Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces

Naseer Shahzad,Aniefiok Udomene

doi:10.1155/fpta/2006/18909

Naseer Shahzad, Aniefiok Udomene

Open Access

https://doi.org/10.1155/fpta/2006/18909

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Abstract

AbstractSuppose "Equation missing" is a nonempty closed convex subset of a real Banach space "Equation missing". Let "Equation missing" be two asymptotically quasi-nonexpansive maps with sequences "Equation missing" such that "Equation missing" and "Equation missing", and "Equation missing". Suppose "Equation missing" is generated iteratively by "Equation missing" where "Equation missing" and "Equation missing" are real sequences in "Equation missing". It is proved that (a) "Equation missing" converges strongly to some "Equation missing" if and only if "Equation missing"; (b) if "Equation missing" is uniformly convex and if either "Equation missing" or "Equation missing" is compact, then "Equation missing" converges strongly to some "Equation missing". Furthermore, if "Equation missing" is uniformly convex, either "Equation missing" or "Equation missing" is compact and "Equation missing" is generated by "Equation missing", where "Equation missing", "Equation missing" are bounded, "Equation missing" are real sequences in "Equation missing" such that "Equation missing" and "Equation missing", "Equation missing" are summable; it is established that the sequence "Equation missing" (with error member terms) converges strongly to some "Equation missing".

Highlights

Let K be a nonempty subset of a real normed linear space E
The class of asymptotically nonexpansive maps was introduced by Goebel and Kirk [3]
As an important generalization of the class of nonexpansive maps. They established that if K is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is an asymptotically nonexpansive self-mapping of K, T has a fixed point

Summary

Introduction

Let K be a nonempty subset of a real normed linear space E. As an important generalization of the class of nonexpansive maps They established that if K is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is an asymptotically nonexpansive self-mapping of K, T has a fixed point. Ghosh and Debnath [2] extended Petryshyn and Williamson’s results and obtained some necessary and sufficient conditions for an Ishikawa-type iterative sequence to converge to a fixed point of a quasi-nonexpansive mapping. In [11], Qihou established sufficient conditions for the strong convergence of the Ishikawa-type iterative sequences with error member for a uniformly (L,γ)-Lipschitzian asymptotically nonexpansive self-mapping of a nonempty compact convex subset of a uniformly convex Banach space. Our results are significant generalizations of the corresponding results of Ghosh and Debnath [2], Petryshyn and Williamson [8], Qihou [9,10,11], and of Khan and Takahashi [6]

Preliminaries

Main results