Unique common fixed points for two generalized expansive mappings on non-normal cone metric spaces with Banach algebras

Yongjie Piao

doi:10.28919/afpt/3780

Abstract

By using some elementary results concerning cone metric spaces over Banach algebras and the related ones about c-sequence on cone metric spaces, some new coincidence point and common fixed point theorems for two generalized expansive mappings were discussed and obtained on cone metric spaces over Banach algebras without the assumption of normality and some unique fixed point theorems were given. Also, One of the main results is supported with a relevant example.

Highlights

By using some elementary results concerning cone metric spaces over Banach algebras and the related ones about c-sequence on cone metric spaces, some new coincidence point and common fixed point theorems for two generalized expansive mappings were discussed and obtained on cone metric spaces over Banach algebras without the assumption of normality and some unique fixed point theorems were given
In 2007, cone metric spaces were reviewed by Huang and Zhang, as a generalization of metric spaces
Some authors investigated the problems of whether cone metric spaces are equivalent to metric spaces in terms of the existence of fixed points of the mappings and successfully established the equivalence between some fixed point results in metric spaces and in cone metric spaces, see [3,4,5,6]. They showed that any cone metric space (X, d) is equivalent to a usual metric space (X, d∗), where the real-metric function d∗ is defined by a nonlinear scalarization function ξe(see [4]) or by a Minkowski function qe(see[5])

Summary

Introduction

In 2007, cone metric spaces were reviewed by Huang and Zhang, as a generalization of metric spaces (see [1]). Some authors investigated the problems of whether cone metric spaces are equivalent to metric spaces in terms of the existence of fixed points of the mappings and successfully established the equivalence between some fixed point results in metric spaces and in (topological vector space-valued) cone metric spaces, see [3,4,5,6] They showed that any cone metric space (X, d) is equivalent to a usual metric space (X, d∗), where the real-metric function d∗ is defined by a nonlinear scalarization function ξe(see [4]) or by a Minkowski function qe(see[5]). We give an example to support the main result

Preliminaries

Unique common fixed points for expansive mappings