Some Fixed Point (Vertex) Theorems in Fuzzy Graph Metric Space

Abstract

In this article, we introduce the concepts of sequence and Cauchy sequence in fuzzy graph metric space (f.g.m.s) and we show that any f.g.m.s. is always complete. We also establish that a contractive mapping from a f.g.m.s. into itself has a unique fixed vertex. Further, under the assumption that T is an ε-contractive mapping, the limit of a subsequence of $\{ T^{n} (\sigma (x_{0} ))\}$ is a periodic vertex of T. Moreover, if we assume T be contractive instead of ε-contractive, then the limit of a subsequence of $\{ T^{n} (\sigma (x_{0} ))\}$ becomes a fixed vertex of T.