On distal flows and common fixed point theorems in Banach spaces

Abstract

A semigroup S consisting of self-mappings defined on a set B of a Banach space X is said to be a distal flow on B provided0∉{Sx−Sy:S∈S}‾,for allx≠y∈B. In this paper, we show that if T is a nonexpansive self-mapping defined on a nonempty compact convex set C of X so that S≡{Tn:n∈N} forms a distal flow, then T:C→C is a surjective isometry; in particular, if C is of finite dimension, then T is a surjective affine isometry. Consequently, i) every distal S of nonexpansive mappings on a convex compact set C consists of surjective self-isometries on C; ii) all S in the distal S have a common fixed point in the Chebyshev center of C; iii) if, in addition, C is finite dimensional, then every S in S is a surjective affine self-isometry on C. Therefore, it can also be understood as a refinement of a recent common fixed point theorem due to Wiśnicki. We also show that the conclusion i) is not true if C is not compact. Finally, we give a localized setting of Lim-Lin's common fixed point theorem for isometries.