Multivalued perturbations of subdifferential type evolution equations in Hilbert spaces

AbstractIn this paper we study the multivalued evolution equation −ẋ(t)ϵ∂ϑ(x(t)) + F(t, x(t)), x(0) = x0, where ϑ: X → R̄ is a proper, convex, lower semicontinuous (l.s.c.) function, F(·, ·) is a multivalued perturbation, and X is an infinite dimensional, separable Hilbert space. We have an existence result for F(·, ·) being nonconvex valued, and another for F(·, ·) being convex valued but not closed valued. When ϑ = δK = indicator function of a compact, convex set K, we obtain some extensions of earlier results by Moreau and Henry. Then using the Kuratowski-Mosco convergence of sets and the τ-convergence of functions, we prove a well posedness result for the evolution inclusion we are studying. Also we consider a random version of it and prove the existence of a random solution. Finally we present applications to problems in partial differential equations