Random fixed points of random multivalued operators on polish spaces

Abstract

RANDOM coincidence point theorems and random fixed point theorems are stochastic generalizations of classical coincidence point theorems and classical fixed point theorems. Random fixed point theorems for contraction mappings in Polish spaces were proved by Spacek [l] and Hans (2, 31. For a complete survey, we refer to Bharucha-Reid [4]. Itoh [5] proved several random fixed point theorems and gave their applications to random differential equations in Banach spaces. Recently, Sehgal and Singh [7], Papageorgiou [8] and Lin [9] have proved different stochastic versions of the well-known Schauder’s fixed point theorem. The aim of this paper is to prove various stochastic versions of Banach type fixed point theorems for multivalued operators. Section 2 is aimed at clarifying the terminology to be used and recalling basic definitions and facts. Section 3 deals with random coincidence point theorems for a pair of compatible random multivalued operators. The structure of common random fixed points of these operators is also studied. In Section 4, the existence of a common random fixed point of two random multivalued operators satisfying the Meir-Keeler type condition in Polish spaces is proved. Section 5 contains a random fixed point theorem for a pair of locally contractive random multivalued operators in .s-chainable Polish spaces. As an application, a theorem on random approximation is also obtained.