## Journal of Applied Mathematics

### An Implicit Iteration Process for Common Fixed Points of Two Infinite Families of Asymptotically Nonexpansive Mappings in Banach Spaces

#### Abstract

Let $K$ be a nonempty, closed, and convex subset of a real uniformly convex Banach space $E$. Let $\left\{{T}_{\lambda }{\right\}}_{\lambda \in \mathrm{\Lambda }}$ and $\left\{{S}_{\lambda }{\right\}}_{\lambda \in \mathrm{\Lambda }}$ be two infinite families of asymptotically nonexpansive mappings from $K$ to itself with $F:=\left\{x\in K:{T}_{\lambda }x=x={S}_{\lambda }x,\lambda \in \mathrm{\Lambda }\right\}\ne \varnothing$. For an arbitrary initial point ${x}_{0}\in K$, $\left\{{x}_{n}\right\}$ is defined as follows: ${x}_{n}={\alpha }_{n}{x}_{n-1}+{\beta }_{n}\left({T}_{n-1}^{*}{\right)}^{{m}_{n-1}}{x}_{n-1}+{\gamma }_{n}\left({T}_{n}^{*}{\right)}^{{m}_{n}}{y}_{n}$, ${y}_{n}={\alpha }_{n}^{\prime }{x}_{n}+{\beta }_{n}^{\prime }\left({S}_{n-1}^{*}{\right)}^{{m}_{n-1}}{x}_{n-1}+{\gamma }_{n}^{\prime }\left({S}_{n}^{*}{\right)}^{{m}_{n}}{x}_{n}$, $n=1,2,3,\dots$, where ${T}_{n}^{*}={T}_{{\lambda }_{{i}_{n}}}$ and ${S}_{n}^{*}={S}_{{\lambda }_{{i}_{n}}}$ with ${i}_{n}$ and ${m}_{n}$ satisfying the positive integer equation: $n=i+\left(m-1\right)m/2$, $m\ge i$; $\left\{{T}_{{\lambda }_{i}}{\right\}}_{i=1}^{\infty }$ and $\left\{{S}_{{\lambda }_{i}}{\right\}}_{i=1}^{\infty }$ are two countable subsets of $\left\{{T}_{\lambda }{\right\}}_{\lambda \in \mathrm{\Lambda }}$ and $\left\{{S}_{\lambda }{\right\}}_{\lambda \in \mathrm{\Lambda }},$ respectively; $\left\{{\alpha }_{n}\right\}$, $\left\{{\beta }_{n}\right\}$, $\left\{{\gamma }_{n}\right\}$, $\left\{{\alpha }_{n}^{\prime }\right\}$, $\left\{{\beta }_{n}^{\prime }\right\}$, and $\left\{{\gamma }_{n}^{\prime }\right\}$ are sequences in $\left[\delta ,1-\delta \right]$ for some $\delta \in \left(0,1\right)$, satisfying ${\alpha }_{n}+{\beta }_{n}+{\gamma }_{n}=1$ and ${\alpha }_{n}^{\prime }+{\beta }_{n}^{\prime }+{\gamma }_{n}^{\prime }=1$. Under some suitable conditions, a strong convergence theorem for common fixed points of the mappings $\left\{{T}_{\lambda }{\right\}}_{\lambda \in \mathrm{\Lambda }}$ and $\left\{{S}_{\lambda }{\right\}}_{\lambda \in \mathrm{\Lambda }}$ is obtained. The results extend those of the authors whose related researches are restricted to the situation of finite families of asymptotically nonexpansive mappings.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 602582, 6 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807843

Digital Object Identifier
doi:10.1155/2013/602582

Mathematical Reviews number (MathSciNet)
MR3029966

Zentralblatt MATH identifier
1266.47090

#### Citation

Deng, Wei-Qi; Bai, Peng. An Implicit Iteration Process for Common Fixed Points of Two Infinite Families of Asymptotically Nonexpansive Mappings in Banach Spaces. J. Appl. Math. 2013 (2013), Article ID 602582, 6 pages. doi:10.1155/2013/602582. https://projecteuclid.org/euclid.jam/1394807843

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