Contractibility of Fixed Point Sets of Mean-Type Mappings

S Iampiboonvatana,Phichet Chaoha

doi:10.1155/2017/3689069

Abstract

We establish a convergence theorem and explore fixed point sets of certain continuous quasi-nonexpansive mean-type mappings in general normed linear spaces. We not only extend previous works by Matkowski to general normed linear spaces, but also obtain a new result on the structure of fixed point sets of quasi-nonexpansive mappings in a nonstrictly convex setting.

Highlights

The theory of mean iteration has been studied long before the 19th century [1]
Johann Carl Friedrich Gauss observed the connection between the arithmetic-geometric mean iteration and an elliptic integral
We are able to establish a convergence theorem for certain continuous quasi-nonexpansive mean-type mappings in general normed linear spaces and conclude the contractibility of their fixed point sets

Summary

Introduction

The theory of mean iteration has been studied long before the 19th century [1]. Johann Carl Friedrich Gauss observed the connection between the arithmetic-geometric mean iteration and an elliptic integral. We are able to establish a convergence theorem for certain continuous quasi-nonexpansive mean-type mappings in general normed linear spaces (which immediately covers the result in [3]) and conclude the contractibility of their fixed point sets. We prove a convergence theorem as well as the contractibility of fixed point sets for certain continuous quasi-nonexpansive meantype mappings.

Results

Conclusion