2011/02/07 by Thabet Abdeljawad, Abdeljawad, Thabet, Shahram Rezapour +1 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Dynamical Systems (math.DS) #FOS: Mathematics #Fixed Point Theorems Analysis #General Topology (math.GN) #Structural Analysis and Optimization #math.DS #math.GN
paper · pdf · doi:10.48550/arxiv.1102.1419
arxiv created 2011/02/07 · openalex publication_date 2011/02/07 · arxiv updated 2011/02/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We prove that every TVS-cone metric space (i.e a cone metric space over a locally convex topological vector space E) is first countable paracompact topological space and by using Du's results in " [A note on cone metric fixed point theory and its equivalence, Nonlinear Analysis,72(5),2259-2261 (2010)]", we conclude that every TVS-cone metric space is topologically isomorphic to a topological metric space. We also show how to construct comparable metric topologies to TVS-cone metric topologies by using the system of seminorms generating the topology of the locally convex topological vector space E. When E is a Banach space these metric topologies turn to be equivalent to the original TVS-cone metric topologies. Even though, we remark that there are still some fixed point theorems to deal nontrivially with them in TVS-cone metric spaces. The nonlinear scalarization is used also to prove some fixed point theorems with nonlinear contractive conditions.