A multivalued history-dependent operator and implicit evolution inclusions. I

On the space of continuous functions from a line segment to a reflexive Banach space, we consider some operator whose values are closed convex subsets of the space. If the values are singletons, the operator becomes a well-known single-valued history-dependent operator. We study the properties of the operator, prove a fixed-point theorem analogous to the fixed-point theorem for single-valued history-dependent operators, and provide some examples. The results are applied to study implicit (unresolved for derivatives) evolution inclusions with maximal monotone operators and with perturbations in a Hilbert space. These perturbations are single-valued and multivalued history-dependent operators.