞, there exists a positive constant M such that |Jx l |≤M. Thus, by F 1 , −M ≤ Jx l  1 δ  1 T  n1  p n  x l n − x l n−1  δ1 − q n  x l n  δ1   T  n1 F  n, x l n  ≤ 1 δ  1 T  n1 p n 2 δ1   x l n  δ1   x l n−1  δ1  − 1 δ  1 T  n1 q n  x l n  δ1  T  n1 F  n, x l n  ≤ 2 δ1 δ  1 T  n1 p n  p n1   x l n  δ1 − 1 δ  1 T  n1 q n  x l n  δ1 − a 1 T  n1    x l n    β  a 2 T  1 δ  1 T  n1  2 δ1 p n  p n1  − q n  x l n  δ1 − a 1   x l   β β  a 2 T. 3.4 Set A 0  max n∈Z1,T 2 δ1 p n  p n1  − q n . 3.5 Then A 0 > 0. Also, by the above inequality, we have −M ≤ Jx l  ≤ A 0 δ  1   x l   δ1 δ1 − a 1   x l   β β  a 2 T. 3.6 X. Cai and J. Yu 5 In view of T  n1   x l n   δ1 ≤ T β−δ−1/β  T  n1   x l n   β  δ1/β , 3.7 we have   x l   β β ≥ T δ1−β/δ1   x l   β δ1 . 3.8 Then we get −M ≤ Jx l  ≤ A 0 δ  1   x l   δ1 δ1 − a 1 T δ1−β/δ1   x l   β δ1  a 2 T. 3.9 Therefore, for any l ∈ N, a 1 T δ1−β/δ1   x l   β δ1 − A 0 δ  1   x l   δ1 δ1 ≤ M  a 2 T. 3.10 Since β>δ1, the above inequality implies that {x l } is a bounded sequence in E T . Thus {x l } possesses convergent subsequences, and the proof is complete. Theorem 3.2. Suppose that F 1  and following conditions hold: F 3  for each n ∈ Z, lim z→0 fn, z z δ  0; 3.11 F 4  q n < 0, ∀n ∈ Z1,T. 3.12 Then there exist at least two nontrivial T-periodic solutions for 1.1. Proof. We will use Lemma 1.1 to prove Theorem 3.2.First,byLemma 3.1, J satisfies P-S condi- tion. Next, we will prove that conditions I 1  and I 2  hold. In fact, by F 3 , there exists ρ>0 such that for any |z| <ρand n ∈ Z1,T, |Fn, z|≤ − q max 2δ  1 z δ1 , 3.13 where q max  max n∈Z1,T q n < 0. Thus for any x ∈ E T , x≤ρ for all n ∈ Z1,T, we have Jx ≥− q max δ  1 T  n1 x δ1 n  q max 2δ  1 T  n1 x δ1 n  − q max 2δ  1 x δ1 δ1 ≥− q max 2δ  1 c δ1 1 x δ1 2 . 3.14 6 Advances in Difference Equations Taking a  −c δ1 1 q max /2δ  1ρ δ1 , we have Jx| ∂B ρ ≥ a>0, 3.15 and the assumption I 1  is verified. Clearly, J00. For any given w ∈ E T with w  1anda constant α>0, Jαw 1 δ  1 T  n1 p n αw n − αw n−1  δ1 − q n αw n  δ1  T  n1 Fn, αw n  ≤ 1 δ  1 T  n1 p n 2α δ1 − q n α δ1  − a 1 T  n1 |αw n | β  a 2 T ≤ 1 δ  1 T  n1 2 δ1 p n − q n α δ1 α δ1 − a 1 T 2−β/2 α β  a 2 T −→ − ∞ , α −→  ∞. 3.16 Thus we can easily choose a sufficiently large α such that α>ρand for x  αw ∈ E T , Jx < 0. Therefore, by Lemma 1.1, there exists at least one critical value c ≥ a>0. We suppose that x is a critical point corresponding to c,thatis,Jxc, and J  x0. By a similar argument to the proof of Lemma 3.1, for any x ∈ E T , there exists x ∈ E T such that J  xc max . Clearly, x /  0. If x /  x, and the proof is complete; otherwise, x  x and c  c max . By Lemma 1.1, c  inf h∈Γ sup s∈0,1 J  hs  , 3.17 where Γ{h ∈ C0, 1,E T  | h00,h1x}. Then for any h ∈ Γ,c max  max s∈0,1 Jhs. By the continuity of Jhs in s, J0 ≤ 0andJ x < 0 show that there exists some s 0 ∈ 0, 1 such that Jhs 0   c max . If we choose h 1 ,h 2 ∈ Γ such that the intersection {h 1 s | s ∈ 0, 1}  {h 2 s | s ∈ 0, 1} is empty, then there exist s 1 ,s 2 ∈ 0, 1 such that Jh 1 s 1   Jh 2 s 2   c max . Thus we obtain two different critical points x 1  h 1 s 1 , x 2  h 2 s 2  of J in E T . In this case, in fact, we may obtain at least two nontrivial critical points which correspond to the critical value c max . The proof of Theorem 3.2 is complete. When fn, x n  ≡ h n ,wehave the following results. Theorem 3.3. Assume that the following conditions hold: G 1  qn < 0, ∀n ∈ Z1,T; 3.18 G 2  1 c δ1 1  T  n1 h 2 n  δ1/2 T  n1 −q n  <  p min λ δ1/2 2 − q max   T  n1 h n  δ1 , 3.19 X. Cai and J. Yu 7 where p min  min n∈Z1,T p n ,q max  max n∈Z1,T q n ,c 1 is a constant in 2.10,andλ 2 is the minimal positive eigenvalue of the matrix A  ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 2 −10··· 0 −1 −12−1 ··· 00 0 −12··· 00 ··· ··· ··· ··· ··· ··· 000··· 2 −1 −10 0··· −12 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ T×T . 3.20 Then equation Δ  p n Δx n−1  δ  q n x δ n  h n ,n∈ Z, 3.21 possesses at least one T-periodic solution. First, we proved the following lemma. Lemma 3.4. Assume that G 1  holds, then the functional Jx 1 δ  1 T  n1 p n  Δx n−1  δ1 − 1 δ  1 T  n1 q n x δ1 n  T  n1 h n x n 3.22 satisfies P-S condition on E T . Proof. For any sequence {x l }⊂E T with Jx l  being bounded and J  x l  → 0asn → ∞, there exists a positive constant M such that |Jx l |≤M. In view of G 3  and T  n1 |h n x l n |≤  T  n1 h 2 n  1/2  T  n1  x l n  2  1/2 , 3.23 we have M ≥ Jx l  1 δ  1 T  n1  p n  Δx l n−1  δ1  − 1 δ  1 T  n1 q n  x l n  δ1  T  n1 h n x l n ≥− 1 δ  1 T  n1 q n  x l n  δ1 − T  n1   h n x l n   ≥− 1 δ  1 q max T  n1  x l n  δ1 −  T  n1 h 2 n  1/2  T  n1  x l n  2  1/2  − q max δ  1   x l   δ1 δ1 −  T  n1 h 2 n  1/2   x l   ≥− q max δ  1 c δ1 1   x l   δ1 −  T  n1 h 2 n  1/2   x l   . 3.24 By δ  1 > 1, the above inequality implies that {x l } is a bounded sequence in E T .Thus{x l } possesses a convergent subsequence, and the proof of Lemma 3.4 is complete. Now we prove Theorem 3.3 by the saddle point theorem. 8 Advances in Difference Equations Proof of Theorem 3.3. For any w z,z, ,z T ∈ W, we have Jw− 1 δ  1 T  n1 q n z δ1  T  n1 h n z. 3.25 Take z   T n1 h n /  T n1 q n  1/δ and ρ  w  T 1/2 |  T n1 h n /  T n1 q n | 1/δ , then Jw δ δ  1   T n1 h n  δ1/δ     T n1 q n    1/δ . 3.26 Set σ  δ δ  1   T n1 h n  δ1/δ     T n1 q n    1/δ , 3.27 then we have Jwσ, ∀w ∈ ∂B ρ  Y. 3.28 On the other hand, for any x ∈ Y, we have Jx 1 δ  1 T  n1 p n  Δx n−1  δ1 − 1 δ  1 T  n1 q n x δ1 n  T  n1 h n x n ≥ p min δ  1 T  n1  Δx n−1  δ1 − q max δ  1 T  n1 x δ1 n − T  n1 |h n x n | ≥ p min δ  1 c δ1 1  T  n1 Δx n−1  2  δ1/2 − q max δ  1 x δ1 δ1 −  T  n1 h 2 n  1/2 x  p min δ  1 c δ1 1 x T Ax δ1/2 − q max δ  1 x δ1 δ1 − T  n1 |h n x n |, 3.29 where x T x 1 ,x 2 , ,x T . Clearly, λ 1  0 is an eigenvalue of the matrix A and ξ v,v, ,v T ∈ E T is an eigen- vector of A corresponding to 0, where v /  0,v∈ R.Letλ 2 ,λ 3 , ,λ T be the other eigenvalues of A. By matrix theory, we have λ j > 0 for all j ∈ Z2,T. Without loss of generality, we may assume that 0  λ 1 <λ 2 ≤ ··· ≤ λ T , then for any x ∈ Y, Jx ≥ p min δ  1 c δ1 1 λ δ1/2 2 x δ1 − q max δ  1 x δ1 δ1 −  T  n1 h 2 n  1/2 x   p min δ  1 c δ1 1 λ δ1/2 2 − q max δ  1 c δ1 1  x δ1 −  T  n1 h 2 n  1/2 x ≥− δ δ  1  T  n1 h 2 n  1/2    T n1 h 2 n  1/2 p min c δ1 1 λ δ1/2 2 − q max c δ1 1  1/δ , 3.30 X. Cai and J. Yu 9 as one finds by minimizing with respect to x. That is Jx ≥− δ δ  1   T n1 h 2 n  δ1/2δ 1/c 1  δ1/δ  p min λ δ1/2 2 − q max  1/δ . 3.31 Set w 0  − δ δ  1   T n1 h 2 n  δ1/2δ 1/c 1  δ1/δ  p min λ δ1/2 2 − q max  1/δ , 3.32 then by G 2 ,wehave Jx ≥ w 0 >σ, ∀x ∈ Y. 3.33 This implies that the assumption of saddle point theorem is satisfied. Thus there exists at least one critical point of J on E T , and the proof is complete. When q n > 0, we have the following result. Theorem 3.5. Assume that the following conditions are satisfied: G 3  2 δ1 p n  p n1  <q n ,q n > 0 for all n ∈ Z1,T; G 4   T n1 h 2 n  δ1/2δ   T n1 q n  1/δ C δ1 1 < −A 0   T n1 h n  δ1/δ , where A 0  max n∈Z1,T 2 δ1 p n  p n1  − q n . Then 3.21 possesses at least one T-periodic solution. Before proving Theorem 3.5, first, we prove the following result. Lemma 3.6. Assume that G 3  holds, then Jx defined by 3.22 satisfies P-S condition. Proof. For any sequence {x l }∈E T with Jx l  being bounded and J  x l  → 0as n → ∞, there exists a positive constant M such that |Jx l |≤M. Thus −M ≤ J  x l  ≤ 1 δ  1 T  n1 p n  Δx l n−1  δ1 − 1 δ  1 T  n1 q n  x l n  δ1  T  n1 h n x l n ≤ 2 δ1 δ  1 T  n1  p n  p n1  x l n  δ1 − 1 δ  1 T  n1 q n  x l n  δ1  T  n1   h n x l n   ≤ 1 δ  1 T  n1  2 δ1  p n  p n1  − q n  x l n  δ1   T  n1 h 2 n  1/2   x l   ≤ 1 δ  1 A 0   x l   δ1 δ1   T  n1 h 2 n  1/2   x l   ≤ A 0 δ  1 c δ1 2   x l   δ1   T  n1 h 2 n  1/2   x l   . 3.34 10 Advances in Difference Equations That is, −c δ1 2 A 0 δ  1   x l   δ1 −  T  n1 h 2 n  1/2   x l   ≤ M, ∀n ∈ N. 3.35 By δ  1 > 1, the above inequality implies that {x l } is a bounded sequence in E T . Thus {x l } possesses convergent subsequences, and the proof is complete. Proof of Theorem 3.5. For any w z,z, ,z T ∈ W, we have Jω− 1 δ  1 T  n1 q n z δ1  T  n1 h n z. 3.36 Take z   T n1 h n /  T n1 q n ,ρ w  T 1/2 |  T n1 h n /  T n1 | 1/δ , then Jw δ δ  1   T n1 h n  δ1/δ     T n1 q n    1/δ , ∀w ∈ ∂B ρ  W. 3.37 Set σ  δ δ  1   T n1 h n  δ1/δ |  T n1 q n | 1/δ , 3.38 then Jwσ for all w ∈ ∂B ρ  W. On the other hand, for any x ∈ Y, we have Jx ≤ 1 δ  1 T  n1 2 δ1 p n  p n1  − q n x δ1 n   T  n1 h 2 n  1/2 x ≤ A 0 δ  1 c δ1 2 x δ1   T  n1 h 2 n  1/2 x ≤− δ δ  1  1 A 0  1/δ  1 c 2  δ1/δ  T  n1 h 2 n  δ1/2δ . 3.39 Set w 0  −δ/δ  11/A 0  1/δ 1/c 2  δ1/δ   T n1 h 2 n  δ1/2δ , then Jx ≤ w 0 <σ.Thus −Jx satisfies the assumption of saddle point theorem, that is, there exists at least one critical point of J on E T . This completes the proof of Theorem 3.5. 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Let X be a real Hilbert space, I ∈ C 1 X, R, which implies that