- 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

Matthews (1994) introduced a new distance on a nonempty set , which is called partial metric. If is a partial metric space, then may not be zero for . In the present paper, we give some fixed point results on these interesting spaces.

## 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

(p_{1})
,

(p_{2})
,

(p_{3})
,

(p_{4})
.

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 .

is a metric on .

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 .

- (iii)
A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

- (iv)
A mapping is said to be continuous at , if for every , there exists such that .

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.

for all with .

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)

and taking limit , we have . This contradicts .

Consequently, has no two fixed points.

## Authors’ Affiliations

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## Copyright

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