Existence theorems for nonlinear evolution inclusions

Abstract

In this paper we obtain an existence theorem for the abstract Cauchy problem for multivalued differential equations of the form u′∈−∂−f(u)+G(u), u(O)=x0, where ∂−f is the Frechet subdifferential of a functionf defined on an open subset Ω of a real separable Hilbert space H, taking its values in R ∪ {+∞} and G is a multifunction from C([0, T], Ω) into the nonempty subsets of L2([0, T], H). As an application we obtain an existence theorem for the multivalued perturbed problem x′∈−∂−f(x)+F(t, x), x(0)=x0, where F:[0, T]×Ω→(H) is a multifunction satisfying some regularity assumptions.