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Inspired by biological phenomena with effects of switching off (maybe just for a while), we investigate non-autonomous reaction-diffusion inclusions whose multi-valued reaction term may depend on the essential supremum over a time interval in the recent past (but) pointwise in space. The focus is on sufficient conditions for the existence of pullback attractors. If the multi-valued reaction term satisfies a form of inclusion principle standard tools for non-autonomous dynamical systems in metric spaces can be applied and provide new results (even) for infinite time intervals of delay. More challenging is the case without assuming such a monotonicity assumption. Then we consider the parabolic differential inclusion with the time interval of delay depending on space and extend the approaches of norm-to-weak semigroups to a purely metric setting. This provides completely new tools for proving pullback attractors of non-autonomous dynamical systems in metric spaces.

In the nineties, Aubin suggested how to formulate ordinary differential equations and their main existence theorems in metric spaces:

*mutational equations*(which are quite similar to the quasidifferential equations of Panasyuk). Now the well-known Antosiewicz-Cellina Theorem is extended to so-called mutational inclusions. It provides new results about nonlocal set evolutions in R

^{ N }.

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