FORWARD OMEGA LIMIT SETS OF NONAUTONOMOUS DYNAMICAL SYSTEMS

The forward omega-limit set omega(B) of a nonautonomous dynamical system phi with a positively invariant absorbing family B = {B(t), t is an element of R} of dosed and bounded subsets of a Banach space X which is asymptotically compact is shown to be asymptotically positive invariant in general and asymptotic negative invariant if phi is also strongly asymptotically compact and eventually continuous in its initial value uniformly on bounded time sets independently of the initial time. In addition, a necessary and sufficient condition for phi-invariant family A = {A(t), t is an element of R} in B of nonempty compact subsets of X to be a forward attractor is generalised to this context.