Abstract
In this paper, a Suzukitype 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 singlevalued selfmapping and a fuzzy mapping. An example is also provided to support the main result presented herein.
MSC: 47H10, 47H04, 47H07.
Keywords:
fixed fuzzy point; fuzzy mapping; fuzzy set; approximate quantity1 Introduction and preliminaries
Let X be a space of points with generic elements of X denoted by x and
for
where
Let
Corresponding to each
For
Let
and
Note that
In a more general sense than that given in [1], a mapping
Definition 1 ([3])
A fuzzy point
Let
A fuzzy point
We denote by
A hybrid pair
A mapping g is called Ffuzzy weakly commuting at some point
Lemma 1 ([4])
LetXbe a nonempty set and
Definition 2 Let X be a nonempty set. Then
Let
Define
An ordered metric space is said to satisfy the order sequential limit property if
A mapping
(a)
(b)
The following lemmas are needed in the sequel.
Lemma 2 (Heilpern [1])
Let
1. if
2.
3. if
Lemma 3 (Lee and Cho [5])
Let
Zadeh [6] introduced the concept of a fuzzy set. Heilpern [1] 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 [7]. Estruch and Vidal [3] 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 [3] were proved in [8,9]. Recently, Suzuki [10] generalized the Banach contraction principle and characterized the metric completeness property of an underlying space. Among many generalizations (see [1113]) of the results given in [10], Dorić and Lazović [14] obtained Suzukitype 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 [15]. 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 [1622].
The aim of this paper is to investigate Suzukitype 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 singlevalued selfmapping and a fuzzy mapping are obtained. We provide an example to support the result.
Throughout this paper, let
2 Main results
The following theorem is the main result of the paper and is a generalization of [[14], Theorem 2.1] for fuzzy mappings in ordered metric spaces.
Theorem 4Let
for all
Then there exists a point
Proof Let
By the given assumption, we have
Also,
That is,
So, we have
Note that
a contradiction. Hence,
By the given assumption, we have
Therefore,
We claim that
a contradiction as
and
Hence,
As
Now, for all
implies that
which on taking limit as
If
then
Hence,
Now, we show that
Now,
Now,
implies that
Hence,
which further implies that
We claim that
a contradiction, so we deduce that
Therefore,
a contradiction. Hence,
Now, when
for all
As
for all
This implies that
Hence, for
On taking the limit as
If
On taking the limit as
By the given assumption, we have
Thus, for any
as
Corollary 5Let
for all
Then there exists a point
Corollary 6Let
for all
and
3 An application
Let
(c)
(d)
(e)
Theorem 7Let
for all
Then
(f) Fandgarewfuzzy compatible,
(g) gisFfuzzy weakly commuting for some
(h) gis continuous atxfor some
Proof By Lemma 1, there exists
As g is onetoone on E,
for all
for all
implies that
On taking limit as
Example 1 Let
and for
Define a selfmap
Note that for all
Also, for all
And
If
hold true, where
and
Hence, all the conditions of Theorem 7 are satisfied. Moreover, for each
4 Conclusion
The Banach contraction principle has become a classical tool to show the existence of solutions of functional equations in nonlinear analysis (see for details [2326]). Suzukitype 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. Suzukitype 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 singlevalued selfmapping 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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
The authors are thankful to the referees for their critical remarks which helped to improve the presentation of this paper.
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