Common Fixed Point Theorems in Fuzzy Metric Spaces Satisfying -Contractive Condition with Common Limit Range Property

Sunny Chauhan,Wutiphol Sintunavarat,M Alamgir Khan

doi:10.1155/2013/735217

Sunny Chauhan, Wutiphol Sintunavarat + Show 1 more

Open Access

https://doi.org/10.1155/2013/735217

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Abstract

The objective of this paper is to emphasize the role of “common limit range property” to ascertain the existence of common fixed point in fuzzy metric spaces. Some illustrative examples are furnished which demonstrate the validity of the hypotheses and degree of utility of our results. We derive a fixed point theorem for four finite families of self-mappings which can be utilized to derive common fixed point theorems involving any finite number of mappings. As an application to our main result, we prove an integral-type fixed point theorem in fuzzy metric space. Our results improve and extend a host of previously known results including the ones contained in Imdad et al. (2012).

Highlights

In 1965, Zadeh [1] studied the concept of a fuzzy set in his seminal paper
We show that the (CLRST) property implies the common property (E.A.), but converse is not true
We show that z is a common fixed point of the pair (A, S)

Summary

Introduction

In 1965, Zadeh [1] studied the concept of a fuzzy set in his seminal paper. Thereafter, it was developed extensively by many researchers, which include interesting applications of this theory in different fields. Many mathematicians used different conditions on self-mappings and proved several fixed point theorems for contractions in fuzzy metric spaces (see [6,7,8,9,10,11,12,13]). Imdad et al [34] extended the notion of common limit range property to two pairs of self-mappings which relaxes the requirement on closedness of the subspaces. We prove some common fixed point theorems for weakly compatible mappings with common limit range property in fuzzy metric spaces which include fuzzy metric spaces of two types, namely, Kramosil and Michalek fuzzy metric spaces and George and Veeramani fuzzy metric spaces. Extend, and generalize a host of previously known results existing in the literature

Preliminaries

Main Results

Fixed Point Theorems in GV-Fuzzy Metric Spaces

Integral Analogue of Related Fixed Point Theorems