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.
Keywords:fixed 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.
is a metric on X.
We recall some definitions of a partial metric space, as follows.
Remark 2 The limit in a partial metric space is not unique.
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 .
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
Aydi et al. also introduced the following properties of mappings and .
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
(c1) T is strictly α-admissible;
We now state and prove our main result.
Theorem 2Letbe a complete partial metric space. Suppose thatis anα-Meir-Keeler contraction with respect to the partial Hausdorff metric ℋ and that there existssuch thatfor all. ThenThas a fixed point inX (that is, there existssuch that).
Using (1) and (2), we obtain
Using (5) and (6), we obtain
Now, from (7) and by the mathematical induction, we obtain
Using (8) and (9), we obtain
By the property (p2) of a partial metric and using (11), we have
Using (12) and (13), we get
Using (16), we get
By the property (p4) of a partial metric, we have
The following theorem, the main result of , is a consequence of Theorem 2 by taking for .
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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