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Hindawi Publishing Corporation Advances in Difference Equations Volume 2007, Article ID 42530, 9 pages doi:10.1155/2007/42530 Research Article Existence of Periodic and Subharmonic Solutions for Second-Order p-Laplacian Difference Equations Peng Chen and Hui Fang Received 26 December 2006; Accepted 13 February 2007 Recommended by Kanishka Perera We obt ai n a sufficient condition for the existence of periodic and subharmonic solutions of second-order p-Laplacian difference equations using the critical point theory. Copyright © 2007 P. Chen and H. Fang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In this paper, we denote by N, Z, R the set of all natural numbers, integers, and real numbers, respectively. For a,b ∈ Z,defineZ(a) ={a,a +1, }, Z(a,b) ={a,a +1, ,b} when a ≤ b. Consider the nonlinear s econd-order difference equation Δ  ϕ p  Δx n−1  + f  n,x n+1 ,x n ,x n−1  = 0, n ∈ Z, (1.1) where Δ is the forward difference operator Δx n = x n+1 − x n , Δ 2 x n = Δ(Δx n ), ϕ p (s)is p-Laplacian operator ϕ p (s) =|s| p−2 s (1 <p<∞), and f : Z × R 3 → R is a continuous functional in the second, the third, and fourth variables and satisfies f (t + m,u,v, w) = f (t,u,v,w) for a given positive integer m. We may think of (1.1) as being a discrete analogue of the second-order functional differential equation  ϕ p (x  )   + f  t,x(t +1),x(t),x(t − 1)  = 0, t ∈ R (1.2) which includes the following equation: c 2 y  (x) = v   y(x +1)− y(x)  − v   y(x) − y(x − 1)  . (1.3) 2AdvancesinDifference Equations Equations similar in structure to (1.3) arise in the study of the existence of solitary waves of lattice differential equations, see [1] and the references cited therein. Some special cases of (1.1) have been studied by many researchers via variational methods, see [2–7]. However, to our best knowledge, no similar results are obtained in the literature for (1.1). Since f in (1.1)dependsonx n+1 and x n−1 , the traditional ways of establishing the functional in [2–7] are inapplicable to our case. The main purpose of this paper is to give some sufficient conditions for the existence of periodic and subharmonic solutions of (1.1) using the critical point theory. 2. Some basic lemmas To apply critical point theory to study the existence of periodic solutions of (1.1), we will state some basic notations and lemmas (see [5, 8]), which will be used in the proofs of our main results. Let S be the set of sequences, x = ( ,x −n , ,x −1 ,x 0 ,x 1 , ,x n , ) ={x n } +∞ −∞ , that is, S ={x ={x n } : x n ∈ R, n ∈ Z}. For a given positive integer q and m, E qm is defined as a subspace of S by E qm =  x ={x n }∈S | x n+qm = x n , n ∈ Z  . (2.1) For any x, y ∈ S, a,b ∈ R, ax + by is defined by ax + by =  ax n + by n  +∞ n=−∞ . (2.2) Then S is a vector space. Clearly, E qm is isomorphic to R qm , E qm can be equipped with inner product x, y E qm = qm  j=1 x j y j , ∀x, y ∈ E qm , (2.3) by which the norm ·can be induced by x=  qm  j=1 x 2 j  1/2 , ∀x ∈ E qm . (2.4) It is obvious that E qm with the inner product in (2.3) is a finite dimensional Hilbert space and linearly homeomorphic to R qm . On the other hand, we define the norm · p on E qm as follows: x p =  qm  i=1   x i   p  1/p , (2.5) for all x ∈ E qm and p>1. Clearly, x=x 2 .Since· p and · 2 are equivalent, there exist constants C 1 , C 2 ,suchthatC 2 ≥ C 1 > 0, and C 1 x p ≤x 2 ≤ C 2 x p , ∀x ∈ E qm . (2.6) P. Chen and H. Fang 3 Define the functional J on E qm as follows: J(x) = qm  n=1  1 p   Δx n   p − F  n,x n+1 ,x n   , (2.7) where f (t,u,v,w) = F  2 (t − 1,v, w)+F  3 (t,u,v), F  2 (t − 1,v, w) = ∂F(t − 1,v, w) ∂v , F  3 (t,u,v) = ∂F(t,u,v) ∂v , (2.8) then f  n,x n+1 ,x n ,x n−1  = F  3  n,x n+1 ,x n  + F  2  n −1,x n ,x n−1  . (2.9) Clearly, J ∈ C 1 (E qm ,R)andforanyx ={x n } n∈Z ∈ E qm , by using x 0 = x qm , x 1 = x qm+1 ,we can compute the partial derivative as ∂J ∂x n =−  Δ  ϕ p  Δx n−1  + f  n,x n+1 ,x n ,x n−1  , n ∈ Z(1,qm). (2.10) By the periodicity of {x n } and f (t,u,v, w) in the first variable t, we reduce the existence of periodic solutions of (1.1) to the existence of critical points of J on E qm . That is, the functional J is just the var iational framework of (1.1). For convenience, we identify x ∈ E qm with x = (x 1 ,x 2 , ,x qm ) T . Let X be a real Hilbert space, I ∈ C 1 (X,R), which means that I is a continuously Fr ´ echet differentiable functional defined on X. I is said to satisfy Palais-Smale condition (P-S condition for short) if any sequence {u n }⊂X for which {I(u n )} is bounded and I  (u n ) → 0, as n →∞, possesses a convergent subsequence in X. Let B ρ be the open ball in X with radius ρ and centered at 0 and let ∂B ρ denote its boundary. Lemma 2.1 (linking theorem) [8, Theorem 5.3]. Let X be a real Hilbert space, X = X 1 ⊕ X 2 , where X 1 is a finite-dimensional subspace of X. Assume that I ∈ C 1 (X,R) satisfies the P-S condition and (A 1 ) there exist constants σ>0 and ρ>0, such that I| ∂B ρ ∩X 2 ≥ σ; (A 2 ) there is an e ∈ ∂B 1 ∩ X 2 and a constant R 1 >ρ, such that I| ∂Q ≤ 0,whereQ = (B R 1 ∩ X 1 ) ⊕{re | 0 <r<R 1 }. Then, I possesses a critical value c ≥ σ,where c = inf h∈Γ max u∈Q I  h(u)  , Γ =  h ∈ C  Q,X  | h| ∂Q = id  (2.11) and id denotes the identity operator. 4AdvancesinDifference Equations 3. Main results Theorem 3.1. Assume that the following conditions are satisfied: (H 1 ) f (t,u,v,w) ∈ C(R 4 ,R) and there exists a positive integer m, such that for every (t,u,v,w) ∈ R 4 , f (t + m,u,v,w) = f (t,u,v,w); (H 2 ) there exists a functional F(t,u,v) ∈ C 1 (R 3 ,R) with F(t,u,v) ≥ 0 and it satisfies F  2 (t − 1,v, w)+F  3 (t,u,v) = f (t,u,v,w), lim ρ→0 F(t,u,v) ρ p = 0, ρ =  u 2 + v 2 ; (3.1) (H 3 ) there exist constants β ≥ p +1, a 1 > 0, a 2 > 0, such that F(t,u,v) ≥ a 1   u 2 + v 2  β − a 2 , ∀(t,u,v) ∈ R 3 . (3.2) Then, for a given positive integer q,(1.1) has at least two nontrivial qm-periodic solutions. First, we prove two lemmas which are useful in the proof of Theorem 3.1. Lemma 3.2. Assume that f (t, u,v,w) satis fies condition (H 3 )ofTheorem 3.1, then the func- tional J(x) =  qm n =1 [1/p|Δx n | p − F(n, x n+1 ,x n )] is bounded from above on E qm . Proof. By (H 3 ), there exist a 1 > 0, a 2 > 0, β>p, such that for all x ∈ E qm , J(x) = qm  n=1  1 p   Δx n   p − F  n,x n+1 ,x n   ≤ qm  n=1  2 p p max    x n+1   p ,   x n   p  − F  n,x n+1 ,x n   ≤ 2 p p qm  n=1    x n+1   p +   x n   p  − a 1 qm  n=1   x 2 n+1 + x 2 n  β + a 2 qm ≤ 2 p+1 p qm  n=1   x n   p − a 1 qm  n=1   x n   β + a 2 qm = 2 p+1 p x p p − a 1 x β β + a 2 qm. (3.3) In view of (2.6), there exist constants C 1 , C 3 ,suchthat x p ≤ 1 C 1 x, x β ≥ 1 C 3 x. (3.4) So J(x) ≤ 2 p+1 p  C 1  p x p − a 1  C 3  β x β + a 2 qm. (3.5) By β>pand the above inequality, there exists a constant M>0, such that for every x ∈ E qm , J(x) ≤ M. The proof is complete.  Lemma 3.3. Assume that f (t, u,v,w) satis fies condition (H 3 )ofTheorem 3.1, then the func- tional J satisfies P-S condition. P. Chen and H. Fang 5 Proof. Let x (k) ∈ E qm ,forallk ∈ N,besuchthat{J(x (k) )} is bounded. Then there exists M 1 > 0, such that −M 1 ≤ J  x (k)  ≤ 2 p+1 pC p 1   x (k)   p − a 1 C 3 β   x (k)   β + a 2 qm, (3.6) that is, a 1 C 3 β   x (k)   β − 2 p+1 pC p 1   x (k)   p ≤ M 1 + a 2 qm. (3.7) By β>p, there exists M 2 > 0suchthatforeveryk ∈ N, x (k) ≤M 2 . Thus, {x (k) } is bounded on E qm .SinceE qm is finite dimensional, there exists a subse- quence of {x (k) }, which is convergent in E qm and the P-S condition is verified.  Proof of Theorem 3.1. The proof of Lemma 3.2 implies lim x→∞ J(x) =−∞,then−J is coercive. Let c 0 = sup x∈E qm J(x). By continuity of J on E qm , there exists x ∈ E qm ,suchthat J( x) = c 0 ,andx is a critical point of J.Weclaimthatc 0 > 0. In fact, we have J(x) = 1 p  qm  n=1   Δx n   p  1/p  p − qm  n=1 F  n,x n+1 ,x n  ≥ 1 p  1 C 2  p  qm  n=1   Δx n   2  1/2  p − qm  n=1 F  n,x n+1 ,x n  = 1 p  1 C 2  p  qm  n=1 2  x 2 n − x n x n+1   p/2 − qm  n=1 F  n,x n+1 ,x n  = 1 p  1 C 2  p  x T Ax  p/2 − qm  n=1 F  n,x n+1 ,x n  , (3.8) where x = (x 1 ,x 2 , ,x qm ) T , A = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 2 −10··· 0 −1 −12−1 ··· 00 0 −12··· 00 . . . . . . . . . . . . . . . . . . −10 0··· −12 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ qm×qm . (3.9) Clearly, 0 is an eigenvalue of A and ξ = (v, v, , v) T ∈ E qm is an eigenvector of A cor- responding to 0, where v = 0, v ∈ R.Letλ 1 , λ 2 , ,λ qm−1 be the other eigenvalues of A.By matrix theory, we have λ j > 0, for all j ∈ Z(1,qm − 1). Denote Z ={(v, v, , v) T ∈ E qm | v ∈ R} and Y = Z ⊥ ,suchthatE qm = Y ⊕ Z. Set λ min = min j∈Z(1,qm−1) λ j > 0, λ max = max j∈Z(1,qm−1) λ j > 0. (3.10) 6AdvancesinDifference Equations By condition (H 2 ), we have lim ρ→0 F(t,u,v) ρ p = 0, ρ =  u 2 + v 2 . (3.11) Choose ε = 2 −p/2−2 (1/p)λ p/2 min (C 1 /C 2 ) p , there exists δ>0, such that   F(t,u,v)   ≤ 2 −p/2−2 1 p λ p/2 min  C 1 C 2  p ρ p , ∀ρ ≤ δ. (3.12) Therefore, for any x = (x 1 ,x 2 , ,x qm ) T with x≤δ, x ∈ Y,wehave J(x) ≥ 1 p  1 C 2  p  x T Ax  p/2 − qm  n=1 F  n,x n+1 ,x n  ≥ 1 p λ p/2 min  1 C 2  p x p − 2 −p/2−2 1 p λ p/2 min  C 1 C 2  p qm  n=1  2 p/2 max    x n+1   p ,   x n   p  ≥ 1 p λ p/2 min  1 C 2  p x p − 2 −p/2−2 1 p λ p/2 min  C 1 C 2  p qm  n=1  2 p/2    x n+1   p +   x n   p  = 1 p λ p/2 min  1 C 2  p x p − 2 −p/2−2 1 p λ p/2 min  C 1 C 2  p 2 p/2+1 x p p ≥ 1 p λ p/2 min  1 C 2  p x p − 1 2p λ p/2 min  C 1 C 2  p  1 C 1  p x p = 1 2p  1 C 2  p λ p/2 min x p . (3.13) Take σ = 1/2p(1/C 2 ) p λ p/2 min δ p ,then J(x) ≥ σ>0, ∀x ∈ Y ∩ ∂B δ . (3.14) So c 0 = sup x∈E qm J(x) ≥ σ>0, (3.15) which implies that J satisfies the condition (A 1 ) of the linking theorem. Noting that Ax = 0, for all x ∈ Z,wehave J(x) ≤ 1 p  1 C 1  p  x T Ax  p/2 − qm  n=1 F  n,x n+1 ,x n  ≤ 0. (3.16) Therefore, the critical point associated to the critical value c 0 of J is a nontrivial qm- periodic solution of (1.1). Now, we need to verify other conditions of the linking theorem. P. Chen and H. Fang 7 By Lemma 3.3, J satisfies P-S condition. So, it suffices to verify condition (A 2 ). Take e ∈ ∂B 1 ∩ Y,foranyz ∈ Z, r ∈ R,letx = re + z,then J(x) = 1 p qm  n=1   Δx n   p − qm  n=1 F  n,x n+1 ,x n  ≤ 1 p  1 C 1  p  qm  n=1   Δx n   2  p/2 − qm  n=1 F  n,x n+1 ,x n  = 1 p  1 C 1  p  x T Ax  p/2 − qm  n=1 F  n,x n+1 ,x n  = 1 p  1 C 1  p  A(re+ z),(re + z)  p/2 − qm  n=1 F  n,re n+1 + z n+1 ,re n + z n  = 1 p  1 C 1  p  Are,re  p/2 − qm  n=1 F  n,re n+1 + z n+1 ,re n + z n  ≤ 1 p  1 C 1  p λ p/2 max r p − a 1 qm  n=1    re n+1 + z n+1  2 +  re n + z n  2  β + a 2 qm ≤ 1 p  1 C 1  p λ p/2 max r p − a 1  1 C 3  β  qm  n=1   re n+1 + z n+1  2 +  re n + z n  2   β/2 + a 2 qm = 1 p λ p/2 max  1 C 1  p r p − a 1  1 C 3  β  2r 2 +2z 2  β/2 + a 2 qm ≤ 1 p λ p/2 max  1 C 1  p r p − a 1  1 C 3  β 2 β/2 r β − a 1  1 C 3  β 2 β/2 z β + a 2 qm. (3.17) Let g 1 (r) = 1 p λ p/2 max  1 C 1  p r p − a 1  1 C 3  β 2 β/2 r β , g 2 (t) =−a 1  1 C 3  β 2 β/2 t β + a 2 qm. (3.18) Then lim r→+∞ g 1 (r) =−∞,lim t→+∞ g 2 (t) =−∞, (3.19) and g 1 (r)andg 2 (t) are bounded from above. Thus, there exists a constant R 2 >δ,suchthatJ(x) ≤ 0, for all x ∈ ∂Q,where Q =  B R 2 ∩ Z  ⊕  re | 0 <r<R 2  . (3.20) By the linking theorem, J possesses a critical value c ≥ σ>0, where c = inf h∈Γ max u∈Q J  h(u)  , Γ =  h ∈ C  Q,E qm  | h| ∂Q = id  . (3.21) The rest of the proof is similar to that of [5, Theorem 1.1], but for the sake of com- pleteness, we give the details. 8AdvancesinDifference Equations Let x ∈ E qm be a critical point associated to the critical value c of J, that is, J(x) = c. If x = x, then the proof is complete; if x = x,thenc 0 = J(x) = J(x) = c,thatis sup x∈E qm J(x) = inf sup h∈Γu∈Q J  h(u)  . (3.22) Choose h = id, we have sup x∈Q J(x) = c 0 . Since the choice of e ∈ ∂B 1 ∩ Y is ar bitrary, we can take −e ∈ ∂B 1 ∩ Y. By a similar argument, there exists a constant R 3 >δ,forany x ∈ ∂Q 1 , J(x) ≤ 0, where Q 1 =  B R 3 ∩ Z  ⊕  − re | 0 <r<R 3  . (3.23) Again, by using the linking theorem, J possesses a critical value c  ≥ σ>0, where c  = inf max h∈Γ 1 u∈Q 1 J  h(u)  , Γ 1 =  h ∈ C  Q 1 ,E qm  | h| ∂Q 1 = id  . (3.24) If c  = c 0 , then the proof is complete. If c  = c 0 ,thensup x∈Q 1 J(x) = c 0 .Duetothefact that J | ∂Q ≤ 0, J| ∂Q 1 ≤ 0, J attains its maximum at some points in the interior of the set Q and Q 1 .Clearly,Q ∩ Q 1 =∅,andforanyx ∈ Z, J(x) ≤ 0. This shows that there must be apoint x ∈ E qm ,suchthatx = x and J(x) = c  = c 0 . The above argument implies that whether or not c = c 0 ,(1.1) possesses at least two nontrivial qm-periodic solutions. Remark 3.4. when qm = 1, (1.1)isreducedtotrivialcase;whenqm = 2, A has the fol- lowing form: A =  2 −2 −22  . (3.25) In this case, it is easy to complete the proof of Theorem 3.1. Finally, we give an example to illustrate Theorem 3.1. Example 3.5. Assume that f (t,u,v,w) = 2(p +1)v  1 +sin 2 πt m   u 2 + v 2  p +  1 +sin 2 π(t − 1) m   v 2 + w 2  p  . (3.26) Take F(t,u,v) =  1 +sin 2 πt m   u 2 + v 2  p+1 . (3.27) Then, F  2 (t − 1,v, w)+F  3 (t,u,v) = 2(p +1)v  1 +sin 2 πt m   u 2 + v 2  p +  1 +sin 2 π(t − 1) m   v 2 + w 2  p  . (3.28) P. Chen and H. Fang 9 It is easy to verify that the assumptions of Theorem 3.1 are satisfied and then (1.1)pos- sesses at least two nontrivial qm-periodic solutions.  Acknowledgment This research is supported by the National Natural Science Foundation of China (no. 10561004). References [1] D. Smets and M. Willem, “Solitary waves with prescribed speed on infinite lattices,” Journal of Functional Analysis, vol. 149, no. 1, pp. 266–275, 1997. [2] R. P. Agarwal, K. Perera, and D. O’Regan, “Multiple positive solutions of singular discrete p- Laplacian problems via variational methods,” Advances in Difference Equations, vol. 2005, no. 2, pp. 93–99, 2005. [3] R. P. Agarwal, K. Perera, and D. O’Regan, “Multiple positive solutions of singular and nonsin- gular discrete problems via variational methods,” Nonlinear Analysis, vol. 58, no. 1, pp. 69–73, 2004. [4] X. Cai and J. Yu, “Existence theorems for second-order discrete boundary value problems,” Jour- nal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 649–661, 2006. [5] Z. Guo and J. Yu, “Existence of periodic and subharmonic solutions for second-order superlin- ear difference equations,” Science in China. Series A. Mathematics, vol. 46, no. 4, pp. 506–515, 2003. [6] Z. Guo and J. Yu, “The existence of periodic and subharmonic solutions of subquadratic second order difference equations,” Journal of the London Mathematical Society. Second Series, vol. 68, no. 2, pp. 419–430, 2003. [7] J. Yu, Y. Long, and Z. Guo, “Subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation,” JournalofDynamicsandDifferential Equations,vol.16,no.2,pp. 575–586, 2004. [8] P. H. Rabinowitz, Minimax Methods in Crit ical Point Theory with Applications to Di fferential Equations, vol. 65 of CBMS Regional Conference Ser ies in Mathematics, American Mathematical Society, Providence, RI, USA, 1986. Peng Chen: Department of Applied Mathematics, Faculty of Science, Kunming University of Science and Technology, Yunnan 650093, China Email address: pengchen729@sina.com Hui Fang: Department of Applied Mathematics, Faculty of Science, Kunming University of Science and Technology, Yunnan 650093, China Email address: huifang@public.km.yn.cn . Equations Volume 2007, Article ID 42530, 9 pages doi:10.1155/2007/42530 Research Article Existence of Periodic and Subharmonic Solutions for Second-Order p-Laplacian Difference Equations Peng Chen and Hui Fang Received. conditions for the existence of periodic and subharmonic solutions of (1.1) using the critical point theory. 2. Some basic lemmas To apply critical point theory to study the existence of periodic solutions. sufficient condition for the existence of periodic and subharmonic solutions of second-order p-Laplacian difference equations using the critical point theory. Copyright © 2007 P. Chen and H. Fang. This