Periodic solutions for a class of differential inclusions in general Banach spaces

Abstract

Let X be a real Banach space, A : D ( A ) ⊆ X ↝ X an m-accretive operator and F : R × D ( A ) ¯ ↝ X a multi-function which is 2 π-periodic with respect to its first argument, has nonempty, closed, convex and weakly compact values and is strongly–weakly upper semicontinuous. In this paper we prove the existence of at least one solution for the problem { u ′ ( t ) + A u ( t ) ∋ f ( t ) , f ( t ) ∈ F ( t , u ( t ) ) , u ( t ) = u ( t + 2 π ) , in the case in which F satisfies an appropriate “sign” condition.