Determining Fuzzy Distance between Sets by Application of Fixed Point Technique Using Weak Contractions and Fuzzy Geometric Notions

Abstract

In the present paper, we solve the problem of determining the fuzzy distance between two subsets of a fuzzy metric space. We address the problem by reducing it to the problem of finding an optimal approximate solution of a fixed point equation. This approach is well studied for the corresponding problem in metric spaces and is known as proximity point problem. We employ fuzzy weak contractions for that purpose. Fuzzy weak contraction is a recently introduced concept intermediate to a fuzzy contraction and a fuzzy non-expansive mapping. Fuzzy versions of some geometric properties essentially belonging to Hilbert spaces are considered in the main theorem. We include an illustrative example and two corollaries, one of which comes from a well-known fixed point theorem. The illustrative example shows that the main theorem properly includes its corollaries. The work is in the domain of fuzzy global optimization by use of fixed point methods.

Best Proximity Point Theorems without Fuzzy P-Property for Several (ψ − ϕ)-Weak Contractions in Non-Archimedean Fuzzy Metric Spaces

“…Saha et al in [28] established the existence and uniqueness of best proximity point for the weak contractions mentioned above in the frame of fuzzy metric spaces, as stated in the following theorem.…”

“…Compared with those results, in Saha's work, control functions are supposed to satisfy some other conditions suitable for fuzzy metric spaces. In 2019, Saha et al [28] investigated the existence of the unique best proximity point for such weak contraction in fuzzy metric spaces by exploring P-property, which provided a way to obtain some proximity points after the unavailability of fixed points and approximate points for non-self mappings, extending and fuzzifying the existing results in metric spaces.…”