A Topological Fixed-Point Index Theory for Evolution Inclusions

Abstract

In the paper we construct a topological fixed-point theory for a class of set-valued maps which appears in natural way in boundary value problems for differential inclusions. Our construction is based upon the notion of ($U,V$)-approximation in the sense of Ben-El-Mechaiekh and Deguire. As applications we consider initial-value problems for nonlinear evolution inclusions of the type

where the operator $A$ satisfies various monotonicity assumptions and $F$ is an upper semicontinuous set-valued perturbation.