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 $\{T_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$ and $\{S_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$ be two infinite families of asymptotically nonexpansive mappings from $K$ to itself with $F:=\{x\in K:T_{\lambda }x=x=S_{\lambda }x,\lambda \in \mathrm{\Lambda }\}\ne \varnothing$. For an arbitrary initial point $x_0\in K$, $\{x_n\}$ is defined as follows: $x_n={\alpha }_n{x}_{n-1}+{\beta }_n(T_{n-1}^{*}{)}^{m_{n-1}}{x}_{n-1}+{\gamma }_n(T_n^{*}{)}^{m_n}{y}_n$, $y_n={\alpha }_n^{\prime }{x}_n+{\beta }_n^{\prime }(S_{n-1}^{*}{)}^{m_{n-1}}{x}_{n-1}+{\gamma }_n^{\prime }(S_n^{*}{)}^{m_n}{x}_n$, $n=\mathrm{1,2},3,\dots$, 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: $n=i+(m-1)m/2$, $m\ge i$; $\{T_{{\lambda }_i}{\}}_{i=1}^{\infty }$ and $\{S_{{\lambda }_i}{\}}_{i=1}^{\infty }$ are two countable subsets of $\{T_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$ and $\{S_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }},$ respectively; $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, $\{{\alpha }_n^{\prime }\}$, $\{{\beta }_n^{\prime }\}$, and $\{{\gamma }_n^{\prime }\}$ are sequences in $[\delta ,1-\delta ]$ for some $\delta \in (\mathrm{0,1})$, 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 $\{T_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$ and $\{S_{\lambda }{\}}_{\lambda \in \mathrm{\Lambda }}$ is obtained. The results extend those of the authors whose related researches are restricted to the situation of finite families of asymptotically nonexpansive mappings.