Common fixed point theorems for a commutative family of nonexpansive mappings in complete random normed modules

TL;DR

This paper develops fixed point theorems for nonexpansive mappings in complete random normed modules, introducing random Chebyshev centers and random normal structure, with applications to strong random nonexpansive operators.

Contribution

It introduces the notions of random Chebyshev centers and random normal structure, establishing fixed point theorems in the setting of complete random normed modules.

Findings

Equivalence of random complete normal structure and random normal structure for $L^0$-convexly compact sets.

Existence of common fixed points for commutative families of nonexpansive mappings.

Fixed point results for strong random nonexpansive operators without measurable selection.

Abstract

In this paper, we first introduce and study the notion of random Chebyshev centers. Further, based on the recently developed theory of stable sets, we introduce the notion of random complete normal structure so that we can prove the two deeper theorems: one of which states that random complete normal structure is equivalent to random normal structure for an $L^0$-convexly compact set in a complete random normed module; the other of which states that if $G$ is an $L^0$-convexly compact subset with random normal structure of a complete random normed module, then every commutative family of nonexpansive mappings from $G$ to $G$ has a common fixed point. We also consider the fixed point problems for isometric mappings in complete random normed modules. Finally, as applications of the fixed point theorems established in random normed modules, when the measurable selection theorems fail to work, we can still prove that a family of strong random nonexpansive operators from $(\Omega,\mathcal{F},P)\times C$ to $C$ has a common random fixed point, where $(\Omega,\mathcal{F},P)$ is a probability space and $C$ is a weakly compact convex subset with normal structure of a Banach space.