Suzuki-type fixed point theorem for fuzzy mappings in ordered metric spaces
© Ali and Abbas; licensee Springer 2013
Received: 8 September 2012
Accepted: 18 December 2012
Published: 10 January 2013
In this paper, a Suzuki-type fixed fuzzy point result for fuzzy mappings in complete ordered metric spaces is obtained. As an application, we establish the existence of coincidence fuzzy points and common fixed fuzzy points for a hybrid pair of a single-valued self-mapping and a fuzzy mapping. An example is also provided to support the main result presented herein.
MSC:47H10, 47H04, 47H07.
1 Introduction and preliminaries
Let be a family of all approximate quantities in X. A fuzzy set A is said to be more accurate than a fuzzy set B denoted by (that is, B includes A) if and only if for each x in X, where and denote the membership function of A and B, respectively. It is easy to see that if , then .
Note that is a nondecreasing function of α and D is a metric on . Let . Define . Let be a metric space and Y be an arbitrary set. A mapping is called a fuzzy mapping, that is, for each y in Y. Thus, if we characterize a fuzzy set Fy in a metric space X by a membership function Fy, then is the grade of membership of x in Fy. Therefore, a fuzzy mapping F is a fuzzy subset of with a membership function .
Definition 1 ()
A fuzzy point in X is called a fixed fuzzy point of the fuzzy mapping F if , that is, or . That is, the fixed degree of x in Fx is at least α. If , then x is a fixed point of a fuzzy mapping F.
Let and .
A fuzzy point in X is called a coincidence fuzzy point of the hybrid pair if , that is, or . That is, the fixed degree of gx in Fx is at least α. A fuzzy point in X is called a common fixed fuzzy point of the hybrid pair if , that is, (the fixed degree of x and gx in Fx is the same and is at least α).
We denote by and the set of all coincidence fuzzy points and the set of all common fixed fuzzy points of the hybrid pair , respectively.
A hybrid pair is called w-fuzzy compatible if whenever .
A mapping g is called F-fuzzy weakly commuting at some point if .
Lemma 1 ()
Let X be a nonempty set and. Then there exists a subsetsuch thatandis one-to-one.
Definition 2 Let X be a nonempty set. Then is called an ordered metric space if is a metric space and is partially ordered.
Let be a partially ordered set. Then are said to be comparable if or holds.
An ordered metric space is said to satisfy the order sequential limit property if for all n, whenever a sequence and for all n.
implies that .
implies that whenever and .
The following lemmas are needed in the sequel.
Lemma 2 (Heilpern )
if , then ;
if , then .
Lemma 3 (Lee and Cho )
Letbe a complete metric space and F be a fuzzy mapping from X intoand. Then there exists ansuch that.
Zadeh  introduced the concept of a fuzzy set. Heilpern  introduced the concept of fuzzy mappings in a metric space and proved a fixed point theorem for fuzzy contraction mappings as a generalization of the fixed point theorem for multivalued mappings given by Nadler . Estruch and Vidal  proved a fixed point theorem for fuzzy contraction mappings in complete metric spaces which in turn generalizes the Heilpern fixed point theorem. Further generalizations of the result given in  were proved in [8, 9]. Recently, Suzuki  generalized the Banach contraction principle and characterized the metric completeness property of an underlying space. Among many generalizations (see [11–13]) of the results given in , Dorić and Lazović  obtained Suzuki-type fixed point results for a generalized multivalued contraction in complete metric spaces.
On the other hand, the existence of fixed points in ordered metric spaces has been introduced and applied by Ran and Reurings . Fixed point theorems in partially ordered metric spaces are hybrid of two fundamental principles: Banach contraction theorem with a contractive condition for comparable elements and a selection of an initial point to generate a monotone sequence. For results concerning fixed points and common fixed points in partially ordered metrics spaces, we refer to [16–22].
The aim of this paper is to investigate Suzuki-type fixed point results for fuzzy mappings in complete ordered metric spaces. As an application, a coincidence fuzzy point and a common fixed fuzzy point of the hybrid pair of a single-valued self-mapping and a fuzzy mapping are obtained. We provide an example to support the result.
2 Main results
The following theorem is the main result of the paper and is a generalization of [, Theorem 2.1] for fuzzy mappings in ordered metric spaces.
Then there exists a pointsuch thatprovided that X satisfies the order sequential limit property.
a contradiction. Hence, .
as , we get . Hence by Lemma 2, . □
Then there exists a pointsuch thatprovided that X satisfies the order sequential limit property.
and, . Then there exists a pointsuch thatprovided that X satisfies the order sequential limit property.
3 An application
implies that .
gives whenever and .
whenever for all .
F and g are w-fuzzy compatible, and for some , and g is continuous at u.
g is F-fuzzy weakly commuting for some and is a fixed point of g, that is, .
g is continuous at x for some and for some such that .
On taking limit as , we get and therefore . By Lemma 2 we obtain . Consequently, . Hence, is a common fixed fuzzy point of F and g. Suppose now that (g) holds. If for some , g is F-fuzzy weakly commuting and , then . Hence, is a common fixed fuzzy point of F and g. Suppose now that (h) holds and assume that for some and for some , and . By the continuity of g at x and y, we get . The result follows. □
Hence, all the conditions of Theorem 7 are satisfied. Moreover, for each , we have and . For , we have .
The Banach contraction principle has become a classical tool to show the existence of solutions of functional equations in nonlinear analysis (see for details [23–26]). Suzuki-type fixed point theorems [10, 14] are the generalizations of the Banach contraction principle that characterize metric completeness of underlying spaces. Fuzzy sets and mappings play important roles in the process of fuzzification of systems. Suzuki-type fixed point theorems for fuzzy mappings obtained in this article can further be used in the process of finding the solutions of functional equations involving fuzzy mappings in fuzzy systems. In the main result, we not only extended the mapping to a fuzzy mapping, but also the underlying metric space has been replaced with ordered metric spaces. In this article, we defined coincidence fuzzy points and common fixed fuzzy points of the hybrid pair of a single-valued self-mapping and a fuzzy mapping and applied our main result to obtain the existence of coincidence fuzzy points and common fixed fuzzy points of the hybrid pair.
The authors are thankful to the referees for their critical remarks which helped to improve the presentation of this paper.
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