Abstract
Coincidence point theorems for hybrid pairs of singlevalued and multivalued mappings on an arbitrary nonempty set with values in a partial metric space using a partial Hausdorff metric have been proved. As an application of our main result, the existence and uniqueness of common and bounded solutions of functional equations arising in dynamic programming are discussed.
MSC: 47H10, 54H25, 54E50.
Keywords:
coincidence point; orbitally complete; common fixed point; partial metric space1 Introduction and preliminaries
Fixed point theory plays a fundamental role in solving functional equations [1] arising in several areas of mathematics and other related disciplines as well. The Banach contraction principle is a key principle that made a remarkable progress towards the development of metric fixed point theory. Markin [2] and Nadler [3] proved a multivalued version of the Banach contraction principle employing the notion of a Hausdorff metric. Afterwards, a number of generalizations (see [49]) were obtained using different contractive conditions. The study of hybrid type contractive conditions involving singlevalued and multivalued mappings is a valuable addition to the metric fixed point theory and its applications (for details, see [1014]). Among several generalizations of the Banach contraction principle, Suzuki’s work [[15], Theorem 2.1] led to a number of results (for details, see [13,1621]).
On the other hand, Matthews [22] introduced the concept of a partial metric space as a part of the study of denotational semantics of dataflow networks. He obtained a modified version of the Banach contraction principle, more suitable in this context (see also [23,24]). Since then, results obtained in the framework of partial metric spaces have been used to constitute a suitable framework to model the problems related to the theory of computation (see [22,2528]). Recently, Aydi et al.[29] initiated the concept of a partial Hausdorff metric and obtained an analogue of Nadler’s fixed point theorem [3] in partial metric spaces.
The aim of this paper is to obtain some coincidence point theorems for a hybrid pair of singlevalued and multivalued mappings on an arbitrary nonempty set with values in a partial metric space. Our results extend, unify and generalize several known results in the existing literature (see [13,15,21,30]). As an application, we obtain the existence and uniqueness of a common and bounded solution for SuzukiZamfirescu class of functional equations under contractive conditions weaker than those given in [1,3134].
Throughout this work, a mapping is defined by
In the sequel, the letters ℝ, and ℕ will denote the set of all real numbers, the set of all nonnegative real numbers and the set of all positive integers, respectively. Consistent with [22,29,35,36], the following definitions and results will be needed in the sequel.
Definition 1.1[22]
Let X be any nonempty set. A mapping is said to be a partial metric if and only if for all the following conditions are satisfied:
The pair is called a partial metric space. If , then (P1) and (P2) imply that . But the converse does not hold in general. A classical example of a partial metric space is the pair , where is defined as (see also [37]).
Example 1.2[22]
defines a partial metric p on X.
For more interesting examples, we refer to [23,27,28,35,38,39]. Each partial metric p on X generates a topology on X which has as a base the family of open balls (pballs) , where
for all and . A sequence in a partial metric space is called convergent to a point with respect to if and only if (for details, see [22]). If p is a partial metric on X, then the mapping given by defines a metric on X. Furthermore, a sequence converges in a metric space to a point if and only if
Definition 1.3[22]
Let be a partial metric space, then
(a) A sequence in X is called Cauchy if and only if exists and is finite.
(b) A partial metric space is said to be complete if every Cauchy sequence in X converges with respect to to a point such that .
Letbe a partial metric space, then
(c) A sequenceinXis Cauchy inif and only if it is Cauchy in.
(d) A partial metric spaceis complete if and only ifis complete.
Consistent with [29], let be the family of all nonempty, closed and bounded subsets of the partial metric space , induced by the partial metric p. Note that closedness is taken from ( is the topology induced by p) and boundedness is given as follows: A is a bounded subset in if there exists an and such that for all , we have , that is, . For and , define and
It can be verified that implies , where .
Lemma B[35]
Letbe a partial metric space andAbe a nonempty subset ofX, thenif and only if.
Proposition 1.4[29]
Letbe a partial metric space. For any, we have the following:
Proposition 1.5[29]
Letbe a partial metric space. For any, we have the following:
The mapping is called a partial Hausdorff metric induced by a partial metric p. Every Hausdorff metric is a partial Hausdorff metric, but the converse is not true (see Example 2.6 in [29]).
Lemma C[29]
Letbe a partial metric space andand, then for any, there exists asuch that.
Theorem 1.6[29]
Letbe a partial metric space. Ifis a multivalued mapping such that for all, we have, where. ThenThas a fixed point.
Definition 1.7 Let be a partial metric space and and . A point is said to be (i) a fixed point off if , (ii) a fixed point ofT if , (iii) a coincidence point of a pair if , (iv) a common fixed point of the pair if .
We denote the set of all fixed points of f, the set of all coincidence points of the pair and the set of all common fixed points of the pair by , and , respectively. Motivated by the work of [4,13], we give the following definitions in partial metric spaces.
Definition 1.8 Let be a partial metric space and and . The pair is called (i) commuting if for all , (ii) weakly compatible if the pair commutes at their coincidence points, that is, whenever , (iii) ITcommuting [11] at if .
Definition 1.9 Let be a partial metric space and Y be any nonempty set. Let and be singlevalued and multivalued mappings, respectively. Suppose that , then the set
is called an orbit for the pair at . A partial metric space X is called orbitally complete if and only if every Cauchy sequence in the orbit for at converges with respect to to a point such that .
Singh and Mishra [13] introduced SuzukiZamfirescu type hybrid contractive condition in complete metric spaces. In the context of partial metric spaces, the condition is given as follows.
Definition 1.10 Let be a partial metric space, and be singlevalued and multivalued mappings, respectively. The hybrid pair is said to satisfy SuzukiZamfirescu hybrid contraction condition if there exists such that implies that
Lemma DLetbe a partial metric space, andbe singlevalued and multivalued mappings, respectively. Then the partial metric spaceisorbitally complete if and only ifisorbitally complete.
Proof Suppose that is orbitally complete and is an arbitrary element of X. If is a Cauchy sequence in in , then it is also Cauchy in . Therefore, by (1.2) we deduce that there exists y in X such that
and converges to y in . Conversely, let be orbitally complete. If is a Cauchy sequence in in , then it is also a Cauchy sequence in . Therefore,
For given , there exists such that
for all . Consequently, we have
2 Coincidence points of a hybrid pair of mappings
In the following theorem, the existence of coincidence points of a hybrid pair of singlevalued and multivalued mappings that satisfy SuzukiZamfirescu hybrid contraction condition in partial metric spaces is established.
Theorem 2.1Letbe a partial metric space andYbe any nonempty set. Assume that a pair of mappingsandsatisfies SuzukiZamfirescu hybrid contraction condition with. If there existssuch thatisorbitally complete at, then. IfandisITcommuting at coincidence points of, thenprovided thatfzis a fixed point offfor some.
Proof Let and be such that . By the given assumption, we have . So, there exists a point such that . As , so by Lemma C, there exists a point such that
Using the fact that , we obtain a point such that . Therefore,
Since
we have
If
then
If
then we obtain
As , we choose such that . Using the fact that , we obtain a point such that and
Since
so we have
Following the arguments similar to those given above, we obtain
which further implies that
Continuing this process, we obtain a sequence such that for any integer , and
for every . This shows that . Since
so we obtain
It follows that is a Cauchy sequence in . By Lemma A, we have is a Cauchy sequence in . Since is orbitally complete at , so again by Lemma D, is orbitally complete at . Hence, there exists an element such that . This implies that
give
Similarly, we can show that
Now, we will claim that
If or , then . This gives , which implies that and we are done. Now from (2.1), there exists a positive integer such that for all,
This implies
On taking limit as n tends to ∞, we obtain
If
then we are done. If
then we obtain
and hence (2.2) holds. Next, we show that
for any . If , then , and the claim follows from (2.2). Suppose that , then . As f is a nonconstant singlevalued mapping, we have
This implies
Therefore,
Hence, (2.3) holds for any . Note that
On taking limit as , we obtain
We obtain , which further implies that . Hence, . Further if and , then due to ITcommutativity of the pair , we have . This shows that fz is a common fixed point of the pair . □
Corollary ALetbe a partial metric space andYbe any nonempty set. Assume that here existssuch that the mappingsandsatisfy
for all, with. If there existssuch thatisorbitally complete at, then. IfandisITcommuting at coincidence points of the pair, thenprovided thatfzis a fixed point offfor some.
Example 2.2 Let and . Define a mapping as follows:
Then p is a partial metric on X. Let be as given in Theorem 2.1 and the mappings and be given as
Note that
If we take and , then for all ,
holds. If we consider , then . Then, for , we have , hence is satisfied trivially. Now consider
implies
Let , . As , there exists a point in Y such that and , we obtain a point in Y such that . Continuing this way, we construct an orbit for at . Also, is orbitally complete at . So, all the conditions of Corollary A are satisfied. Moreover, .
On the other hand, the metric induced by the partial metric p is given by
Now, we show that Corollary A is not applicable (in the case of a metric induced by a partial metric p) in this case. Since
is satisfied for any , x and y in X, so it must imply . But
and
Corollary BLetbe a partial metric space, Ybe any nonempty set andbe such that. Suppose that there existssuch thatisorbitally complete at. Assume further that there exists ansuch that
implies that
for all. Then. Further, ifand the pairis commuting atxwhere, thenis a singleton.
Proof It follows from Theorem 2.1, that . If , then . Further, if and is commuting at u, then . Now,
implies that
As , we obtain , which further implies that . Hence, fu is a common fixed point of f and T.
For uniqueness, assume there exist , such that and . Then
which implies
3 An application
In this section, we assume that U and V are Banach spaces, and . Suppose that
Considering W and D as the state and decision spaces respectively, the problem of dynamic programming reduces to the problem of solving the functional equations:
Then equations (3.1) and (3.2) can be reformulated as
For more on the multistage process involving such functional equations, we refer to [23,3134]. Now, we study the existence and uniqueness of a common and bounded solution of the functional equations (3.3)(3.4) arising in dynamic programming in the setup of partial metric spaces.
Let denote the set of all bounded realvalued functions on W. For an arbitrary , define . Then is a Banach space endowed with the metric d defined as . Now, consider
where , and is a partial metric on . Let be defined as in Section 1. Suppose that the following conditions hold:
(C1): G, F, g, and are bounded.
Moreover, assume that there exists such that for every , and ,
implies
where
(C3): For any , there exists such that for ,
Theorem 3.1Assume that the conditions (C1)(C4) are satisfied. Ifis a closed convex subspace of, then the functional equations (3.3) and (3.4) have a unique, common and bounded solution.
Proof Note that is a complete partial metric space. By (C1), J, K are selfmaps of . The condition (C3) implies that . It follows from (C4) that J and K commute at their coincidence points. Let λ be an arbitrary positive number and . Choose and such that
where , . Further, from (3.5) and (3.6), we have
Therefore, (3.8) in (C2) becomes
Then (3.13) together with (3.10) and (3.12) implies
Now, (3.10), (3.11) and (3.13) imply
From (3.14) and (3.15), we have
As the above inequality is true for any and is taken arbitrarily, so from (3.13) we obtain
implies
Therefore, by Corollary B, the pair has a common fixed point , that is, is a unique, bounded and common solution of (3.3) and (3.4). □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the editor and anonymous reviewers for their useful comments that helped to improve the presentation of this paper.
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