Fixed point theorems and convergence theorems for a generalized nonexpansive mapping in uniformly convex Banach spaces

TL;DR

This paper establishes fixed point existence and convergence theorems for generalized nonexpansive mappings in uniformly convex Banach spaces, demonstrating faster convergence of a specific iterative scheme compared to previous methods.

Contribution

It introduces new fixed point and convergence theorems for a class of generalized nonexpansive mappings in uniformly convex Banach spaces, with an improved iterative scheme.

Findings

Proved fixed point existence under condition (Da)

Established convergence theorems for iterative schemes

Demonstrated faster convergence of the new scheme

Abstract

In this paper, we prove the existence of fixed points of mappings satisfying the condition (Da), a kind of generalized nonexpansive mappings, on a weakly compact convex subset in a Banach space satisfying Opial's condition. And we use Sahu([6]) and Thakur([10])'s iterative scheme to establish several convergence theorems in uniformly convex Banach spaces and give an example to show that this scheme converges faster than the scheme in [1]