Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings (Q2750850)

Let \(X\) be a Banach space; \(C\) a weakly compact convex subset of \(X\). A map \(T: C \rightarrow C\) is said to be asymptotically nonexpansive if there exists a sequence \(k_n \rightarrow 1\) such that \(\|T^n(x) - T^n(y)\|\leq k_n\|x - y\|\) for \(x,y \in C\) and \(n = 1,2,\dots\) Under some additional conditions it is proved that the fixed point set of \(T\) is a nonexpansive retract of \(C\). This result is extended to the common fixed point set of a commuting family of asymptotically nonexpansive maps.