Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality

Shaoyuan Xu,Stojan Radenović

doi:10.1186/1687-1812-2014-102

Shaoyuan Xu, Stojan Radenović

Open Access

https://doi.org/10.1186/1687-1812-2014-102

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Abstract

In this paper, we first present some elementary results concerning cone metric spaces over Banach algebras. Next, by using these results and the related ones about c-sequence on cone metric spaces we obtain some new fixed point theorems for the generalized Lipschitz mappings on cone metric spaces over Banach algebras without the assumption of normality. As a consequence, our main results improve and generalize the corresponding results in the recent paper by Liu and Xu (Fixed Point Theory Appl. 2013:320, 2013).

Highlights

In, cone metric spaces were reviewed by Huang and Zhang, as a generalization of metric spaces
They gave the version of the Banach contraction principle and other basic theorems in the setting of cone metric spaces
Many authors have been interested in the study of fixed point results in the setting of cone metric spaces

Summary

Introduction

In , cone metric spaces were reviewed by Huang and Zhang, as a generalization of metric spaces (see [ ]). In [ ], the authors proved that there exists a unique fixed point for contractive mappings in complete cone metric spaces. We delete the superfluous assumption of normality of the paper [ ] and obtain the existence and uniqueness of the fixed point for the generalized Lipschitz mappings in the setting of cone metric spaces over Banach algebras.

Results

Conclusion