Global solvability for nonlinear nonautonomous evolution inclusions of Volterra-type and its applications

Abstract

We study a nonlinear nonautonomous evolution inclusion of Volterra-type with time delay involving a multivalued nonlinearity. The family of operators (possibly unbounded) in the principal part of the evolution inclusion generates a noncompact evolution family. Global solvability for local and nonlocal initial value problems corresponding to the evolution inclusion is considered. The investigation in the present paper is based on two topological fixed point theorems. We impose a condition of Hausdorff-measure of noncompactness on the nonlinearity, which allows us to overcome the lack of compactness of the evolution family. Finally, we present an example to illustrate the abstract results.