Random approximations and random fixed point theorems for continuous $1$-set-contractive random maps

Abstract

Recently the author [Proc. Amer. Math. Soc. 103 (1988), 1129-1135] proved random versions of an interesting theorem of Ky Fan [Theorem 2, Math. Z. 112 (1969), 234-240] for continuous condensing random maps and nonexpansive random maps defined on a closed convex bounded subset in a separable Hilbert space. In this paper, we prove that it is still true for (more general) continuous 1-set-contractive random maps, which include condensing, nonexpansive, locally almost nonexpansive (LANE), semicontractive maps, etc. Then we use these theorems to obtain random fixed points theorems for the above-mentioned maps satisfying weakly inward conditions. In order to obtain these results, we first need to prove a random fixed point theorem for 1-set-contractive self-maps in a separable Banach space. This leads to the discovery of some new random fixed point theorems in a separable uniform convex Banach space.