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
In the sequel, the letters ℝ,
Definition 1.1[22]
Let X be any nonempty set. A mapping
(P1)
(P2)
(P3)
(P4)
The pair
Example 1.2[22]
If
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
for all
Definition 1.3[22]
Let
(a) A sequence
(b) A partial metric space
Let
(c) A sequence
(d) A partial metric space
Consistent with [29], let
It can be verified that
Lemma B[35]
Let
Proposition 1.4[29]
Let
(i)
(ii)
(iii)
(iv)
Proposition 1.5[29]
Let
(h1)
(h2)
(h3)
(h4)
The mapping
Lemma C[29]
Let
Theorem 1.6[29]
Let
Definition 1.7 Let
We denote the set of all fixed points of f, the set of all coincidence points of the pair
Definition 1.8 Let
Definition 1.9 Let
is called an orbit for the pair
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
for all
Lemma DLet
Proof Suppose that
and
For given
for all
whenever
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 2.1Let
Proof Let
Using the fact that
Since
we have
If
then
If
then we obtain
As
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
for every
so we obtain
Now, for
It follows that
Let
give
Similarly, we can show that
Now, we will claim that
If
So, for any
Hence, for any
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
This implies
Therefore,
Hence, (2.3) holds for any
On taking limit as
We obtain
Corollary ALet
for all
Example 2.2 Let
Then p is a partial metric on X. Let
Note that
If we take
holds. If we consider
Hence, for all
implies
Let
On the other hand, the metric
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
and
Hence, for any
Corollary BLet
implies that
for all
Proof It follows from Theorem 2.1, that
implies that
As
For uniqueness, assume there exist
which implies
We obtain
3 An application
In this section, we assume that U and V are Banach spaces,
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
where
(C1): G, F, g, and
(C2): For
Moreover, assume that there exists
implies
where
(C3): For any
(C4): There exists
Theorem 3.1Assume that the conditions (C1)(C4) are satisfied. If
Proof Note that
where
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
implies
Therefore, by Corollary B, the pair
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.
References

Baskaran, R, Subrahmanyam, PV: A note on the solution of a class of functional equations. Appl. Anal.. 22(34), 235–241 (1986). Publisher Full Text

Markin, J: A fixed point theorem for set valued mappings. Bull. Am. Math. Soc.. 74, 639–640 (1968). Publisher Full Text

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

Ćirić, L: Fixed points for generalized multivalued contractions. Mat. Vesn.. 9, 265–272 (1972)

Ćirić, L: Multivalued nonlinear contraction mappings. Nonlinear Anal.. 71, 2716–2723 (2009). Publisher Full Text

Covitz, H, Nadler, SB: Multivalued contraction mappings in generalized metric spaces. Isr. J. Math.. 8, 5–11 (1970). Publisher Full Text

Daffer, PZ, Kaneko, H: Fixed points of generalized contractive multivalued mappings. J. Math. Anal. Appl.. 192, 655–666 (1995). Publisher Full Text

Reich, S: Fixed points of contractive functions. Boll. Unione Mat. Ital.. 5, 26–42 (1972)

Semenov, PV: Fixed points of multivalued contractions. Funct. Anal. Appl.. 36, 159–161 (2002). Publisher Full Text

Naimpally, SA, Singh, SL, Whitfield, JHM: Coincidence theorems for hybrid contractions. Math. Nachr.. 127, 177–180 (1986). Publisher Full Text

Singh, SL, Mishra, SN: Nonlinear hybrid contractions. J. Natur. Phys. Sci.. 5/8, 191–206 (1991/1994)

Singh, SL, Mishra, SN: On a Ljubomir Ćirić fixed point theorem for nonexpansive type maps with applications. Indian J. Pure Appl. Math.. 33, 531–542 (2002)

Singh, SL, Mishra, SN: Coincidence theorems for certain classes of hybrid contractions. Fixed Point Theory Appl.. 2010, Article ID 898109 (2010)

Singh, SL, Mishra, SN: Remarks on recent fixed point theorems. Fixed Point Theory Appl. doi:10.1155/2010/452905 (2010)

Suzuki, T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc.. 136, 1861–1869 (2008)

Ali, B, Abbas, M: Suzuki type fixed point theorem for fuzzy mappings in ordered metric spaces. Fixed Point Theory Appl.. 2013, Article ID 9 (2013)

Ćirić, L, Abbas, M, Rajović, M, Ali, B: Suzuki type fixed point theorems for generalized multivalued mappings on a set endowed with two bmetrics. Appl. Math. Comput.. 219, 1712–1723 (2012). Publisher Full Text

Dhompongsa, S, Yingtaweesittikul, H: Fixed points for multivalued mappings and the metric completeness. Fixed Point Theory Appl.. 2009, Article ID 972395 (2009)

Kikkawa, M, Suzuki, T: Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Anal.. 69, 2942–2949 (2008). Publisher Full Text

Kikkawa, M, Suzuki, T: Some similarity between contractions and Kannan mappings. Fixed Point Theory Appl.. 2008, Article ID 649749 (2008)

Moţ, G, Petruşel, A: Fixed point theory for a new type of contractive multivalued operators. Nonlinear Anal.. 70, 3371–3377 (2009). Publisher Full Text

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

Bari, CD, Vetro, P: Fixed points for weak φcontractions on partial metric spaces. Int. J. Eng. Contemp. Math. Sci.. 1, 5–13 (2011)

Paesano, D, Vetro, P: Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topol. Appl.. 159, 911–920 (2012). Publisher Full Text

Ćirić, L, Samet, B, Aydi, H, Vetro, C: Common fixed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput.. 218, 2398–2406 (2011). Publisher Full Text

Heckmann, R: Approximation of metric spaces by partial metric spaces. Appl. Categ. Struct.. 7, 71–83 (1999). Publisher Full Text

Romaguera, S: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl.. 2010, Article ID 493298 (2010)

Schellekens, MP: The correspondence between partial metrics and semivaluations. Theor. Comput. Sci.. 315, 135–149 (2004). 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

Zamfirescu, T: Fixed point theorems in metric spaces. Arch. Math.. 23, 292–298 (1972). Publisher Full Text

Bellman, R: Methods of Nonlinear Analysis. Vol. II, Academic Press, New York (1973)

Bellman, R, Lee, ES: Functional equations in dynamic programming. Aequ. Math.. 17, 1–18 (1978). Publisher Full Text

Bhakta, PC, Mitra, S: Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl.. 98, 348–362 (1984). Publisher Full Text

Pathak, HK, Cho, YJ, Kang, SM, Lee, BS: Fixed point theorems for compatible mappings of type P and applications to dynamic programming. Matematiche. 50, 15–33 (1995)

Altun, I, Simsek, H: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud.. 1, 1–8 (2008)

Altun, I, Sola, F, Simsek, H: Generalized contractions on partial metric spaces. Topol. Appl.. 157, 2778–2785 (2010). Publisher Full Text

Abbas, M, Nazir, T: Fixed point of generalized weakly contractive mappings in ordered partial metric spaces. Fixed Point Theory Appl.. 2012, Article ID 1 (2012)

Bukatin, MA, Shorina, SY: Partial metrics and cocontinuous valuations. In: Nivat M (ed.) Foundations of Software Science and Computation Structure, pp. 125–139. Springer, Berlin (1998)

Romaguera, S, Valero, O: A quantitative computational model for complete partial metric spaces via formal balls. Math. Struct. Comput. Sci.. 19, 541–563 (2009). Publisher Full Text