Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

Abstract

Let $C$ be a nonempty closed convex subset of a real uniformly smooth Banach space $X$, $\{{T}_{k}{\}}_{k=1}^{\infty }:C\to C$ an infinite family of nonexpansive mappings with the nonempty set of common fixed points ${\bigcap }_{k=1}^{\infty }\mathrm{Fix}({T}_{k})$, and $f:C\to C$ a contraction. We introduce an explicit iterative algorithm ${x}_{n+1}={\alpha }_{n}f({x}_{n})+(1-{\alpha }_{n}){L}_{n}{x}_{n}$, where ${L}_{n}={\sum }_{k=1}^{n}\left({\omega }_{k}/{\text{s}}_{n}\right){T}_{k},{S}_{n}={\sum }_{k=1}^{n}{\omega }_{k},\mathrm{ }$ and ${w}_{k}>0$ with ${\sum }_{k=1}^{\infty }{\omega }_{k}=1$. Under certain appropriate conditions on $\{{\alpha }_{n}\}$, we prove that $\{{x}_{n}\}$ converges strongly to a common fixed point ${x}^{*}$ of $\{{T}_{k}{\}}_{k=1}^{\infty }$, which solves the following variational inequality: $〈{x}^{*}-f({x}^{*}),J({x}^{*}-p)〉\le 0,\mathrm{ }\mathrm{ }p\in {\bigcap }_{k=1}^{\infty }\text{Fix}({T}_{k})$, where $J$ is the (normalized) duality mapping of $X$. This algorithm is brief and needs less computational work, since it does not involve $W$-mapping.