Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces

TL;DR

This paper investigates the strong convergence of iterative schemes for finding fixed points of nonexpansive mappings in Banach spaces, extending and improving previous results under certain continuity conditions.

Contribution

It provides a general convergence theorem for explicit iterative schemes involving nonexpansive mappings in Banach spaces with specific duality mapping properties.

Findings

Established strong convergence under new conditions

Improved existing convergence results in the literature

Unified framework for various iterative schemes

Abstract

Let $X$ be a real Banach space with a normalized duality mapping uniformly norm-to-weak$^\star$ continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping $J_{\Phi}$ with gauge $\phi$. Let $f$ be an {\em $\alpha$-contraction} and $\{T_n\}$ a sequence of nonexpansive mapping, we study the strong convergence of explicit iterative schemes x_{n+1} = \alpha_n f(x_n) + (1-\alpha_n) T_n x_n with a general theorem and then recover and improve some specific cases studied in the literature