Existence of solutions of multi-valued Urysohn integral equations

Abstract

We prove the existence of a solution of an integral inclusion of Urysohn type with delay. By imposing standard boundedness, convexity, and semicontinuity conditions on the set-valued mapping defining the integral inclusion, we show that the right-hand side of the relation constitutes a mapping defined on a suitable Banach space and satisfying the conditions of Kakutani's theorem for the existence of a fixed point of a set-valued mapping. By introducing the notion of generalized or chattering state solutions, we show how the convexity requirements may be relaxed.