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

Parbati Saha,Manuel De La Sen,Binayak S Choudhury,Shantau Guria

doi:10.3390/sym11060812

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.

Highlights

Introduction and Mathematical PreliminariesFixed point methods in mathematics are well known for their potentials of applications
A new tenet in mathematics appeared with the introduction of fuzzy set theory by Zadeh [4]
Let A and B be two closed subsets of X and f : A → B be a non-self weak contraction mapping such that the following conditions are satisfied: (i) ( A, B) satisfies the fuzzy P-property, (ii) f ( A0 ) ⊆ B0, (iii) A0 is nonempty

Summary

Introduction

Fixed point methods in mathematics are well known for their potentials of applications. There are several uses of the Banach contraction mapping principle itself. The idea of contraction was generalized and extended in several directions. Other types of contractive mappings having very different features like discontinuity, etc. Have appeared in a large way in the context of fixed point theory. Books like [1,2,3] provide an account of this line of study along with applications that are experiencing rapid expansion even today. A new tenet in mathematics appeared with the introduction of fuzzy set theory by Zadeh [4]

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