Resumen de Existence ofixed points for pointwise eventually asymptotically nonexpansive mappings

Mohanasundaram Radhakrishnan, S. Rajesh

• Kirk introduced the notion of pointwise eventually asymptotically nonexpansive mappings and proved that uniformly convex Banach spaces have the fixed point property for pointwise eventually asymptotically nonexpansive maps. Further, Kirk raised the following question: ``Does a Banach space $X$ have the fixed point property for pointwise eventually asymptotically nonexpansive mappings whenever $X$ has the fixed point property for nonexpansive mappings?". In this paper, we prove that a Banach space $X$ has the fixed point property for pointwise eventually asymptotically nonexpansive maps if $X$ has uniform normal sturcture or $X$ is uniformly convex in every direction with the Maluta constant $D(X)<1.$ Also, we study the asymptotic behavior of the sequence $\{T^nx\}$ for a pointwise eventually asymptotically nonexpansive map $T$ defined on a nonempty weakly compact convex subset $K$ of a Banach space $X$ whenever $X$ satisfies the uniform Opial condition or $X$ has a weakly continuous duality map.