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.
Keywordsanalysis fuzzy set fuzzy mapping α-fuzzy fixed point
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.
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