Existence ofixed points for pointwise eventually asymptotically nonexpansive mappings

M. Radhakrishnan, S. Rajesh

Abstract

Kirk introduced the notion of pointwise eventually asymptotically non-expansive mappings and proved that uniformly convex Banach spaces have the fixed point property for pointwise eventually asymptotically non expansive maps. Further, Kirk raised the following question: “Does a Banach space X have the fixed point property for pointwise eventually asymptotically nonexpansive mappings when ever 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 structure or X is uniformly convex in every direction with the Maluta constant D(X) < 1. Also, we study the asymptotic behavior of the sequence {Tnx} 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.


Keywords

fixed points; pointwise eventually asymptotically nonexpansive mappings; uniform normal structure; uniform Opial condition; duality mappings

Subject classification

47H10; 47H09.

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References

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