Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 508730, 10 pages
doi:10.1155/2011/508730
Research Article
Fixed Point Theorems for Monotone Mappings on
Partial Metric Spaces
Ishak Altun and Ali Erduran
Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan,
Kirikkale, Turkey
Correspondence should be addressed to Ishak Altun, [email protected]
Received 12 November 2010; Accepted 24 December 2010
Academic Editor: S. Al-Homidan
Copyright q 2011 I. Altun and A. Erduran. 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.
Matthews 1994 introduced a new distance p on a nonempty set X, which is called partial metric.
If X, p is a partial metric space, then px, x may not be zero for x ∈ X. In the present paper, we
give some fixed point results on these interesting spaces.
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,andAltunetal.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 X is a function p : X × X →
Each partial metric p on X generates a T
0
topology τ
p
on X, which has as a base the
family open p-balls {B
p
x, ε : x ∈ X, ε > 0},whereB
p
x, ε{y ∈ X : px, y <px, xε}
for all x ∈ X and ε>0.
If p is a partial metric on X, then the function p
s
: X × X →
given by
p
s
x, y
2p
x, y
− p
x, x
− p
lim
n,m →∞
px
n
,x
m
.
iv A mapping F : X → X is said to be continuous at x
0
∈ X, if for every ε>0, there
exists δ>0suchthatFB
p
x
0
,δ ⊆ B
p
Fx
0
,ε.
Lemma 1.1 see 1, 3. Let X, p be a partial metric space.
a {x
n
} is a Cauchy sequence in X, p if and only if it is a Cauchy sequence in the metric
space X, p
s
.
b A partial metric space X, p is complete if and only if the metric space X, p
s
is complete.
Furthermore, lim
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.
Fixed Point Theory and Applications 3
2. Main Result
Theorem 2.1. Let X, be partially ordered set, and suppose that there is a p artial metric p on
X such that X, p is a complete partial metric space. Suppose F : X → X is a continuous and
nondecreasing mapping such that
p
Fx,Fy
≤ Ψ
max
p
x, y
,p
x, Fx
,p
exists x ∈ X such that x Fx.Moreover,px, x0.
Proof. From the conditions on Ψ, it is clear that lim
n →∞
Ψ
n
t0fort>0andΨt <t.If
Fx
0
x
0
, then the proof is finished, so suppose x
0
/
Fx
0
.Now,letx
n
Fx
n−1
for n 1, 2,
If x
n
0
x
n
0
1
for some n
0
∈ , then it is clear that x
x
n1
,x
n
p
Fx
n
,Fx
n−1
≤ Ψ
max
p
x
n
,x
n−1
,p
x
n
,Fx
n
max
p
x
n
,x
n−1
,p
x
n
,x
n1
,
1
2
p
x
n−1
,x
n
p
x
n
,x
n
p
x
n−1
,x
n1
≤ p
x
n−1
,x
n
p
x
n
,x
n1
2.4
and Ψ is nondecreasing. Now, if
max
p
≤ Ψ
p
x
n
,x
n1
<p
x
n
,x
n1
, 2.6
4 Fixed Point Theory and Applications
which is a contradiction since px
n
,x
n1
> 0. Thus
max
p
x
n
x
n
,x
n−1
, 2.8
and so
p
x
n1
,x
n
≤ Ψ
n
p
x
1
,x
0
. 2.9
On the other hand, since
max
n
,x
n
,p
x
n1
,x
n1
≤ Ψ
n
p
x
1
,x
0
. 2.11
Therefore,
p
s
x
n
p
x
n
,x
n
p
x
n1
,x
n1
≤ 4Ψ
n
p
x
1
,x
0
.
2.12
This shows that lim
n →∞
≤ 4Ψ
nk−1
p
x
1
,x
0
··· 4Ψ
n
p
x
1
,x
0
.
2.13
Since
∞
n1
Ψ
lim
n,m →∞
p
x
n
,x
m
.
2.14
Moreover, since {x
n
} is a Cauchy sequence in the metric space X, p
s
,wehave
lim
n,m →∞
p
s
x
n
,x
m
0, and, from 2.11, we have lim
n →∞
px
n
,x
n
0.
2.15
Fixed Point Theory and Applications 5
Now, we claim that Fx x. Suppose px, Fx > 0. Since F is continuous, then, given ε>0,
there exists δ>0suchthatFB
p
x, δ ⊆ B
p
Fx,ε.Sincepx, xlim
n →∞
px
n
,x0, then
there exists k ∈
such that px
n
,x <px, xδ for all n ≥ k. Therefore, we have x
n
∈ B
p
x, δ
for all n ≥ k.Thus,Fx
n
∈ FB
p
x, δ ⊆ B
p
Fx,ε,andsopFx
n
. 2.16
Therefore, we obtain
p
x, Fx
≤ p
x, x
n1
p
x
n1
,Fx
− p
x
n1
,x
n1
≤ p
x, x
n1
Fx,Fx
≤ Ψ
p
x, Fx
<p
x, Fx
,
2.18
which is a contradiction since px, Fx > 0. Thus, px, Fx0, and so x Fx.
In the following theorem, we remove the continuity of F. Also, The contractive
condition 2.1 does not have to be satisfied for x y, but we add a condition on X.
Theorem 2.2. Let X, be a partially ordered set, and suppose that there is a partial metric p on X
such that X, p is a complete partial metric space. Suppose F : X → X is a nondecreasing mapping
such that
p
Fx,Fy
≤ Ψ
max
p
∞
n1
Ψ
n
t is convergent for each t>0. Also, the condition
If
{
x
n
}
⊂ X is a increasing sequence with x
n
−→ x in X, then x
n
≺ x, ∀n 2.20
holds. If there exists an x
0
∈ X with x
0
Fx
0
, then there exists x ∈ X such that x Fx.Moreover,
px, x0.
6 Fixed Point Theory and Applications
Proof. As in the proof of Theorem 2 .1, we can construct a sequence {x
n
} in X by x
n
Fx
n−1
x
n
,x
lim
n,m →∞
p
x
n
,x
m
0.
2.22
Now, we claim that Fx x. Suppose px, Fx > 0. Since the condition 2.20 is satisfied, then
we can use 2.19 for y x
n
. Therefore, we obtain
p
Fx,Fx
n
≤ Ψ
max
p
≤ Ψ
max
p
x, x
n
,p
x, Fx
,p
x
n
,x
n1
,
1
2
p
x, x
n1
x
n
,x
n1
,
1
2
p
x, x
n1
p
x
n
,x
p
x, Fx
,
2.23
using the continuity of Ψ and letting n →∞, we have lim
lim
n →∞
p
Fx
n
,Fx
≤ Ψ
p
x, Fx
<p
x, Fx
,
2.24
which is a contradiction. Thus, px, Fx0, and so x Fx.
Example 2.3. Let X 0, ∞ and px, ymax{x, y}, then it is clear that X, p is a complete
partial metric space. We can define a partial order on X as follows:
x y ⇐⇒ x y or
x, y ∈
0, 1
n
t ≤ tt/1 t
n
, and so we have
∞
n1
Ψ
n
t
that is convergent. Also, F is nondecreasing with respect to ,andfory ≺ x,we
have
p
Fx,Fy
max
x
2
1 x
,
y
2
1 y
x
2
1 x
p
y, Fx
,
2.27
that is, the condition 2.19 of Theorem 2.2 is satisfied. Also, it is clear t hat the condition
2.20 is satisfied, and for x
0
0, we have x
0
Fx
0
. Therefore, all conditions of
Theorem 2.2 are satisfied, and so F has a fixed point in X.Notethatifx 1andy 2,
then
p
Fx,Fy
4
/
≤
16
5
Ψ
max
p
Fx,Fy
≤ Ψ
max
p
x, y
,
1
2
p
x, Fx
p
y, Fy
,
1
2
p
F
n
z, F
n
y
≤ Ψ
max
p
F
n−1
z, F
n−1
y
,
1
2
p
F
n−1
z, F
n
z
Ψ
max
p
z, y
,
1
2
p
z, z
p
y, y
Ψ
p
z, y
<p
z, y
p
F
n−1
z, F
n−1
x
,
1
2
p
F
n−1
z, F
n
z
p
F
n−1
x, F
n
x
,
1
1
2
p
z, z
p
F
n−1
x, F
n
x
,
1
2
p
z, F
n
x
p
F
n−1
x, z
1
2
p
z, F
n
x
p
F
n−1
x, z
Ψ
max
p
z, F
n−1
x
,
1
2
p
z, F
n
x
<p
z, F
n
x
, 2.34
which is a contradiction. Thus, pz, F
n−1
x ≥ pz, F
n
x for all n,andso
p
z, F
n
x
≤ Ψ
p
z, F
z, F
n
x
p
F
n
x, y
− p
F
n
x, F
n
x
≤ p
z, F
n
x
p
F
n
x, y
11 I. Altun and H. Simsek, “Some fixed point theorems on ordered metric spaces and application,” Fixed
Point Theory and Applications, vol. 2010, Article ID 621469, 17 pages, 2010.
12 I. Beg and A. R. Butt, “Fixed point for set-valued mappings satisfying an implicit relation in partially
ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 3699–
3704, 2009.
10 Fixed Point Theory and Applications
13 L. Ciric, N. Caki
´
c, M. Rajovi
´
c, and J. S . Ume, “Monotone generalized nonlinear c ontractions in
partially ordered metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 131294, p.
11, 2008.
14 J. Harjani and K. Sadarangani, “Generalized co ntractions in partially ordered metric spaces and
applications to ordinary differential equations,” Nonlinear Analysis: Theory, Methods & Applications,
vol. 72, no. 3-4, pp. 1188–1197, 2010.
15 J. J. Nieto and R. Rodr
´
ıguez-L
´
opez, “Contractive mapping theorems in partially ordered sets and
applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005.
16 I. Altun, “Some fixed point theorems for single a nd multi valued mappings on ordered non-
Archimedean fuzzy metric spaces,” Iranian Journal of Fuzzy Systems, vol. 7, no. 1, pp. 91–96, 2010.
17 I. Altun and D. Mihet¸, “Ordered non-Archimedean fuzzy metric spaces and some fixed p oint results,”
Fixed Point Theory and Applications, vol. 2010, Article ID 782680, 11 pages, 2010.
18 I. Altun and M. Imdad, “Some fixed point theorems on ordered uniform spaces,” Filomat, vol. 23, pp.
15–22, 2009.
Copyright: Tài liệu đại học ©