Cone Metric Spaces over Topological Modules and Fixed Point Theorems for Lipschitz Mappings

Adrian Nicolae Branga,Ion Marian Olaru

doi:10.3390/math8050724

Abstract

In this paper, we introduce the concept of cone metric space over a topological left module and we establish some coincidence and common fixed point theorems for self-mappings satisfying a condition of Lipschitz type. The main results of this paper provide extensions as well as substantial generalizations and improvements of several well known results in the recent literature. In addition, the paper contains an example which shows that our main results are applicable on a non-metrizable cone metric space over a topological left module. The article proves that fixed point theorems in the framework of cone metric spaces over a topological left module are more effective and more fertile than standard results presented in cone metric spaces over a Banach algebra.

Highlights

The concept of metric space was defined by the mathematician Fréchet [1,2]
The distance between two elements x and y in a cone metric space X is defined to be a vector in a ordered Banach space E, and a mapping T : X → X is said to be a contraction if there is a positive constant k < 1 such that d( Tx, Ty) ≤ k · d( x, y), for all x, y ∈ X
This paper introduced the concept of cone metric space over a topological left module and established some coincidence and common fixed point theorems for self-mappings satisfying a condition of Lipschitz type

Summary

Introduction

The concept of metric space was defined by the mathematician Fréchet [1,2]. Afterwards, Kurepa [3] introduced more abstract metric spaces, where the metric values are given in an ordered vector space. The distance between two elements x and y in a cone metric space X is defined to be a vector in a ordered Banach space E, and a mapping T : X → X is said to be a contraction if there is a positive constant k < 1 such that d( Tx, Ty) ≤ k · d( x, y), for all x, y ∈ X. It was proved that any cone metric space ( X, d) is equivalent with the usual metric space where the real-valued metric d∗ is defined by a nonlinear scalarization function ξ e [4] or by a Minkowski functional qe [8]. The above results have been extended by Olaru and Secelean [9] to nonlinear contractive condition on TVS-cone metric space. Liu and Xu [10]

Methods

Results

Conclusion