Isbell convexity in fuzzy quasi-metric spaces

Abstract

The concept of hyperconvexity in metric spaces was introduced by Aronszajn and Panichpakdi in 1956. This concept was then generalised to the framework of quasi-metric spaces by John Isbell in 1964, which he called Isbell convexity. In 2019, Yi git and Efe generalised this concept of hyperconvexity to the framework of fuzzy metric spaces and they called this new concept fuzzy hyperconvexity. In this MSc thesis, we introduce the concept of Isbell convexity in fuzzy quasi-metric spaces, which we call fuzzy Isbell convexity. This idea extends Isbell convexity in quasi-metric spaces to fuzzy quasi-metric spaces. We prove that a fuzzy quasi-metric space is fuzzy Isbell convex if and only if it is fuzzy metrically convex and has a mixed binary intersection property. Furthermore, we present the concept of a compatible quasi-metric, which generalises the concept of the compatible metric introduced by Radu, to the assymetric setting. We then use this new concept to generalise some xed point theorems in quasi-metric spaces to the framework of fuzzy quasi-metric spaces. Finally, we introduce a t-nonexpansive map and show that the xed point set of a t-nonexpansive map in an F-bounded fuzzy Isbell convex space is fuzzy Isbell convex.