Second Order Delay Lattice Dynamical System Of Global Attractor

In this paper, we are concerned with the long-time behavior of the following second-order retarded lattice differential equations given by with the initial date: where Z denotes the integer set, u =( u i)i∈Z∈l2 ,λis a positive constant, f,h are nonlinear smooth functions satisfying some conditions, g=( g i) i∈Z∈l 2is given , the delay term u it = ui(t +s) with t >0 in (1) is a continuous function of s which maps the interval of delay time [?υ,0] to R andυis a positive constant.The main purpose of this paper is to establish the existence of a compact global attractor. The uniqueness and existence is first proved for the solutions of the second-order retarded lattice dynamical systems in the Hilbert space Eλ= Xυλ×Xυ, and a priori estimate is obtained on the solutions. The existence of {sυ(t)}t≥0and an absorbing set B 0υ= B ( 0,R0) is then discussed for the systems. In the following, an estimate on tails of the solutions is derived when the time is large enough, which ensures the asymptotic compactness of solutions. Finally, we prove the existence of the global attractor.In Section 1, the background and history about the related are given.In Section 2, we introduce some preliminary results, including basic concepts and some notations, meanwhile we present a simple description of the systems.In Section 3, we devote to the existence of an infinite-dimensional lattice dynamical system {sυ(t)}t≥0 generated by equations (1),(2) under the given assumptions.In Section 4, we get the main results, i.e., the existence of a global attractor for the second-order retarded lattice dynamical systems. The proof is mainly composed of two important results, one is the existence of an absorbing set, and the other is the asymptotic compactness of solutions.