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 x'(t) \in –A(t,x(t)) + F(t, x(t)) |x(0) = x_0 where the operator A satisfies various monotonicity assumptions and F is an upper semicontinuous set-valued perturbation.