On the Ryll-Nardzewski fixed point property in convex uniform spaces

Abstract

In this paper, we introduce the concept of a convex uniform space as a natural generalization of locally convex spaces, and a fixed point property as a semigroup formulation of the famous Ryll-Nardzewski fixed point theorem, and study its non-linear counterpart in this framework. The Nardzewski's fixed point theorem is a beautiful result in fixed point theory that ensures the existence of a common fixed point for any noncontracting semigroup of continuous affine self-mappings on a non-void weakly compact convex set in a separated locally convex space. We establish that in a separated convex uniform space satisfying a certain property (P), any noncontracting finite semigroup of contraction mappings of a non-empty bounded closed convex subset possesses a common fixed point. In case where the underlying space is admissible, we prove that this result is extendable to arbitrary noncontracting semigroups without condition (P) if we assume that the phase space is weakly compact and convex.