Common fuzzy fixed points for fuzzy mappings
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RESEARCH
Open Access
Common fuzzy fixed points for fuzzy mappings Akbar Azam1 and Ismat Beg2* *
Correspondence: [email protected] 2 Centre for Applicable Mathematics and Statistics, University of Central Punjab, Lahore, 54770, Pakistan Full list of author information is available at the end of the article
Abstract Let (X, d) be a metric space and S, T be mappings from X to a set of all fuzzy subsets of X. We obtained sufficient conditions for the existence of a common α -fuzzy fixed point of S and T. Keywords: analysis; fuzzy set; fuzzy mapping; α -fuzzy fixed point
1 Introduction Fixed point theorems play a fundamental role in demonstrating the existence of solutions to a wide variety of problems arising in mathematics, physics, engineering, medicine and social sciences. The study of fixed point theorems in fuzzy mathematics was instigated by Weiss [], Butnariu [], Singh and Talwar [], Estruch and Vidal [], Wang et al. [], Mihet [], Qiu et al. [] and Beg and Abbas []. Heilpern [] introduced the concept of fuzzy contraction mappings and established the fuzzy Banach contraction principle on a complete metric linear spaces with the d∞ -metric for fuzzy sets. Azam and Beg [] proved common fixed point theorems for a pair of fuzzy mappings satisfying Edelstein, Alber and Guerr-Delabriere type contractive conditions in a metric linear space. Azam et al. [] presented some fixed point theorems for fuzzy mappings under Edelstein locally contractive conditions on a compact metric space with the d∞ -metric for fuzzy sets. Frigon and Regan [] generalized the Heilpern theorem under a contractive condition for -level sets (i.e., [Tx] ) of a fuzzy contraction T on a complete metric space, where -level sets are not assumed to be convex and compact. Amemiya and Takahashi [] studied some mathematical properties of contractive type set-valued and fuzzy mappings to obtain fixed points of fuzzy mappings by using the concept of w-distance (see []) in complete metric spaces. Recently, Zhang et al. [] proved some common fixed point theorems for contraction mappings in a new fuzzy metric space. The aim of this paper is to obtain a common α-fuzzy fixed point of a pair of fuzzy mappings S and T on a complete metric space under a generalized contractive condition for α-level sets (i.e., [Sx]α , [Tx]α ) of S and T in connection with Hausdorff metric for fuzzy sets. Our result (Theorem ) generalizes the results proved by Azam and Arshad [, Theorem ], Bose and Sahani [] and Vijayaraju and Marudai [, Theorem .] among others. © 2013 Azam and Beg; 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.
Azam and Beg Fixed Point Theory and Applications 2013, 2013:14 http://www.fixedpointtheoryandapplications.com/content/2013/1/14
2 Preliminaries Let (X, d) be a metric space and CB(X) be the
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