2013 Approximating Common Fixed Points for a Finite Family of Asymptotically Nonexpansive Mappings Using Iteration Process with Errors Terms
Seyit Temir, Adem Kiliçman
Abstr. Appl. Anal. 2013(SI27): 1-8 (2013). DOI: 10.1155/2013/974317

## Abstract

Let $X$ be a real Banach space and $K$ a nonempty closed convex subset of $X$. Let ${T}_{i}:K\to K\mathrm{}\mathrm{}\mathrm{}\mathrm{} (i=\mathrm{1},\mathrm{}\mathrm{}\mathrm{2},\mathrm{}\dots ,\mathrm{}\mathrm{}m)$ be $m$ asymptotically nonexpansive mappings with sequence $\{{k}_{n}\}\subset [\mathrm{1}, \mathrm{\infty })$, ${\sum }_{n=\mathrm{1}}^{\mathrm{\infty }}({k}_{n}-\mathrm{1})<\mathrm{\infty }$, and $\scr F={\bigcap }_{i=\mathrm{1}}^{m}\mathrm{‍}F({T}_{i})\ne \mathrm{\varnothing }$, where $F$ is the set of fixed points of ${T}_{i}$. Suppose that $\{{a}_{in}{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$, $\{{b}_{in}{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$, $i=\mathrm{1,2},\dots ,m$ are appropriate sequences in $[\mathrm{0,1}]$ and $\{{u}_{in}{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$, $i=\mathrm{1,2},\dots ,m$ are bounded sequences in $K$ such that ${\sum }_{n=\mathrm{1}}^{\mathrm{\infty }}{b}_{in}<\mathrm{\infty }$ for $i=\mathrm{1,2},\dots ,m$. We give $\{{x}_{n}\}$ defined by ${x}_{\mathrm{1}}\in K,$ ${x}_{n+\mathrm{1}}=(\mathrm{1}-{a}_{\mathrm{1}n}-{b}_{\mathrm{1}n}){y}_{n+m-\mathrm{2}}+{a}_{\mathrm{1}n}{T}_{\mathrm{1}}^{n}{y}_{n+m-\mathrm{2}}+{b}_{\mathrm{1}n}{u}_{\mathrm{1}n},\mathrm{}{y}_{n+m-\mathrm{2}}=(\mathrm{1}-{a}_{\mathrm{2}n}-{b}_{\mathrm{2}n}){y}_{n+m-\mathrm{3}}+{a}_{\mathrm{2}n}{T}_{\mathrm{2}}^{n}{y}_{n+m-\mathrm{3}} +$ ${b}_{\mathrm{2}n}{u}_{\mathrm{2}n},\dots ,$ ${y}_{n+\mathrm{2}}=(\mathrm{1}-{a}_{(m-\mathrm{2})n}-{b}_{(m-\mathrm{2})n}){y}_{n+\mathrm{1}}+{a}_{(m-\mathrm{2})n}{T}_{m-\mathrm{2}}^{n}{y}_{n+\mathrm{1}}+{b}_{(m-\mathrm{2})n}{u}_{(m-\mathrm{2})n},$ ${y}_{n+\mathrm{1}}=(\mathrm{1}-{a}_{(m-\mathrm{1})n} -$ ${b}_{(m-\mathrm{1})n}){y}_{n}+{a}_{(m-\mathrm{1})n}{T}_{m-\mathrm{1}}^{n}{y}_{n}+{b}_{(m-\mathrm{1})n}{u}_{(m-\mathrm{1})n},$ ${y}_{n}=(\mathrm{1}-{a}_{mn}-{b}_{mn}){x}_{n}+{a}_{mn}{T}_{m}^{n}{x}_{n}+{b}_{mn}{u}_{mn},\mathrm{}\mathrm{}\mathrm{}\mathrm{}m\ge \mathrm{2},\mathrm{}\mathrm{}\mathrm{}n\ge \mathrm{1}.$ The purpose of this paper is to study the above iteration scheme for approximating common fixed points of a finite family of asymptotically nonexpansive mappings and to prove weak and some strong convergence theorems for such mappings in real Banach spaces. The results obtained in this paper extend and improve some results in the existing literature.

## Citation

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Seyit Temir. Adem Kiliçman. "Approximating Common Fixed Points for a Finite Family of Asymptotically Nonexpansive Mappings Using Iteration Process with Errors Terms." Abstr. Appl. Anal. 2013 (SI27) 1 - 8, 2013. https://doi.org/10.1155/2013/974317

## Information

Published: 2013
First available in Project Euclid: 26 February 2014

zbMATH: 07095548
MathSciNet: MR3147818
Digital Object Identifier: 10.1155/2013/974317

Rights: Copyright © 2013 Hindawi

JOURNAL ARTICLE
8 PAGES

Vol.2013 • No. SI27 • 2013