Characterization of bipolar ultrametric spaces and fixed point theorems

Abstract

Ultrametricity condition on bipolar metric spaces is considered and a geometric characterization of bipolar ultrametric spaces is given. Also embedding a bipolar ultrametric space into a pseudo-ultrametric space is discussed and some conditions are found to be able to embed them into an ultrametric space. Finally, some fixed point theorems on bipolar ultrametric spaces are proven.

\'C}iri{\'c} Contraction with Graphical Structure of Bipolar Metric Spaces and Related Fixed Point Theorems

“…{(0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (0, 6), (0, 7), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (2,3), (2,4), (2,5), (2,6), (2,7), (3,4), (3,5), (3,6), (3,7), (4,5), (4,6), (4, 7), (5,6), (5,7), (6, 7) Hence, we conclude that graphical bipolar metric space is not necessarily bipolar metric space. Now, we discuss Cauchy, complete, covariant and contravariant mappings as follows:…”

“…Keeping in view the applicability of the above mentioned theory, and motivated by the innovative work done in [4,14,15,22,23,26], in this paper, we aim to introduce the notion of graphical structure of bipolar metric spaces, which is the generalization of bipolar metric spaces and graphical metric spaces. Appropriate example equipped with suitable graphs are provided to endorse the validity of our findings.…”