- Research Article
- Open Access

# Fixed Point Theorems for Monotone Mappings on Partial Metric Spaces

- Ishak Altun
^{1}Email author and - Ali Erduran
^{1}

**2011**:508730

https://doi.org/10.1155/2011/508730

© I. Altun and A. Erduran. 2011

**Received:**12 November 2010**Accepted:**24 December 2010**Published:**28 December 2010

## Abstract

## Keywords

- Partial Order
- Point Theorem
- Differential Geometry
- Matrix Equation
- Monotone Mapping

## 1. Introduction

There are a lot of fixed and common fixed point results in different types of spaces. For example, metric spaces, fuzzy metric spaces, and uniform spaces. One of the most interesting is partial metric space, which is defined by Matthews [1]. In partial metric spaces, the distance of a point in the self may not be zero. After the definition of partial metric space, Matthews proved the partial metric version of Banach fixed point theorem. Then, Valero [2], Oltra and Valero [3], and Altun et al. [4] gave some generalizations of the result of Matthews. Again, Romaguera [5] proved the Caristi type fixed point theorem on this space.

First, we recall some definitions of partial metric spaces and some properties of theirs. See [1–3, 5–7] for details.

A partial metric on a nonempty set is a function such that for all

A partial metric space is a pair
such that
is a nonempty set and
is a partial metric on
. It is clear that if
, then from (p_{1}) and (p_{2})
. But if
,
may not be 0. A basic example of a partial metric space is the pair
, where
for all
. Other examples of partial metric spaces, which are interesting from a computational point of view, may be found in [1, 8].

Each partial metric on generates a topology on , which has as a base the family open -balls , where for all and .

Let be a partial metric space, then we have the following.

(i)A sequence in a partial metric space converges to a point if and only if .

Let be a partial metric space.

(a) is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space .

On the other hand, existence of fixed points in partially ordered sets has been considered recently in [9], and some generalizations of the result of [9] are given in [10–15] in a partial ordered metric spaces. Also, in [9], some applications to matrix equations are presented; in [14, 15], some applications to ordinary differential equations are given. Also, we can find some results on partial ordered fuzzy metric spaces and partial ordered uniform spaces in [16–18], respectively.

The aim of this paper is to combine the above ideas, that is, to give some fixed point theorems on ordered partial metric spaces.

## 2. Main Result

Theorem 2.1.

for all with , where is a continuous, nondecreasing function such that is convergent for each . If there exists an with , then there exists such that . Moreover, .

Proof.

which is a contradiction since . Thus, , and so .

In the following theorem, we remove the continuity of . Also, The contractive condition (2.1) does not have to be satisfied for , but we add a condition on .

Theorem 2.2.

holds. If there exists an with , then there exists such that . Moreover, .

Proof.

which is a contradiction. Thus, , and so .

Example 2.3.

This shows that the contractive condition of Theorem 1 of [4] is not satisfied.

Theorem 2.4.

Theorem 2.5.

Adding condition (2.31) to the hypotheses of Theorem 2.4, one obtains uniqueness of the fixed point of .

Proof.

- (i)

- (ii)

## Authors’ Affiliations

## References

- Matthews SG:
**Partial metric topology.**In*Proceedings of the 8th Summer Conference on General Topology and Applications, 1994*.*Volume 728*. Annals of the New York Academy of Sciences; 183–197.Google Scholar - Valero O:
**On Banach fixed point theorems for partial metric spaces.***Applied General Topology*2005,**6**(2):229–240.MATHMathSciNetView ArticleGoogle Scholar - Oltra S, Valero O:
**Banach's fixed point theorem for partial metric spaces.***Rendiconti dell'Istituto di Matematica dell'Università di Trieste*2004,**36**(1–2):17–26.MATHMathSciNetGoogle Scholar - Altun I, Sola F, Simsek H:
**Generalized contractions on partial metric spaces.***Topology and Its Applications*2010,**157**(18):2778–2785. 10.1016/j.topol.2010.08.017MATHMathSciNetView ArticleGoogle Scholar - Romaguera S:
**A Kirk type characterization of completeness for partial metric spaces.***Fixed Point Theory and Applications*2010,**2010:**-6.MathSciNetView ArticleMATHGoogle Scholar - Altun I, Simsek H:
**Some fixed point theorems on dualistic partial metric spaces.***Journal of Advanced Mathematical Studies*2008,**1**(1–2):1–8.MATHMathSciNetGoogle Scholar - Heckmann R:
**Approximation of metric spaces by partial metric spaces.***Applied Categorical Structures*1999,**7**(1–2):71–83.MATHMathSciNetView ArticleGoogle Scholar - Escardó MH:
**PCF extended with real numbers.***Theoretical Computer Science*1996,**162**(1):79–115. 10.1016/0304-3975(95)00250-2MATHMathSciNetView ArticleGoogle Scholar - Ran ACM, Reurings MCB:
**A fixed point theorem in partially ordered sets and some applications to matrix equations.***Proceedings of the American Mathematical Society*2004,**132**(5):1435–1443. 10.1090/S0002-9939-03-07220-4MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, El-Gebeily MA, O'Regan D:
**Generalized contractions in partially ordered metric spaces.***Applicable Analysis*2008,**87**(1):109–116. 10.1080/00036810701556151MATHMathSciNetView ArticleGoogle Scholar - Altun I, Simsek H:
**Some fixed point theorems on ordered metric spaces and application.***Fixed Point Theory and Applications*2010,**2010:**-17.Google Scholar - Beg I, Butt AR:
**Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(9):3699–3704. 10.1016/j.na.2009.02.027MATHMathSciNetView ArticleGoogle Scholar - Ciric L, Cakić N, Rajović M, Ume JS:
**Monotone generalized nonlinear contractions in partially ordered metric spaces.***Fixed Point Theory and Applications*2008,**2008:**11.View ArticleMathSciNetMATHGoogle Scholar - Harjani J, Sadarangani K:
**Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2010,**72**(3–4):1188–1197. 10.1016/j.na.2009.08.003MATHMathSciNetView ArticleGoogle Scholar - Nieto JJ, Rodríguez-López R:
**Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations.***Order*2005,**22**(3):223–239. 10.1007/s11083-005-9018-5MATHMathSciNetView ArticleGoogle Scholar - Altun I:
**Some fixed point theorems for single and multi valued mappings on ordered non-Archimedean fuzzy metric spaces.***Iranian Journal of Fuzzy Systems*2010,**7**(1):91–96.MATHMathSciNetGoogle Scholar - Altun I, Miheţ D:
**Ordered non-Archimedean fuzzy metric spaces and some fixed point results.***Fixed Point Theory and Applications*2010,**2010:**-11.Google Scholar - Altun I, Imdad M:
**Some fixed point theorems on ordered uniform spaces.***Filomat*2009,**23:**15–22. 10.2298/FIL0903015AMATHView ArticleMathSciNetGoogle Scholar

## Copyright

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