Fixed point theorems in fuzzy metric spaces for mappings with Bγ,µ condition

Nilakshi Goswami,Bijoy Patir

doi:10.22199/issn.0717-6279-3333

Abstract

In this paper we prove some fixed point theorems in fuzzy metric spaces for a class of generalized nonexpansive mappings satisfying Bγ,µ condition. We introduce a type of convexity in fuzzy metric spaces with respect to an altering distance function and prove convergence results for some iteration schemes to the fixed point. The results are supported by suitable examples.

Highlights

Introduction and preliminariesIn 1988, Grabiec [7] introduced fixed point theory in fuzzy metric spacesby extending different existing results to such spaces
In 2008, Suzuki[2] defined a class of mappings satisfying condition (C) in a Banach space X, which is wider than the class of nonexpansive mappings
In 2011, García-Falsetet al. [5] and in 2018 Patir et al [18] introduced some new classes ofgeneralized nonexpansive mappings which contain the mappings satisfying(C) condition as a subclass. We have extended these generalized classes of mappings with (C) condition by Suzuki and Bγ,μ condition by Patir et al to fuzzy metric spaces and prove some fixed point theorems

Summary

Introduction and preliminaries

In 1988, Grabiec [7] introduced fixed point theory in fuzzy metric spacesby extending different existing results to such spaces. (iii) A fuzzy metric space X is complete if and only if every Cauchy sequence converges in X. [20] A mapping φ : [0, 1] −→ [0, 1] is said to be an alteringdistance function if the following conditions are satisfied: (i) φ is strictly decreasing and left continuous. A mapping f : E −→ X is said to satisfy fuzzy (C) condition with respect to φ if. [21] A complete fuzzy metric space (X, M, T ) is said tosatisfy fuzzy Opial property with respect to φ if for every sequence {an} inX with an −→ u, we have for each t > 0, lim n−→∞.

Results and discussion

Then f satisfies

Define f on X by