Fixed points for modified fuzzy ψ-contractive set-valued mappings in fuzzy metric spaces

Shihuang Hong

doi:10.1186/1687-1812-2014-12

Abstract

In this paper, we introduce a new concept of fuzzy α-ψ-contractive type set-valued mappings and establish fixed-point theorems for such mappings in complete fuzzy metric spaces. Starting from the fuzzy version of the Banach contraction principle, the presented theorems extend, generalize and improve many existing results in the literature. Moreover, the results are supported by examples.

Highlights

1 Introduction In metric fixed-point theory, the contractive conditions on underlying functions play an important role for ensuring the existence of fixed points
The Banach contraction principle is a remarkable result in metric fixed-point theory
Motivated by the works mentioned above, in this paper we will further modify the type of the ψ -contraction and establish fixed-point theorems for such set-valued mappings on certain complete fuzzy metric spaces

Summary

Introduction

In metric fixed-point theory, the contractive conditions on underlying functions play an important role for ensuring the existence of fixed points. Some authors established fixed-point theorems for such mappings in complete fuzzy metric spaces. Afterwards, Hong and Peng [ ] modified the notion of the fuzzy ψ-contraction via a so-called fw-distance P instead of the fuzzy metric M and provided the sufficient conditions for the existence of fixed points for such contraction set-valued mappings. Motivated by the works mentioned above, in this paper we will further modify the type of the ψ -contraction and establish fixed-point theorems for such set-valued mappings on certain complete fuzzy metric spaces. (ii) A fuzzy metric space (X, M, ∗) in which every Cauchy sequence is convergent is said to be complete. Let (X, M, ∗) be a complete fuzzy metric space and T : X → CB(X) be a fuzzy α-ψ -contractive and α∗-η∗-admissible set-valued mapping.

In the light of

We get

This shows that