Fixed point results for the α-Meir-Keeler contraction on partial Hausdorff metric spaces
© Chen and Karapınar; licensee Springer 2013
Received: 4 March 2013
Accepted: 8 August 2013
Published: 23 August 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.
Keywordsfixed point α-Meir-Keeler contraction partial metric space partial Hausdorff metric space
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.
is a metric on X.
We recall some definitions of a partial metric space, as follows.
Definition 2 
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.
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 .
It is immediate to get that if , then , where .
Remark 3 
Aydi et al.  also introduced the following properties of mappings and .
Proposition 1 
implies that ;
Proposition 2 
implies that .
Aydi et al.  proved the following important result.
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 .
The following definition was introduced in .
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 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;
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 ).
which implies that by Remark 3. □
The following theorem, the main result of , is a consequence of Theorem 2 by taking for .
Theorem 3 
where . Then T has a fixed point.
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