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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 12, no. 2, 2011

pp. 187-192

Some fixed point theorems on the class of
comparable partial metric spaces

Erdal Karapinar

Abstract

In this paper we present existence and uniqueness criteria of a fixed

point for a self mapping on a non-empty set endowed with two compa-

rable partial metrics.

2010 MSC: 46N40,47H10,54H25,46T99.

Keywords: partial metric space, fixed point theory, comparable metrics.

1. Introduction and Preliminaries

In 1992, Matthews [10, 11] introduced the notion of a partial metric space
which is a generalization of usual metric spaces in which d(x, x) are no longer
necessarily zero. After this remarkable contribution, many authors focused on
partial metric spaces and its topological properties (see e.g. [15, 16, 2, 1, 3,
4, 5, 6]). Partial metric spaces have extensive potential applications in the
research area of computer domains and semantics (see e.g. [7, 12, 8, 13, 14]).
Consequently, the attention paid to such spaces rapidly increases.

A partial metric space (see e.g.[10, 11]) is a pair (X, p) such that X is non-
empty set and p : X × X → R+ (where R+ denotes the set of all non negative
real numbers) satisfies:

(PM1) p(x, y) = p(y, x) (symmetry)
(PM2) If p(x, x) = p(x, y) = p(y, y) then x = y (equality)
(PM3) p(x, x) ≤ p(x, y) (small self-distances)
(PM4) p(x, z) + p(y, y) ≤ p(x, y) + p(y, z) (triangle inequality)

for all x, y, z ∈ X. We use the abbreviation PMS for the partial metric space
(X, p).



188 E. Karapinar

Notice that for a partial metric p on X, the function dp : X ×X → R
+ given

by

(1.1) dp(x, y) = 2p(x, y) − p(x, x) − p(y, y)

is a (usual) metric on X. Observe that each partial metric p on X generates a T0
topology τp on X with a base the family open p-balls {Bp(x, ε) : x ∈ X, ε > 0},
where Bp(x, ε) = {y ∈ X : p(x, y) < p(x, x) + ε} for all x ∈ X and ε > 0.
Similarly, closed p-ball is defined as Bp[x, ε] = {y ∈ X : p(x, y) ≤ p(x, x) + ε}

Definition 1.1 (see e.g.[10, 11, 1]).

(i) A sequence {xn} in a PMS (X, p) converges to x ∈ X if p(x, x) =
limn→∞ p(x, xn),

(ii) a sequence {xn} in a PMS (X, p) is called a Cauchy sequence if
limn,m→∞ p(xn, xm) exists (and finite),

(iii) A PMS (X, p) is said to be complete if every Cauchy sequence {xn} in
X converges, with respect to τp, to a point x ∈ X such that p(x, x) =
limn,m→∞ p(xn, xm).

Lemma 1.2 (see e.g.[10, 11, 1] ).

(A) A sequence {xn} in a PMS (X, p) is Cauchy if and only if {xn} is
Cauchy in a metric space (X, dp),

(B) A PMS (X, p) is complete if and only if a metric space (X, dp) is com-
plete. Moreover,

(1.2) lim
n→∞

dp(x, xn) = 0 ⇔ p(x, x) = lim
n→∞

p(x, xn) = lim
n,m→∞

p(xn, xm)

In this manuscript, we present some new fixed point theorems on a non-
empty set on which there exists two partial metrics with certain conditions.

2. Main Results

The following two lemmas will be used in the proof of the main theorem.

Lemma 2.1 (see e.g. [3]). Let (X, p) be a complete PMS. Then

(A) If p(x, y) = 0 then x = y,
(B) If x 6= y, then p(x, y) > 0.

Lemma 2.2 (see e.g. [1, 3]). Assume xn → z as n → ∞ in a PMS (X, p) such
that p(z, z) = 0. Then limn→∞ p(xn, y) = p(z, y) for every y ∈ X.

The following theorem is an extension of the result of Maia [9].

Theorem 2.3. Let X be a non-empty set endowed with two partial metrics
p1, p2, and let T be a mapping of X into itself. Suppose that

(i) (X, p1) is complete,
(ii) p1(x, y) ≤ p2(x, y) for all x, y ∈ X,
(iii) T is continuous with respect to τp1,
(iv) T is a contraction with respect to p2, that is, p2(T x, T y) ≤ kp2(x, y)

for all x, y ∈ X, where 0 ≤ k < 1.



Some fixed point theorems on the class of comparable partial metric spaces 189

Then T has a unique fixed point in X.

Proof. Fix x ∈ X. We construct a sequence {xn} in the following way:

(S1) x0 = x,
(S2) xn = T xn−1 = T

nx0 for each n ∈ N.

Then, by assumption (iv) we have

p2(xn+1, xn) = p2(T xn, T xn−1) ≤ kp2(xn, xn−1) ≤ · · · ≤ k
n
p2(T x0, x0).

Hence, by standard calculations, we get that limn,m→∞ p2(xn, xm) = 0, and
by assumption (ii), limn,m→∞ p1(xn, xm) = 0, i.e., {xn} is a Cauchy sequence
in (X, p1). So, by assumption (i) and Lemma 1.2, it converges in (X, dp1) to a
point z ∈ X. Again by Lemma 1.2,

(2.1) p1(z, z) = lim
n→∞

p1(xn, z) = lim
n,m→∞

p1(xn, xm)

Since limn,m→∞ p1(xn, xm) = 0, then by (2.1) we have p1(z, z) = 0.
By the continuity of T and also Lemma 2.2, one can get

p1(z, z) = limn→∞ p1(z, xn+1) = limn→∞ p(z, T
n+1x0)

= p1(z, T (limn→∞ T
nx0)) = p1(z, T (limn→∞ xn)) = p1(z, T z).

Hence P(T z, z) = p(z, z) = 0. Due to Lemma 2.1 the point z is a unique fixed
point of T . Suppose not, that is, there exist z, y ∈ X such that T z = z and
T y = y. Then, p2(z, y) = p2(T z, T y) ≤ kp2(z, y). Thus, p2(z, y) = 0.Regarding
Lemma 2.1, z = y. �

Theorem 2.4. Let (X, p1) be a PMS and T : X → X a mapping. Consider
the series:

(2.2)

∞
∑

n=0

tnp1(T
nx, T ny)

Suppose that for some t > 1, the series (2.2) converges for every x, y ∈ X.
Then, for such a point t, the function p2 : X × X → R

+ defined by

p2(x, y) =

∞
∑

n=0

tnp1(T
nx, T ny)

is a partial metric on X, moreover,

(i) p2 is an upper bound partial metric for p1,
(ii) T is a contraction with respect to p2.

Proof. Since t > 1 and p1(T
nx, T ny) ≥ 0 for all x, y ∈ X and n ∈ N, then

p2(x, y) ≥ 0. It is clear that p2 satisfies (PM1). For the proof of (PM2),
assume p2(x, x) = p2(x, y) = p2(x, y) which is equivalent to

∞
∑

n=0

tnp1(T
nx, T nx) =

∞
∑

n=0

tnp1(T
nx, T ny) =

∞
∑

n=0

tnp1(T
ny, T ny)



190 E. Karapinar

Hence
∞
∑

n=0

tn (p1(T
ny, T nx) − p1(T

nx, T nx)) =

∞
∑

n=0

tn (p1(T
ny, T nx) − p1(T

ny, T ny)) = 0,

so p1(T
ny, T nx) = p1(T

ny, T ny) = p1(T
nx, T nx) for all n ∈ N ∪ {0}. In

particular, p1(x, y) = p1(x, x) = p1(y, y), and hence, x = y. Moreover, (PM3)
and (PM4) are obtained by definition.

Let us prove (i) and (ii).

p2(x, y) =
∑

∞

n=0
tnp1(T

nx, T ny) = p1(x, y) +
∑

∞

n=1
tnp1(T

nx, T ny)
= p1(x, y) + t

(
∑

∞

n=0
tnp1(T

n+1x, T n+1y)
)

= p1(x, y) + tp2(T x, T y)

Thus,

p2(T x, T y) =
1

t
(p2(x, y) − p1(x, y)) ≤

1

t
p2(x, y).

�

Theorem 2.5. Suppose (X, p1) is a PMS and T : X → X is a mapping such
that p1(T

mx, T my) ≤ kp1(x, y) for some m ∈ N, where 0 ≤ k < 1. Then the
series p2(x, y) =

∑

∞

n=0
tnp1(T

nx, T ny) converges for t > 1, whatever the points
x, y ∈ X.

Proof. By assumption,

p1(T
mx, T my) ≤ kp1(x, y)

for some m ∈ N, and 0 ≤ k < 1. It yields that

p1(T
mnx, T mny) ≤ knp1(x, y)

for every n integer. Then,

p2(x, y) =
∑

∞

n=0
tnp1(T

nx, T ny)
=

∑

∞

n=0
tmnp1(T

nx, T ny) +
∑

∞

n=0
tmn+1p1(T

mn+1x, T mn+1y)
+ · · · +

∑

∞

n=0
tmn+n−1p1(T

mn+n−1x, T mn+n−1y)
≤

∑

∞

n=0
tmnknp1(x, y) + t

∑

∞

n=0
tmnknp1(T x, T y)

+ · · · + tn−1
∑

∞

n=0
tmnknp1(T

n−1x, T n−1y)

Just then take t such that: 1 < tn < 1
k
, because the series converges regard-

less of the points x, y ∈ X. �

Theorem 2.6. Let X be a non-empty set endowed with two partial metrics
p1, p2, and let T be a mapping of X into itself. Suppose that

(i) There exists a point x0 ∈ X such that the sequence of iterates {T
n(x0)}

has a subsequence {T ni(x0)} converging to a point z ∈ X for τp1,
(ii) p1(x, y) ≤ p2(x, y) for all x, y ∈ X,
(iii) T is continuous at z with respect to p1,
(iv) T is contraction with respect to p2, that is, p2(T x, T y) ≤ kp2(x, y) for

all x, y ∈ X, where 0 ≤ k < 1.

Then T has a unique fixed point in X.



Some fixed point theorems on the class of comparable partial metric spaces 191

Proof. Fix x0 ∈ X and define xn+1 = T xn for n ∈ N ∪ {0}. As it shown in
the proof of Theorem 2.3, this sequence {xn} is Cauchy with respect to p2.
By (ii), the sequence {xn} is also Cauchy with respect to p1. By (i), Cauchy
sequence {xn} has a subsequence {xni} which converges z ∈ X for τp1. Thus,
{xn} converges to z for τp1. By the continuity of T and also Lemma 2.2 one
can get

p1(z, z) = limn→∞ p1(z, xn+1) = limn→∞ p(z, T
n+1x0)

= p1(z, T (limn→∞ T
nx0)) = p1(z, T (limn→∞ xn)) = p1(z, T z).

Hence P(T z, z) = p(z, z) = 0. Due to Lemma 2.1 the point z is a unique fixed
point of T . To show uniqueness, assume the contrary. Let z and w be two
different fixed points. Then, by (iv),

p2(z, w) = p2(T z, T w) ≤ kp2(z, w)

Since 0 ≤ k < 1, one can get a contradiction. Thus, T has a unique fixed
point. �

Remark 2.7. Consider the following condition:

(iv)∗ There is a point x0 ∈ X such that the iterated sequence {T
n(x0)} is a

Cauchy sequence with respect to p2.

If the condition (iv) is replaced by (iv)∗ in Theorem 2.6, the theorem will still
guarantee the existence of fixed point.

Acknowledgements. The author express his gratitude to the referee for
constructive and useful remarks and suggestions.

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(Received March 2011 – Accepted July 2011)

Erdal Karapinar (erdalkarapinar@yahoo.com, ekarapinar@atilim.edu.tr)

Department of Mathematics, Atılım University, 06836, İncek, Ankara, Turkey


	Some fixed point theorems on the class of comparable partial metric spaces. By E. Karapinar