Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions

Abstract

We consider a nonautonomous impulsive Cau\-chy 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.