Local Approximate Forward Attractors of Nonautonomous Dynamical Systems

Pullback dynamics of nonautonomous dynamical systems has been considerably developed. However, it is still a tough job to study forward dynamics of nonautonomous dynamical systems, since forward attractors were only obtained in some particular cases. In the paper, under some reasonable conditions, it is shown that closing to a local pullback attractor, there is an approximate forward attractor. Specifically, let ϕ be a cocycle semiflow on a Banach space X with driving system θ on a base space P. Suppose that the base space P is compact and ϕ is uniformly asymptotically compact. Let A(∙) be a local pullback attractor with being compact. We prove that every e-extended neighborhood Ae(∙) of A(∙) will forward attract every bounded set B(∙) that is pullback attracted by A(∙). We then call Ae(∙) an approximate forward attractor of ϕ.