Multiple periodic solutions for resonant difference equations

Jianming Zhang 0 1 Shuli Wang 0 1 Jinsheng Liu 0 1 Yurong Cheng 0 1 0 University of Technology , Taiyuan 1 College of Mathematics , Taiyuan In this paper, we study the existence of multiple periodic solutions for nonlinear second-order difference equations with resonance at origin. The approach is based on critical point theory, minimax methods, homological linking and Morse theory. 1 Introduction In this paper, we consider the existence of multiple periodic solutions for the following nonlinear difference equations: u(k ) = mu(k) + f (k, u(k)), k Z[, N ], u() = u(N ), u() = u(N + ), is a differential function satisfying u(k) = ( u(k)), f (k, ) : R R f (k, ) = , k Z[, N ], N = u(k ) = u(k), k Z[, N ], u() = u(N ), u() = u(N + ). if N is odd, if N is even. in finding nontrivial periodic solutions for (P). It follows from [] that all the eigenvalues be the corresponding orthonormal eigenvectors; if N is even then all eigenvalues are multiplicity two except and N , and let be the corresponding orthonormal eigenvectors. Now we establish the variational framework associated with (P). Set u = u(), u(), . . . , u(N ) T , f(u) = f , u() , f , u() , . . . , f N , u(N ) T , Then we can rewrite (P) and (P) as respectively. Let E = RN with inner product u , v = u E. kN= u(k)v(k) and norm u = u , u . Then For p , define u p = ( kN= |u(k)|p)/p, then there exist positive numbers ap, bp such that apu u p bpu , u E. Define the functional J : E R by J(u) = Au , u mu F k, u(k) , u E, u E, u E, where IN is the identity matrix of order N , f (u) = diag{f (, u()), . . . , f (N , u(N ))}. Hence the periodic solutions of (P) are exactly the critical points of J or J in E. We assume that the nonlinearity f satisfies the following conditions: (f) f (k, ) = for k Z[, N ]. (f+) There exists > such that F(k, x) , k Z[, N ], |x| . F(k, x) , k Z[, N ], |x| . (f+ ) There exist r > and > such that F(k, x) f (k, x)x, k Z[, N ], |x| > r. (f ) There exist r > and > such that F(k, x) f (k, x)x, k Z[, N ], |x| > r. F(k, x) C + |x| , k Z[, N ], x R. Therefore we regard the problem (P) as resonance at origin under the assumption (f). Critical point theory has been widely used to study the existence of periodic solutions and solutions for nonlinear difference boundary value problems since the first result was established by using variational methods in (see []). Since then, by using critical point theory, minimax methods and Mores theory, the existence of solutions for nonresonant difference equations has been extensively investigated (see [] and the references therein). As for resonant cases, Zhu and Yu [] applied critical point theory to study the existence of positive solutions for a second order nonlinear discrete Dirichlet boundary value problem u(k ) = f (k, u(k)), k Z[, N ], u() = , u(N + ) = when nonlinearity f is odd and resonant at infinity. Zheng and Xiao [] employed critical groups and the mountain pass theorem to study the existence of nontrivial solutions for (.) when the nonlinearity f (k, u) = V (u) and is resonant at infinity. Liu et al. [] used Morse theory, critical point theory and minimax methods to study the existence of multiple solutions for (.) with resonance at both infinity and origin, one can refer to [, ]. However, we note that only a few papers concern the existence of periodic solutions for difference equations with resonance. In , Zhang and Wang [] used variational methods and Morse theory to study the multiplicity of periodic solutions for (P) with double resonance between two consecutive eigenvalues at infinity. The main aim of this paper is to study the multiplicity of nonzero periodic solutions for (P) with resonance at origin. The approach is based on critical theory, Morse theory and homological linking. The rest of this paper is organized as follows. In Section , we collect some useful preliminary results about Morse theory. In Section , we give some auxiliary results. Our main results and proofs will be given in Section . 2 Preliminaries about Morse theory In this section, we recall some facts about Morse theory and critical groups [, ]. Let E be a real Hilbert space. We say that J satisfies the (PS) condition if every sequence {un} E such that J(un) is bounded and J (un) as n has a convergent subsequence. Suppose that J C(E, R) is a functional satisfying the (PS) condition. Let u be an isolated critical point of J with J(u) = c R, and let U be a neighborhood of u. The group is called the qth critical group of J at u, where Jc = {u E | J(u) c}, H(A, B) denotes the singular relative homology group of the topological pair (A, B) with coefficient field F. Define K = u E | J (u) = . Cq(J, ) := Hq E, Ja , q Z Assume that K is a finite number. Take a < inf J(K). The group is called the qth critical group of J at infinity (see []). The Morse type numbers of the pair (E, Ja) are defined by Mq := uK dim Cq(J, u). Denote by q := dim Cq(J, ) the Betti numbers of the pair (E, Ja). By Morse theory, the relationship between Mq and q is described by ()qjMj ()qMq = From Mq q, for each q N, it follows that if Cl(J, ) for some l N, then J must have a critical point u with Cl(J, u) . If K = {u}, then Cq(J, ) = Cq(J, u) for all q N. Thus if Cl(J, ) Cl(J, u) for some l N, then J must have a new critical point. For some k Z, define Cq(J, u) = q,F if u is an isolated local minimum of J, q,lF if u is an isolated local maximum of J and l = dim E < . Suppose that J C(E, R) and u K. Then J (u) is a self-adjoint linear operator on E. The dimension of the largest negative space of J (u) is called the Morse index of J at u, and the dimension of the kernel of J (u) is called the nullity of J at u. We say that u is nondegenerate if the nullity of J at u is zero, i.e., J (u) has a bounded inverse. For an isolated critical point, the following important result is valid. Proposition . ([]) Let be an isolated critical point of J C(E, R) with finite Morse index () and nullity (). Assume that J has a local linking at with respect to a direct sum decomposition E = E E+, l = dim E < , i.e., there exists > such that J(z) for z E, u , J(u) for u E+, u . Proposition . ([, ]) Let E be a real Banach space with E = X Y and suppose that l = dim X is finite. Assume that J C(E, R) satisfies the (PS) condition and where B = {u E | u }, (H) there exist R > > and Y with = such that u Q, where Q = {u = v + s | v X, v R , s R}. Cl+(J, u) 3 Auxiliary results We first show that the functional J satisfies the (PS) condition. Lemma . Assume that f satisfies (f+ ) or (f ), then J defined by (.) satisfies the (PS) condition. J(un) C, n N, J (un) as n . We only need to show that {un} is bounded. Taking positive number (/ , /), it follows from (.) that there exists K N such that C + u n J (un) J (un), un , n > K . By (f+ ), there exist C, C > such that F(k, x) C (...truncated)