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Fixed point results for the α-Meir-Keeler contraction on partial Hausdorff metric spaces
Journal of Inequalities and Applications volume 2013, Article number: 410 (2013)
The purpose of this paper is to study fixed point theorems for a multi-valued mapping satisfying the α-Meir-Keeler contraction with respect to the partial Hausdorff metric ℋ in complete partial metric spaces. Our result generalizes and extends some results in the literature.
MSC:47H10, 54C60, 54H25, 55M20.
1 Introduction and preliminaries
Fixed point theory is one of the most crucial tools in nonlinear functional analysis, and it has application in distinct branches of mathematics and in various sciences, such as economics, engineering and computer science. The most impressed fixed point result was given by Banach  in 1922. He concluded that each contraction has a unique fixed point in the complete metric space. Since then, this pioneer work has been generalized and extended in different abstract spaces. One of the interesting generalization of Banach fixed point theorem was given by Matthews  in 1994. In this remarkable paper, the author introduced the following notion of partial metric spaces and proved the Banach fixed point theorem in the context of complete partial metric space.
For the sake of completeness, we recall basic definitions and fundamental results from the literature.
Throughout this paper, by , we denote the set of all nonnegative real numbers, while ℕ is the set of all natural numbers.
Definition 1 
A partial metric on a nonempty set X is a function such that for all
(p1) if and only if ;
A partial metric space is a pair such that X is a nonempty set, and p is a partial metric on X.
Remark 1 It is clear that if , then from (p1) and (p2), we have . But if , the expression may not be 0.
Each partial metric p on X generates a topology on X, which has as a base the family of open p-balls , where for all and . If p is a partial metric on X, then the function given by
is a metric on X.
We recall some definitions of a partial metric space, as follows.
Definition 2 
Let be a partial metric space. Then
a sequence in a partial metric space converges to if and only if ;
a sequence in a partial metric space is called a Cauchy sequence if and only if exists (and is finite);
a partial metric space is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that ;
a subset A of a partial metric space is closed if whenever is a sequence in A such that converges to some , then .
Remark 2 The limit in a partial metric space is not unique.
is a Cauchy sequence in a partial metric space if and only if it is a Cauchy sequence in the metric space ;
a partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if .
Recently, fixed point theory has developed rapidly on partial metric spaces, see, e.g., [3–12] and the reference therein. Very recently, Haghi et al.  proved that some fixed point results in partial metric space results are equivalent to the results in the context of a usual metric space. On the other hand, this case is not valid for our main results, that is, the recent result of Haghi et al.  is not applicable to the main theorems.
Let be a metric space, and let denote the collection of all nonempty, closed and bounded subsets of X. For , we define
where , and it is well known that ℋ is called the Hausdorff metric induced the metric d. A multi-valued mapping is called a contraction if
for all and . The study of fixed points for multi-valued contractions using the Hausdorff metric was introduced in Nadler .
Theorem 1 
Let be a complete metric space, and let be a multi-valued contraction. Then there exists such that .
Very recently, Aydi et al.  established the notion of partial Hausdorff metric induced by the partial metric p. Let be a partial metric space, and let be the collection of all nonempty, closed and bounded subset of the partial metric space . Note that closedness is taken from , and boundedness is given as follows: A is a bounded subset in if there exist and such that for all , we have , that is, . For and , they define
It is immediate to get that if , then , where .
Remark 3 
Let be a partial metric space, and let A be a nonempty subset of X. Then
Aydi et al.  also introduced the following properties of mappings and .
Proposition 1 
Let be a partial metric space. For , the following properties hold:
implies that ;
Proposition 2 
Let be a partial metric space. For , the following properties hold:
implies that .
Aydi et al.  proved the following important result.
Lemma 2 Let be a partial metric space, and . For any , there exists such that
In this study, we also recall the Meir-Keeler-type contraction  and α-admissible . In 1969, Meir and Keeler  introduced the following notion of Meir-Keeler-type contraction in a metric space .
Definition 3 Let be a metric space, . Then f is called a Meir-Keeler-type contraction whenever for each , there exists such that
The following definition was introduced in .
Definition 4 Let be a self-mapping of a set X and . Then f is called an α-admissible if
2 Main results
We first introduce the following notions of a strictly α-admissible and and an α-Meir-Keeler contraction with respect to the partial Hausdorff metric .
Definition 5 Let be a partial metric space, and . We say that T is strictly α-admissible if
Definition 6 Let be a partial metric space and . We call an α-Meir-Keeler contraction with respect to the partial Hausdorff metric if the following conditions hold:
(c1) T is strictly α-admissible;
(c2) for each , there exists such that
Remark 4 Note that if is a α-Meir-Keeler contraction with respect to the partial Hausdorff metric , then we have that for all
Further, if , then . On the other hand, if , then .
We now state and prove our main result.
Theorem 2 Let be a complete partial metric space. Suppose that is an α-Meir-Keeler contraction with respect to the partial Hausdorff metric ℋ and that there exists such that for all . Then T has a fixed point in X (that is, there exists such that ).
Proof Let . Since is an α-Meir-Keeler contraction with respect to the partial Hausdorff metric , by Remark 4, we have that
Put , and let . From Lemma 2 with , we have that
Using (1) and (2), we obtain
So, we can obtain a sequence recursively as follows:
Since T is strictly α-admissible, we deduce that . Continuing this process, we have that
Since is an α-Meir-Keeler contraction with respect to the partial Hausdorff metric , by Remark 4, we have that
From Lemma 2 with , we have that
Using (5) and (6), we obtain
Now, from (7) and by the mathematical induction, we obtain
Since for all , we get
Using (8) and (9), we obtain
Letting in (10). Then
By the property (p2) of a partial metric and using (11), we have
Using (10) and the property (p4) of a partial metric, for any , we have
Using (12) and (13), we get
By the definition of , we get that for any ,
This yields that is a Cauchy sequence in . Since is complete, from Lemma 1, is a complete metric space. Therefore, converges to some with respect to the metric , and we also have
Since is an α-Meir-Keeler contraction with respect to the partial Hausdorff metric ℋ, by Remark 4, we have that
By the definition of the mapping α, we have that . Using (15), we get
Now gives that
Using (16), we get
By the property (p4) of a partial metric, we have
Taking limit as , and using (12), (15) and (17), we obtain
Therefore, from (15), , we obtain
which implies that by Remark 3. □
The following theorem, the main result of , is a consequence of Theorem 2 by taking for .
Theorem 3 
Let be a complete partial metric space. If is a multi-valued mapping such that for all , we have
where . Then T has a fixed point.
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The authors declare that there is no conflict of interests regarding the publication of this article.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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Chen, CM., Karapınar, E. Fixed point results for the α-Meir-Keeler contraction on partial Hausdorff metric spaces. J Inequal Appl 2013, 410 (2013). https://doi.org/10.1186/1029-242X-2013-410
- fixed point
- α-Meir-Keeler contraction
- partial metric space
- partial Hausdorff metric space