Forward omega limit sets of nonautonomous dynamical systems

The forward $ \omega $-limit set $ \omega_{\mathcal{B}} $ of a nonautonomous dynamical system $ \varphi $ with a positively invariant absorbing family $ \mathcal{B} $ $ = $ $ \{ B(t), t \in \mathbb{R}\} $ of closed 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 $ \varphi $ 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 a $ \varphi $-invariant family $ \mathcal{A} $ $ = $ $ \left\{A(t), t \in \mathbb{R}\right\} $ in $ \mathcal{B} $ of nonempty compact subsets of $ X $ to be a forward attractor is generalised to this context.