Lower Semicontinuity of Invariant Sets for Nonautonomous Processes and Skew Products

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We give general sufficient conditions for lower semicontinuity of invariant sets for some generalizations of classical dynamical systems, such as nonautonomous processes and skew products. It is shown that under these conditions, the invariant set is trajectorically lower semicontinuous, i.e., for any of its trajectory, there exists a close trajectory of the perturbed process. We also give conditions under which the mapping which takes a trajectory of the invariant set to a close trajectory of the perturbed process is continuous.