Abstract
The purpose of this paper is to study fixed point theorems for a multivalued mapping satisfying the αMeirKeeler 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; αMeirKeeler contraction; partial metric space; partial Hausdorff metric space1 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 [1] 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 [2] 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[2]
A partial metric on a nonempty set X is a function such that for all
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 (p_{1}) and (p_{2}), 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 pballs , 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[2]
Let be a partial metric space. Then
(1) a sequence in a partial metric space converges to if and only if ;
(2) a sequence in a partial metric space is called a Cauchy sequence if and only if exists (and is finite);
(3) a partial metric space is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that ;
(4) 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.
(1) is a Cauchy sequence in a partial metric spaceif and only if it is a Cauchy sequence in the metric space;
(2) a partial metric spaceis complete if and only if the metric spaceis complete. Furthermore, if and only if.
Recently, fixed point theory has developed rapidly on partial metric spaces, see, e.g., [312] and the reference therein. Very recently, Haghi et al.[13] 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.[13] 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 multivalued mapping is called a contraction if
for all and . The study of fixed points for multivalued contractions using the Hausdorff metric was introduced in Nadler [14].
Theorem 1[14]
Letbe a complete metric space, and letbe a multivalued contraction. Then there existssuch that.
Very recently, Aydi et al.[15] 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[15]
Let be a partial metric space, and let A be a nonempty subset of X. Then
Aydi et al.[15] also introduced the following properties of mappings and .
Proposition 1[15]
Letbe a partial metric space. For, the following properties hold:
Proposition 2[15]
Letbe a partial metric space. For, the following properties hold:
Aydi et al.[15] proved the following important result.
Lemma 2Letbe a partial metric space, and. For any, there existssuch that
In this study, we also recall the MeirKeelertype contraction [16] and αadmissible [17]. In 1969, Meir and Keeler [16] introduced the following notion of MeirKeelertype contraction in a metric space .
Definition 3 Let be a metric space, . Then f is called a MeirKeelertype contraction whenever for each , there exists such that
The following definition was introduced in [17].
Definition 4 Let be a selfmapping of a set X and . Then f is called an αadmissible if
2 Main results
We first introduce the following notions of a strictly αadmissible and and an αMeirKeeler contraction with respect to the partial Hausdorff metric .
Definition 5 Let be a partial metric space, and . We say that T is strictly αadmissible if
Definition 6 Let be a partial metric space and . We call an αMeirKeeler contraction with respect to the partial Hausdorff metric if the following conditions hold:
(c_{1}) T is strictly αadmissible;
(c_{2}) for each , there exists such that
Remark 4 Note that if is a αMeirKeeler 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 2Letbe a complete partial metric space. Suppose thatis anαMeirKeeler 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).
Proof Let . Since is an αMeirKeeler 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 αadmissible, we deduce that . Continuing this process, we have that
Since is an αMeirKeeler 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
Put
Using (8) and (9), we obtain
By the property (p_{2}) of a partial metric and using (11), we have
Using (10) and the property (p_{4}) 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 αMeirKeeler 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
Using (16), we get
By the property (p_{4}) 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 [15], is a consequence of Theorem 2 by taking for .
Theorem 3[15]
Letbe a complete partial metric space. Ifis a multivalued mapping such that for all, we have
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this article.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
References

Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math.. 3, 133–181 (1922)

Matthews, SG: Partial metric topology. Proc. 8th Summer of Conference on General Topology and Applications. 183–197 (1994)

Oltra, S, Valero, O: Banach’s fixed point theorem for partial metric spaces. Rend. Ist. Mat. Univ. Trieste. 36, 17–26 (2004)

Abdeljawad, T: Fixed points for generalized weakly contractive mappings in partial metric spaces. Math. Comput. Model.. 54, 2923–2927 (2011). Publisher Full Text

Agarwal, RP, Alghamdi, MA, Shahzad, N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl.. 2012, (2012) Article ID 40

Altun, I, Erduran, A: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl.. 2011, (2011) Article ID 508730

Aydi, H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud.. 4(2), 1–12 (2011)

Aydi, H, Vetro, C, Sintunavarat, W, Kumam, P: Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces. Fixed Point Theory Appl.. 2012, (2012) Article ID 124

Chi, KP, Karapinar, E, Thanh, TD: A generalized contraction principle in partial metric spaces. Math. Comput. Model.. 55, 1673–1681 (2012). Publisher Full Text

Karapinar, E: Weak ϕcontraction on partial metric spaces. J. Comput. Anal. Appl.. 14(2), 206–210 (2012)

Karapinar, E: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl.. 2011, (2011) Article ID 4

Karapinar, E, Erhan, IM, Yıldız, UA: Fixed point theorem for cyclic maps on partial metric spaces. Appl. Math. Inf. Sci.. 6, 239–244 (2012)

Haghi, RH, Rezapour, S, Shahzad, N: Be careful on partial metric fixed point results. Topol. Appl.. 160(3), 450–454 (2013). Publisher Full Text

Nadler, SB: Multivalued contraction mappings. Pac. J. Math.. 30, 475–488 (1969). Publisher Full Text

Aydi, H, Abbas, M, Vetro, C: Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces. Topol. Appl.. 159, 3234–3242 (2012). Publisher Full Text

Meir, A, Keeler, E: A theorem on contraction mappings. J. Math. Anal. Appl.. 28, 326–329 (1969). Publisher Full Text

Samet, B, Vetro, C, Vetro, P: Fixed point theorems for αψcontractive type mappings. Nonlinear Anal.. 75, 2154–2165 (2012). Publisher Full Text