Random Fixed Point Theorems for Asymptotic 1-set Contraction Operators

Random fixed point theorems for condensing 1-set contraction selfmaps are known. But no random fixed point theorem for more general asymptotic 1-set contraction selfmaps is yet available. The purpose of this paper is to prove random fixed point theorems for such maps. 1. Introduction. Random fixed point theorems are stochastic versions of (classical or deterministic) fixed point theorems and are required for the theory of random equations. The study of random fixed point theorems for contraction maps in separable complete metric spaces was initiated by Spacek [8] and Hans [3]. In separable Banach spaces, Itoh [4] proved a random fixed point theorem for condensing self-maps. This Itoh's result was extended by Xu [11] to condensing non-selfmap T , by assuming an additional condition that T satisfies either weakly inward (see [11]) or the Leray-Schauder condition (see [11]). Lin [5] proved a random fixed point theorem for 1-set contraction selfmap T , by assuming that I − T is demiclosed at zero, where I denotes the identity map.