On the structure of the global attractor for non-autonomous difference equations with weak convergence

Abstract

Abstract. The aim of this paper is to describe the structure of global attractors for non-autonomous difference systems of equations with recurrent (in particular, almost periodic) coefficients. We consider a special class of this type of systems (the so-called weak convergent systems). We study this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our early papers to study the almost periodic (almost automorphic, recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent) solutions of difference equations.

On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence

“…If it is not Lyapunov stable, then the Levinson center contains a minimal almost periodic set, but it is not minimal (this means, in particular, that equation (1) admits a family (more than one) of solutions which are bounded on R). In [7] we generalize this result to the case of difference equations.…”

“…In order not to be repetitive with our previous papers on this topic, especially [6,7], we will skip to recall preliminary definitions and results which are necessary for our analysis and refer the reader to these already published papers.…”

“…Thus I y = {u ∈ C r : such that (u, y) ∈ J X }. It is easy to see that condition (7) means that the non-autonomous dynamical system (X, R + , π), (Y, R, σ), h is weak convergent. Now, to finish the proof of the theorem, it is sufficient to apply Theorem 3.…”