On Ray's theorem for weak firmly nonexpansive mappings in Hilbert Spaces

Tamer Nabil,Ahmed Hussein Soliman

doi:10.24297/jam.v12i1.610

Abstract

In this work, we introduce notions of generalized firmly nonexpansive (G-firmly non expansive) and fundamentally firmly nonexpansive (F-firmly nonexpansive) mappings and utilize to the same to prove Ray's theorem for G-firmly and F-firmly nonexpansive mappings in Hilbert Spaces. Our results extend the result due to F. Kohsaka [ Ray's theorem revisited: a fixed point free firmly nonexpansive mapping in Hilbert spaces, Journal of Inequalities and Applications (2015) 2015:86 ].

Highlights

INTRODUCTION and PRELIMINARIESLet H be a real Hilbert space
We present new two versions of Ray’s theorem for mappings satisfying the conditions of weaker firmly nonexpansive
If every firmly nonexpansive self-mapping on K has a fixed point, K is bounded

Summary

Introduction

INTRODUCTION and PRELIMINARIESLet H be a real Hilbert space. The inner product and the induced norm on H are denoted by .,. and . respectively.The dual space of a Banach space X is denoted X *. In 1980, Ray [2] showed that the converse of Browder’s theorem is true, i.e. every nonexpansive self mapping on K has a fixed point, K is bounded. There are many versions of Ray’s theorem for nonexpansive mapping.

Results

Conclusion