Unbounded perturbation to evolution problems with time-dependent subdifferential operators

Abstract

In this paper, we consider a nonlinear evolution inclusion governed by the subdifferential of a proper convex lower semicontinuous function in a separable Hilbert space. The right-hand side contains a set-valued perturbation with nonempty closed convex and not necessary bounded values. The existence of absolutely continuous solution is stated under different assumptions on the perturbation. The main purpose in this paper is to study, in the setting of infinite dimensional Hilbert space H, the perturbed problem (P), under various assumptions. Throughout the paper, H is a separable Hilbert space whose inner product is denoted by <.,.> and the associated norm by II. II and [0, T] is an interval of R. We will denote by B the closed unit ball of H; P_c(H) the family of all nonempty closed sets of H and P_cc(H) (resp. P_ck(H)) the set of nonempty closed (resp. compact) convex subsets of H. We give some preliminaries and we recall some results which will be used in the paper. We establish the existence theorem for the considered problem (P) for a globally upper hemicontinuous perturbation, then we extend the result obtained in [0, T] to the whole interval R+. Finally, we weaken the result by taking the perturbation G measurable in the time t and upper semicontinuous in the state x.

Keywords

• Evolution equations
• differential inclusion
• subdifferential operator
• unbounded perturbation

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