# A Kirk Type Characterization of Completeness for Partial Metric Spaces

- Salvador Romaguera
^{1}Email author

**2010**:493298

**DOI: **10.1155/2010/493298

© Salvador Romaguera. 2010

**Received: **1 October 2009

**Accepted: **25 November 2009

**Published: **7 December 2009

## Abstract

We extend the celebrated result of W. A. Kirk that a metric space is complete if and only if every Caristi self-mapping for has a fixed point, to partial metric spaces.

## 1. Introduction and Preliminaries

Caristi proved in [1] that if is a selfmapping of a complete metric space such that there is a lower semicontinuous function satisfying

for all then has a fixed point.

This classical result suggests the following notion. A selfmapping of a metric space for which there is a function satisfying the conditions of Caristi's theorem is called a Caristi mapping for

There exists an extensive and well-known literature on Caristi's fixed point theorem and related results (see, e.g., [2–10], etc.).

In particular, Kirk proved in [7] 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 [17] 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 [17] on a set is a function such that for all : (i) ; (ii) ; (iii) ; (iv)

A partial metric space is a pair where is a partial metric on

Each partial metric on induces a topology on which has as a base the family of open balls , where for all and

Next we give some pertinent concepts and facts on completeness for partial metric spaces.

If is a partial metric on , then the function given by is a metric on

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 ]).

It is well known and easy to see that a partial metric space is complete if and only if the metric space is complete.

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.

(i)A selfmapping of a partial metric space is called a -Caristi mapping on if there is a function which is lower semicontinuous for and satisfies , for all

(ii)A selfmapping of a partial metric space is called a -Caristi mapping on if there is a function which is lower semicontinuous for and satisfies , for all

It is clear that every -Caristi mapping is -Caristi but the converse is not true, in general.

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.

Example 1.1.

Note that is not complete, because the metric induces the discrete topology on , and is a Cauchy sequence in . However, there is no -Caristi mappings on as we show in the next.

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

Unfortunately, the existence of fixed point for each -Caristi mapping on a partial metric space neither characterizes completeness of as follows from our discussion in the next section.

## 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.

Definition 2.1.

A sequence in a partial metric space is called 0-Cauchy if We say that is 0-complete if every 0-Cauchy sequence in converges, with respect to to a point such that

Note that every 0-Cauchy sequence in is Cauchy in and that every complete partial metric space is 0-complete.

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.

In the proof of the "only if" part of our main result we will use ideas from [11, 26], whereas the following auxiliary result will be used in the proof of the "if" part.

Lemma 2.2.

Let be a partial metric space. Then, for each the function given by is lower semicontinuous for

Proof.

This yields because

Theorem 2.3.

A partial metric space is 0-complete if and only if every -Caristi mapping on has a fixed point.

Proof.

for all

Now, for each define

Observe that because Moreover is closed in the metric space since is lower semicontinuous for .

Fix Take such that Clearly . Hence, for each we have

Following this process we construct a sequence in such that its associated sequence of closed subsets in satisfies

(i) for all

(ii) for all

Since and, by (i) and (ii), for all it follows that so is a 0-Cauchy sequence in and by our hypothesis, there exists such that and thus Therefore

Finally, we show that To this end, we first note that

for all Consequently so by (ii), for all Since and it follows that Hence since so

Conversely, suppose that there is a 0-Cauchy sequence of distinct points in which is not convergent in Construct a subsequence of such that for all

Put and define by if and for all

Observe that is closed in

Now define by if and for all

Note that for all and that for all

From this fact and the preceding lemma we deduce that is lower semicontinuous for

Moreover, for each we have

Therefore is a Caristi -mapping on without fixed point, a contradiction. This concludes the proof.

## Declarations

### Acknowledgments

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.

## Authors’ Affiliations

## References

- 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.

## Copyright

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.