Existence of Periodic and Subharmonic Solutions for Second-Order p-Laplacian Difference Equations

Advances in Difference Equations, Mar 2007

We obtain a sufficient condition for the existence of periodic and subharmonic solutions of second-order p-Laplacian difference equations using the critical point theory.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://www.advancesindifferenceequations.com/content/pdf/1687-1847-2007-042530.pdf

Existence of Periodic and Subharmonic Solutions for Second-Order p-Laplacian Difference Equations

Peng Chen 0 1 2 Hui Fang 0 1 2 p xn 0 1 2 0 In this paper, we denote by N, Z, R the set of all natural numbers , integers, and real numbers, respectively. For a, b 1 Hui Fang: Department of Applied Mathematics, Faculty of Science, Kunming University of Science and Technology , Yunnan 650093 , China 2 Peng Chen: Department of Applied Mathematics, Faculty of Science, Kunming University of Science and Technology , Yunnan 650093 , China We obtain a sufficient condition for the existence of periodic and subharmonic solutions of second-order p-Laplacian difference equations using the critical point theory. where is the forward difference operator xn = xn+1 xn, 2xn = (xn), p(s) is p-Laplacian operator p(s) = |s|p2s (1 < p < ), and f : Z R3 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 1. Introduction + f n, xn+1, xn, xn1 = 0, n Z, which includes the following equation: 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 [27]. However, to our best knowledge, no similar results are obtained in the literature for (1.1). Since f in (1.1) depends on xn+1 and xn1, the traditional ways of establishing the functional in [27] 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 = (. . . , xn, . . . , x1, x0, x1, . . . , xn, . . . ) = {xn}+, that is, S = {x = {xn} : xn R, n Z}. For a given positive integer q and m, Eqm is defined as a subspace of S by + ax + b y = axn + b yn n=. by which the norm can be induced by j=1 j=1 x, y Eqm = x, y Eqm, x = x Eqm. p on Eqm as follows: x p = i=1 for all x Eqm and p > 1. Clearly, x = x 2. Since exist constants C1, C2, such that C2 C1 > 0, and 2 are equivalent, there C1 x p x 2 C2 x p, x Eqm. Define the functional J on Eqm as follows: J (x) = F n, xn+1, xn , F2(t 1, v, w) = F3(t, u, v) = Clearly, J C1(Eqm, R) and for any x = {xn}nZ Eqm, by using x0 = xqm, x1 = xqm+1, we can compute the partial derivative as xn + f n, xn+1, xn, xn1 , n Z(1, qm). By the periodicity of {xn} 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 Eqm. That is, the functional J is just the variational framework of (1.1). For convenience, we identify x Eqm with x = (x1, x2, . . . , xqm)T . Let X be a real Hilbert space, I C1(X, R), which means that I is a continuously Frechet differentiable functional defined on X. I is said to satisfy Palais-Smale condition (P-S condition for short) if any sequence {un} X for which {I(un)} is bounded and I (un) 0, as n , possesses a convergent subsequence in X. Let B be the open ball in X with radius and centered at 0 and let B denote its boundary. c = inf max I h(u) , h uQ = h C Q, X |h|Q = id and id denotes the identity operator. F2(t 1, v, w) + F3(t, u, v) = f (t, u, v, w), lim0 F(t,up, v) = 0, = u2 + v2; F(t, u, v) a1 a2, (t, u, v) R3. 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) satisfies condition (H3) of Theorem 3.1, then the functional J (x) = nqm=1[1/ p|xn|p F(n, xn+1, xn)] is bounded from above on Eqm. 3. Main results Theorem 3.1. Assume that the following conditions are satisfied: (H1) f (t, u, v, w) C(R4, R) and there exists a positive integer m, such that for every (t, u, v, w) R4, f (t + m, u, v, w) = f (t, u, v, w); (H2) there exists a functional F(t, u, v) C1(R3, R) with F(t, u, v) 0 and it satisfies By > p and the above inequality, there exists a constant M > 0, such that for every x Eqm, J (x) M. The proof is complete. Lemma 3.3. Assume that f (t, u, v, w) satisfies condition (H3) of Theorem 3.1, then the functional J satisfies P-S condition. J (x) = F n, xn+1, xn a1 n=1 n=1 n=1 In view of (2.6), there exist constants C1, C3, such that p F n, xn+1, xn + a2qm a1 x pp a1 x + a2qm. 2p+1 Ca31 x(k) pC1p x(k) p M1 + a2qm. By > p, there exists M2 > 0 such that for every k N, x(k) M2. Thus, {x(k)} is bounded on Eqm. Since Eqm is finite dimensional, there exists a subsequence of {x(k)}, which is convergent in Eqm and the P-S condition is verified. where x = (x1, x2, . . . , xqm)T , n=1 = p C2 F n, xn+1, xn n=1 F n, xn+1, xn , 2 xn2 xnxn+1 n=1 xT Ax p/2 n=1 F n, xn+1, xn F n, xn+1, xn 1 1 min = jZm(1,qinm1) j > 0, max = jZm(1,aqmx1) j > 0. By condition (H2), we have = 0, u2 + v2. xT Ax p/2 n=1 F n, xn+1, xn x p 2p/22 1p mp/i2n CC12 x p 2p/22 1p mp/i2n CC21 x p 2p/22 1p mp/i2n CC12 x p 21p mp/i2n CC12 p n=1 2p/2 max Take = 1/2p(1/C2)pmp/i2n p, then which implies that J satisfies the condition (A1) of the linking theorem. Noting that Ax = 0, for all x Z, we have xT Ax p/2 n=1 F n, xn+1, xn 0. By Lemma 3.3, J satisfies P-S condition. So, it suffices to verify condition (A2). Take e B1 Y , for any z Z, r R, let x = re + z, then n=1 1 1 p qm F n, xn+1, xn p C1 n=1 n=1 F n, xn+1, xn n=1 n=1 F n, xn+1, xn n=1 F n, ren+1 + zn+1, ren + zn F n, ren+1 + zn+1, ren + zn ren+1 + zn+1 2 + ren + zn 2! + a2qm n=1 and g1(r) and g2(t) are bounded from above. Thus, there exists a constant R2 > , such that J(x) 0, for all x Q, where Q = BR2 Z re | 0 < r < R2 . c = inf max J h(u) , h uQ = h C Q, Eqm |h|Q = id . Let x" Eqm 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, then c0 = J (x) = J (x") = c, that is sup J (x) = inf sup J h(u) . xEqm huQ Choose h = id, we have supxQ J (x) = c0. Since the choice of e B1 Y is arbitrary, we can take e B1 Y . By a similar argument, there exists a constant R3 > , for any x Q1, J (x) 0, where Q1 = BR3 Z re | 0 < r < R3 . c = inf max J h(u) , h1uQ1 1 = h C Q1, Eqm |h|Q1 = id . Remark 3.4. when qm = 1, (1.1) is reduced to trivial case; when qm = 2, A has the following form: A = 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 F(t, u, v) = F2(t 1, v, w) + F3(t, u, v) = 2(p + 1)v It is easy to verify that the assumptions of Theorem 3.1 are satisfied and then (1.1) possesses at least two nontrivial qm-periodic solutions. Acknowledgment This research is supported by the National Natural Science Foundation of China (no. 10561004).


This is a preview of a remote PDF: http://www.advancesindifferenceequations.com/content/pdf/1687-1847-2007-042530.pdf

Peng Chen, Hui Fang. Existence of Periodic and Subharmonic Solutions for Second-Order p-Laplacian Difference Equations, Advances in Difference Equations, 2007, 042530,