Generalized Contraction and Invariant Approximation Results on Nonconvex Subsets of Normed Spaces

Mujahid Abbas,Basit Ali,Salvador Romaguera

doi:10.1155/2014/391952

Abstract

Wardowski (2012) introduced a new type of contractive mapping and proved a fixed point result in complete metric spaces as a generalization of Banach contraction principle. In this paper, we introduce a notion of generalizedF-contraction mappings which is used to prove a fixed point result for generalized nonexpansive mappings on star-shaped subsets of normed linear spaces. Some theorems on invariant approximations in normed linear spaces are also deduced. Our results extend, unify, and generalize comparable results in the literature.

Highlights

Introduction and PreliminariesOne of the most basic and important results in metric fixed point theory is the Banach contraction principle due to Banach [1]
Wardowski (2012) introduced a new type of contractive mapping and proved a fixed point result in complete metric spaces as a generalization of Banach contraction principle
We introduce a notion of generalized F-contraction mappings which is used to prove a fixed point result for generalized nonexpansive mappings on star-shaped subsets of normed linear spaces

Summary

Introduction

Introduction and PreliminariesOne of the most basic and important results in metric fixed point theory is the Banach contraction principle due to Banach [1]. Wardowski (2012) introduced a new type of contractive mapping and proved a fixed point result in complete metric spaces as a generalization of Banach contraction principle. Dotson [19] extended Banach’s contraction principle for nonexpansive mappings on star-shaped subsets of Banach spaces and proved Brosowski-Meinardus type theorems on invariant approximations.

Results

Conclusion