Estimate of Number of Periodic Solutions of Second-Order Asymptotically Linear Difference System

Honghua Bin,Zhenkun Huang

doi:10.1155/2013/707686

Abstract

We investigate the number of periodic solutions of second-order asymptotically linear difference system. The main tools are Morse theory and twist number, and the discussion in this paper is divided into three cases. As the system is resonant at infinity, we use perturbation method to study the compactness condition of functional. We obtain some new results concerning the lower bounds of the nonconstant periodic solutions for discrete system.

Highlights

IntroductionLet ηk denote the real eigenvector corresponding to the eigenvalues λk = 4sin2(kπ/p), k ∈ Z[0, r], and r = [p/2], where [⋅] stands for the greatest-integer function
In this paper we are interested in the lower bound of the number of periodic solutions for second-order autonomous difference systemΔ2xn−1 + f = 0, n ∈ Z, (1)where xn ∈ RN, f = (f1, f2, . . . , fN)T ∈ C1(RN, RN), Δxn = xn+1 − xn, Δ2xn = Δ(Δxn), and N is a fixed positive integer.Discrete systems have been investigated by many authors using various methods, and many interesting results have obtained; see [1,2,3,4,5,6,7] and references therein
Let ηk denote the real eigenvector corresponding to the eigenvalues λk = 4sin2(kπ/p), k ∈ Z[0, r], and r = [p/2], where [⋅] stands for the greatest-integer function

Summary

Introduction

Let ηk denote the real eigenvector corresponding to the eigenvalues λk = 4sin2(kπ/p), k ∈ Z[0, r], and r = [p/2], where [⋅] stands for the greatest-integer function. A p-periodic solution {xn} of (1) is called p-resonance, if there exists λk = 4sin2(kπ/p) ∈ σ(F󸀠󸀠(xn)), where F󸀠󸀠 denotes the Hessian matrix of F and σ(⋅) is the spectrum of matrix. If we assume that G(t), g(t) are bounded and system (1) is p-resonant at ∞, functional J does not satisfy the (PS) condition. To the proof of Lemma 5, we need only to prove that {w(j)} is bounded in Ep. Taking φ = w(j) in (19), it follows that o(‖w(j)‖) = ⟨JR󸀠 (x(j)), w(j)⟩ ≥ −c‖w‖+2‖w‖2φR󸀠 (‖w‖2). There is a M > 0 such that ‖ u + V ‖≤ M, and the proof is completed

Preliminaries

Proof of Main Results