@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 12, no. 2, 2011

pp. 213-220

Matkowski’s type theorems for generalized
contractions on (ordered) partial metric spaces

Salvador Romaguera
∗

Abstract

We obtain extensions of Matkowski’s fixed point theorem for generalized

contractions of Ćirić’s type on 0-complete partial metric spaces and on

ordered 0-complete partial metric spaces, respectively.

2010 MSC: 54H25, 47H10, 54E50

Keywords: Matkowski’s fixed point theorem; Generalized contraction; 0-

complete partial metric space; Ordered partial metric space

1. Introduction and preliminaries

Partial metric spaces, and their equivalent weightable quasi-metric spaces,
were introduced by Matthews [14] in the context of his studies on denotational
semantics for dataflow networks. In fact, this class of spaces provides a suitable
framework to construct computational models for metric spaces and related
structures (see e.g. [10, 14, 21, 22, 24, 25, 26]). It also provides an appropriate
setting to discuss, with the help of techniques of fixed points of denotational
semantics, the complexity analysis of several algorithms which can be defined
by recurrence equations (see e.g. [9, 20, 23]).

These facts explain, in part, the recent extensive research on fixed points for
self maps in partial metric spaces ([1, 2, 3, 5, 8, 11, 13, 19], etc).

In this note we try to reach a new advance in this direction by obtaining
extensions of Matkowski’s fixed point theorem [15, Theorem 1.2] for generalized
contractions on 0-complete partial metric space and on ordered 0-complete
partial metric spaces, respectively. These results extend, generalize and unify
some theorems from the current literature.

∗The author thanks the support of the Ministry of Science and Innovation of Spain, under
grant MTM2009-12872-C02-01



214 S. Romaguera

Throughout this paper the letter ω will denote the set of all nonnegative
integer numbers.

Let us recall [14] that a partial metric on a set X is a function p : X × X →
[0,∞) such that for all x,y,z ∈ X : (i) x = y ⇔ p(x,x) = p(x,y) = p(y,y); (ii)
p(x,x) ≤ p(x,y); (iii) p(x,y) = p(y,x); (iv) p(x,z) ≤ p(x,y) + p(y,z) − p(y,y).

A partial metric space is a pair (X,p) where p is a partial metric on X.
Each partial metric p on X induces a T0 topology τp on X which has as a

base the family of open 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.

Matthews observed in [14, p. 187] that a sequence (xn)n∈ω in a partial
metric space (X,p) converges to some x ∈ X with respect to τp if and only if
limn→∞ p(x,xn) = p(x,x).

Next we recall some useful concepts and facts on completeness of partial
metric spaces.

If p is a partial metric on X, then the function ps : X × X → [0,∞) given
by ps(x,y) = 2p(x,y) − p(x,x) − p(y,y), is a metric on X.

Furthermore, a sequence (xn)n∈ω in X converges to some x ∈ X with respect
to τps if and only if limn,m→∞ p(xn,xm) = limn→∞ p(x,xn) = p(x,x).

A sequence (xn)n∈ω in a partial metric space (X,p) is called a Cauchy se-
quence if there exists (and is finite) limn,m p(xn,xm) [14, Definition 5.2].

A partial metric space (X,p) is said to be complete if every Cauchy sequence
(xn)n∈ω in X converges, with respect to τp, to a point x ∈ X such that p(x,x) =
limn,m p(xn,xm) [14, Definition 5.3].

It is well known (see, for instance, [14, p. 194]) that a sequence in a partial
metric space (X,p) is a Cauchy sequence in (X,p) if and only if it is a Cauchy
sequence in the metric space (X,ps), and that a partial metric space (X,p) is
complete if and only the metric space (X,ps) is complete.

Romaguera introduced in [18] the notions of a 0-Cauchy sequence in a partial
metric space and of a 0-complete partial metric space.

A sequence (xn)n∈ω in a partial metric space (X,p) is called 0-Cauchy if
limn,m p(xn,xm) = 0.

We say that a partial metric space (X,p) is 0-complete if every 0-Cauchy
sequence in X converges, with respect to τp, to a point x ∈ X such that
p(x,x) = 0. In this case, p is said to be a 0-complete partial metric on X.

Note that every 0-Cauchy sequence in (X,p) is a Cauchy sequence in (X,p),
and that every complete partial metric space is 0-complete.

On the other hand, the partial metric space (Q ∩ [0,∞),p), where Q denotes
the set of rational numbers and the partial metric p is given by p(x,y) =
max{x,y}, provides an easy example of a 0-complete partial metric space which
is not complete.



generalized contractions on (ordered) partial metric spaces 215

2. Fixed points for 0-complete partial metric spaces

Given a partial metric space (X,p) and f : X → X a map, we define

M(x,y) := max

{

p(x,y),p(x,fx),p(y,fy),
1

2
[p(x,fy) + p(y,fx)]

}

,

for all x,y ∈ X (compare e.g. [5, 19]).

The proof of [19, Lemma 2] shows the following.

Lemma 2.1. Let (X,p) be a partial metric space, f : X → X a map and
x0 ∈ X such that f

nx0 6= f
n+1x0 and

p(fn+1x0,f
n+2x0) ≤ ϕ(M(f

nx0,f
n+1x0)),

for all n ∈ ω, where ϕ : [0,∞) → [0,∞) satisfies ϕ(t) < t for all t > 0. Then
the following hold:

(a) M(fnx0,f
n+1x0) = p(f

nx0,f
n+1x0 ) for all n ∈ ω.

(b) p(fn+1x0,f
n+2x0) ≤ ϕ(p(f

nx0,f
n+1x0)) < p(f

nx0,f
n+1x0) for all

n ∈ ω.

Remark 2.2. Recall [15, 16] that if ϕ : [0,∞) → [0,∞) is a nondecreasing
function such that limn→∞ ϕ

n(t) = 0 for all t > 0, then ϕ(t) < t for all t > 0
and thus ϕ(0) = 0.

Now we prove the main result of this section.

Theorem 2.3. Let (X,p) be a 0-complete partial metric space and f : X → X
a map such that

p(fx,fy) ≤ ϕ(M(x,y)),

for all x,y ∈ X, where ϕ : [0,∞) → [0,∞) is a nondecreasing function such
that limn→∞ ϕ

n(t) = 0 for all t > 0. Then f has a unique fixed point z ∈ X.
Moreover p(z,z) = 0.

Proof. Let x0 ∈ X. For each n ∈ ω put xn = f
nx0. Thus xn+1 = fxn for all

n ∈ ω.
If there is k ∈ ω such that xk = xk+1, then xk is a fixed point of f. Moreover

if fz = z for some z ∈ X, it follows that

p(z,xk) = p(fz,fxk) ≤ ϕ(M(z,xk)) = ϕ(p(z,xk)),

so, p(xk,z) = 0, i.e., z = xk. So xk is the unique fixed point of f, and, clearly,
p(xk,xk) = 0 by the contraction condition.

Hence, we shall assume that fnx0 6= f
n+1x0 for all n ∈ ω. Thus p(xn,xn+1) >

0 for all n ∈ ω. By Lemma 2.1 (b), p(xn,xn+1) ≤ ϕ(p(xn−1,xn)) for all n ∈ N,
and then

p(xn,xn+1) ≤ ϕ
n(p(x0,x1)),

for all n ∈ ω. So

lim
n→∞

p(xn,xn+1) = 0.



216 S. Romaguera

Now choose an arbitrary ε > 0. Since, by Remark 2.2, ϕ(ε) < ε, then there
is nε ∈ N such that

p(xn,xn+1) < ε − ϕ(ε),

for all n ≥ nε. Therefore

p(xn,xn+2) ≤ p(xn,xn+1) + p(xn+1,xn+2)

< ε − ϕ(ε) + ϕ(p(xn,xn+1))

≤ ε − ϕ(ε) + ϕ(ε) = ε,

for all n ≥ nε. So

p(xn,xn+3) ≤ p(xn,xn+1) + p(xn+1,xn+3)

< ε − ϕ(ε) + ϕ(M(xn,xn+2)),

for all n ≥ nε.

Now suppose that there is n ≥ nε such that M(xn,xn+2) > ε. Then, from

M(xn,xn+2) = max{p(xn,xn+2),p(xn,xn+1),p(xn+1,xn+2),

1

2
[p(xn,xn+3) + p(xn+1,xn+2)}

≤ max{ε,
1

2
[p(xn,xn+3) + ϕ(ε)]}.

it follows that

M(xn,xn+2) ≤
1
2
[p(xn,xn+3) + ϕ(ε)],

so

p(xn,xn+3) < ε − ϕ(ε) + ϕ(M(xn,xn+2))

< ε − ϕ(ε) + M(xn,xn+2)

≤ ε − ϕ(ε) +
1

2
[p(xn,xn+3) + ϕ(ε)].

We deduce that M(xn,xn+2) < ε, a contradiction.

Therefore p(xn,xn+3) < ε, and following this process,

p(xn,xn+k) < ε,

for all n ≥ nε and k ∈ N. Consequently

lim
n,m→∞

p(xn,xm) = 0,

and thus (xn)n∈ω is a 0-Cauchy sequence in the 0-complete partial metric space
(X,p). Hence there is z ∈ X such that

lim
n,m→∞

p(xn,xm) = lim
n→∞

p(z,xn) = p(z,z) = 0.

Finally, the fact that z is the unique fixed point of f follows similarly to the
last part of the proof of [19, Theorem 4]. �



generalized contractions on (ordered) partial metric spaces 217

In a recent paper [12], Jachymski showed the equivalence between several
generalized contractions on (ordered) metric spaces. Since the key of his study
is Lemma 1 of the cited paper, then Jachymski’s approach also holds in the
partial metric framework. As an instance, we shall combine this lemma with
Theorem 2.3 above to deduce the following (compare [1, Corollary 2.1]).

Corollary 2.4. Let (X,p) be a 0-complete partial metric space and f : X → X
a map such that

ψ(p(fx,fy)) ≤ ψ(M(x,y)) − φ(M(x,y)),

for all x,y ∈ X, where ψ,φ : [0,∞) → [0,∞) are nondecreasing functions such
that ψ is continuous on [0,∞) and ψ−1(0) = φ−1(0) = {0}. Then f has a
unique fixed point z ∈ X. Moreover p(z,z) = 0.

Proof. By [12, Lemma 1 (ii)⇒(viii)], there exists a continuous and nonde-
creasing function ϕ : [0,∞) → [0,∞) such that ϕ(t) < t for all t > 0, and
p(fx,fy) ≤ ϕ(M(x,y)) for all x,y ∈ X. From continuity of ϕ it follows that
limn→∞ ϕ

n(t) = 0 for all t > 0. Theorem 2.3 concludes the proof. �

Remark 2.5. Theorem 2.3 and Corollary 2.4 extend several fixed point theo-
rems for complete metric spaces due to Dutta and Choudhury, Khan, Swaleh
and Sessa, and Rhoades, among others (see [12, Theorems 1 and 3, and the
bibliography]). Theorem 2.3 also improves [11, Theorem 3.2], [5, Theorem 1]
and [19, Theorem 4].

3. Fixed points for ordered 0-complete partial metric spaces

Our main purpose in this section is to prove an ordered counterpart of The-
orem 1. In this way, we shall extend the main result of Agarwal, El-Gebeily
and O’Regan in [6] (see also [17, Theorem 3.11]).

By an ordered (0-complete) partial metric space we mean a triple (X,�,p)
such that � is a partial order on X and p is a (0-complete) partial metric on
X.

An ordered partial metric space (X,�,p) is called regular if for any nonde-
creasing sequence (xn)n∈ω for �, which converges to some z ∈ X with respect
to τp, it follows xn � z for all n ∈ ω.

We say that a self map f of a partial metric space (X,p) is continuous if it
is continuous from (X,τp) into itself.

Theorem 3.1. Let (X,�,p) be an ordered 0-complete partial metric space and
f : X → X a nondecreasing map for �, such that

p(fx,fy) ≤ ϕ(M(x,y)),

for all x,y ∈ X with x � y, where ϕ : [0,∞) → [0,∞) is a nondecreasing
function such that limn→∞ ϕ

n(t) = 0 for all t > 0.
If there is x0 ∈ X such that x0 � fx0, and f is continuous or (X,�,p) is

regular, then f has a fixed point z ∈ X such that p(z,z) = 0. Moreover, the set
of fixed points of f is a singleton if and only it is well-ordered.



218 S. Romaguera

Proof. For each n ∈ ω put xn = f
nx0. Thus xn+1 = fxn for all n ∈ ω. Since

x0 � fx0 and f is nondecreasing for �, it follows that xn � xn+1 for all n ∈ ω,
so (xn)n∈ω is a nondecreasing sequence in (X,�).

If there is k ∈ ω such that xk = xk+1, then xk is a fixed point of f.

Hence, we shall assume that fnx0 6= f
n+1x0 for all n ∈ ω. Thus p(xn,xn+1) >

0 for all n ∈ ω. Then, the proof of Theorem 2.3 shows (note, in particular, that
xn � xn+k for all n,k ∈ ω) that (xn)n∈ω is a 0-Cauchy sequence in (X,p).
Hence there is z ∈ X such that

lim
n,m→∞

p(xn,xm) = lim
n→∞

p(z,xn) = p(z,z) = 0.

We show that z is a fixed point of f.
Indeed, if f is continuous, we deduce that limn→∞ p(fz,xn) = p(fz,fz).

Since

p(z,fz) ≤ p(z,xn) + p(xn,fz),

for all n ∈ ω, it follows, taking limits as n → ∞, that p(z,fz) ≤ p(fz,fz), so
p(z,fz) = p(fz,fz). Hence, since z � z, we have

p(z,fz) ≤ ϕ(M(z,z)) = ϕ(0) = 0,

and thus z = fz, and p(z,z) = 0.
If f is not continuous, it follows from regularity of (X,�,p) that xn � z for

all n ∈ ω. Assume p(z,fz) > 0. Then, there is n0 ∈ N such that M(z,xn−1) =
p(z,fz) for all n ≥ n0. Thus

p(z,fz) ≤ p(z,xn) + p(xn,fz) ≤ p(z,xn) + ϕ(M(z,xn−1))

= p(z,xn) + ϕ(p(z,xn−1)) ≤ p(z,xn) + p(z,xn−1),

for all n ≥ n0. Taking limits as n → ∞, we deduce that p(z,fz) = 0, a
contradiction. We conclude that z = fz.

Finally, if the set of fixed point is well-ordered and u ∈ X is a fixed point of
f,we deduce, assuming u � z, that

p(u,z) = p(fu,fz) ≤ ϕ(M(u,z)) = ϕ(p(u,z)),

so p(u,z) = 0, i.e., u = z. This concludes the proof. �

From [12, Lemma 1] and Theorem 3.1 we deduce the following ordered coun-
terpart of Corollary 2.4.

Corollary 3.2. Let (X,�,p) be an ordered 0-complete partial metric space and
f : X → X a nondecreasing map for �, map such that

ψ(p(fx,fy)) ≤ ψ(M(x,y)) − φ(M(x,y)),

for all x,y ∈ X with x � y, where ψ,φ : [0,∞) → [0,∞) are nondecreasing
functions such that ψ is continuous on [0,∞) and ψ−1(0) = φ−1(0) = {0}.

If there is x0 ∈ X such that x0 � fx0, and f is continuous or (X,�,p) is
regular, then f has a fixed point z ∈ X such that p(z,z) = 0. Moreover, the set
of fixed points of f is a singleton if and only it is well-ordered.



generalized contractions on (ordered) partial metric spaces 219

Remark 3.3. Theorem 3.1 improves [4, Theorems 2.1 and 2.2]. Note also that
regularity of (X,�,p) can be replaced, in Theorem 3.1, by the more general
condition that for any nondecreasing sequence (xn)n∈ω for �, which converges
to some z ∈ X with respect to τps, it follows xn � z for all n ∈ ω. Moreover,
continuity of f from (X,τp) into itself can be replaced by continuity from
(X,τps) into itself.

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(Received September 2011 – Accepted November 2011)

S. Romaguera (sromague@mat.upv.es)
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica
de València, Camı́ de Vera s/n, 46022 Valencia, Spain


	Matkowski's type theorems for generalized contractions on (ordered) partial metric spaces. By S. Romaguera