Common fuzzy fixed points for fuzzy mappings
© Azam and Beg; licensee Springer. 2013
Received: 17 August 2012
Accepted: 9 January 2013
Published: 22 January 2013
Let 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.
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 -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 -metric for fuzzy sets. Frigon and Regan  generalized the Heilpern theorem under a contractive condition for 1-level sets (i.e., ) of a fuzzy contraction T on a complete metric space, where 1-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., , ) of S and T in connection with Hausdorff metric for fuzzy sets. Our result (Theorem 5) generalizes the results proved by Azam and Arshad [, Theorem 4], Bose and Sahani  and Vijayaraju and Marudai [, Theorem 3.1] among others.
We denote the collection of all approximate quantities in a metric linear space X by . Let X be an arbitrary set, Y be a metric space. A mapping T is called a fuzzy mapping if T is a mapping from X into . A fuzzy mapping T is a fuzzy subset on with a membership function . The function is the grade of membership of y in .
Definition 1 Let S, T be fuzzy mappings from X into . A point z in X is called an α-fuzzy fixed point of T if . The point z is called a common α-fuzzy fixed point of S and T if . When , it is called a common fixed point of fuzzy maps.
For the sake of convenience, we first state some known results for subsequent use in the next section.
Lemma 2 
Lemma 3 
Let be a metric space and , then for each , , there exists an element such that .
Lemma 4 
Let V be a metric linear space, be a fuzzy mapping and . Then there exists such that .
3 Common fuzzy fixed points
In this section, we establish Theorem 5 on the existence of an α-fuzzy fixed point of a fuzzy mapping and also obtain a fixed point of fuzzy mappings (see Corollaries 8 and 9) and multivalued mappings (see Corollary 7).
for all , where , , , , are nonnegative real numbers and and either or . Then there exists such that .
we can show that , which implies that .
and is a crisp set. □
In the following, we furnish an illustrative example to highlight the utility of Theorem 5.
if . Hence, for , , the conditions of Theorem 5 are satisfied to obtain . Since , the results proved by Vijayaraju and Marudai [, Theorem 3.1] and Azam and Arshad [, Theorem 4] are also not applicable.
where , , , , are nonnegative real numbers and and either or . Then there exists such that .
Corollary 8 
where , , , , are nonnegative real numbers and and either or . Then there exists such that , .
Now, by Theorem 5, we obtain such that , i.e., , . □
for all , where , , , , are nonnegative real numbers and and either or . Then there exists a point such that and for all .
for all . □
In this paper, we obtained fixed point results for fuzzy set-valued mappings under a generalized contractive condition related to the -metric which is useful for computing Hausdorff dimensions. These dimensions help us to understand -space which is used in high energy physics. Our results are also useful in geometric problems arising in high energy physics. This is because events in this case are mostly fuzzy sets.
Authors are grateful to the editor and referees for their valuable suggestions and critical remarks for improving this paper.
- Weiss MD: Fixed points and induced fuzzy topologies for fuzzy sets. J. Math. Anal. Appl. 1975, 50: 142–150. 10.1016/0022-247X(75)90044-XMathSciNetView ArticleGoogle Scholar
- Butnariu D: Fixed point for fuzzy mapping. Fuzzy Sets Syst. 1982, 7: 191–207. 10.1016/0165-0114(82)90049-5MathSciNetView ArticleGoogle Scholar
- Singh SL, Talwar R: Fixed points of fuzzy mappings. Soochow J. Math. 1993, 19(1):95–102.MathSciNetGoogle Scholar
- Estruch, VD, Vidal, A: A note on fixed fuzzy points for fuzzy mappings. Proceedings of the II Italian-Spanish Congress on General Topology and Its Applications (Italian) (Trieste, 1999). Rend. Ist. Mat. Univ. Trieste 32(suppl. 2), 39–45 (2001)Google Scholar
- Wang G, Wu C, Wu C: Fuzzy α -almost convex mappings and fuzzy fixed point theorems for fuzzy mappings. Ital. J. Pure Appl. Math. 2005, 17: 137–150.Google Scholar
- Mihet D: On fuzzy ϵ -contractive mappings in fuzzy metric spaces. Fixed Point Theory Appl. 2007., 2007: Article ID 87471Google Scholar
- Qiu D, Shu L, Guan J: Common fixed point theorems for fuzzy mappings under φ -contraction condition. Chaos Solitons Fractals 2009, 41(1):360–367. 10.1016/j.chaos.2008.01.003MathSciNetView ArticleGoogle Scholar
- Beg I, Abbas M: Invariant approximation for fuzzy nonexpansive mappings. Math. Bohem. 2011, 136(1):51–59.MathSciNetGoogle Scholar
- Heilpern S: Fuzzy mappings and fixed point theorems. J. Math. Anal. Appl. 1981, 83: 566–569. 10.1016/0022-247X(81)90141-4MathSciNetView ArticleGoogle Scholar
- Azam A, Beg I: Common fixed points of fuzzy maps. Math. Comput. Model. 2009, 49: 1331–1336. 10.1016/j.mcm.2008.11.011MathSciNetView ArticleGoogle Scholar
- Azam A, Arshad M, Beg I: Fixed points of fuzzy contractive and fuzzy locally contractive maps. Chaos Solitons Fractals 2009, 42: 2836–2841. 10.1016/j.chaos.2009.04.026MathSciNetView ArticleGoogle Scholar
- Frigon M, O’Regan D: Fuzzy contractive maps and fuzzy fixed points. Fuzzy Sets Syst. 2002, 129: 39–45. 10.1016/S0165-0114(01)00171-3MathSciNetView ArticleGoogle Scholar
- Amemiya M, Takahashi W: Fixed point theorems for fuzzy mappings in complete metric spaces. Fuzzy Sets Syst. 2002, 125: 253–260. 10.1016/S0165-0114(01)00046-XMathSciNetView ArticleGoogle Scholar
- Zhang W, Qiu D, Li Z, Xiong G: Common fixed point theorems in a new fuzzy metric space. J. Appl. Math. 2012., 2012: Article ID 890678Google Scholar
- Azam A, Arshad M: A note on ‘Fixed point theorems for fuzzy mappings’ by P. Vijayaraju and M. Marudai. Fuzzy Sets Syst. 2010, 161: 1145–1149. 10.1016/j.fss.2009.10.016MathSciNetView ArticleGoogle Scholar
- Bose RK, Sahani D: Fuzzy mappings and fixed point theorems. Fuzzy Sets Syst. 1987, 21: 53–58. 10.1016/0165-0114(87)90152-7MathSciNetView ArticleGoogle Scholar
- Vijayaraju P, Marudai M: Fixed point theorems for fuzzy mappings. Fuzzy Sets Syst. 2003, 135: 401–408. 10.1016/S0165-0114(02)00367-6MathSciNetView ArticleGoogle Scholar
- Nadler SB: Multivalued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475MathSciNetView ArticleGoogle Scholar
- Arora SC, Sharma V: Fixed points for fuzzy mappings. Fuzzy Sets Syst. 2000, 110: 127–130. 10.1016/S0165-0114(97)00366-7MathSciNetView ArticleGoogle Scholar
- Azam A, Arshad M, Vetro P: On a pair of fuzzy ϕ -contractive mappings. Math. Comput. Model. 2010, 52: 207–214. 10.1016/j.mcm.2010.02.010MathSciNetView ArticleGoogle Scholar
- Shi-sheng Z: Fixed point theorems for fuzzy mappings (II). Appl. Math. Mech. 1986, 7(2):147–152. 10.1007/BF01897057MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.