We consider a nonautonomous impulsive Cauchy problem of parabolic type involving a nonlocal initial condition in a Banach space X, where the operators in linear part (possibly unbounded) depend on time t and generate an evolution family. New existence theorems of mild solutions to such a problem, in the absence of compactness and Lipschitz continuity of the impulsive item and nonlocal item, are established. The non-autonomous impulsive Cauchy problem of neutral type with nonlocal initial condition is also considered. Comparisons with available literature are also given. Finally, as a sample of application, these results are applied to a system of partial differential equations with impulsive condition and nonlocal initial condition. Our results essentially extend some existing results in this area.
We consider a nonlinear delay evolution equation with multivalued perturbation on a noncompact interval. The nonlinearity, having convex and closed values, is upper hemicontinuous with respect to the solution variable. A basic question on whether there exists a solution set carrying $R_{\delta }$-structure remains unsolved when the operator families generated by the principal part lack compactness. One of our main goals is to settle this question in the affirmative. Moreover, we prove that the solution map, having compact values, is an $R_{\delta }$-map, which maps any connected set into a connected set. It is then exploited to deal with the existence in the large for a nonlocal problem. Finally, several examples are worked out in detail, illustrating the applicability of our general results.
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