A Kirk Type Characterization of Completeness for Partial Metric Spaces
© Salvador Romaguera. 2010
Received: 1 October 2009
Accepted: 25 November 2009
Published: 7 December 2009
1. Introduction and Preliminaries
Caristi proved in  that if is a selfmapping of a complete metric space such that there is a lower semicontinuous function satisfying
In particular, Kirk proved in  that a metric space is complete if and only if every Caristi mapping for has a fixed point. (For other characterizations of metric completeness in terms of fixed point theory see [11–14], etc., and also [15, 16] for recent contributions in this direction.)
In this paper we extend Kirk's characterization to a kind of complete partial metric spaces.
Let us recall that partial metric spaces were introduced by Matthews in  as a part of the study of denotational semantics of dataflow networks. In fact, it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation (see [18–25], etc.).
A partial metric  on a set is a function such that for all : (i) ; (ii) ; (iii) ; (iv)
Next we give some pertinent concepts and facts on completeness for partial metric spaces.
A sequence in a partial metric space is called a Cauchy sequence if there exists (and is finite) ([17, Definition ])
Note that is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space (see, e.g., [17, page 194]).
A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to to a point such that ([17, Definition ]).
In order to give an appropriate notion of a Caristi mapping in the framework of partial metric spaces, we naturally propose the following two alternatives.
In a first attempt to generalize Kirk's characterization of metric completeness to the partial metric framework, one can conjecture that a partial metric space is complete if and only if every -Caristi mapping on has a fixed point.
The following easy example shows that this conjecture is false.
Indeed, let and suppose that there is a lower semicontinuous function from into such that for all If we have , which means that for any so by lower semicontinuity of which contradicts condition Therefore which again contradicts condition We conclude that is not a -Caristi mapping on
2. The Main Result
In this section we characterize those partial metric spaces for which every -Caristi mapping has a fixed point in the style of Kirk's characterization of metric completeness. This will be done by means of the notion of a 0-complete partial metric space which is introduced as follows.
On the other hand, the partial metric space where denotes the set of rational numbers and the partial metric is given by provides a paradigmatic example of a 0-complete partial metric space which is not complete.
The author is very grateful to the referee for his/her useful suggestions. This work was partially supported by the Spanish Ministry of Science and Innovation, and FEDER, Grant MTM2009-12872-C02-01.
- Caristi J: Fixed point theorems for mappings satisfying inwardness conditions. Transactions of the American Mathematical Society 1976, 215: 241–251.MathSciNetView ArticleMATHGoogle Scholar
- Beg I, Abbas M: Random fixed point theorems for Caristi type random operators. Journal of Applied Mathematics & Computing 2007,25(1–2):425–434. 10.1007/BF02832367MathSciNetView ArticleMATHGoogle Scholar
- Downing D, Kirk WA: A generalization of Caristi's theorem with applications to nonlinear mapping theory. Pacific Journal of Mathematics 1977,69(2):339–346.MathSciNetView ArticleMATHGoogle Scholar
- Feng Y, Liu S: Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. Journal of Mathematical Analysis and Applications 2006,317(1):103–112. 10.1016/j.jmaa.2005.12.004MathSciNetView ArticleMATHGoogle Scholar
- Jachymski JR: Caristi's fixed point theorem and selections of set-valued contractions. Journal of Mathematical Analysis and Applications 1998,227(1):55–67. 10.1006/jmaa.1998.6074MathSciNetView ArticleMATHGoogle Scholar
- Khamsi MA: Remarks on Caristi's fixed point theorem. Nonlinear Analysis: Theory, Methods & Applications 2009,71(1–2):227–231. 10.1016/j.na.2008.10.042MathSciNetView ArticleMATHGoogle Scholar
- Kirk WA: Caristi's fixed point theorem and metric convexity. Colloquium Mathematicum 1976,36(1):81–86.MathSciNetMATHGoogle Scholar
- Latif A: Generalized Caristi's fixed point theorems. Fixed Point Theory and Applications 2009, Article ID 170140 2009:-7 Pages.Google Scholar
- Kirk WA, Caristi J: Mappings theorems in metric and Banach spaces. Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 1975,23(8):891–894.MathSciNetMATHGoogle Scholar
- Suzuki T: Generalized Caristi's fixed point theorems by Bae and others. Journal of Mathematical Analysis and Applications 2005,302(2):502–508. 10.1016/j.jmaa.2004.08.019MathSciNetView ArticleMATHGoogle Scholar
- Park S: Characterizations of metric completeness. Colloquium Mathematicum 1984,49(1):21–26.MathSciNetMATHGoogle Scholar
- Reich S: Kannan's fixed point theorem. Bollettino dell'Unione Matematica Italiana 1971, 4: 1–11.MATHGoogle Scholar
- Subrahmanyam PV: Completeness and fixed-points. Monatshefte für Mathematik 1975,80(4):325–330. 10.1007/BF01472580MathSciNetView ArticleMATHGoogle Scholar
- Suzuki T, Takahashi W: Fixed point theorems and characterizations of metric completeness. Topological Methods in Nonlinear Analysis 1996,8(2):371–382.MathSciNetMATHGoogle Scholar
- Dhompongsa S, Yingtaweesittikul H: Fixed points for multivalued mappings and the metric completeness. Fixed Point Theory and Applications 2009, Article ID 972395 2009:-15 Pages.Google Scholar
- Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proceedings of the American Mathematical Society 2008,136(5):1861–1869.MathSciNetView ArticleMATHGoogle Scholar
- Matthews SG: Partial metric topology. In Proceedings of the 8th Summer Conference on General Topology and Applications (Flushing, NY, 1992), Annals of the New York Academy of Sciences. Volume 728. The New York Academy of Sciences, New York, NY, USA; 1994:183–197.Google Scholar
- Heckmann R: Approximation of metric spaces by partial metric spaces. Applied Categorical Structures 1999,7(1–2):71–83.MathSciNetView ArticleMATHGoogle Scholar
- O'Neill SJ: Partial metrics, valuations, and domain theory. In Proceedings of the 11th Summer Conference on General Topology and Applications (Gorham, ME, 1995), Annals of the New York Academy of Sciences. Volume 806. The New York Academy of Sciences, New York, NY, USA; 1996:304–315.Google Scholar
- Romaguera S, Schellekens M: Partial metric monoids and semivaluation spaces. Topology and Its Applications 2005,153(5–6):948–962. 10.1016/j.topol.2005.01.023MathSciNetView ArticleMATHGoogle Scholar
- Romaguera S, Valero O: A quantitative computational model for complete partial metric spaces via formal balls. Mathematical Structures in Computer Science 2009,19(3):541–563. 10.1017/S0960129509007671MathSciNetView ArticleMATHGoogle Scholar
- Schellekens M: The Smyth completion: a common foundation for denotational semantics and complexity analysis. Electronic Notes in Theoretical Computer Science 1995, 1: 535–556.MathSciNetView ArticleMATHGoogle Scholar
- Schellekens MP: A characterization of partial metrizability: domains are quantifiable. Theoretical Computer Science 2003,305(1–3):409–432.MathSciNetView ArticleMATHGoogle Scholar
- Waszkiewicz P: Quantitative continuous domains. Applied Categorical Structures 2003,11(1):41–67. 10.1023/A:1023012924892MathSciNetView ArticleMATHGoogle Scholar
- Waszkiewicz P: Partial metrisability of continuous posets. Mathematical Structures in Computer Science 2006,16(2):359–372. 10.1017/S0960129506005196MathSciNetView ArticleMATHGoogle Scholar
- Penot J-P: Fixed point theorems without convexity. Bulletin de la Société Mathématique de France 1979, (60):129–152.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.