On random approximations and a random fixed point theorem for set valued mappings

Abstract

Abstract.We prove a random fixed point theorem in a Banach space for set valued mappings and then derive a corollary that yields a fixed point theorem of BharuchaReid and Mukherjea, as a special case.In a recent paper [1], Bharucha-Reid and Mukherjea proved the following stochastic analogue of the well-known Schauder's fixed point theorem.Theorem 1. Let S be a compact and convex subset of a Banach space E and T: Ü X S -> S be a continuous random operator. Then T admits a random fixed point.In this paper, we shall show that Theorem 1 can be derived from a more general result. For detailed definitions and terminologies we refer to Bharucha-Reid [1] or a recent paper of Itoh [3]. Throughout, this paper, (ß, 2) is a measurable space with 2 a sigma algebra of subsets of ß. The symbol 2E denotes the class of nonempty subsets of a Banach space E. A mapping F: ß -» 2E is called measurable iff for each open set G of E,

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