Common Fixed Points of Two Weakly Compatible Mappings in Fuzzy Metric Spaces

Abstract

In this paper, we investigate the notions of convexity, strict convexity, and normal structure in the framework of fuzzy metric spaces in the sense of George and Veeramani. These concepts are developed to extend classical geometric ideas from metric spaces to fuzzy settings. Using these structures, we establish the existence of a common fixed point for a pair of weakly compatible non-expansive self-mappings defined on compact convex subsets of strictly convex fuzzy metric spaces. The main result relies on the existence of normal structure and employs Zorn’s lemma to ensure minimal invariant subsets. Our theorem generalizes Takahashi’s fixed point theorem from convex metric spaces to fuzzy metric spaces and extends it from a single mapping to a pair of mappings. The results obtained unify and extend several known fixed point theorems in fuzzy and probabilistic metric spaces, providing a broader framework for further developments in nonlinear analysis within fuzzy environments.

2020 Mathematics Subject Classifications: 47H09, 47H10, 47H30, 54E40