@
Appl. Gen. Topol. 17, no. 2(2016), 199-209

doi:10.4995/agt.2016.5920

c© AGT, UPV, 2016

Some fixed point results for dualistic rational

contractions

Muhammad Nazam a, Muhammad Arshad a and Mujahid Abbas b

a Department of Mathematics and Statistics, International Islamic University, Islamabad Pak-

istan. (nazim.phdma47@iiu.edu.pk, marshadzia@iiu.edu.pk)
b Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road,

Pretoria 0002, South Africa. (abbas.mujahid@gmail.com)

Abstract

In this paper, we introduce a new contraction called dualistic contrac-
tion of rational type and used it to obtain some fixed point results in
ordered dualistic partial metric spaces. These results generalize various
comparable results appeared in the literature. We provide an example
to show the usefulness of our results among corresponding fixed point
results proved in metric spaces.

2010 MSC: 47H09; 47H10; 54H25.

Keywords: fixed point; dualistic partial metric; dualistic contraction of ra-
tional type.

1. Introduction

Matthews [3] introduced the concept of partial metric space as a suitable
mathematical tool for program verification and proved an analogue of Banach
fixed point theorem in complete partial metric spaces. O’Neill [7] introduced
the notion of dualistic partial metric, which is more general than partial met-
ric and established a robust relationship between dualistic partial metric and
quasi metric. In [9], Oltra and Valero presented a Banach fixed point theorem
on complete dualistic partial metric spaces and in this way presented a gen-
eralization of famous Banach fixed point theorem. They also showed that the

Received 11 June 2016 – Accepted 28 August 2016

http://dx.doi.org/10.4995/agt.2016.5920


M. Nazam, M. Arshad, M. Abbas

contractive condition in Banach fixed point theorem in complete dualistic par-
tial metric spaces cannot be replaced by the contractive condition of Banach
fixed point theorem for complete partial metric spaces. Later on, Nazam et al.
[4] established a fixed point theorem for Geraghty type contractions in ordered
dualistic partial metric spaces and applied this result to show the existence of
solution of integral equations .

Harjani et al. [1] extended Banach fixed point principle as follows:

Theorem 1.1 ([1]). Let M be complete ordered metric space and T : M → M
a continuous and non decreasing mapping satisfying,

d(T(j),T(k)) ≤
αd(j,T(j)) ·d(k,T(k))

d(j,k)
+ βd(j,k),

for all comparable j,k ∈ M with j 6= k and 0 < α + β < 1. Then T has a
unique fixed point m∗ ∈ M. Moreover, the Picard iterative sequence {Tn(j)}n∈N
converges to m∗ for every j ∈ M.

Isik and Tukroglu [2] presented an ordered partial metric space version of
Theorem 1.1, stated below:

Theorem 1.2 ([2]). Let M be complete ordered partial metric space and T :
M → M a continuous and non decreasing mapping satisfying,

d(T(j),T(k)) ≤
αd(j,T(j)) ·d(k,T(k))

d(j,k)
+ βd(j,k),

for all comparable j,k ∈ M with j 6= k and 0 < α + β < 1. Then T has a
unique fixed point m∗ ∈ M. Moreover, the Picard iterative sequence {Tn(j)}n∈N
converges to m∗ for every j ∈ M.

In this paper, we obtain some fixed point theorems for dualistic contractions
of rational type. These results extend the comparable results in [2]. We give
examples to show that existing results in partial metric space cannot be applied
to obtain fixed points of mappings involved herein.

2. Preliminaries

Following mathematical basics will be needed in the sequel. Throughout
this paper, we denote (0,∞) by R+, [0,∞) by R+0 , (−∞, +∞) by R and set of
natural numbers by N.

Let T be a self mapping on a nonempty set M. An element m∗ ∈ M is
called a fixed point of T if it remains invariant under the action of T . If j0 is
a given point in M, then a sequence {jn} in M by jn = T(jn−1) = Tn(j0),
n ∈ N is called sequence is called Picard iterative sequence with initial guess j0.

O’Neill [7] introduced the notion of a dualistic partial metric space as a gen-
eralization of partial metric space in order to expand the connections between
partial metrics and semantics via valuation spaces. According to O’Neill, a
dualistic partial metric can be defined as follow.

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Dualistic contractions of rational type

Definition 2.1 ([7]). Let M be a nonempty set. A function D : M ×M → R
is called a dualistic partial metric if for any j,k, l ∈ M, the following conditions
hold:

(D1) j = k ⇔ D(j,j) = D(k,k) = D(j,k).
(D2) D(j,j) ≤ D(j,k).
(D3) D(j,k) = D(k,j).
(D4) D(j, l) ≤ D(j,k) + D(k,l) −D(k,k).

We observe that, as in the metric case, if D is a dualistic partial metric then
D(j,k) = 0 implies j = k. In case D(j,k) ∈ R+0 for all j,k ∈ M, then D is
a partial metric on M. If (M,D) is a dualistic partial metric space, then the
function dD : M ×M → R+0 defined by

dD(j,k) = D(j,k) −D(j,j),

is a quasi metric on M such that τ(D) = τ(dD). In this case, d
s
D(j,k) =

max{dD(j,k),dD(k,j)} defines a metric on M.

Remark 2.2. It is obvious that every partial metric is dualistic partial metric
but converse is not true. To support this comment, define D∨ : R×R → R by

D∨(j,k) = j ∨k = sup{j,k} ∀ j,k ∈ R.

It is easy to check that D∨ is a dualistic partial metric. Note that D∨ is not a
partial metric, because D∨(−1,−2) = −1 /∈ R+0 . Nevertheless, the restriction
of D∨ to R+0 , D∨|R+0 , is a partial metric.

Example 2.3. If (M,d) is a metric space and c ∈ R is an arbitrary constant,
then D : M ×M → R given by

D(j,k) = d(j,k) + c.

defines a dualistic partial metric on M.

Following [7], each dualistic partial metric D on M generates a T0 topology
τ(D) on M. The elements of the topology τ(D) are open balls of the form
{BD(j,�) : j ∈ M,� > 0} where BD(j,�) = {k ∈ M : D(j,k) < � + D(j,j)}.
A sequence {jn}n∈N in (M,D) converges to a point j ∈ M if and only if
D(j,j) = limn→∞D(j,jn).

Definition 2.4 ([7]). Let (M,D) be a dualistic partial metric space, then

(1) A sequence {jn}n∈N in (M,D) is called a Cauchy sequence if
limn,m→∞D(jn,jm) exists and is finite.

(2) A dualistic partial metric space (M,D) is said to be complete if every
Cauchy sequence {jn}n∈N in M converges, with respect to τ(D), to a
point j ∈ M such that D(j,j) = limn,m→∞D(jn,jm).

Following lemma will be helpful in the sequel.

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M. Nazam, M. Arshad, M. Abbas

Lemma 2.5 ([7, 9]).

(1) A dualistic partial metric (M,D) is complete if and only if the metric
space (M,dsD) is complete.

(2) A sequence {jn}n∈N in M converges to a point j ∈ M, with respect to
τ(dsD) if and only if limn→∞D(j,jn) = D(j,j) = limn→∞D(jn,jm).

(3) If limn→∞ jn = υ such that D(υ,υ) = 0 then limn→∞D(jn,k) =
D(υ,k) for every k ∈ M.

Oltra and Valero ([6]) extended partial metric space version of the Banach
contraction principle to dualistic partial metric spaces.

Theorem 2.6 ([6]). Let (M,D) be a complete dualistic partial metric space
and T : M → M. If there exists α ∈ [0, 1[ such that

|D(T(j),T(k))| ≤ α|D(j,k)|,

for any j,k ∈ M. Then T has a unique fixed point m∗ ∈ M. Moreover,
D(m∗,m∗) = 0 and for every j ∈ M, the Picard iterative sequence {Tn(j)}n∈N
converges with respect to τ(dsD) to m

∗.

3. The results

In this section, we shall show that, the dualistic contractions of rational
type along with certain conditions have unique fixed point. We will support
obtained results by some concrete examples. We introduce the following,

Definition 3.1. Let (M,�,D) be an ordered dualistic partial metric space. A
self-mapping T defined on M is called dualistic contraction of rational type if
for any j,k ∈ M, we have

(3.1) |D(T(j),T(k))| ≤ α
∣∣∣∣D(j,T(j)) ·D(k,T(k))D(j,k)

∣∣∣∣ + β|D(j,k)|,
for all comparable j,k ∈ M and 0 < α + β < 1.

We start with the following result.

Theorem 3.2. Let (M,�,D) be a complete ordered dualistic partial metric
space and T : M → M be a continuous and non decreasing dualistic contraction
of rational type. Then T has a fixed point m∗ in M provided there exists j0 ∈ M
such that j0 � T(j0). Moreover, D(m∗,m∗) = 0.

Proof. Let j0 be a given point ∈ M and jn = T(jn−1) , n ≥ 1 an iterative
sequence starting with j0. If there exists a positive integer r such that jr+1 = jr,
then jr is a fixed point of T and D(jr,jr) = 0. Suppose that jn 6= jn+1 for
any n ∈ N. As j0 � T(j0) = j1 , that is, j0 � j1 which further implies that
j1 = T(j0) � T(j1) = j2. Continuing this way, we obtain that

j0 � j1 � j2 � j3 � ···� jn � jn+1....

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Dualistic contractions of rational type

since jn � jn+1 by (3.1), we have

|D(jn,jn+1)| = |D(T(jn−1),T(jn))|

≤ α
∣∣∣∣D(jn−1,jn) ·D(jn,jn+1)D(jn−1,jn)

∣∣∣∣ + β|D(jn−1,jn)|,
≤ α|D(jn,jn+1)| + β|D(jn−1,jn)|,

|D(jn,jn+1)|−α|D(jn,jn+1)| ≤ β|D(jn−1,jn)|,
(1 −α)|D(jn,jn+1)| ≤ β|D(jn−1,jn)|,

|D(jn,jn+1)| ≤ (
β

1 −α
)|D(jn−1,jn)|.

If γ =
β

1 −α
, then 0 < γ < 1 and we have

(3.2) |D(jn,jn+1)| ≤ γ|D(jn−1,jn)|.

Thus
(3.3)
|D(jn,jn+1)| ≤ γ|D(jn−1,jn)| ≤ γ2|D(jn−2,jn−1)| ≤ ·· · ≤ γn|D(j0,j1)|.

As jn � jn, for each n ∈ N, by (3.1) we have

|D(jn,jn)| = |D(T(jn−1),T(jn−1))| ≤
α|D(jn−1,jn)|2

|D(jn−1,jn−1)|
+ β|D(jn−1,jn−1)|

≤ |D(jn−1,jn−1)|

{
α

∣∣∣∣ D(jn−1,jn)D(jn−1,jn−1)
∣∣∣∣2 + β

}

≤ (α + β)|D(jn−1,jn−1)|. Indeed
∣∣∣∣ D(jn−1,jn)D(jn−1,jn−1)

∣∣∣∣2 = 1.
Thus we obtain that

(3.4) |D(jn,jn)| ≤ (α + β)n|D(j0,j0)|.

Now we show that {jn} is a Cauchy sequence in (M,dsD). Note that, dD(jn,jn+1) =
D(jn,jn+1) − D(jn,jn), that is, dD(jn,jn+1) + D(jn,jn) = D(jn,jn+1) ≤
|D(jn,jn+1)|. Thus, we have

dD(jn,jn+1) + D(jn,jn) ≤ γn|D(j0,j1)|.
dD(jn,jn+1) ≤ γn|D(j0,j1)| + |D(jn,jn)|

≤ γn|D(j0,j1)| + (α + β)n|D(j0,j0)|

Continuing this way, we obtain that

dD(jn+k−1,jn+k) ≤ γn+k−1|D(j0,j1)| + (α + β)n+k−1|D(j0,j0)|

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M. Nazam, M. Arshad, M. Abbas

Now

dD(jn,jn+k) ≤ dD(jn,jn+1) + dD(jn+1,jn+2) + · · · + dD(jn+k−1,jn+k).
≤ {γn + γn+1 + · · · + γn+k−1}|D(j0,j1)|
+ {(α + β)n + (α + β)n+1 + · · · + (α + β)n+k−1}|D(j0,j0)|.

Thus for n + k = m > n

(3.5) dD(jn,jm) ≤
γn

1 −γ
|D(j0,j1)| +

(α + β)n

1 − (α + β)
|D(j0,j0)|.

Similarly, we have

(3.6) dD(jm,jn) ≤
γn

1 −γ
|D(j1,j0)| +

(α + β)n

1 − (α + β)
|D(j0,j0)|.

On taking limit as n,m →∞, we have
lim

n,m→∞
dD(jm,jn) = 0 = lim

n,m→∞
dD(jn,jm) and hence lim

n,m→∞
dsD(jm,jn) = 0,

we get that {jn} is a Cauchy sequence in (M,dsD). Since (M,D) is a complete
dualistic partial metric space, so by Lemma 2.5 (M,dsD) is also a complete
metric space. Thus, there exists m∗ in M such that limn→∞d

s
D(jn,m

∗) = 0,
again from Lemma 2.5, we get
(3.7)

lim
n→∞

dsD(jn,m
∗) = 0 ⇐⇒ D(m∗,m∗) = lim

n→∞
D(jn,m

∗) = lim
n,m→∞

D(jm,jn).

Now limn,m→∞dD(jm,jn) = 0 implies that limn,m→∞[D(jm,jn)−D(jn,jn)] =
0 and hence limn,m→∞D(jn,jm) = limn→∞D(jn,jn). By (3.4), we have
limn→∞D(jn,jn) = 0. Consequently, limn,m→∞D(jn,jm) = 0. Thus

(3.8) D(m∗,m∗) = lim
n→∞

D(jn,m
∗) = 0.

Now

dD(m
∗,T(m∗)) = D(m∗,T(m∗)) −D(m∗,m∗) = D(m∗,T(m∗)).

implies that D(m∗,T(m∗)) ≥ 0. Since T is continuous, for a given � > 0, There
exists δ > 0 such that T(BD(m

∗,δ)) ⊆ BD(T(m∗),�). Since limn→∞D(jn+1,m∗) =
D(m∗,m∗) = 0, so there exists r ∈ N such that D(jn,m∗) < D(m∗,m∗) + δ
∀ n ≥ r, therefore {jn} ⊂ BD(m∗,δ) ∀ n ≥ r. This implies that T(jn) ∈
T(BD(m

∗,δ) ⊆ BD(T(m∗),�) and so D(T(jn),T(m∗)) < D(T(m∗),T(m∗)) +�
∀ n ≥ r. Now for any � > 0, we know that

−� + D(T(m∗),T(m∗)) < D(T(m∗),T(m∗)) ≤ D(jn+1,T(m∗))
Which yields that

|D(jn+1,T(m∗)) −D(T(m∗),T(m∗))| < �
That is D(T(m∗),T(m∗)) = limn→∞D(jn+1,T(m

∗)), finally uniqueness of
limit in R implies
(3.9) lim

n→∞
D(jn+1,T(m

∗)) = D(T(m∗),T(m∗)) = D(m∗,T(m∗)).

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Dualistic contractions of rational type

Finally, we have D(T(m∗),m∗) = limn→∞D(T(jn),jn) = limn→∞D(jn+1,jn) =
0. This shows that D(m∗,T(m∗)) = 0. So from (3.8) and (3.9) we deduce that

D(m∗,T(m∗)) = D(T(m∗),T(m∗)) = D(m∗,m∗).

This leads us to conclude that m∗ = T(m∗) and hence m∗ is a fixed point of T
�

In order to prove the uniqueness of fixed point of a mapping T in the above
theorem, we need an additional assumption.

Theorem 3.3. Let (M,D,�) be complete ordered dualistic partial metric space
and T : M → M a mapping which satisfy all conditions of theorem (3.2). Then
T has a unique fixed point provided that for each fixed point m∗, n∗ of T, there
exists ω ∈ M which is comparable to both m∗ and n∗.

Proof. From theorem (3.2), it follows that the set of fixed points of T is
nonempty. To prove the uniqueness: Let n∗ be another fixed point of T, that
is, n∗ = T(n∗) and D(n∗,n∗) = 0. If m∗ and n∗ are comparable (m∗ � n∗),
then we have,

|D(m∗,n∗)| = |D(T(m∗),T(n∗))|,

≤ α
∣∣∣∣D(m∗,T(m∗)) ·D(n∗,T(n∗))D(m∗,n∗)

∣∣∣∣ + β|D(m∗,n∗)|.
≤ α

∣∣∣∣D(m∗,m∗) ·D(n∗,n∗)D(m∗,n∗)
∣∣∣∣ + β|D(m∗,n∗)|.

That is, (1 −β)|D(m∗,n∗)| ≤ 0 which implies that |D(m∗,n∗)| ≤ 0 and hence
D(m∗,n∗) = 0 = D(m∗,m∗) = D(n∗,n∗). The result follows. Suppose that m∗

and n∗ are incomparable, there exists ω which is comparable to both m∗, n∗.
Without any loss of generality, we assume that m∗ � ω, and n∗ � ω. As T
is non decreasing, T(m∗) � T(ω) and T(n∗) � T(ω) imply that Tn−1(m∗) �
Tn−1(ω) and Tn−1(n∗) � Tn−1(ω). Thus

|D(Tn(m∗),Tn(ω))| ≤ α
∣∣∣∣D(Tn−1(m∗),Tn(m∗)) ·D(Tn−1(ω),Tn(ω))D(Tn−1(m∗),Tn−1(ω))

∣∣∣∣
+ β|D(Tn−1(m∗),Tn−1(ω))|.

That is, |D(m∗,Tn(ω))| ≤ β|D(m∗,Tn−1(ω))|. Thus, limn→∞D(m∗,Tn(ω)) =
0.
Similarly, we can have limn→∞D(n

∗,Tn(ω)) = 0. Note that

D(n∗,m∗) ≤ D(n∗,Tn(ω)) + D(Tn(ω),m∗) −D(Tn(ω),Tn(ω)),
≤ D(n∗,Tn(ω)) + D(Tn(ω),m∗) −D(Tn(ω),m∗) −D(m∗,Tn(ω)) + D(m∗,m∗).

On taking limit as n →∞ we obtain that D(n∗,m∗) ≤ 0 . Now dD(m∗,m∗) =
D(n∗,m∗)−D(n∗,n∗) implies that D(n∗,m∗) ≥ 0. Hence D(n∗,m∗) = 0 which
gives that n∗ = m∗ �

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M. Nazam, M. Arshad, M. Abbas

Example 3.4. Let M = R2. Define D∨ : M ×M → R by D∨(j,k) = j1 ∨k1,
where j = (j1,j2) and k = (k1,k2). Note that (M,D∨) is a complete dualistic
partial metric space. Let T : M → M be given by

T(j) =
j

2
for all j ∈ M.

In M, we define the relation � in the following way:
j � k if and only if j1 ≥ k1, where j = (j1,j2) and k = (k1,k2).

Clearly, � is a partial order on M and T is continuous, non decreasing mapping
with respect to �. Moreover, T(−1, 0) � (−1, 0). We shall show that for all
j,k ∈ M, (3.1) is satisfied. For this, note that

|D∨(T(j),T(k))| =
∣∣∣∣D∨

(
j

2
,
k

2

)∣∣∣∣ =
∣∣∣∣j12
∣∣∣∣ for all j1 ≥ k1,

|D∨(j,T(j))| =
∣∣∣∣D∨

(
j,
j

2

)∣∣∣∣ =
{ ∣∣j1

2

∣∣ if j1 ≤ 0
|j1| if j1 ≥ 0,

|D∨(k,T(k))| =
∣∣∣∣D∨

(
k,
k

2

)∣∣∣∣ =
{ ∣∣k1

2

∣∣ if k1 ≤ 0
|k1| if k1 ≥ 0,

and |D∨(j,k)| = |j1| for all j1 ≥ k1.

Now for α =
1

3
, β =

1

2
. If j1 ≤ 0, k1 ≤ 0, then

|D∨(T(j),T(k))| ≤ α
∣∣∣∣D∨(j,T(j)) ·D∨(k,T(k))D∨(j,k)

∣∣∣∣ + β|D∨(j,k)| for all j � k
holds if and only if 6|j1| ≤ |k1| + 6|j1|.

For if j1 ≥ 0, k1 ≥ 0, then contractive condition

|D∨(T(j),T(k))| ≤ α
∣∣∣∣D∨(j,T(j)) ·D∨(k,T(k))D∨(j,k)

∣∣∣∣ + β|D∨(j,k)| for all j � k
holds if and only if j1 ≤ 23k1 + j1.

Finally, if j1 ≥ 0, k1 ≤ 0, then

|D∨(T(j),T(k))| ≤ α
∣∣∣∣D∨(j,T(j)) ·D∨(k,T(k))D∨(j,k)

∣∣∣∣ + β|D∨(j,k)| ∀ j � k
holds if and only if 3j1 ≤ |k1| + 3j1. Thus, all the conditions of Theorem 3.2
are satisfied. Moreover, (0, 0) is a fixed point of T .

Remark 3.5. As every dualistic partial metric D is an extension of partial
metric p, therefore, Theorem 3.2 is an extension of Theorem 1.2.

There arises the following natural question:
Whether the contractive condition in the statement of Theorem 3.2 can be

replaced by the contractive condition in Theorem 1.2.

The following example provides a negative answer to the above question.

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Dualistic contractions of rational type

Example 3.6. Define the mapping T0 : R+ → R by

T0(j) =

{
0 if j > 1
−5 if j = 1 .

Clearly, for any j,k ∈ R, following contractive condition is satisfied

D∨(T0(j),T0(k)) ≤
αD∨(j,T0(j)) ·D∨(k,T0(k))

D∨(j,k)
+ βD∨(j,k)

where D∨ is a complete dualistic partial metric on R. Here, T has no fixed
point. Thus a fixed point free mapping satisfies the contractive condition of
Theorem 1.2. On the other hand, for all 0 < α + β < 1, we have

5 = |D∨(−5,−5)| = |D∨(T0(1),T0(1))| > α
∣∣∣∣D∨(1,T0(1)) ·D(1,T0(1))D∨(1, 1)

∣∣∣∣+β|D∨(1, 1)|.
Thus contractive condition of Theorem 3.2 does not hold.

Theorem 3.2 remains true if we replace the continuity hypothesis by the
following property:

(H): If {jn} is a non decreasing sequence in M such that jn → υ, then

(3.10) jn � υ for all n ∈ N.
This statement is given as follows:

Theorem 3.7. Let (M,�,D) be a complete ordered dualistic partial metric
space and if,

(i) T : M → M be a non decreasing dualistic contraction of rational type.
(ii) there exists j0 ∈ M such that j0 � T(j0).
(ii) (H) holds.

Then T has a fixed point m∗ in M. Moreover, D(m∗,m∗) = 0.

Proof. Following the proof of Theorem 3.2, we know that {jn} is non decreasing
sequence in M such that jn → m∗. By (H), we have jn � m∗. As T is non
decreasing, we have T(jn) � T(m∗), that is, jn+1 � T(m∗). Also, j0 � j1 �
T(m∗) and jn � m∗ , n ≥ 1 imply that
(3.11) m∗ � T(m∗).
From the proof of Theorem 3.2, we deduce that {Tn(m∗)} is non decreasing
sequence. Suppose that limn→+∞T

n(m∗) = µ for some µ ∈ M. Now j0 � m∗
gives Tn(j0) � Tn(m∗), that is, jn � Tn(m∗) for all n ≥ 1. Thus we have

jn � m∗ � T(m∗) � Tn(m∗) n ≥ 1.
By (3.1), we have

|D(jn+1,Tn+1(m∗))| = |D(T(jn),T(Tn(m∗)))|,

≤ α
∣∣∣∣D(jn,jn+1) ·D(Tn(m∗),Tn+1(m∗))D(jn,Tn(m∗))

∣∣∣∣ + β|D(jn,Tn(m∗))|.
c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 207



M. Nazam, M. Arshad, M. Abbas

On taking limit as n approaches to plus infinity, we obtain that

|D(m∗,µ)| ≤ β|D(m∗,µ)|.
which implies that m∗ = µ. Thus limn→+∞T

n(m∗) = µ implies that limn→+∞T
n(m∗) =

m∗. Hence

(3.12) T(m∗) � m∗

From (3.11) and (3.12), it follows that m∗ = T(m∗). �

Some deductions are given below.

Corollary 3.8. Let (M,�,D) be a complete ordered dualistic partial metric
space and T : M → M be a non decreasing mapping such that,

(1)

|D(T(j),T(k))| ≤ α
∣∣∣∣D(j,T(j)) ·D(k,T(k))D(j,k)

∣∣∣∣ , where 0 < α < 1.
(2) there exists j0 ∈ M such that j0 � T(j0).

(3) either T is continuous or (H) holds.

Then T has a fixed point m∗ in M. Moreover, D(m∗,m∗) = 0.

Proof. Set β = 0 in Theorem 3.2. �

The next deduction generalizes Theorem 2.6 presented by Valero in [6]

Corollary 3.9. Let (M,�,D) be a complete ordered dualistic partial metric
space and T : M → M be a non decreasing mapping such that,

(1) |D(T(j),T(k))| ≤ β|D(j,k)|, where 0 < β < 1.

(2) there exists j0 ∈ M such that j0 � T(j0).

(3) either T is continuous or (H) holds.

Then T has a fixed point m∗ in M. Moreover, D(m∗,m∗) = 0.

Proof. Set α = 0 in Theorem 3.2. �

Remark 3.10.

(1) If we set D(j,k) ∈ R+0 for all j,k ∈ M in Corollary 3.8 and in Corollary
3.9, we obtain results in partial metric spaces.

(2) If we set D(j,k) ∈ R+0 for all j,k ∈ M and D(j,j) = 0 for all j ∈ M in
Corollary 3.8 and in Corollary 3.9, we obtain results in metric spaces.

c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 208



Dualistic contractions of rational type

Acknowledgements. The authors are grateful to the anonymous referee for
a careful checking of the details and for helpful comments that improved this
paper.

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