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Forward omega limit sets of nonautonomous dynamical systems

Dedicated to Professor Jürgen Scheurle on his 65th birthday

HC was partially funded by China Postdoctoral Science Foundation 2017M612430. PEK and MY were partially supported by the Chinese NSF grant 11571125

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  • 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.

    Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L30.


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