Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces

Kiany Fatemeh, Amini-Harandi, Amini-Harandi Alireza

Description

In this paper, we first present a fixed point theorem for set-valued fuzzy contraction type maps in complete fuzzy metric spaces which extends and improves some well-know results in literature. Then by presenting an endpoint result we initiate endpoint theory for fuzzy contraction maps in fuzzy metric spaces. 0 2000 Mathematics Subject Classification: 47H10, 54H25.

Sujets

• Fixed point
• Endpoint
• Topology

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 11
• Langue: English

Extrait

Kiany and AminiHarandiFixed Point Theory and Applications2011,2011:94 http://www.fixedpointtheoryandapplications.com/content/2011/1/94

R E S E A R C H

Open Access

Fixed point and endpoint theorems for setvalued fuzzy contraction maps in fuzzy metric spaces 1* 2,3 Fatemeh Kiany and Alireza AminiHarandi

* Correspondence: [e-mail] 1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran Full list of author information is available at the end of the article

Abstract In this paper, we first present a fixed point theorem for setvalued fuzzy contraction type maps in complete fuzzy metric spaces which extends and improves some well know results in literature. Then by presenting an endpoint result we initiate endpoint theory for fuzzy contraction maps in fuzzy metric spaces. 0 2000 Mathematics Subject Classification: 47H10, 54H25. Keywords:Fixed point, Endpoint, Setvalued fuzzy contraction map, Fuzzy metric space, Topology

1. Introduction and preliminaries Many authors have introduced the concept of fuzzy metric spaces in different ways [14]. Kramosil and Michalek [5] introduced the fuzzy metric space by generalizing the concept of the probabilistic metric space to fuzzy situation. George and Veeramani [6,7] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [5] and obtained a Hausdorff topology for this kind of fuzzy metric spaces. Recently, the fixed point theory in fuzzy metric spaces has been studied by many authors [818]. In [11], the following definition is given. Definition 1.1. A sequence (t n ) of positive real numbers is said to be ansincreasing sequence if there existsm 0 ÎNsuch thatt m + 1≤t m+1 , for allm≥m 0 . Gregori and Sapena [11] proved the following fixed point theorem. Theorem 1.2.Let(X, M, *)be a complete fuzzy metric space such that for every s increasing sequence(t n )and every x, yÎX ∞ ∗(x lim i=n M,y,t n ) = 1 n→∞

Suppose f:X®X is a map such that for each x, yÎX and t> 0,we have

M fx,fy,kt

≥M x,y,t

where0 <k< 1.Then, f has a unique fixed point. In this article, we first give a fixed point theorem for setvalued contraction maps which improve and generalize the abovementioned result of Gregori and Sapena. Then, in Section 2, we initiate endpoint theory in fuzzy metric spaces by presenting an endpoint result for setvalued maps.

© 2011 Kiany and AminiHarandi; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.