Common fixed point property through analysis of retractions (Q889908)

In this article, the author deals with the problem of the existence of common fixed points for a commuting family of nonexpansive mappings. The main result proved in that paper states that, if \(C\) is a nonempty, \(\sigma(E, \Gamma)\)-compact and convex subset of a Banach space \(E\) (\(\Gamma\) denotes here a nonempty subspace of the dual \(E^{*}\) which is a norming set for \(E\)) which has the fixed point property as well as the fixed point property for spheres and the \(\sigma(E, \Gamma)\)-sequential closure of any convex subset of \(C\) coincides with its \(\sigma(E, \Gamma)\)-closure, then for any commuting family \(S\) of self-mappings of \(C\), the set \(\mathrm{Fix}(S)\) of common fixed points of \(S\) is a nonempty nonexpansive retract of \(C\). As a corollary of that result, the author gives the well-known Bruck's theorem stating that if \(C\) is a nonempty weakly compact and convex subset of a Banach space \(E\) which possesses the fixed point property as well as the conditional fixed point property, then for any commuting family \(S\) of nonexpansive self-mappings of \(C\), the set \(\mathrm{Fix}(S)\) of common fixed points of \(S\) is a nonempty nonexpansive retract of \(C\).