- Open Access
Fixed point of Suzuki-Zamfirescu hybrid contractions in partial metric spaces via partial Hausdorff metric
© Abbas and Ali; licensee Springer 2013
- Received: 9 October 2012
- Accepted: 12 January 2013
- Published: 31 January 2013
Coincidence point theorems for hybrid pairs of single-valued and multi-valued mappings on an arbitrary non-empty 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.
- coincidence point
- orbitally complete
- common fixed point
- partial metric space
Fixed point theory plays a fundamental role in solving functional equations  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  and Nadler  proved a multi-valued version of the Banach contraction principle employing the notion of a Hausdorff metric. Afterwards, a number of generalizations (see [4–9]) were obtained using different contractive conditions. The study of hybrid type contractive conditions involving single-valued and multi-valued mappings is a valuable addition to the metric fixed point theory and its applications (for details, see [10–14]). Among several generalizations of the Banach contraction principle, Suzuki’s work [, Theorem 2.1] led to a number of results (for details, see [13, 16–21]).
On the other hand, Matthews  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, 25–28]). Recently, Aydi et al.  initiated the concept of a partial Hausdorff metric and obtained an analogue of Nadler’s fixed point theorem  in partial metric spaces.
The aim of this paper is to obtain some coincidence point theorems for a hybrid pair of single-valued and multi-valued mappings on an arbitrary non-empty 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 Suzuki-Zamfirescu class of functional equations under contractive conditions weaker than those given in [1, 31–34].
In the sequel, the letters ℝ, and ℕ will denote the set of all real numbers, the set of all non-negative 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 
if and only if ;
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 ).
Example 1.2 
defines a partial metric p on X.
Definition 1.3 
A sequence in X is called Cauchy 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 sequence in X is Cauchy in if and only if it is Cauchy in .
A partial metric space is complete if and only if is complete.
It can be verified that implies , where .
Lemma B 
Let be a partial metric space and A be a non-empty subset of X, then if and only if .
Proposition 1.4 
Proposition 1.5 
implies that .
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 ).
Lemma C 
Let be a partial metric space and and , then for any , there exists a such that .
Theorem 1.6 
Let be a partial metric space. If is a multi-valued mapping such that for all , we have , where . Then T has a fixed point.
Definition 1.7 Let be a partial metric space and and . A point is said to be (i) a fixed point of f if , (ii) a fixed point of T 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) IT-commuting  at if .
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  introduced Suzuki-Zamfirescu type hybrid contractive condition in complete metric spaces. In the context of partial metric spaces, the condition is given as follows.
Lemma D Let be a partial metric space, and be single-valued and multi-valued mappings, respectively. Then the partial metric space is -orbitally complete if and only if is -orbitally complete.
whenever . The result follows. □
In the following theorem, the existence of coincidence points of a hybrid pair of single-valued and multi-valued mappings that satisfy Suzuki-Zamfirescu hybrid contraction condition in partial metric spaces is established.
Theorem 2.1 Let be a partial metric space and Y be any non-empty set. Assume that a pair of mappings and satisfies Suzuki-Zamfirescu hybrid contraction condition with . If there exists such that is -orbitally complete at , then . If and is IT-commuting at coincidence points of , then provided that fz is a fixed point of f for some .
We obtain , which further implies that . Hence, . Further if and , then due to IT-commutativity of the pair , we have . This shows that fz is a common fixed point of the pair . □
for all , with . If there exists such that is -orbitally complete at , then . If and is IT-commuting at coincidence points of the pair , then provided that fz is a fixed point of f for some .
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, .
for all . Then . Further, if and the pair is commuting at x where , then is a singleton.
As , we obtain , which further implies that . Hence, fu is a common fixed point of f and T.
We obtain , which further implies that . Hence, . □
For more on the multistage process involving such functional equations, we refer to [23, 31–34]. 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.
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.
Theorem 3.1 Assume that the conditions (C1)-(C4) are satisfied. If is a closed convex subspace of , then the functional equations (3.3) and (3.4) have a unique, common and bounded solution.
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). □
The authors would like to thank the editor and anonymous reviewers for their useful comments that helped to improve the presentation of this paper.
- Baskaran R, Subrahmanyam PV: A note on the solution of a class of functional equations. Appl. Anal. 1986, 22(3–4):235–241. 10.1080/00036818608839621MathSciNetView ArticleGoogle Scholar
- Markin J: A fixed point theorem for set valued mappings. Bull. Am. Math. Soc. 1968, 74: 639–640. 10.1090/S0002-9904-1968-11971-8MathSciNetView ArticleGoogle Scholar
- Nadler SB: Multi-valued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475MathSciNetView ArticleGoogle Scholar
- Ćirić L: Fixed points for generalized multi-valued contractions. Mat. Vesn. 1972, 9: 265–272.Google Scholar
- Ćirić L: Multi-valued nonlinear contraction mappings. Nonlinear Anal. 2009, 71: 2716–2723. 10.1016/j.na.2009.01.116MathSciNetView ArticleGoogle Scholar
- Covitz H, Nadler SB: Multi-valued contraction mappings in generalized metric spaces. Isr. J. Math. 1970, 8: 5–11. 10.1007/BF02771543MathSciNetView ArticleGoogle Scholar
- Daffer PZ, Kaneko H: Fixed points of generalized contractive multi-valued mappings. J. Math. Anal. Appl. 1995, 192: 655–666. 10.1006/jmaa.1995.1194MathSciNetView ArticleGoogle Scholar
- Reich S: Fixed points of contractive functions. Boll. Unione Mat. Ital. 1972, 5: 26–42.Google Scholar
- Semenov PV: Fixed points of multi-valued contractions. Funct. Anal. Appl. 2002, 36: 159–161. 10.1023/A:1015682926496MathSciNetView ArticleGoogle Scholar
- Naimpally SA, Singh SL, Whitfield JHM: Coincidence theorems for hybrid contractions. Math. Nachr. 1986, 127: 177–180. 10.1002/mana.19861270112MathSciNetView ArticleGoogle Scholar
- Singh SL, Mishra SN: Nonlinear hybrid contractions. J. Natur. Phys. Sci. 1991/1994, 5/8: 191–206.MathSciNetGoogle Scholar
- Singh SL, Mishra SN: On a Ljubomir Ćirić fixed point theorem for nonexpansive type maps with applications. Indian J. Pure Appl. Math. 2002, 33: 531–542.MathSciNetGoogle Scholar
- Singh SL, Mishra SN: Coincidence theorems for certain classes of hybrid contractions. Fixed Point Theory Appl. 2010., 2010: Article ID 898109Google Scholar
- Singh SL, Mishra SN: Remarks on recent fixed point theorems. Fixed Point Theory Appl. 2010. doi:10.1155/2010/452905Google Scholar
- Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 2008, 136: 1861–1869.View ArticleGoogle Scholar
- Ali B, Abbas M: Suzuki type fixed point theorem for fuzzy mappings in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 9Google Scholar
- Ćirić L, Abbas M, Rajović M, Ali B: Suzuki type fixed point theorems for generalized multi-valued mappings on a set endowed with two b-metrics. Appl. Math. Comput. 2012, 219: 1712–1723. 10.1016/j.amc.2012.08.011MathSciNetView ArticleGoogle Scholar
- Dhompongsa S, Yingtaweesittikul H: Fixed points for multi-valued mappings and the metric completeness. Fixed Point Theory Appl. 2009., 2009: Article ID 972395Google Scholar
- Kikkawa M, Suzuki T: Three fixed point theorems for generalized contractions with constants in complete metric spaces. Nonlinear Anal. 2008, 69: 2942–2949. 10.1016/j.na.2007.08.064MathSciNetView ArticleGoogle Scholar
- Kikkawa M, Suzuki T: Some similarity between contractions and Kannan mappings. Fixed Point Theory Appl. 2008., 2008: Article ID 649749Google Scholar
- Moţ G, Petruşel A: Fixed point theory for a new type of contractive multi-valued operators. Nonlinear Anal. 2009, 70: 3371–3377. 10.1016/j.na.2008.05.005MathSciNetView ArticleGoogle Scholar
- Matthews SG: Partial metric topology. Ann. New York Acad. Sci. 728. Proc. 8th Summer Conference on General Topology Appl. 1994, 183–197.Google Scholar
- Bari CD, Vetro P: Fixed points for weak φ -contractions on partial metric spaces. Int. J. Eng. Contemp. Math. Sci. 2011, 1: 5–13.Google Scholar
- Paesano D, Vetro P: Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topol. Appl. 2012, 159: 911–920. 10.1016/j.topol.2011.12.008MathSciNetView ArticleGoogle Scholar
- Ćirić L, Samet B, Aydi H, Vetro C: Common fixed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput. 2011, 218: 2398–2406. 10.1016/j.amc.2011.07.005MathSciNetView ArticleGoogle Scholar
- Heckmann R: Approximation of metric spaces by partial metric spaces. Appl. Categ. Struct. 1999, 7: 71–83. 10.1023/A:1008684018933MathSciNetView ArticleGoogle Scholar
- Romaguera S: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 493298Google Scholar
- Schellekens MP: The correspondence between partial metrics and semivaluations. Theor. Comput. Sci. 2004, 315: 135–149. 10.1016/j.tcs.2003.11.016MathSciNetView ArticleGoogle Scholar
- Aydi H, Abbas M, Vetro C: Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces. Topol. Appl. 2012, 159: 3234–3242. 10.1016/j.topol.2012.06.012MathSciNetView ArticleGoogle Scholar
- Zamfirescu T: Fixed point theorems in metric spaces. Arch. Math. 1972, 23: 292–298. 10.1007/BF01304884MathSciNetView ArticleGoogle Scholar
- Bellman R Mathematics in Science and Engineering 61. In Methods of Nonlinear Analysis. Vol. II. Academic Press, New York; 1973.Google Scholar
- Bellman R, Lee ES: Functional equations in dynamic programming. Aequ. Math. 1978, 17: 1–18. 10.1007/BF01818535MathSciNetView ArticleGoogle Scholar
- Bhakta PC, Mitra S: Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl. 1984, 98: 348–362. 10.1016/0022-247X(84)90254-3MathSciNetView ArticleGoogle Scholar
- Pathak HK, Cho YJ, Kang SM, Lee BS: Fixed point theorems for compatible mappings of type P and applications to dynamic programming. Matematiche 1995, 50: 15–33.MathSciNetGoogle Scholar
- Altun I, Simsek H: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud. 2008, 1: 1–8.MathSciNetGoogle Scholar
- Altun I, Sola F, Simsek H: Generalized contractions on partial metric spaces. Topol. Appl. 2010, 157: 2778–2785. 10.1016/j.topol.2010.08.017MathSciNetView ArticleGoogle Scholar
- Abbas M, Nazir T: Fixed point of generalized weakly contractive mappings in ordered partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 1Google Scholar
- Bukatin MA, Shorina SY, et al.: Partial metrics and co-continuous valuations. Lecture Notes in Comput. Sci. 1378. In Foundations of Software Science and Computation Structure. Edited by: Nivat M. Springer, Berlin; 1998:125–139.View ArticleGoogle Scholar
- Romaguera S, Valero O: A quantitative computational model for complete partial metric spaces via formal balls. Math. Struct. Comput. Sci. 2009, 19: 541–563. 10.1017/S0960129509007671MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.