Approximating Common Fixed Points for a Finite Family of Asymptotically Nonexpansive Mappings Using Iteration Process with Errors Terms

Abstract

Let $X$ be a real Banach space and $K$ a nonempty closed convex subset of $X$. Let $T_i:K\to K\mathrm{}\mathrm{}\mathrm{}\mathrm{}\hspace{0.17em}(i=\mathrm{1},\mathrm{}\mathrm{}\mathrm{2},\mathrm{}\dots ,\mathrm{}\mathrm{}m)$ be $m$ asymptotically nonexpansive mappings with sequence $\{k_n\}\subset [\mathrm{1},\hspace{0.17em}\mathrm{\infty })$, , and $ℱ={\bigcap }_{i=\mathrm{1}}^m\mathrm{‍}F(T_i)\ne \mathrm{\varnothing }$, 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 for $i=\mathrm{1,2},\dots ,m$. We give $\{x_n\}$ defined by $x_{\mathrm{1}}\in K,$ $x_{n+\mathrm{1}}=(\mathrm{1}-a_{\mathrm{1}n}-b_{\mathrm{1}n})y_{n+m-\mathrm{2}}+a_{\mathrm{1}n}{T}_{\mathrm{1}}^n{y}_{n+m-\mathrm{2}}+b_{\mathrm{1}n}{u}_{\mathrm{1}n},\mathrm{}{y}_{n+m-\mathrm{2}}=(\mathrm{1}-a_{\mathrm{2}n}-b_{\mathrm{2}n})y_{n+m-\mathrm{3}}+a_{\mathrm{2}n}{T}_{\mathrm{2}}^n{y}_{n+m-\mathrm{3}}\hspace{0.17em}+$ $b_{\mathrm{2}n}{u}_{\mathrm{2}n},\dots ,$ $y_{n+\mathrm{2}}=(\mathrm{1}-a_{(m-\mathrm{2})n}-b_{(m-\mathrm{2})n})y_{n+\mathrm{1}}+a_{(m-\mathrm{2})n}{T}_{m-\mathrm{2}}^n{y}_{n+\mathrm{1}}+b_{(m-\mathrm{2})n}{u}_{(m-\mathrm{2})n},$ $y_{n+\mathrm{1}}=(\mathrm{1}-a_{(m-\mathrm{1})n}\hspace{0.17em}-$ $b_{(m-\mathrm{1})n})y_n+a_{(m-\mathrm{1})n}{T}_{m-\mathrm{1}}^n{y}_n+b_{(m-\mathrm{1})n}{u}_{(m-\mathrm{1})n},$ $y_n=(\mathrm{1}-a_{mn}-b_{mn})x_n+a_{mn}{T}_m^n{x}_n+b_{mn}{u}_{mn},\mathrm{}\mathrm{}\mathrm{}\mathrm{}m\ge \mathrm{2},\mathrm{}\mathrm{}\mathrm{}n\ge \mathrm{1}.$ 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.