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

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.

Keywords

• cone metric spaces
• topological left modules
• fixed point theorems
• points of coincidence
• weakly compatible self-maps
• Lipschitz mappings