International Journal of Analysis and Applications
ISSN 2291-8639
Volume 12, Number 1 (2016), 38-48
http://www.etamaths.com

ON FIXED POINTS OF GENERALIZED α-ψ CONTRACTIVE TYPE MAPPINGS

IN PARTIAL METRIC SPACES

PRIYA SHAHI∗, JATINDERDEEP KAUR AND S. S. BHATIA

Abstract. Recently, Samet et al. (B. Samet, C. Vetro and P. Vetro, Fixed point theorem for α-ψ

contractive type mappings, Nonlinear Anal. 75 (2012), 2154–2165) introduced a very interesting new
category of contractive type mappings known as α-ψ contractive type mappings. The results obtained

by Samet et al. generalize the existing fixed point results in the literature, in particular the Banach

contraction principle. Further, Karapinar and Samet (E. Karapinar and B. Samet, Generalized α-ψ-
contractive type mappings and related fixed point theorems with applications, Abstract and Applied

Analysis 2012 Article ID 793486, 17 pages doi:10.1155/2012/793486) generalized the α-ψ contractive

type mappings and established some fixed point theorems for this generalized class of contractive
mappings. In (G. S. Matthews, Partial metric topology, Ann. New York Acad. Sci. 728 (1994),

183–197), the author introduced and studied the concept of partial metric spaces, and obtained a

Banach type fixed point theorem on complete partial metric spaces. In this paper, we establish the
fixed point theorems for generalized α-ψ contractive mappings in the context of partial metric spaces.

As consequences of our main results, we obtain fixed point theorems on partial metric spaces endowed

with a partial order and that for cyclic contractive mappings. Our results extend and strengthen
various known results. Some examples are also given to show that our generalization from metric

spaces to partial metric spaces is real.

1. Introduction

The notion of metric space was introduced by Fréchet in 1906. Later, many authors attempted to
generalize the notion of metric space such as pseudometric space, quasimetric space, semimetric space
etc. In this paper, we consider another generalization of a metric space, so called partial metric space.
When compared to metric spaces, the innovation of partial metric spaces is that the self distance of a
point is not necessarily zero. Initially, Matthews discussed not only the general topological properties
of partial metric spaces but also some properties of convergence of sequences. Matthews also stated
and proved the fixed point theorem of contractive mapping on partial metric spaces: Any mapping
T of a complete partial metric space X into itself that satisfies, for some 0 ≤ k < 1, the inequality
d(Tx,Ty) ≤ kd(x,y), for all x,y ∈ X, has a unique fixed point. Very recently, many authors have
focussed on this subject and have generalized some fixed point theorems from the class of metric spaces
to the class of partial metric spaces.

The purpose of this work is to establish the fixed point theorems for generalized α-ψ-contractive
mappings in the context of partial metric spaces. As consequences of our main results, we obtain fixed
point theorems on partial metric spaces endowed with a partial order and that for cyclic contractive
mappings. Presented theorems are generalizations of very recent fixed point theorems due to Samet
et al.[23] and Karapinar and Samet [12]. Some examples are given to show that presented results are
real generalizations.

2. Preliminaries

Throughout this work the letters R, R+, Q, N will denote the sets of real numbers, nonnegative real
numbers, rational numbers and natural numbers, respectively.

Before presenting our results, we collect relevant definitions and results which will be needed in the
proof of our main results.

2010 Mathematics Subject Classification. 54H25, 47H10, 54E50.

Key words and phrases. fixed point; partial metric space; partial order; contractive mapping.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

38



ON FIXED POINTS OF GENERALIZED α-ψ CONTRACTIVE TYPE MAPPINGS 39

Definition 2.1. (See e.g. [14, 9]). Let X be a nonempty set. The mapping p : X × X → [0,∞) is
said to be a partial metric on X if the following conditions hold:
(P1) x = y if and only if p(x,x) = p(y,y) = p(x,y),
(P2) p(x,x) ≤ p(x,y),
(P3) p(x,y) = p(y,x),
(P4) p(x,z) ≤ p(x,y) + p(y,z) −p(y,y),
for any x,y,z ∈ X. The pair (X,p) is then called a partial metric space (in short PMS).

Let (X,p) be a partial metric space. Then, the functions dp,dm : X ×X → [0,∞) given by

dp(x,y) = 2p(x,y) −p(x,x) −p(y,y)
and

dm(x,y) = max p(x,y) −p(x,x),p(x,y) −p(y,y)
are well-known metrics on X. It is easy to check that dp and dm are equivalent. Note that each partial
metric p on X generates a T0-topology τp with a base of the family of open p-balls {Bp(x,�) : x ∈
X,� > 0}, where Bp(x,�) = {y ∈ X : p(x,y) < p(x,x) + �}.

Definition 2.2. (See e.g. [2, 9]). Let (X,p) be a partial metric space.
(1) A sequence {xn} in X converges to x ∈ X if and only if p(x,x) = lim

n→∞
p(xn,x).

(2) A sequence {xn} in X is called a Cauchy sequence if and only if lim
m,n→∞

p(xn,xm) exists (and

finite).
(3) (X,p) is said to be complete if every Cauchy sequence {xn} in X converges to x ∈ X.
(4) A mapping f : X → X is said to be continuous at x0 ∈ X if for every � > 0, there exists δ > 0
such that f(Bp(x0,δ)) ⊂ Bp(f(x0),�).

Example 2.3. Let X = [0, +∞) and define p(x,y) = max{x,y}, for all x,y ∈ X. Then (X,p) is a
complete partial metric space. It is clear that p is not a (usual) metric.

Definition 2.4. Let (X,p) be a partial metric space and T : X → X be a given mapping. We say that
T is continuous at x0 ∈ X, if for every � > 0, there exists η > 0 such that T(Bp(x0,η)) ⊆ Bp(Tx0,�).

The following lemmas have an important role in the proof of our main results.

Lemma 2.1. (See e.g. [2, 9]). Let (X,p) be a partial metric space.
(1) A sequence {xn} is a Cauchy sequence in (X,p) if and only if {xn} is a Cauchy sequence in (X,dp).
(2) (X,p) is complete if and only if (X,dp) complete. Moreover,

lim
n→∞

dp(xn,x) = 0 ⇔ p(x,x) = lim
n→∞

p(xn,x) = lim
n,m→∞

p(xm,xn).

Lemma 2.2. (See e.g. [2]). Assume that xn → z as n → ∞ in a PMS (X,p) such that p(z,z) = 0.
Then lim

n→∞
p(xn,y) = p(z,y) for every y ∈ X.

Lemma 2.3. (Sequential characterization of continuity)Let (X,p) be a partial metric space and T :
X → X be a given mapping. T is said to be continuous at x0 ∈ X if it is sequentially continuous at
x0, that is, if and only if

∀{xn}⊂ X : lim
n→+∞

xn = x0 ⇒ lim
n→+∞

Txn = Tx0

Let Ψ be the family of functions ψ : [0,∞) → [0,∞) satisfying the following conditions:
(i) ψ is nondecreasing.

(ii)

+∞∑
n=1

ψn(t) < ∞ for all t > 0, where ψn is the nth iterate of ψ.

These functions are known as (c)-comparison functions in the literature. It can be easily verified that
if ψ is a (c)-comparison function, then ψ(t) < t for any t > 0.

Recently, Samet et al. [23] introduced the following new concepts of α-ψ-contractive type mappings
and α-admissible mappings:



40 SHAHI, KAUR AND BHATIA

Definition 2.5. Let (X,d) be a metric space and T : X → X be a given self mapping. T is said to be
an α-ψ-contractive mapping if there exists two functions α : X ×X → [0, +∞) and ψ ∈ Ψ such that

α(x,y)d(Tx,Ty) ≤ ψ(d(x,y))
for all x,y ∈ X.

Definition 2.6. Let T : X → X and α : X ×X → [0, +∞). T is said to be α-admissible if
x,y ∈ X, α(x,y) ≥ 1 ⇒ α(Tx,Ty) ≥ 1.

The following fixed point theorems are the main results in [23]:

Theorem 2.4. Let (X,d) be a complete metric space and T : X → X be an α-ψ-contractive mapping
satisfying the following conditions:
(i) T is α-admissible;
(ii) there exists x0 ∈ X such that α(x0,Tx0) ≥ 1;
(iii) T is continuous.
Then, T has a fixed point, that is, there exists x∗ ∈ X such that Tx∗ = x∗.

Theorem 2.5. Let (X,d) be a complete metric space and T : X → X be an α-ψ-contractive mapping
satisfying the following conditions:
(i) T is α-admissible;
(ii) there exists x0 ∈ X such that α(x0,Tx0) ≥ 1;
(iii) if {xn} is a sequence in X such that α(xn,xn+1) ≥ 1 for all n and xn → x ∈ X as n → +∞,
then α(xn,x) ≥ 1 for all n.
Then, T has a fixed point.

Samet et al. [23] added the following condition to the hypotheses of Theorem 2.4 and Theorem 2.5
to assure the uniqueness of the fixed point:
(C): For all x,y ∈ X, there exists z ∈ X such that α(x,z) ≥ 1 and α(y,z) ≥ 1.

Recently, Karapinar and Samet [12] introduced the following concept of generalized α-ψ-contractive
type mappings:

Definition 2.7. Let (X,d) be a metric space and T : X → X be a given mapping. We say that T is a
generalized α-ψ-contractive type mapping if there exists two functions α : X ×X → [0,∞) and ψ ∈ Ψ
such that for all x,y ∈ X, we have

α(x,y)d(Tx,Ty) ≤ ψ(M(x,y)),

where M(x,y) = max

{
d(x,y),

d(x,Tx) + d(y,Ty)

2
,
d(x,Ty) + d(y,Tx)

2

}
.

Further, Karapinar and Samet [12] established fixed point theorems for this new class of contractive
mappings. Also, they obtained fixed point theorems on metric spaces endowed with a partial order
and fixed point theorems for cyclic contractive mappings.

3. Main results

Firstly, we present the concept of generalized α-ψ contractive type mappings in the context of partial
metric spaces as follows:

Definition 3.1. Let (X,p) be a partial metric space and T : X → X be a given mapping. We say that
T is a generalized α-ψ-contractive type mapping if there exists two functions α : X ×X → [0,∞) and
ψ ∈ Ψ such that for all x,y ∈ X, we have

α(x,y)p(Tx,Ty) ≤ ψ(M(x,y)),(1)

where M(x,y) = max

{
p(x,y),

p(x,Tx) + p(y,Ty)

2
,
p(x,Ty) + p(y,Tx)

2

}
.

Now, we present our main results as follows.



ON FIXED POINTS OF GENERALIZED α-ψ CONTRACTIVE TYPE MAPPINGS 41

Theorem 3.1. Let (X,d) be a complete partial metric space, α : X × X → [0,∞) be a function,
ψ ∈ Ψ and T be a generalized α-ψ contractive type mapping on X. Suppose that T is α-admissible and
continuous. Also, assume that there exists x0 ∈ X such that α(x0,Tx0) ≥ 1. Then there exists u ∈ X
such that Tu = u.

Proof. Take x0 ∈ X such that α(x0,Tx0) ≥ 1 and define the sequence {xn} in X by xn+1 = Txn for
all n ≥ 0. If xn = xn+1 for some n, then x∗ = xn is a fixed point of T . Assume that xn 6= xn+1 for all
n. Owing to α-admissible property of T, we have

α(x0,Tx0) = α(x0,x1) ≥ 1 ⇒ α(Tx0,Tx1) = α(x1,x2) ≥ 1

Continuing this process inductively, we obtain

α(xn,xn+1) ≥ 1(2)

for all n = 0, 1, 2, ... . Thus for each n, we have

p(xn,xn+1) = p(Txn−1,Txn) ≤ α(xn−1,xn)p(Txn−1,Txn)
≤ ψ(M(xn−1,xn))(3)

On the other hand, we have

M(xn−1,xn) = max

{
p(xn−1,xn),

p(xn−1,xn) + p(xn,xn+1)

2
,
p(xn−1,xn+1) + p(xn,xn)

2

}
≤ max

{
p(xn−1,xn),

p(xn−1,xn) + p(xn,xn+1)

2
,
p(xn−1,xn) + p(xn,xn+1)

2

}
≤ max{p(xn−1,xn),p(xn,xn+1)} .(4)

From (3), (4) and using the fact that ψ is a nondecreasing function, we get that

p(xn+1,xn) ≤ ψ(max{p(xn−1,xn),p(xn,xn+1)})

for all n. If max{p(xn−1,xn),p(xn,xn+1)} = p(xn,xn+1), then

p(xn,xn+1) ≤ ψ(p(xn,xn+1)) < p(xn,xn+1),

which is a contradiction. Thus, max{p(xn−1,xn),p(xn,xn+1)} = p(xn−1,xn) for all n. Hence,

p(xn,xn+1) ≤ ψ(p(xn−1,xn))

for all n. Continuing this process inductively, we obtain

p(xn,xn+1) ≤ ψn(p(x0,x1)),(5)

for all n. Now, using the definition of partial metric, we have

max{p(xn,xn),p(xn+1,xn+1)}≤ p(xn,xn+1)(6)

which in view of (5) gives rise to

max{p(xn,xn),p(xn+1,xn+1)}≤ ψn((p(x0,x1))(7)

Therefore, owing to (4) and (5), we have

ps(xn,xn+1) = 2p(xn,xn+1) −p(xn,xn) −p(xn+1,xn+1)
≤ 2p(xn,xn+1) + p(xn,xn) + p(xn+1,xn+1)
≤ 4ψn(p(x0,x1)).(8)

Now, using inequality (8), we have

ps(xn+k,xn) ≤ ps(xn+k,xn+k−1) + ... + ps(xn+1,xn)
≤ 4ψn+k−1(p(x0,x1)) + ... + 4ψn(p(x0,x1))

≤ 4
n+k−1∑
i=n

ψi(p(x0,x1))(9)



42 SHAHI, KAUR AND BHATIA

and as

∞∑
i=0

ψi(p(x0,x1)) is convergent, from the last inequality, using Cauchy’s criteria for convergent

series, we obtain that {xn} is a Cauchy sequence in the metric space (X,ps). Now, in view of Lemma
2.1 and the completeness of (X,p), we conclude the completeness of (X,ps). Therefore, the sequence
{xn} is convergent in the space (X,ps), say lim

n→∞
ps(xn,u) = 0. Again from Lemma 2.1, we get

p(u,u) = lim
n→∞

p(xn,u) = lim
m,n→∞

p(xm,xn)(10)

Moreover, since {xn} is a Cauchy sequence in the metric space (X,ps), we have

lim
m,n→∞

ps(xm,xn) = 0(11)

and in view of (7), one gets

lim
n→∞

p(xn,xn) = 0(12)

Notice that in view of (11), (12) and definition of ps, we conclude that

lim
m,n→∞

p(xm,xn) = 0(13)

On using (10), we have

p(u,u) = lim
n→∞

p(xn,u) = lim
m,n→∞

p(xm,xn) = 0(14)

Now, we proceed to show that Tu = u. Due to the continuity of T, we infer from Lemma 2.3 that

p(Tu,Tu) = lim
n→∞

p(Txn,Tu) = lim
m,n→∞

p(Txm,Txn)(15)

that is,

p(Tu,Tu) = lim
m,n→∞

p(xm+1,xn+1).(16)

Notice that in view of (14) and (16),

p(u,u) = p(Tu,Tu) = 0(17)

Owing to Lemma 2.2, we have

lim
n→∞

p(xn,Tu) = p(u,Tu)(18)

Therefore, using (15), (17) and (18), we obtain

p(Tu,Tu) = p(u,u) = p(u,Tu) = 0

implying thereby Tu = u. Thus, we conclude that u is a fixed point of T . This completes the proof. �

In the next theorem, we omit the continuity hypothesis of T .

Theorem 3.2. Let (X,d) be a complete partial metric space, α : X × X → [0,∞) be a function,
ψ ∈ Ψ and T be a generalized α-ψ contractive type mapping on X. Suppose that T is α-admissible
and that there exists x0 ∈ X such that α(x0,Tx0) ≥ 1. Assume that if {xn} is a sequence in X such
that α(xn,xn+1) ≥ 1 for all n and {xn}→ x ∈ X as n →∞, then there exists a subsequence {xn(k)}
of {xn} such that α(xn(k),x) ≥ 1 for all k. Then there exists u ∈ X such that Tu = u.

Proof. Following the proof of Theorem 3.1, we know that the sequence {xn} given by xn+1 = Txn
for all n ≥ 0, converges to some u ∈ X. From (2) and given hypotheses, there exists a subsequence
{xn(k)} of {xn} such that

α(xn(k),u) ≥ 1(19)



ON FIXED POINTS OF GENERALIZED α-ψ CONTRACTIVE TYPE MAPPINGS 43

for all k. Now, we proceed to show that u is a fixed point of T . Suppose the contrary, then p(u,Tu) > 0.
Therefore, from (1) and (19), we infer that

p(u,Tu) ≤ p(u,xn(k)+1) + p(xn(k)+1,Tu) −p(xn(k)+1,xn(k)+1)
≤ p(u,xn(k)+1) + p(xn(k)+1,Tu)
= p(u,xn(k)+1) + p(Txn(k),Tu)

≤ p(u,xn(k)+1) + α(xn(k),u)p(Txn(k),Tu)
≤ p(u,xn(k)+1) + ψ(M(xn(k),u)))(20)

On the other hand, we have

M(xn(k),u)) = max

{
p(xn(k),u),

p(xn(k),xn(k)+1) + p(u,Tu)

2
,
p(xn(k),Tu) + p(u,xn(k)+1)

2

}
(21)

Letting k →∞ in (21) and using the above equality, we get

p(u,Tu) ≤ ψ
(
p(u,Tu)

2

)
<
p(u,Tu)

2
,(22)

which is a contradiction. Therefore, p(u,Tu) = 0 and Tu = u. �

We demonstrate the use of Theorem 3.1 and Theorem 3.2 with the help of the following examples.
These examples also show that our theorems are more general than some other known fixed point
results.

Example 3.2. Let X = R+, where (X,p) is a complete partial metric space with partial metric p

given by p(x,y) = max{x,y}. The mapping T(x) =
x2

1 + x
∀x ∈ X is continuous. Let us define the

function α by

α(x,y) =

{
1 x ≥ y
0 otherwise

Clearly, T is a generalized α-ψ contractive type mapping with ψ(t) =
t2

1 + t
for all t ≥ 0. In fact, for

all x,y ∈ X, we have
α(x,y)p(Tx,Ty) ≤ ψ(M(x,y))(23)

Moreover, there exists x0 ∈ X such that α(x0,Tx0) ≥ 1. In fact, for x0 = 1, we have
α(1,T1) = α(1, 1/2) = 1(24)

Now we proceed to show that T is α-admissible. For this, we have

α(x,y) ≥ 1 ⇒ x ≥ y ⇒ Tx ≥ Ty ⇒ α(Tx,Ty) ≥ 1(25)
Thus, T is α-admissible. Now, all the hypotheses of Theorem 3.1 are satisfied. Consequently, T has a
fixed point. In this case, 0 is a fixed point.
The same conclusion cannot be obtained by Theorem 2.3 from [12]. Indeed, using ps(a,b) = 2p(a,b) −
p(a,a) −p(b,b), and then taking ps instead p, x = 3, y = 2 in (1), we obtain

α(3, 2)ps(T3,T2) =
11

12



1

2
= ψ(1).(26)

Therefore, this example shows that our generalization from metric spaces to partial metric spaces is
real.

Example 3.3. Let X = {0, 1, 2, 3} and the function p : X×X → [0, +∞) defined by p(1, 2) = p(2, 3) =
1,p(1, 3) = 3

2
,p(1, 1) = p(3, 3) = 1

2
,p(2, 2) = 0 and p(x,y) = p(y,x). Obviously, p is a partial metric

on X but not a metric (since p(x,x) 6= 0 for x = 1 and x = 3). Let us define the self-mapping T on
X by T0 = 1, T1 = 2, T2 = 2, T3 = 1 for each x ∈ X. Clearly, T is a generalized α-ψ contractive
type mapping with ψ(t) = 2

3
t for t ≥ 0. In fact, for all x,y ∈ X, we have

α(x,y)p(Tx,Ty) ≤ ψ(M(x,y)),(27)



44 SHAHI, KAUR AND BHATIA

where

α(x,y) =

{
1 (x,y) 6= (0, 0)
0 otherwise

Moreover, there exists x0 ∈ X such that α(x0,Tx0) ≥ 1. In fact, for x0 = 1, we have

α(1,T1) = α(1, 2) = 1(28)

Let {xn} be a sequence in X such that α(xn,xn+1) ≥ 1 for all n and xn → x as n → +∞ for some
x ∈ X. From the definition of α, for all n, we have xn 6= 0 for all n. Thus, x 6= 0 and we have
α(xn,x) ≥ 1 for all n. Now we proceed to show that T is α-admissible. For this, we have

α(x,y) ≥ 1 ⇒ x 6= 0,y 6= 0 ⇒ Tx 6= 0,Ty 6= 0 ⇒ α(Tx,Ty) ≥ 1(29)

Thus, T is α-admissible. Now, all the hypotheses of Theorem 3.2 are satisfied. Consequently, T has a
fixed point. In this case, 2 is a fixed point.
The same conclusion cannot be obtained by Theorem 2.4 from [12]. Indeed, using ps(a,b) = 2p(a,b) −
p(a,a) −p(b,b), and then taking ps instead p, x = 1, y = 3 in (1), we obtain

α(1, 3)ps(T1,T3) =
3

2



4

3
= ψ(2).(30)

Therefore, this example shows that our generalization from metric spaces to partial metric spaces is
real.

4. Fixed Point Theorems on Partial Metric Spaces Endowed with a Partial Order

Fixed point theory has developed rapidly in partially ordered metric spaces. The first result in this
direction was given by Turinici [24], where he extended the Banach contraction principle in partially
ordered sets. Some applications of Turinici’s theorem to matrix equations were presented by Ran
and Reurings [20]. Subsequently, Nieto and Rodŕiguez-López [17] extended this result and applied
it to obtain a unique solution for periodic boundary value problems. Further results were obtained
by several authors (see, for example, [4, 8, 13, 7, 16] and the references cited therein). Altun and
Erduran [5] used the idea of partial order and established fixed point theorems to the frame of ordered
partial metric spaces. Aydi [6], Samet et al. [22], Abbas and Nazir [1] also studied fixed point results
on partially ordered partial metric spaces. Before presenting our result, we collect relevant concepts
which will be needed in the proof of our results.

Definition 4.1. Let (X,�) be a partially ordered set and T : X → X be a given mapping. We say
that T is nondecreasing with respect to � if

x,y ∈ X,x � y ⇒ Tx � Ty.

Definition 4.2. Let (X,�) be a partially ordered set. A sequence {xn}⊂ X is said to be nondecreasing
with respect to � if xn � xn+1 for all n.

Definition 4.3. [12] Let (X,�) be a partially ordered set and p be a partial metric on X. We say that
(X,�,p) is regular if for every nondecreasing sequence {xn} ⊂ X such that xn → x ∈ X as n → ∞,
there exists a subsequence {xn(k)} of {xn} such that xn(k) � x for all k.

Now, we have the following result.

Corollary 4.1. Let (X,�) be a partially ordered set and p be a partial metric on X such that (X,p)
is complete. Let T : X → X be a nondecreasing mapping with respect to �. Suppose that there exists
a function ψ ∈ Ψ such that

p(Tx,Ty) ≤ ψ(M(x,y)),(31)

for all x,y ∈ X with x � y. Suppose also that the following conditions hold:
(i) there exists x0 ∈ X such that x0 � Tx0;
(ii) T is continuous or (X,�,d) is regular.
Then, T has a fixed point.



ON FIXED POINTS OF GENERALIZED α-ψ CONTRACTIVE TYPE MAPPINGS 45

Proof. Let us define the mapping α : X ×X → [0,∞) by

α(x,y) =

{
1 if x � y or x � y
0 otherwise

(32)

for all x,y ∈ X. In view of condition (i), we obtain α(x0,Tx0) ≥ 1. Moreover, for all x,y ∈ X, from
the monotone property of T, we get

α(x,y) ≥ 1 ⇒ x � y or x � y ⇒ Tx � Ty or Tx � Ty ⇒ α(Tx,Ty) ≥ 1.(33)

Thus, T is α-admissible. Now, if T is continuous, the existence of a fixed point follows from Theorem
3.1. Suppose now that (X,�,d) is regular. Let {xn} be a sequence in X such that α(xn,xn+1) ≥ 1
for all n and xn → x ∈ X as n → ∞. So, from the regularity hypothesis, there exists a subsequence
{xn(k)} of {xn} such that xn(k) � x for all k. Notice that in view of definition of α, we obtain that
α(xn(k),x) ≥ 1 for all k. Thus, we get the existence of a fixed point from Theorem 3.2. �

The following corollaries can be straightway derived from Corollary 4.1

Corollary 4.2. Let (X,�) be a partially ordered set and p be a partial metric on X such that (X,p)
is complete. Let T : X → X be a nondecreasing mapping with respect to �. Suppose that there exists
a function ψ ∈ Ψ such that

p(Tx,Ty) ≤ ψ(p(x,y)),(34)

for all x,y ∈ X with x � y. Suppose also that the following conditions hold:
(i) there exists x0 ∈ X such that x0 � Tx0;
(ii) T is continuous or (X,�,p) is regular.
Then T is a fixed point.

Corollary 4.3. Let (X,�) be a partially ordered set and p be a partial metric on X such that (X,p)
is complete. Let T : X → X be a nondecreasing mapping with respect to �. Suppose that there exists
a constant λ ∈ (0, 1) such that

p(Tx,Ty) ≤ max
{
p(x,y),

p(x,Tx) + p(y,Ty)

2
,
p(x,Ty) + p(y,Tx)

2

}
,(35)

for all x,y ∈ X with x � y. Suppose also that the following conditions hold:
(i) there exists x0 ∈ X such that x0 � Tx0;
(ii) T is continuous or (X,�,p) is regular.
Then T is a fixed point.

Corollary 4.4. Let (X,�) be a partially ordered set and p be a partial metric on X such that (X,p)
is complete. Let T : X → X be a nondecreasing mapping with respect to �. Suppose that there exists
constants A,B,C ≥ 0 with (A + 2B + 2C) ∈ (0, 1) such that

p(Tx,Ty) ≤ Ap(x,y) + B[p(x,Tx) + p(y,Ty)] + C[p(x,Ty) + p(y,Tx)],(36)

for all x,y ∈ X with x � y. Suppose also that the following conditions hold:
(i) there exists x0 ∈ X such that x0 � Tx0;
(ii) T is continuous or (X,�,p) is regular.
Then T is a fixed point.

Corollary 4.5. Let (X,�) be a partially ordered set and p be a partial metric on X such that (X,p)
is complete. Let T : X → X be a nondecreasing mapping with respect to �. Suppose that there exists
a constant λ ∈ (0, 1/2) such that

p(Tx,Ty) ≤ λ[p(x,Tx) + p(y,Ty)],(37)

for all x,y ∈ X with x � y. Suppose also that the following conditions hold:
(i) there exists x0 ∈ X such that x0 � Tx0;
(ii) T is continuous or (X,�,p) is regular.
Then T is a fixed point.



46 SHAHI, KAUR AND BHATIA

Corollary 4.6. Let (X,�) be a partially ordered set and p be a partial metric on X such that (X,p)
is complete. Let T : X → X be a nondecreasing mapping with respect to �. Suppose that there exists
a constant λ ∈ (0, 1/2) such that

p(Tx,Ty) ≤ λ[p(x,Ty) + p(y,Tx)],(38)

for all x,y ∈ X with x � y. Suppose also that the following conditions hold:
(i) there exists x0 ∈ X such that x0 � Tx0;
(ii) T is continuous or (X,�,p) is regular.
Then T is a fixed point.

5. Fixed Point Theorems for Cyclic Contractive Mappings

As a generalization of the Banach contraction mapping principle, Kirk-Srinivasan-Veeramani devel-
oped the cyclic contraction. A contraction T : A∪B → A∪B on non-empty sets A,B is called cyclic
if T(A) ⊂ B and T(B) ⊂ A hold for closed subsets A,B of a complete metric space X. In the last
decade, several authors have used the cyclic representations and cyclic contractions to obtain various
fixed point results. See e.g., ([3, 10, 11, 18, 19, 21]). In this section, we will show that, from our
Theorem 3.1 and 3.2, we can deduce some fixed point theorems for cyclic contractive mappings.
Now, we have the following result.

Corollary 5.1. Let {Ai} be nonempty closed subsets of a complete partial metric space (X,p) and
T : Y → Y be a given mapping, where Y = A1 ∪A2. Suppose that the following conditions hold:
(i) T(A1) ⊆ A2 and T(A2) ⊆ A1;
(ii) there exists a function ψ ∈ Ψ such that

p(Tx,Ty) ≤ ψ(M(x,y)), ∀(x,y) ∈ A1 ×A2.(39)

Then T has a fixed point that belongs to A1 ∩A2.

Proof. Due to the fact that A1 and A2 are closed subsets of the complete metric space (X,d), we get
completeness of the space (Y,d). Let us define the mapping α : Y ×Y → [0,∞) by

α(x,y) =

{
1 if (x,y) ∈ (A1 ×A2) ∪ (A2 ×A1),
0 otherwise

(40)

Notice that in view of definition α and condition (ii), we infer that

α(x,y)p(Tx,Ty) ≤ ψ(M(x,y)),(41)

for all x,y ∈ Y . Thus T is a generalized α-ψ contractive mapping. Now, we proceed to show that T
is α-admissible. For thus, let (x,y) ∈ Y × Y such that α(x,y) ≥ 1. If (x,y) ∈ A1 × A2, then from
(i), we have (Tx,Ty) ∈ A2 × A1, thereby implying α(Tx,Ty) ≥ 1. Again from (i), we obtain that
(x,y) ∈ A2 ×A1 implies that (Tx,Ty) ∈ A1 ×A2, which further implies that α(Tx,Ty) ≥ 1. Thus,
we have α(Tx,Ty) ≥ 1 in all the cases. Therefore, we obtain that T is α-admissible.
Also, in view of (i), for any u ∈ A1, we have (u,Tu) ∈ A1 ×A2, which suggest that α(u,Tu) ≥ 1.
Now, we consider that {xn} be a sequence in X such that α(xn,xn+1) ≥ 1 for all n and xn → x ∈ X
as n →∞. This suggest from the definition of α that

(xn,xn+1) ∈ (A1 ×A2) ∪ (A2 ×A1),(42)

for all n. Since (A1 ×A2) ∪ (A2 ×A1) is a closed set with respect to the Euclidean metric, we obtain
that

(x,x) ∈ (A1 ×A2) ∪ (A2 ×A1),(43)

which refer that x ∈ A1 ∩A2. Consequently, we get from the definition of α that α(xn,x) ≥ 1 for all
n. Now, all the hypotheses of Theorem 3.2 are satisfied, and we conclude that T has a fixed point that
belongs to A1 ∩A2(from (i)). �

The following results are immediate consequences of Corollary 5.1.



ON FIXED POINTS OF GENERALIZED α-ψ CONTRACTIVE TYPE MAPPINGS 47

Corollary 5.2. Let {Ai}2i=1 be nonempty closed subsets of a complete partial metric space (X,p) and
T : Y → Y be a given mapping, where Y = A1 ∪A2. Suppose that the following conditions hold:
(i) T(A1) ⊆ A2 and T(A2) ⊆ A1;
(ii) there exists a function ψ ∈ Ψ such that

p(Tx,Ty) ≤ ψ(p(x,y)), ∀(x,y) ∈ A1 ×A2.(44)

Then T has a fixed point that belongs to A1 ∩A2.

Corollary 5.3. Let {Ai}2i=1 be nonempty closed subsets of a complete partial metric space (X,p) and
T : Y → Y be a given mapping, where Y = A1 ∪A2. Suppose that the following conditions hold:
(i) T(A1) ⊆ A2 and T(A2) ⊆ A1;
(ii) there exists a constant λ ∈ (0, 1) such that

p(Tx,Ty) ≤ λ max
{
p(x,y),

d(x,Tx) + d(y,Ty)

2
,
d(x,Ty) + d(y,Tx)

2

}
,

∀(x,y) ∈ A1 ×A2.(45)

Then T has a fixed point that belongs to A1 ∩A2.

Corollary 5.4. Let {Ai}2i=1 be nonempty closed subsets of a complete partial metric space (X,p) and
T : Y → Y be a given mapping, where Y = A1 ∪A2. Suppose that the following conditions hold:
(i) T(A1) ⊆ A2 and T(A2) ⊆ A1;
(ii) there exists constants A,B,C ≥ 0 with (A + 2B + 2C) ∈ (0, 1) such that

p(Tx,Ty) ≤ Ap(x,y) + B[d(x,Tx) + d(y,Ty)] + C[d(x,Ty) + d(y,Tx)],
∀(x,y) ∈ A1 ×A2.(46)

Then T has a fixed point that belongs to A1 ∩A2.

Corollary 5.5. Let {Ai}2i=1 be nonempty closed subsets of a complete partial metric space (X,p) and
T : Y → Y be a given mapping, where Y = A1 ∪A2. Suppose that the following conditions hold:
(i) T(A1) ⊆ A2 and T(A2) ⊆ A1;
(ii) there exists a constant λ ∈ (0, 1) such that

p(Tx,Ty) ≤ λp(x,y), ∀(x,y) ∈ A1 ×A2.(47)

Then T has a fixed point that belongs to A1 ∩A2.

Corollary 5.6. Let {Ai}2i=1 be nonempty closed subsets of a complete partial metric space (X,p) and
T : Y → Y be a given mapping, where Y = A1 ∪A2. Suppose that the following conditions hold:
(i) T(A1) ⊆ A2 and T(A2) ⊆ A1;
(ii) there exists a constant λ ∈ (0, 1/2) such that

p(Tx,Ty) ≤ λ[p(x,Tx) + p(y,Ty)], ∀(x,y) ∈ A1 ×A2.(48)

Then T has a fixed point that belongs to A1 ∩A2.

Corollary 5.7. Let {Ai}2i=1 be nonempty closed subsets of a complete partial metric space (X,p) and
T : Y → Y be a given mapping, where Y = A1 ∪A2. Suppose that the following conditions hold:
(i) T(A1) ⊆ A2 and T(A2) ⊆ A1;
(ii) there exists a constant λ ∈ (0, 1/2) such that

p(Tx,Ty) ≤ λ[p(x,Ty) + p(y,Tx)], ∀(x,y) ∈ A1 ×A2.(49)

Then T has a fixed point that belongs to A1 ∩A2.

Acknowledgments

The first author gratefully acknowledges the University Grants Commission, Government of India for
financial support during the preparation of this manuscript.



48 SHAHI, KAUR AND BHATIA

References

[1] M. Abbas and T. Nazir, Fixed points of generalized weakly contractive mappings in ordered partial metric spaces,

Fixed Point Theory Appl. 2012 (2012), Art. ID 1.
[2] T. Abedelljawad, E. Karapinar and K.Tas, Existence and uniqueness of common fixed point on partial metric spaces,

Appl. Math. Lett. 24 (2011), 1894–1899.

[3] R. P. Agarwal, M. A. Alghamdi and N. Shahzad, Fixed point theory for cyclic generalized contractions in partial
metric spaces, Fixed Point Theory Appl. (2012) (2012), Art. ID 40.

[4] R. P. Agarwal, M. A. El-Gebeily and D. O’ Regan, Generalized contractions in partially ordered metric spaces,

Applicable Analysis 87 (2008), 1–8.
[5] I. Altun and A. Erduran, Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory

Appl. 2011 (2011), Art. ID 508730.
[6] H. Aydi, Some fixed point results in ordered partial metric spaces,J. Nonlinear Sci. Appl. 4 (2011), 210217.

[7] H. Aydi, H.K. Nashine, B. Samet and H. Yazidi, Coincidence and common fixed point results in partially ordered

cone metric spaces and applications to integral equations, Nonlinear Analysis 74 (2011), 6814–6825.
[8] T. G. Bhaskar and V. Lakshmikantham, Fixed Point Theory in partially ordered metric spaces and applications,

Nonlinear Analysis 65 (2006), 1379–1393.

[9] D. Ilić, V. Pavlović and V. Rakocević, Some new extensions of Banach’s contraction principle to partial metric
space, Appl. Math. Lett. 24 (2011), 1326–1330.

[10] E. Karapinar, Fixed point theory for cyclic weak φ-contraction, Appl. Math. Lett. 24 (2011), 822–825.

[11] E. Karapinar and K. Sadaranagni, Fixed point theory for cyclic (φ−ψ)-contractions, Fixed point theory Appl. 2011
(2011), Art. ID 69.

[12] E. Karapinar and B. Samet, Generalized α-ψ-contractive type mappings and related fixed point theorems with

applications, Abstract and Applied Analysis 2012(2012), Article ID 793486.

[13] V. Lakshmikantham and L. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered
metric spaces, Nonlinear Analysis 70 (2009), 4341–4349.

[14] G. S. Matthews, Partial metric topology, Research Report 212, Dept. of Computer Science University of Warwick,

1992.
[15] G. S. Matthews, Partial metric topology, Ann. New York Acad. Sci. 728 (1994), 183–197.

[16] H. K. Nashine, Z. Kadelburg and S. Radenović, Common fixed point theorems for weakly isotone increasing mappings

in ordered partial metric spaces, Math. Comput. Modelling (2012), doi:10.1016/j.mcm.2011.12.019.
[17] J. J. Nieto and R. R. Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary

differential equations, Order 22 (2005), 223–239.

[18] M. Pacurar and I. A. Rus, Fixed point theory for cyclic ϕ-contractions, Nonlinear Anal. 72 (2010), 1181–1187.
[19] M. A. Petric, Some results concerning cyclic contractive mappings, General Mathematics 18 (2010), 213–226.

[20] A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to

matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435–1443.
[21] I. A. Rus, Cyclic representations and fixed points, Ann. T. Popovicin. Seminar Funct. Eq. Approx. Convexity 3

(2005), 171–178.
[22] B. Samet, M. Rajović, R. Lazović and R. Stoiljković, Common fixed point results for nonlinear contractions in

ordered partial metric spaces, Fixed Point Theory Appl. 2011 (2011), Art. ID 71.

[23] B. Samet, C. Vetro and P. Vetro, Fixed point theorem for α-ψ contractive type mappings, Nonlinear Anal. 75
(2012), 2154–2165.

[24] M. Turinici, Abstract comparison principles and multivariable Gronwall-Bellman inequalities, J. Math. Anal. Appl.
117 (1986), 100–127.

School of Mathematics and Computer Applications, Thapar University, Patiala-147004, India

∗Corresponding author: priya.thaparian@gmail.com