Abstract
In this paper, we obtain some fixed point theorems for fuzzy mappings in a left K-sequentially complete quasi-pseudo-metric space and in a right K-sequentially complete quasi-pseudo-metric space, respectively. Our analysis is based on the fact that fuzzy fixed point results can be obtained from the fixed point theorem of multivalued mappings with closed values. It is observed that there are many situations in which the mappings are not contractive on the whole space but they may be contractive on its subsets. We feel that this feature of finding the fuzzy fixed points via closed balls was overlooked, and our paper will re-open the research activity into this area.
MSC:6S40, 47H10, 54H25.
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1 Introduction
The fixed points of fuzzy mappings were initially studied by Weiss [1] and Butnariu [2]. Then Heilpern [3] initiated the idea of fuzzy contraction mappings and proved a fixed point theorem for fuzzy contraction mappings which is a fuzzy analogue of Nadler’s [4] fixed point theorem for multivalued mappings. Afterward many authors [5–8] explored the fixed points for generalized fuzzy contractive mappings.
Gregori and Pastor [9] proved a fixed point theorem for fuzzy contraction mappings in left K-sequentially complete quasi-pseudo-metric spaces. Their result is a generalization of the result of Heilpern [3]. In [10] the authors considered a generalized contractive-type condition involving fuzzy mappings in left K-sequentially complete quasi-metric spaces and established the fixed point theorem which is an extension of [[11], Theorem 2]. Moreover, the main result of [10] is a quasi-metric version of [[11], Theorem 1]. Subsequently, several other authors studied the fixed points of fuzzy contractive mappings in quasi-pseudo-metric space.
In this paper, we establish some local versions of fixed point theorems involving fuzzy contractive mappings in left K-sequentially complete quasi-pseudo-metric spaces and right K-sequentially complete quasi-pseudo-metric spaces, respectively.
2 Preliminaries
Throughout this paper, the letter ℕ denotes the set of positive integers. If A is a subset of a topological space , we will denote by the closure of A in .
A quasi-pseudo-metric on a nonempty set X is a nonnegative real-valued function d on such that, for all ,
-
(i)
and
-
(ii)
.
A set X along with a quasi-pseudo-metric d is called a quasi-pseudo-metric space.
Each quasi-pseudo-metric d on X induces a topology which has as a base the family of all d-balls , where .
If d is a quasi-pseudo-metric on X, then the function , defined on by , is also a quasi-pseudo-metric on X. By and we denote and , respectively.
Let d be a quasi-pseudo-metric on X. A sequence in X is said to be
-
(i)
left K-Cauchy [12] if for each , there is a such that for all with .
-
(ii)
right K-Cauchy [12] if for each , there is a such that for all with .
A quasi-pseudo-metric space is said to be left (right) K-sequentially complete [12] if each left (right) K-Cauchy sequence in converges to some point in X (with respect to the topology ).
Now let be a quasi-pseudo-metric space and let A and B be nonempty subsets of X. Then the Hausdorff distance between subsets A and B is defined by
where .
Note that with iff , and for any nonempty subset A, B and C of X. Clearly, H is the usual Hausdorff distance if d is a metric on X.
A fuzzy set on X is an element of where . The α-level set of a fuzzy set A, denoted by , is defined by for , and when and T is a contraction.
Definition 2.1 Let be a quasi-pseudo-metric space and be a metric linear space. The families and of fuzzy sets on and on are defined by
Note that for a metric linear space ,
Definition 2.2 [13]
Let be a quasi-pseudo-metric space and let (or ) and . Then we define
where the Hausdorff metric H is deduced from the quasi-pseudo-metric d on X,
It is easy to see that is a non-decreasing function of α, and .
Definition 2.3 Let X be an arbitrary set and Y be any quasi-pseudo-metric space. F is said to be a fuzzy mapping if F is a mapping from X into (or ).
Definition 2.4 We say that x is a fixed point of the mapping if .
Before establishing our main results, we require the following lemmas recorded from ([9, 13]).
Lemma 2.5 Let be a quasi-pseudo-metric space and let and (or ). Then if and only if
Lemma 2.6 Let be a quasi-pseudo-metric space and let (or ). Then
for any and .
Lemma 2.7 Let be a quasi-pseudo-metric space and let . Then
for each (or ) and .
Lemma 2.8 Suppose is compact in the quasi-pseudo-metric space (or ). If , then there exists such that
3 Fixed point theorems for fuzzy contractive maps
In the present section, we prove the local versions of fixed point results for fuzzy contraction mappings in a left (right) K-sequentially complete quasi-pseudo-metric space.
Theorem 3.1 Let be a left K-sequentially complete quasi-pseudo-metric space, , and be a fuzzy mapping. If there exists such that
and
then there exists such that .
Proof We apply Lemma 2.8 to the nonempty -compact set and to find such that
It also implies that .
We can write
By Lemma 2.8, choose such that
We can show that since
By Lemma 2.8, choose such that
We can show that since
We follow the same procedure to obtain such that
Now, to verify that is a left K-Cauchy sequence, for , we have
As and is a left K-sequentially complete quasi-pseudo-metric space, this implies that is a left K-Cauchy sequence in X. Therefore, there exists such that .
Now, from Lemma 2.6 and Lemma 2.7, we get
since and as . Thus we have
Lemma 2.5 yields that . □
We will furnish the following example in the support of the above result.
Example 3.2 Let , where . Define by , for all , ,
and
then is a left K-sequentially complete quasi-pseudo-metric space. Now defined as
is a fuzzy mapping. For ,
Define by
where
Now, for ,
and
Then such that .
Note that the fuzzy mapping defined in the above example is not contractive on the whole space; for example, whenever and , then
When is a right K-sequentially complete quasi-pseudo-metric space, using Lemmas 2.5, 2.6, 2.7 and 2.8, for , we get the following result.
Theorem 3.3 Let be a right K-sequentially complete quasi-pseudo-metric space, , and be a fuzzy mapping. If there exists such that
and
then T has a fuzzy fixed point such that .
The proof of Theorem 3.3 is similar to the proof of Theorem 3.1 and therefore omitted.
Remark 3.4 If is a left K-sequentially complete quasi-pseudo-metric space, by imposing the contractive condition on the whole space X in Theorem 3.1, we get the following result of Gregori and Pastor [9].
Corollary 3.5 Let be a left K-sequentially complete quasi-pseudo-metric space and be a fuzzy mapping. If there exists such that
then there exists such that .
Theorem 3.6 Let be a left K-sequentially complete quasi-pseudo-metric space, , and be a fuzzy mapping. If there exists such that
and
then there exists such that .
Proof We apply Lemma 2.8 to the nonempty -compact set and to find such that
It also implies that .
We can write
Now choose such that . By Lemma 2.8, we get
Now we consider the following cases.
Case 1: If we consider as a maximum in above inequality (2) and use inequality (1), we get
Case 2: If we consider as a maximum in inequality (2), we have
Note that , using inequality (1), we get
It follows from the above two cases that
We can show that since
We follow the same procedure to obtain such that
Now, to verify that is a left K-Cauchy sequence, for , we have
As and is a left K-sequentially complete quasi-pseudo-metric space, this implies that is a left K-Cauchy sequence in X. Therefore, there exists such that .
Now, from Lemma 2.6 and Lemma 2.7, we get
since and and as . Thus we have
Lemma 2.5 yields that . □
Example 3.7 Let be the left K-sequentially complete quasi-pseudo-metric space of Example 3.2. Now defined as
is a fuzzy mapping. For ,
Define by
where
Now, for ,
and
Then such that .
If is a right K-sequentially complete quasi-pseudo-metric space, using Lemmas 2.5, 2.6, 2.7 and 2.8 for , we get the following result.
Theorem 3.8 Let be a right K-sequentially complete quasi-pseudo-metric space, , and be a fuzzy mapping. If there exists such that
and
then T has a fuzzy fixed point such that .
The proof of Theorem 3.8 is similar to the proof of Theorem 3.6 and therefore omitted.
Theorem 3.9 Let be a left K-sequentially complete quasi-pseudo-metric space, , and be a fuzzy mapping. If there exists such that
and
then there exists such that .
Proof We apply Lemma 2.8 to the nonempty -compact set and to find such that
It also implies that .
We can write
Now choose such that . By Lemma 2.8, we get
Now we consider the following cases.
Case 1: If we consider as a maximum in above inequality (4) and use inequality (3), we get
Case 2: If we consider as a maximum in inequality (4), we have
Note that , using inequality (3), we get
It follows from the above two cases that
We can show that since
We follow the same procedure to obtain such that
Now, to verify that is a left K-Cauchy sequence, for , we have
As and is a left K-sequentially complete quasi-pseudo-metric space, this implies that is a left K-Cauchy sequence in X. Therefore, there exists such that .
Now, from Lemma 2.6 and Lemma 2.7, we get
since and and as . Thus we have
Lemma 2.5 yields that . □
Example 3.10 Let be the left K-sequentially complete quasi-pseudo-metric space of Example 3.2. Now defined as
is a fuzzy mapping. For ,
Now, for ,
and
Then such that .
If is a right K-sequentially complete quasi-pseudo-metric space, using Lemmas 2.5, 2.6, 2.7 and 2.8 for , we get the following result.
Theorem 3.11 Let be a right K-sequentially complete quasi-pseudo-metric space, , and be a fuzzy mapping. If there exists such that
and
then T has a fuzzy fixed point such that .
The proof of Theorem 3.11 is similar to the proof of Theorem 3.9 and therefore omitted.
4 Conclusion
From the application point of view, it often happens that a mapping T is a fuzzy contraction on a subset Y of X but not on the entire space X. However, if Y is closed, then it is complete, so that T has a fuzzy fixed x in Y, provided we impose a restriction on the choice of , so that the sequence remains in the closed subset Y. In this paper, we used this method to find fixed points of fuzzy mappings on a left (right) K-sequentially complete quasi-pseudo-metric space X.
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Azam, A., Waseem, M. & Rashid, M. Fixed point theorems for fuzzy contractive mappings in quasi-pseudo-metric spaces. Fixed Point Theory Appl 2013, 27 (2013). https://doi.org/10.1186/1687-1812-2013-27
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DOI: https://doi.org/10.1186/1687-1812-2013-27