Thông tin tài liệu
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 Δp n Δx n−1 δ q n x δ n fn, 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 Za{a, a 1, }, Za, 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 n1 − x n and Δ 2 x n−1 ΔΔx n−1 Δx n − Δx n−1 . 1.2 In 1.1, the given real sequences {p n }, {q n } satisfy p nT p n > 0,q nT q n for any n ∈ Z, f : Z × R → R is continuous in the second variable, and fn T, zfn, 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 Δp n Δx n q n x n 0 1.3 2 Advances in Difference Equations has been investigated. In 3, the nonlinear difference equation Δp n Δx n−1 q n x n fn, 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 ptϕu ft, u0. 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 {Iu 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 Ix ≥ a for all x ∈ ∂B ρ ,whereB ρ {x ∈ X : x X <ρ}; I 2 I0 ≤ 0 and there exists x 0 ∈ B ρ such that Ix 0 ≤ 0. Then c inf h∈Γ sup s∈0,1 Ihs is a positive critical value of I,where Γ h ∈ C 0, 1,X : h00,h1x 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| eX 2 ≥ ω. Then I possesses a critical value c ≥ ω and c inf h∈Γ max u∈B ρ X 1 Ihu, 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 nT x n ,n∈ Z . 2.4 Clearly, E T is isomorphic to R T , and can be equipped with inner product x, y T i1 x i y i , ∀x, y ∈ E T , 2.5 by which the norm · can be induced by x : T i1 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: Jx 1 δ 1 T n1 p n Δx n−1 δ1 − 1 δ 1 T n1 q n x δ1 n T n1 Fn, x n , ∀x ∈ E T , 2.7 where Ft, z z 0 ft, sds. 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 T1 , we can compute the partial derivative as ∂J ∂x n −Δp n Δx n−1 δ − q n x δ n fn, x n ,n∈ Z1,T. 2.8 Thus x {x n } n∈Z is a critical point of J on E T i.e., J x0 if and only if Δp n Δx n−1 δ q n x δ n fn, x n ,n∈ Z1,T. 2.9 By the periodicity of x n and fn, 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∈ Z1,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 i1 |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 δ1 ≤ 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 fn, sds ≤−a 1 |z| β a 2 , ∀z ∈ R; 3.1 F 2 q n ≤ 0, ∀n ∈ Z. 3.2 Then the functional Jx 1 δ 1 T n1 p n Δx n−1 δ1 − 1 δ 1 T n1 q n x δ1 n T n1 Fn, x n 3.3 satisfies P-S condition. Proof. For any sequence {x l }⊂E T , with Jx l being bounded and J x l → 0as l → ∞, there exists a positive constant M such that |Jx l |≤M. Thus, by F 1 , −M ≤ Jx l 1 δ 1 T n1 p n x l n − x l n−1 δ1 − q n x l n δ1 T n1 F n, x l n ≤ 1 δ 1 T n1 p n 2 δ1 x l n δ1 x l n−1 δ1 − 1 δ 1 T n1 q n x l n δ1 T n1 F n, x l n ≤ 2 δ1 δ 1 T n1 p n p n1 x l n δ1 − 1 δ 1 T n1 q n x l n δ1 − a 1 T n1 x l n β a 2 T 1 δ 1 T n1 2 δ1 p n p n1 − q n x l n δ1 − a 1 x l β β a 2 T. 3.4 Set A 0 max n∈Z1,T 2 δ1 p n p n1 − q n . 3.5 Then A 0 > 0. Also, by the above inequality, we have −M ≤ Jx l ≤ A 0 δ 1 x l δ1 δ1 − a 1 x l β β a 2 T. 3.6 X. Cai and J. Yu 5 In view of T n1 x l n δ1 ≤ T β−δ−1/β T n1 x l n β δ1/β , 3.7 we have x l β β ≥ T δ1−β/δ1 x l β δ1 . 3.8 Then we get −M ≤ Jx l ≤ A 0 δ 1 x l δ1 δ1 − a 1 T δ1−β/δ1 x l β δ1 a 2 T. 3.9 Therefore, for any l ∈ N, a 1 T δ1−β/δ1 x l β δ1 − A 0 δ 1 x l δ1 δ1 ≤ M a 2 T. 3.10 Since β>δ1, 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 fn, z z δ 0; 3.11 F 4 q n < 0, ∀n ∈ Z1,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 ∈ Z1,T, |Fn, z|≤ − q max 2δ 1 z δ1 , 3.13 where q max max n∈Z1,T q n < 0. Thus for any x ∈ E T , x≤ρ for all n ∈ Z1,T, we have Jx ≥− q max δ 1 T n1 x δ1 n q max 2δ 1 T n1 x δ1 n − q max 2δ 1 x δ1 δ1 ≥− q max 2δ 1 c δ1 1 x δ1 2 . 3.14 6 Advances in Difference Equations Taking a −c δ1 1 q max /2δ 1ρ δ1 , we have Jx| ∂B ρ ≥ a>0, 3.15 and the assumption I 1 is verified. Clearly, J00. For any given w ∈ E T with w 1anda constant α>0, Jαw 1 δ 1 T n1 p n αw n − αw n−1 δ1 − q n αw n δ1 T n1 Fn, αw n ≤ 1 δ 1 T n1 p n 2α δ1 − q n α δ1 − a 1 T n1 |αw n | β a 2 T ≤ 1 δ 1 T n1 2 δ1 p n − q n α δ1 α δ1 − 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 , Jx < 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,Jxc, and J x0. By a similar argument to the proof of Lemma 3.1, for any x ∈ E T , there exists x ∈ E T such that J xc 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 hs , 3.17 where Γ{h ∈ C0, 1,E T | h00,h1x}. Then for any h ∈ Γ,c max max s∈0,1 Jhs. By the continuity of Jhs in s, J0 ≤ 0andJ x < 0 show that there exists some s 0 ∈ 0, 1 such that Jhs 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 Jh 1 s 1 Jh 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 fn, x n ≡ h n ,wehave the following results. Theorem 3.3. Assume that the following conditions hold: G 1 qn < 0, ∀n ∈ Z1,T; 3.18 G 2 1 c δ1 1 T n1 h 2 n δ1/2 T n1 −q n < p min λ δ1/2 2 − q max T n1 h n δ1 , 3.19 X. Cai and J. Yu 7 where p min min n∈Z1,T p n ,q max max n∈Z1,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 Jx 1 δ 1 T n1 p n Δx n−1 δ1 − 1 δ 1 T n1 q n x δ1 n T n1 h n x n 3.22 satisfies P-S condition on E T . Proof. For any sequence {x l }⊂E T with Jx l being bounded and J x l → 0asn → ∞, there exists a positive constant M such that |Jx l |≤M. In view of G 3 and T n1 |h n x l n |≤ T n1 h 2 n 1/2 T n1 x l n 2 1/2 , 3.23 we have M ≥ Jx l 1 δ 1 T n1 p n Δx l n−1 δ1 − 1 δ 1 T n1 q n x l n δ1 T n1 h n x l n ≥− 1 δ 1 T n1 q n x l n δ1 − T n1 h n x l n ≥− 1 δ 1 q max T n1 x l n δ1 − T n1 h 2 n 1/2 T n1 x l n 2 1/2 − q max δ 1 x l δ1 δ1 − T n1 h 2 n 1/2 x l ≥− q max δ 1 c δ1 1 x l δ1 − T n1 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 Jw− 1 δ 1 T n1 q n z δ1 T n1 h n z. 3.25 Take z T n1 h n / T n1 q n 1/δ and ρ w T 1/2 | T n1 h n / T n1 q n | 1/δ , then Jw δ δ 1 T n1 h n δ1/δ T n1 q n 1/δ . 3.26 Set σ δ δ 1 T n1 h n δ1/δ T n1 q n 1/δ , 3.27 then we have Jwσ, ∀w ∈ ∂B ρ Y. 3.28 On the other hand, for any x ∈ Y, we have Jx 1 δ 1 T n1 p n Δx n−1 δ1 − 1 δ 1 T n1 q n x δ1 n T n1 h n x n ≥ p min δ 1 T n1 Δx n−1 δ1 − q max δ 1 T n1 x δ1 n − T n1 |h n x n | ≥ p min δ 1 c δ1 1 T n1 Δx n−1 2 δ1/2 − q max δ 1 x δ1 δ1 − T n1 h 2 n 1/2 x p min δ 1 c δ1 1 x T Ax δ1/2 − q max δ 1 x δ1 δ1 − T n1 |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 ∈ Z2,T. Without loss of generality, we may assume that 0 λ 1 <λ 2 ≤ ··· ≤ λ T , then for any x ∈ Y, Jx ≥ p min δ 1 c δ1 1 λ δ1/2 2 x δ1 − q max δ 1 x δ1 δ1 − T n1 h 2 n 1/2 x p min δ 1 c δ1 1 λ δ1/2 2 − q max δ 1 c δ1 1 x δ1 − T n1 h 2 n 1/2 x ≥− δ δ 1 T n1 h 2 n 1/2 T n1 h 2 n 1/2 p min c δ1 1 λ δ1/2 2 − q max c δ1 1 1/δ , 3.30 X. Cai and J. Yu 9 as one finds by minimizing with respect to x. That is Jx ≥− δ δ 1 T n1 h 2 n δ1/2δ 1/c 1 δ1/δ p min λ δ1/2 2 − q max 1/δ . 3.31 Set w 0 − δ δ 1 T n1 h 2 n δ1/2δ 1/c 1 δ1/δ p min λ δ1/2 2 − q max 1/δ , 3.32 then by G 2 ,wehave Jx ≥ 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 δ1 p n p n1 <q n ,q n > 0 for all n ∈ Z1,T; G 4 T n1 h 2 n δ1/2δ T n1 q n 1/δ C δ1 1 < −A 0 T n1 h n δ1/δ , where A 0 max n∈Z1,T 2 δ1 p n p n1 − 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 Jx defined by 3.22 satisfies P-S condition. Proof. For any sequence {x l }∈E T with Jx l being bounded and J x l → 0as n → ∞, there exists a positive constant M such that |Jx l |≤M. Thus −M ≤ J x l ≤ 1 δ 1 T n1 p n Δx l n−1 δ1 − 1 δ 1 T n1 q n x l n δ1 T n1 h n x l n ≤ 2 δ1 δ 1 T n1 p n p n1 x l n δ1 − 1 δ 1 T n1 q n x l n δ1 T n1 h n x l n ≤ 1 δ 1 T n1 2 δ1 p n p n1 − q n x l n δ1 T n1 h 2 n 1/2 x l ≤ 1 δ 1 A 0 x l δ1 δ1 T n1 h 2 n 1/2 x l ≤ A 0 δ 1 c δ1 2 x l δ1 T n1 h 2 n 1/2 x l . 3.34 10 Advances in Difference Equations That is, −c δ1 2 A 0 δ 1 x l δ1 − T n1 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 Jω− 1 δ 1 T n1 q n z δ1 T n1 h n z. 3.36 Take z T n1 h n / T n1 q n ,ρ w T 1/2 | T n1 h n / T n1 | 1/δ , then Jw δ δ 1 T n1 h n δ1/δ T n1 q n 1/δ , ∀w ∈ ∂B ρ W. 3.37 Set σ δ δ 1 T n1 h n δ1/δ | T n1 q n | 1/δ , 3.38 then Jwσ for all w ∈ ∂B ρ W. On the other hand, for any x ∈ Y, we have Jx ≤ 1 δ 1 T n1 2 δ1 p n p n1 − q n x δ1 n T n1 h 2 n 1/2 x ≤ A 0 δ 1 c δ1 2 x δ1 T n1 h 2 n 1/2 x ≤− δ δ 1 1 A 0 1/δ 1 c 2 δ1/δ T n1 h 2 n δ1/2δ . 3.39 Set w 0 −δ/δ 11/A 0 1/δ 1/c 2 δ1/δ T n1 h 2 n δ1/2δ , then Jx ≤ w 0 <σ.Thus −Jx 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. Acknowledgment This project is supported by specialized research fund for the doctoral program of higher edu- cation, Grant no. 20020532014. [...]... 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... 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. .. 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... 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 Doˇ l´ , and M Marini, “Positive decreasing solutions of quasi-linear difference equasa... 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, II 11 P J Y Wong and R P Agarwal, “Oscillation and monotone solutions of second order quasilinear difference equations,” Funkcialaj Ekvacioj, vol 39,... 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, . 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. 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. 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
Ngày đăng: 22/06/2014, 19:20
Xem thêm: Báo cáo hóa học: " Research Article Existence Theorems of Periodic Solutions for Second-Order Nonlinear Difference Equations" doc
