International Journal of Analysis and Applications
ISSN 2291-8639
Volume 11, Number 2 (2016), 168-182
http://www.etamaths.com

ON MULTI-VALUED WEAKLY PICARD OPERATORS IN HAUSDORFF

METRIC-LIKE SPACES

ABDELBASSET FELHI1,2,∗

Abstract. In this paper, we study multi-valued weakly Picard operators on Hausdorff metric-like

spaces. Our results generalize some recent results and extend several theorems in the literature.

Some examples are presented making effective our results.

1. Introduction and preliminaries

Let (X,d) be a metric space and CB(X) denotes the collection of all nonempty, closed and bounded
subsets of X. Also, CL(X) denotes the collection of nonempty closed subsets of X. For A,B ∈ CB(X),
define

H(A,B) := max

{
sup
a∈A

d(a,B), sup
b∈B

d(b,A)

}
,

where d(x,A) := inf{d(x,a) : a ∈ A} is the distance of a point x to the set A. It is known that H
is a metric on CB(X), called the Hausdorff metric induced by d. Throughout the paper, N, R, and
R+ denote the set of positive integers, the set of all real numbers and the set of all non-negative real
numbers, respectively.

Definition 1.1. ([1]) Let (X,d) be a metric space and T : X → CL(X) be a multi-valued operator.
We say that T is a multi-valued weakly Picard operator (MWP operator) if for all x ∈ X and y ∈ Tx,
there exists a sequence {xn} such that:

(i) x0 = x,x1 = y;
(ii) xn+1 ∈ Txn for all n = 0, 1, 2, . . . ;
(iii) the sequence {xn} is convergent and its limit is a fixed point of T.

The theory of MWP operators is studied by several authors (see for instance [1, 2]). In 2008 Suzuki
[3] generalizes the Banach contraction principle by introducing a new type of mapping. Very recently,
Jleli et al. [4] established Kikkawa-Suzuki type fixed point theorems for a new type of generalized
contractive conditions on partial Hausdorff metric spaces. The purpose of this paper is to discuss
multi-valued weakly Picard operators on partial Hausdorff metric spaces and on Hausdorff metric-like
spaces. We will establish the above fixed point theorems for a new type of generalized contractive
conditions which generalizes that of Jleli et al.

We recall that the study of fixed points for multi-valued contractions using the Hausdorff metric
was initiated by Nadler [18] who proved the following theorem.

Theorem 1.2. ([18]) Let (X,d) be a complete metric space and T : X → CB(X) be a multi-valued
mapping satisfying H(Tx,Ty) ≤ kd(x,y) for all x,y ∈ X and for some k in [0, 1). Then there exists
x ∈ X such that x ∈ Tx.

We recall that the notion of partial metric spaces was introduced by Matthews [8] in 1994 as a part
to study the denotational semantics of dataflow networks which play an important role in constructing
models in the theory of computation. Moreover, the notion of metric-like spaces has been discovered
by Amini-Harandi [12] which is an interesting generalization of the notion of partial metric spaces. For
more fixed point results on metric-like spaces, see [7], [10], [11], [13], [15], [16], [17], [19], [20], [21], [22].

2010 Mathematics Subject Classification. 47H10, 54H25.

Key words and phrases. Hausdorff metric-like: multi-valued operator; partial metric space; metric-like space.

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.

168



ON MULTI-VALUED WEAKLY PICARD OPERATORS 169

Note that, every partial metric space is a metric-like space but the converse is not true in general. In
what follows, we recall some definitions and results we will need in the sequel.

Definition 1.3. ([8]) A partial metric on a nonempty set X is a function p : X ×X → [0,∞) such
that for all x,y,z ∈ X
(PM1) p(x,x) = p(x,y) = p(y,y), then x = y;
(PM2) p(x,x) ≤ p(x,y);
(PM3) p(x,y) = p(y,x);
(PM4) p(x,z) + p(y,y) ≤ p(x,y) + p(y,z).
The pair (X,p) is then called a partial metric space (PMS).

According to [8], each partial metric p on X generates a T0 topology τp on X which has as a base the
family of 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. Following [8], several topological concepts can be defined as follows. A sequence
{xn} in a partial metric space (X,p) converges to a point x ∈ X if and only if p(x,x) = limn→∞p(xn,x)
and is called a Cauchy sequence if limn,m→∞p(xn,xm) exists and is finite. Moreover, a partial metric
space (X,p) is called 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). It is known [8] that if p is a partial metric
on X, then the function ps : X ×X → R+ defined by

ps(x,y) = 2p(x,y) −p(x,x) −p(y,y)
for all x,y ∈ X, is a metric on X.

Note that if a sequence converges in a partial metric space (X,p) with respect to τps, then it
converges with respect to τp.

Also, a sequence {xn} in a partial metric space (X,p) is Cauchy if and only if it is a Cauchy sequence
in the metric space (X,ps). Consequently, a partial metric space (X,p) is complete if and only if the
metric space (X,ps) is complete. Moreover, if {xn} is a sequence in a partial metric space (X,p) and
x ∈ X, one has that

lim
n→∞

ps(xn,x) = 0 ⇔ p(x,x) = lim
n→∞

p(xn,x) = lim
n,m→∞

p(xn,xm).

We have the following lemmas.

Lemma 1.4. Let (X,p) be a partial metric space. Then,
(1) if p(x,y) = 0 then, x = y,
(2) if x 6= y then, p(x,y) > 0.

Following [9], let (X,p) be a partial metric space and CBp(X) be the family of all nonempty, closed
and bounded subsets of the partial metric space (X,p), induced by the partial metric p.

For A,B ∈ CBp(X) and x ∈ X, we define
p(x,A) = inf{p(x,a) : a ∈ A}, Hp(A,B) = max{sup

a∈A
p(a,B), sup

b∈B
p(b,A)}.

Lemma 1.5. ([5]) Let (X,p) be a partial metric space and A any nonempty set in (X,p), then a ∈ A
if and only if p(a,A) = p(a,a), where A denotes the closure of A with respect to the partial metric p.

Proposition 1.6. ([9]) Let (X,p) be a partial metric space. For all A,B,C ∈ CBp(X), we have
(h1) Hp(A,A) ≤ Hp(A,B);
(h2) Hp(B,A) = Hp(A,B);
(h3) Hp(A,B) ≤ Hp(A,C) + Hp(C,B) − infc∈C p(c,c);
(h4) Hp(A,B) = 0 ⇒ A = B.

Definition 1.7. Let X be a nonempty set. A function σ : X × X → R+ is said to be a metric-like
(dislocated metric) on X if for any x,y,z ∈ X, the following conditions hold:

(P1) σ(x,y) = 0 =⇒ x = y;
(P2) σ(x,y) = σ(y,x);
(P3) σ(x,z) ≤ σ(x,y) + σ(y,z).

The pair (X,σ) is then called a metric-like (dislocated metric) space.



170 FELHI

In the following example, we give a metric-like which is neither a metric nor a partial metric.

Example 1.8. Let X = {0, 1, 2} and σ : X ×X → R+ defined by

σ(0, 0) = σ(1, 1) = 0, σ(2, 2) = 3, σ(0, 1) = 1, σ(0, 2) = σ(1, 2) = 2,

and σ(x,y) = σ(y,x) for all x ∈ X. Then, (X,σ) is a metric-like space. Note that σ is nor a metric as,
σ(2, 2) > 0 and not a partial metric on X as, σ(2, 2) > σ(0, 2).

Each metric-like σ on X generates a T0 topology τσ on X which has as a base the family open
σ-balls {Bσ(x,ε) : x ∈ X,ε > 0}, where Bσ(x,ε) = {y ∈ X : |σ(x,y)−σ(x,x)| < ε}, for all x ∈ X and
ε > 0.

Observe that a sequence {xn} in a metric-like space (X,σ) converges to a point x ∈ X, with respect
to τσ, if and only if σ(x,x) = lim

n→∞
σ(x,xn).

Definition 1.9. Let (X,σ) be a metric-like space.

(a) A sequence {xn} in X is said to be a Cauchy sequence if lim
n,m→∞

σ(xn,xm) exists and is finite.

(b) (X,σ) is said to be complete if every Cauchy sequence {xn} in X converges with respect to τσ
to a point x ∈ X such that lim

n→∞
σ(x,xn) = σ(x,x) = lim

n,m→∞
σ(xn,xm).

We have the following trivial inequality:

(1.1) σ(x,x) ≤ 2σ(x,y) for all x,y ∈ X.

Very recently, Aydi et al. [6] introduced the concept of Hausdorff metric-like. Let CBσ(X) be the
family of all nonempty, closed and bounded subsets of the metric-like space (X,σ), induced by the
metric-like σ. Note that the boundedness is given as follows: A is a bounded subset in (X,σ) if there
exist x0 ∈ X and M ≥ 0 such that for all a ∈ A, we have a ∈ Bσ(x0,M), that is,

|σ(x0,a) −σ(a,a)| < M.

The Closeness is taken in (X,τσ) (where τσ is the topology induced by σ).
For A,B ∈ CBσ(X) and x ∈ X, define

σ(x,A) = inf{σ(x,a), a ∈ A}, δσ(A,B) = sup{σ(a,B) : a ∈ A} and
δσ(B,A) = sup{σ(b,A) : b ∈ B}.

We have the the following useful lemmas.

Lemma 1.10. [6]
Let (X,σ) be a metric-like space and A any nonempty set in (X,σ), then

if σ(a,A) = 0, then a ∈ Ā,

where A denotes the closure of A with respect to the metric-like σ. Also, if {xn} is a sequence in (X,σ)
that is τσ-convergent to x ∈ X, then

lim
n→∞

|σ(xn,A) −σ(x,A)| = σ(x,x).

Lemma 1.11. Let A,B ∈ CBσ(X) and a ∈ A. Suppose that σ(a,B) > 0. Then, for each h > 1, there
exists b = b(a) ∈ B such that σ(a,b) < hσ(a,B).

Proof. We argue by contradiction, that is, there exists h > 1, such that for all b ∈ B, there is
σ(a,b) ≥ hσ(a,B). Then, σ(a,B) = inf{σ(a,B) : b ∈ B} ≥ hσ(a,B). Hence, h ≤ 1, which is a
contradiction. �

Let (X,σ) be a metric-like space. For A,B ∈ CBσ(X), define

Hσ(A,B) = max{δσ(A,B),δσ(B,A)} .

We have also some properties of Hσ : CB
σ(X) ×CBσ(X) → [0,∞).



ON MULTI-VALUED WEAKLY PICARD OPERATORS 171

Proposition 1.12. [6] Let (X,σ) be a metric-like space. For any A,B,C ∈ CBσ(X), we have the
following:

(i) : Hσ(A,A) = δσ(A,A) = sup{σ(a,A) : a ∈ A};
(ii) : Hσ(A,B) = Hσ(B,A);

(iii) : Hσ(A,B) = 0 implies that A = B;

(iv) : Hσ(A,B) ≤ Hσ(A,C) + Hσ(C,B).

The mapping Hσ : CB
σ(X) × CBσ(X) → [0, +∞) is called a Hausdorff metric-like induced by

σ. Note that each partial hausdorff metric is a Hausdorff metric-like but the converse is not true in
general as it is clear from the following example.

Example 1.13. Going back to Example 1.8, taking A = {2}, B = {0} we have Hσ(A,A) = σ(2, 2) =
3 > 2 = σ(0, 2) = Hσ(A,B).

We denote by Ψ the class of all functions ψ : R+ → R+ satisfying
(ψ1) ψ is nondecreasing;

(ψ2)
∑
n

ψn(t) < ∞ for each t ∈ R+, where ψn is the n−th iterate of ψ.

Also, we denote by Φ the class of all functions ϕ : R+ → R+ satisfying
(ϕ1) ϕ is nondecreasing;
(ϕ2) t ≤ ϕ(t) for each t ∈ R+.

Lemma 1.14. (i) If ψ ∈ Ψ, then ψ(t) < t for any t > 0 and ψ(0) = 0.
(ii) If ϕ ∈ Φ, then t ≤ ϕn(t) for all n ∈ N∪{0} and for any t ∈ R+.

We have the following useful lemma.

Lemma 1.15. Let (X,σ) be a metric-like space, B ∈ CBσ(X) and c > 0. If a ∈ X and σ(a,B) < c
then there exists b = b(a) ∈ B such that σ(a,b) < c.

Proof. We argue by contradiction, that is, σ(a,b) ≥ c for all b ∈ B, then σ(a,B) = inf{σ(a,b) : b ∈
B}≥ c, which is a contradiction. Hence there exists b = b(a) ∈ B such that σ(a,b) < c. �

2. Main Results

In this section, we give some fixed point results on metric-like spaces first and next we give some
fixed point results on partial metric spaces.

Now, we need the following definition.

Definition 2.1. Let (X,σ) be a metric-like space. A multi-valued mapping T : X → CBσ(X) is said
to be (ϕ,ψ)−contractive multi-valued operator if there exist ϕ ∈ Φ and ψ ∈ Ψ such that

(2.1) σ(y,Tx) ≤ ϕ(σ(y,x)) ⇒ Hσ(Tx,Ty) ≤ ψ(Mσ(x,y))

for all x,y ∈ X, where

Mσ(x,y) = max{σ(x,y),σ(x,Tx),σ(y,Ty),
1

4
[σ(x,Ty) + σ(Tx,y)]}.

Now, we state and prove our first main result.

Theorem 2.2. Let (X,σ) be a complete metric-like space and T : X → CBσ(X) be (ϕ,ψ)−contractive
multi-valued operator.

If 2t ≤ ϕ(t) for each t ∈ R+, then T is an MWP operator.

Proof. Let x0 ∈ X and x1 ∈ Tx0. Let c a given real number such that σ(x0,x1) < c.
Clearly, if x1 = x0 or x1 ∈ Tx1, we conclude that x1 is a fixed point of T and so the proof is finished.

Now, we assume that x1 6= x0 and x1 6∈ Tx1. So then, σ(x0,x1) > 0 and σ(x1,Tx1) > 0.
Since x1 ∈ Tx0 and 2t ≤ ϕ(t), we get

σ(x1,Tx0) ≤ σ(x1,x1) ≤ 2σ(x1,x0) ≤ ϕ(σ(x1,x0)).



172 FELHI

Hence by (2.1) and triangular inequality, we have

0 < σ(x1,Tx1) ≤ Hσ(Tx0,Tx1) ≤ ψ(Mσ(x0,x1))

≤ ψ(max{σ(x0,x1),σ(x0,Tx0),σ(x1,Tx1),
1

4
[σ(x0,Tx1) + σ(x1,Tx0)]})

≤ ψ(max{σ(x0,x1),σ(x0,x1),σ(x1,Tx1),
1

4
[σ(x0,Tx1) + σ(x1,x1)]})

≤ ψ(max{σ(x0,x1),σ(x0,x1),σ(x1,Tx1),
1

4
[σ(x1,Tx1) + 3σ(x0,x1)]})

= ψ(max{σ(x0,x1),σ(x1,Tx1)}).

If max{σ(x0,x1),σ(x1,Tx1)} = σ(x1,Tx1), then we obtain

0 < σ(x1,Tx1) ≤ ψ(σ(x1,Tx1)) < σ(x1,Tx1)

wish is a contradiction. Then

0 < σ(x1,Tx1) ≤ ψ(σ(x0,x1)) < ψ(c).

Thus, by Lemma 1.15, there exist x2 ∈ Tx1 such that

σ(x1,x2) < ψ(c).(2.2)

If x1 = x2 or x2 ∈ Tx2, we conclude that x2 is a fixed point of T and so the proof is finished. Now,
we assume that x2 6= x1 and x2 6∈ Tx2. Then we have

σ(x2,Tx1) ≤ σ(x2,x2) ≤ 2σ(x2,x1) ≤ ϕ(σ(x2,x1)).

Hence by (2.1), triangular inequality and (2.2), we have

0 < σ(x2,Tx2) ≤ Hσ(Tx1,Tx2) ≤ ψ(Mσ(x1,x2)) ≤ ψ(max{σ(x1,x2),σ(x2,Tx2)})
= ψ(σ(x1,x2)) < ψ

2(c).

Then, by Lemma 1.15, there exist x3 ∈ Tx2 such that

σ(x2,x3) < ψ
2(c).(2.3)

Continuing in this fashion, we construct a sequence {xn} in X such that for all n ∈ N
(i) xn 6∈ Txn, xn 6= xn+1, xn+1 ∈ Txn;
(ii)

(2.4) σ(xn,xn+1) ≤ ψn(c).

Now, for m > n, we have

σ(xn,xm) ≤
m−1∑
i=n

σ(xi,xi+1) ≤
m−1∑
i=n

ψi(c) ≤
∞∑
i=n

ψi(c) → 0 as n →∞.

Thus,

lim
n,m→∞

σ(xn,xm) = 0.(2.5)

Hence, {xn} is σ−Cauchy. Moreover since (X,σ) is complete, it follows there exists ν ∈ X such that

lim
n→∞

σ(xn,ν) = σ(ν,ν) = lim
n,m→∞

σ(xn,xm) = 0.(2.6)

We will show that ν is a fixed point of T. First, we should prove that there exits a subsequence {xn(k)}
of {xn} such that

σ(ν,Txn(k)) ≤ ϕ(σ(ν,xn(k))), for all k = 0, 1, 2, . . .(2.7)

Arguing by contradiction, that is, there exists N ∈ N such that σ(ν,Txn) > ϕ(σ(ν,xn)) for all n ≥ N.
Since xn+1 ∈ Txn, it follows that σ(ν,xn+1) > ϕ(σ(ν,xn)) for all n ≥ N. Having ϕ nondecreasing, so
by induction we get

σ(ν,xn+m) > ϕ
m(σ(ν,xn)), for all n ≥ N and m = 1, 2, 3, . . .(2.8)



ON MULTI-VALUED WEAKLY PICARD OPERATORS 173

Now, for all n ≥ N and m ∈ N, we have

σ(xn,xn+m) ≤
n+m−1∑
i=n

σ(xi,xi+1) ≤
m−1∑
i=n

ψi(c) ≤
∞∑
i=n

ψi(c).

Then for all n ≥ N and m ∈ N, we obtain

σ(ν,xn) ≤ σ(ν,xn+m) + σ(xn+m,xn) ≤ σ(ν,xn+m) +
∞∑
i=n

ψi(c).

Passing to the limit as m →∞, we get

σ(ν,xn) ≤
∞∑
i=n

ψi(c).

This implies that for all n ≥ N and m ∈ N,

σ(ν,xn+m) ≤
∞∑

i=n+m

ψi(c).(2.9)

Combining (2.8) and (2.9), we have

σ(ν,xn) ≤ ϕm(σ(ν,xn)) < σ(ν,xn+m) ≤
∞∑

i=n+m

ψi(c).

Then for all n ≥ N and m ∈ N, we obtain

σ(ν,xn) <

∞∑
i=n+m

ψi(c).(2.10)

Letting m →∞ in (2.10), we get σ(ν,xn) = 0 for all n ≥ N and so, σ(ν,xn+m) = 0 for all n ≥ N and
m ∈ N. Using (2.8), we have 0 ≤ ϕm(0) < 0, which is a contradiction. Therefore, (2.7) holds.

Now, we will show that σ(ν,Tν) = 0. Suppose in the contrary, that is σ(ν,Tν) > 0.
By (2.1) and (2.7), we have for all k ∈ N
σ(ν,Tν) ≤ σ(ν,xn(k)+1) + σ(xn(k)+1,Tν) ≤ σ(ν,xn(k)+1) + Hσ(Txn(k),Tν)

≤ σ(ν,xn(k)+1) + ψ(Mσ(xn(k),ν))
≤ σ(ν,xn(k)+1)

+ ψ(max{σ(xn(k),ν),σ(xn(k),Txn(k)),σ(ν,Tν),
1

4
[σ(xn(k),Tν) + σ(ν,Txn(k))]})

≤ σ(ν,xn(k)+1)

+ ψ(max{σ(xn(k),ν),σ(xn(k),xn(k)+1),σ(ν,Tν),
1

4
[σ(xn(k),Tν) + σ(ν,xn(k)+1)]}).

We know that

lim
k→∞

σ(xn(k),ν) = lim
k→∞

σ(xn(k),xn(k)+1) = lim
k→∞

σ(xn(k)+1,ν) = 0, lim
k→∞

σ(xn(k),Tν) = σ(ν,Tν).

Then there exists N ∈ N such that for all k ≥ N

max{σ(xn(k),ν),σ(xn(k),xn(k)+1),σ(ν,Tν),
1

4
[σ(xn(k),Tν) + σ(ν,xn(k)+1)]}) = σ(ν,Tν).

It follows that for all k ≥ N
0 < σ(ν,Tν) ≤ σ(ν,xn(k)+1) + ψ(σ(ν,Tν)).

Passing to the limit as k →∞, we get
0 < σ(ν,Tν) ≤ ψ(σ(ν,Tν)) < σ(ν,Tν)

which is a contradiction. Hence σ(ν,Tν) = 0 and so, by Lemma 1.10 we have ν ∈ Tν = Tν, that is ν
is a fixed point of T. �

We give an example to illustrate the utility of Theorem 2.2.



174 FELHI

Example 2.3. Let X = {0, 1, 2} and σ : X ×X → R+ defined by:
σ(0, 0) = 0, σ(1, 1) = 3, σ(2, 2) = 1

σ(0, 1) = σ(1, 0) = 7, σ(0, 2) = σ(2, 0) = 3, σ(1, 2) = σ(2, 1) = 4.

Then (X,σ) is a complete metric-like space. Note that σis not a partial metric on X because σ(0, 1) �
σ(2, 0) + σ(2, 1) −σ(2, 2).
Define the map T : X → CBσ(X) by

T0 = T2 = {0} and T1 = {0, 2}.
Note that Tx is bounded and closed for all x ∈ X in metric-like space (X,σ). Take ϕ(t) = st with
s ≥ 7 and ψ(t) = rt with r ∈ [ 3

4
, 1).

It is easy tho show that

max{σ(y,Tx), x,y ∈ X} = σ(1, 0) = 7 ≤ 7 min{σ(y,x), x,y ∈ X, (x,y) 6= (0, 0)}
≤ ϕ(min{σ(y,x), x,y ∈ X, (x,y) 6= (0, 0)}).

This implies that, for all x,y ∈ X with (x,y) 6= (0, 0)
σ(y,Tx) ≤ ϕ(σ(y,x)).

Now, we shall show that for all x,y ∈ X with (x,y) 6= (0, 0)
Hσ(Tx,Ty) ≤ ψ(Mσ(x,y)).(2.11)

For this, we consider the following cases:
case1 : x,y ∈{0, 2}. We have

Hσ(Tx,Ty) = σ(0, 0) = 0 ≤ ψ(Mσ(x,y)).
case2 : x ∈{0, 2}, y = 1. We have

Hσ(Tx,Ty) = Hσ({0},{0, 2}) = max{σ(0,{0, 2}),max{σ(0, 0),σ(0, 2)}}

= max{0, 3} = 3 ≤
3

4
σ(x,y) ≤ ψ(Mσ(x,y)).

case3 : x = y = 1. We have

Hσ(Tx,Ty) = Hσ({0, 2},{0, 2}) = max{σ(0,{0, 2}),σ(2,{0, 2})}

= min{σ(0, 2),σ(2, 2)} = 1 ≤
3

4
σ(1, 1) ≤ ψ(Mσ(x,y)).

Note that (2.11) is also true for (x,y) = (0, 0). Then, all the required hypotheses of Theorem 2.2 are
satisfied. Here, x = 0 is the unique fixed point of T

We state the following corollaries as consequences of Theorem 2.2.

Corollary 2.4. Let (X,σ) be a complete metric-like space and T : X → CBσ(X) be a multi-valued
mapping. Assume that there exist ϕ ∈ Φ and ψ ∈ Ψ such that, for all x,y ∈ X

Hσ(Tx,Ty) ≤ ψ(Mσ(x,y)) −ϕ(σ(y,x)) + σ(y,Tx),(2.12)
where Mσ(x,y) = max{σ(x,y),σ(x,Tx),σ(y,Ty), 14 [σ(x,Ty) + σ(Tx,y)]}.

If 2t ≤ ϕ(t) for each t ∈ R+, then T is an MWP operator.

Proof. Let x,y ∈ X such that σ(y,Tx) ≤ ϕ(σ(y,x)). Then, if (2.12) holds, we have
Hσ(Tx,Ty) ≤ ψ(Mσ(x,y)) −ϕ(σ(y,x)) + σ(y,Tx) ≤ ψ(Mσ(x,y)).

Thus, the proof is concluded by Theorem 2.2. �

Corollary 2.5. Let (X,σ) be a complete metric-like space and T : X → CBσ(X) be a multi-valued
mapping. Assume that there exist r ∈ [0, 1) and s ≥ 2 such that, for all x,y ∈ X

σ(y,Tx) ≤ sσ(y,x) ⇒ Hσ(Tx,Ty) ≤ rMσ(x,y),
where Mσ(x,y) = max{σ(x,y),σ(x,Tx),σ(y,Ty), 14 [σ(x,Ty) + σ(Tx,y)]}.

Then T is an MWP operator.

Proof. It suffice to take ϕ(t) = st and ψ(t) = rt in Theorem 2.2. �



ON MULTI-VALUED WEAKLY PICARD OPERATORS 175

Corollary 2.6. Let (X,σ) be a complete metric-like space and T : X → CBσ(X) be a multi-valued
mapping. Assume that there exist r ∈ [0, 1) and s ≥ 2 such that, for all x,y ∈ X

σ(y,Tx) ≤ sσ(y,x) ⇒ Hσ(Tx,Ty) ≤ r max{σ(x,y),σ(x,Tx),σ(y,Ty)}.

Then T is an MWP operator.

Corollary 2.7. Let (X,σ) be a complete metric-like space and T : X → CBσ(X) be a multi-valued
mapping. Assume that there exist r ∈ [0, 1) and s ≥ 2 such that, for all x,y ∈ X

σ(y,Tx) ≤ sσ(y,x) ⇒ Hσ(Tx,Ty) ≤
r

3
{σ(x,y) + σ(x,Tx) + σ(y,Ty)}.

Then T is an MWP operator.

Corollary 2.8. [6] Let (X,σ) be a complete metric-like space. If T : X → CBσ(X) is a multi-valued
mapping such that for all x,y ∈ X, we have

(2.13) Hσ(Tx,Ty) ≤ k M(x,y),

where k ∈ [0, 1) and

M(x,y) = max

{
σ(x,y),σ(x,Tx),σ(y,Ty),

1

4
(σ(x,Ty) + σ(y,Tx))

}
.

Then T has a fixed point.

Proof. Let ϕ(t) = 2t and ψ(t) = kt. Then, if (2.13) holds, we have

Hσ(Tx,Ty) ≤ ψ(M(x,y)),

for all x,y ∈ X satisfying σ(y,Tx) ≤ 2σ(y,x). Thus, the proof is concluded by Theorem 2.2. �

If T is a single-valued mapping, we deduce the following results.

Corollary 2.9. Let (X,σ) be a complete metric-like space and T : X → X be a mapping. Assume
that there exist ϕ ∈ Φ and ψ ∈ Ψ such that, for all x,y ∈ X

σ(y,Tx) ≤ ϕ(σ(y,x)) ⇒ σ(Tx,Ty) ≤ ψ(Mσ(x,y)),

where Mσ(x,y) = max{σ(x,y),σ(x,Tx),σ(y,Ty), 14 [σ(x,Ty) + σ(Tx,y)]}.
If 2t ≤ ϕ(t) for each t ∈ R+ and if ψ(2t) < t for each t > 0, then T has a unique fixed point.

Proof. The existence follows immediately from Theorem 2.2. Thus, we need to prove uniqueness of
fixed point. We assume that there exist x,y ∈ X such that x = Tx and y = Ty with x 6= y.

Since σ(y,Tx) = σ(y,x) ≤ ϕ(σ(y,x)), then by (2.1) and since ψ(2t) < t , we get

0 < σ(x,y) = σ(Tx,Ty) ≤ ψ(max{σ(x,y),σ(x,Tx),σ(y,Ty),
1

4
[σ(x,Ty) + σ(Tx,y)]})

= ψ(max{σ(x,y),σ(x,x),σ(y,y),
1

2
σ(x,y)})

≤ ψ(2σ(x,y)) < σ(x,y).

which is a contradiction. Hence x = y, so the uniqueness of the fixed point of T.
�

Corollary 2.10. Let (X,σ) be a complete metric-like space and T : X → X be a mapping. Assume
that there exist r ∈ [0, 1

2
) and s ≥ 2 such that, for all x,y ∈ X

σ(y,Tx) ≤ s(σ(y,x)) ⇒ σ(Tx,Ty) ≤ r(Mσ(x,y)),

where Mσ(x,y) = max{σ(x,y),σ(x,Tx),σ(y,Ty), 14 [σ(x,Ty) + σ(Tx,y)]}.
Then T has a unique fixed point.

Now, we need the following definition.



176 FELHI

Definition 2.11. Let (X,σ) be a metric-like space. A multi-valued mapping T : X → CBσ(X) is
said to be (r,s)−contractive multi-valued operator if there exist r,s ∈ [0, 1), such that

(2.14)
1

1 + r
σ(x,Tx) ≤ σ(y,x) ≤

1

1 −s
σ(x,Tx) ⇒ Hσ(Tx,Ty) ≤ rMσ(x,y)

for all x,y ∈ X, where

Mσ(x,y) = max{σ(x,y),σ(x,Tx),σ(y,Ty),
1

4
[σ(x,Ty) + σ(Tx,y)]}.

We give the following result.

Theorem 2.12. Let (X,σ) be a complete metric-like space and T : X → CBσ(X) be (r,s)−contractive
multi-valued operator with r < s. Then T is an MWP operator.

Proof. Let r1 be a real number such that 0 ≤ r ≤ r1 < s. Let x0 ∈ X. Clearly, if x0 ∈ Tx0, then x0 is
a fixed point of T and so, the proof is finished. Now, we assume that x0 6∈ Tx0. Then σ(x0,Tx0) > 0.
By Lemma 1.11, there exists x1 ∈ Tx0 such that

σ(x0,x1) ≤
1 −r1
1 −s

σ(x0,Tx0).

If x1 ∈ Tx1, then x1 is a fixed point of T and also, the proof is finished. Now, we assume that x1 6∈ Tx1.
Then σ(x1,Tx1) > 0. Since

1

1 + r
σ(x0,Tx0) ≤ σ(x0,x1)) ≤

1 −r1
1 −s

σ(x0,Tx0),

then, by (2.14), we have

σ(x1,Tx1) ≤ Hσ(Tx0,Tx1) ≤

r max{σ(x0,x1),σ(x0,Tx0),σ(x1,Tx1),
1

4
[σ(x0,Tx1) + σ(x1,Tx0)]}

≤ r max{σ(x0,x1),σ(x0,x1),σ(x1,Tx1),
1

4
[σ(x1,Tx1) + 3σ(x0,x1)]}

≤ r max{σ(x0,x1),σ(x1,Tx1)}.

If max{σ(x0,x1),σ(x1,Tx1)} = σ(x1,Tx1), then we obtain σ(x1,Tx1) ≤ rσ(x1,Tx1) < σ(x1,Tx1),
which is a contradiction. Thus, we get

σ(x1,Tx1) ≤ rσ(x0,x1).

By Lemma 1.11, there exists x2 ∈ Tx1 such that

σ(x1,x2) ≤
r1
r
σ(x1,Tx1) and σ(x1,x2) ≤

1 −r1
1 −s

σ(x1,Tx1).

This implies that

σ(x1,x2) ≤ r1σ(x0,x1) and σ(x1,x2) ≤
1 −r1
1 −s

σ(x1,Tx1).

It follows that

1

1 + r
σ(x1,Tx1) ≤ σ(x1,Tx2) ≤

1

1 −s
σ(x1,Tx1).

Then, by (2.14), we get σ(x2,Tx2) ≤ rσ(x1,x2). Continuing this process, we construct a sequence
{xn} in X such that

(i) xn+1 ∈ Txn;
(ii) σ(xn,Txn) ≤ rσ(xn−1,xn);
(iii) σ(xn,xn+1) ≤ r1σ(xn−1,xn);
(iv) σ(xn,xn+1) ≤ 1−r11−s σ(xn,Txn)



ON MULTI-VALUED WEAKLY PICARD OPERATORS 177

for all n = 1, 2, . . .
Since σ(xn,xn+1) ≤ r1σ(xn−1,xn), by induction we obtain

σ(xn,xn+1) ≤ rn1 σ(x0,x1) for all n = 1, 2, . . .
Now, for m > n, we have

σ(xn,xm) ≤
m−1∑
i=n

σ(xi,xi+1) ≤ σ(x0,x1)
m−1∑
i=n

ri1 ≤ σ(x0,x1)
∞∑
i=n

ri1 → 0 as n →∞.

Thus,

lim
n,m→∞

σ(xn,xm) = 0.(2.15)

Hence, {xn} is σ−Cauchy. Moreover since (X,σ) is complete, it follows there exists z ∈ X such that
lim
n→∞

σ(xn,z) = σ(z,z) = lim
n,m→∞

σ(xn,xm) = 0.(2.16)

For all m,n ∈ N, we have
σ(xn,xn+m) ≤ σ(xn,xn+1) + σ(xn+1,xn+2) + . . . + σ(xn+m−1,xn+m)

≤ [1 + r1 + r21 + . . . + r
m−1
1 ]σ(xn,xn+1) =

1 −rm1
1 −r1

σ(xn,xn+1).

It follows that for all m,n ∈ N

σ(xn,z) ≤ σ(xn,xn+m) + σ(xn+m,z) ≤ σ(xn+m,z) +
1 −rm1
1 −r1

σ(xn,xn+1).

Passing to limit as m →∞, we get for all n ∈ N

σ(xn,z) ≤
1

1 −r1
σ(xn,xn+1) ≤

1

1 −r1
·

1 −r1
1 −s

σ(xn,Txn) =
1

1 −s
σ(xn,Txn).

Thus, we have for all n ∈ N

σ(xn,z) ≤
1

1 −s
σ(xn,Txn).(2.17)

Now, we assume that there exists N ∈ N such that
1

1 + r
σ(xn,Txn) > σ(xn,z)

for all n ≥ N. Then we have

σ(xn,xn+1) ≤ σ(xn,z) + σ(z,xn+1) <
1

1 + r
[σ(xn,Txn) + σ(xn+1,Txn+1)]

<
1

1 + r
[σ(xn,xn+1) + rσ(xn,xn+1)] = σ(xn,xn+1).

which is a contradiction. Thus, there exists a subsequence {xn(k)} of {xn} such that
1

1 + r
σ(xn(k),Txn(k)) ≤ σ(xn(k),z)(2.18)

for all k ∈ N. Now, we should show that z is a fixed point of T.
Using (2.17), (2.18) and (2.14), we have for all k ∈ N

σ(xn(k)+1,Tz) ≤ Hσ(Txn(k),Tz) ≤

r max{σ(xn(k),z),σ(xn(k),Txn(k)),σ(z,Tz),
1

4
[σ(xn(k),Tz) + σ(z,Txn(k))]}≤

r max{σ(xn(k),z),σ(xn(k),xn(k)+1),σ(z,Tz),
1

4
[σ(xn(k),Tz) + σ(z,xn(k)+1)]}.

Passing to limit as k →∞, we get
σ(z,Tz) ≤ rσ(z,Tz).

Since r < 1, it follows that σ(z,Tz) = 0. Thus, by Lemma 1.10 we obtain z ∈ Tz, that is, z is a fixed
point of T. �



178 FELHI

Corollary 2.13. Let (X,σ) be a complete metric-like space and T : X → X be a mapping. Assume
that there exist r ∈ [0, 1) such that, for all x,y ∈ X

1

1 + r
σ(x,Tx) ≤ σ(x,y) ≤

1

1 −r
σ(x,Tx) ⇒ σ(Tx,Ty) ≤ rMσ(x,y),

where Mσ(x,y) = max{σ(x,y),σ(x,Tx),σ(y,Ty), 14 [σ(x,Ty) + σ(Tx,y)]}.
Then T has a fixed point.

Proof. Let x0 ∈ X. Define the sequence {xn} by xn+1 = Txn for all n = 0, 1, 2, . . . We have for all
n = 0, 1, 2, . . .

1

1 + r
σ(xn,Txn) ≤ σ(xn,xn+1) ≤

1

1 −r
σ(xn,Txn)

It follows that for all n = 0, 1, 2, . . .

σ(xn+1,xn+2) = σ(Txn,Txn+1) ≤ rσ(xn,xn+1).
Thus the sequence {xn} is Cauchy in (X,σ). By completeness of (X,σ) there exists z ∈ X such that

lim
n→∞

σ(xn,z) = σ(z,z) = lim
n,m→∞

σ(xn,xm) = 0.(2.19)

We have for all n,m ∈ N

σ(xn,xn+m) ≤
1 −rm

1 −r
σ(xn,xn+1).

It follows that

σ(xn,z) ≤ σ(xn,xn+m) + σ(xn+m,z) ≤
1 −rm

1 −r
σ(xn,xn+1) + σ(xn+m,z).

Passing to limit as m →∞, we get

σ(xn,z) ≤
1

1 −r
σ(xn,xn+1)

Proceeding as in the proof of Theorem 2.12, we can find a subsequence {xn(k)} of {xn} such that
1

1 + r
σ(xn(k),Txn(k)) ≤ σ(xn(k),z)(2.20)

for all k ∈ N. Then as in the proof of Theorem 2.12 we get z is a fixed point of T. �

We give the following illustrative example inspired from [4].

Example 2.14. Let X = {0, 1, 2} and σ : X ×X → R+ defined by:

σ(0, 0) = σ(2, 2) =
1

4
, σ(1, 1) = 0, σ(0, 1) = σ(1, 0) =

1

3
,

σ(0, 2) = σ(2, 0) =
2

5
, σ(1, 2) = σ(2, 1) =

11

15
.

Then (X,σ) is a complete metric-like space. Note that σis not a partial metric on X as σ(1, 2) >
σ(1, 0) + σ(0, 2) −σ(0, 0).
Define the map T : X → CBσ(X) by

T0 = T1 = {1} and T2 = {0, 1}.
Note that Tx is bounded and closed for all x ∈ X in metric-like space (X,σ).

We have

max{σ(x,Tx), x ∈ X} = max{σ(0, 1),σ(1, 1),σ(2, 0)} =
2

5
,

min{σ(x,Tx), x ∈ X −{1}} =
1

3
.

Therefore, we have

1

4
≤ σ(x,y) ≤

11

15



ON MULTI-VALUED WEAKLY PICARD OPERATORS 179

for all x,y ∈ X with (x,y) 6= (1, 1). It follows that
1

1 + r
σ(x,Tx) ≤ σ(x,y) ≤

1

1 −s
σ(x,Tx)

for all x,y ∈ X with x 6= 1 and for some 3
5
≤ r < s < 1. Observe that the above inequalities are also

true for x = y = 1 but not hold for x = 1 and y ∈{0, 2}.
Now, we shall show that

Hσ(Tx,Ty) ≤ rMσ(x,y)(2.21)
for all x,y ∈ X for some 5

6
≤ r < 1. For this, we consider the following cases:

case1 : x,y ∈{0, 1}, with (x,y) 6= (1, 0). We have
Hσ(Tx,Ty) = σ(1, 1) = 0 ≤ rMσ(x,y).

case2 : x = 0, y = 2. We have

Hσ(Tx,Ty) = Hσ({1},{0, 1}) = max{σ(1,{0, 1}),max{σ(1, 1),σ(1, 0)}}

=
1

3
≤

5

6
σ(x,y) ≤ rMσ(x,y)

case3 : x = y = 2. We have

Hσ(Tx,Ty) = Hσ({0, 1},{0, 1}) = max{σ(0,{0, 1}),σ(1,{0, 1})}

= min{σ(0, 0),σ(0, 1)} =
1

4

Moreover, we have Mσ(2, 2) = max{σ(2, 2),σ(2,T2)} = max{14,
2
5
} = 2

5
. Then for x = y = 2 we get

Hσ(T2,T2) =
1

4
≤

5

6
.
2

5
≤ rMσ(2, 2).

Then, all the required hypotheses of Theorem 2.12 are satisfied. Here, x = 1 is the unique fixed point
of T.

Now, we need the following definition.

Definition 2.15. Let (X,p) be a partial metric space. A multi-valued mapping T : X → CBp(X) is
said to be (ϕ,ψ)−contractive multi-valued operator if there exist ϕ ∈ Φ and ψ ∈ Ψ such that
(2.22) p(y,Tx) ≤ ϕ(p(y,x)) ⇒ Hp(Tx,Ty) ≤ ψ(Mp(x,y))
for all x,y ∈ X, where

Mp(x,y) = max{p(x,y),p(x,Tx),p(y,Ty),
1

2
[p(x,Ty) + p(Tx,y)]}.

We give the following result.

Theorem 2.16. Let (X,p) be a complete partial metric space and T : X → CBp(X) be (ϕ,ψ)−contractive
multi-valued operator. Then T is an MWP operator.

Proof. Let x0 ∈ X and x1 ∈ Tx0. Let c a given real number such that p(x0,x1) < c.
Clearly, if x1 = x0 or x1 ∈ Tx1, we conclude that x1 is a fixed point of T and so the proof is finished.

Now, we assume that x1 6= x0 and x1 6∈ Tx1. So then, p(x0,x1) > 0 and p(x1,Tx1) > 0.
Since x1 ∈ Tx0, we get

p(x1,Tx0) ≤ p(x1,x1) ≤ p(x1,x0) ≤ ϕ(p(x1,x0)).
Hence by (2.22) and triangular inequality, we have

0 < p(x1,Tx1) ≤ Hp(Tx0,Tx1) ≤ ψ(Mp(x0,x1))

≤ ψ(max{p(x0,x1),p(x0,Tx0),p(x1,Tx1),
1

2
[p(x0,Tx1) + p(x1,Tx0)]})

≤ ψ(max{p(x0,x1),p(x1,Tx1),
1

2
[p(x0,Tx1) + p(x1,x1)]})

≤ ψ(max{σ(x0,x1),σ(x0,x1),σ(x1,Tx1),
1

2
[p(x1,Tx1) + p(x0,x1)]})

= ψ(max{p(x0,x1),p(x1,Tx1)}) = ψ(p(x0,x1)) < ψ(c).



180 FELHI

Proceeding as in the proof of Theorem 2.2, we construct a sequence {xn} in X such that for all n ∈ N
(i) xn 6∈ Txn, xn 6= xn+1, xn+1 ∈ Txn;

(ii)

(2.23) p(xn,xn+1) ≤ ψn(c).

Now, for m > n, we have

p(xn,xm) ≤
m−1∑
i=n

p(xi,xi+1) −
m−1∑
i=n+1

p(xi,xi) ≤
m−1∑
i=n

ψi(c) ≤
∞∑
i=n

ψi(c) → 0 as n →∞.

Thus,

lim
n,m→∞

p(xn,xm) = 0.(2.24)

Hence, {xn} is Cauchy in (X,p). Moreover since (X,p) is complete, it follows there exists z ∈ X such
that

lim
n→∞

p(xn,z) = p(z,z) = lim
n,m→∞

p(xn,xm) = 0.(2.25)

Proceeding again as in the proof of Theorem 2.2, we prove that z is a fixed point of T. �

Analogously, we can derive the following results.

Corollary 2.17. Let (X,p) be a complete partial metric space and T : X → CBp(X) be a multi-valued
mapping. Assume that there exist ϕ ∈ Φ and ψ ∈ Ψ such that, for all x,y ∈ X

Hp(Tx,Ty) ≤ ψ(Mp(x,y)) + p(y,Tx) −ϕ(p(y,x)),

where Mp(x,y) = max{p(x,y),p(x,Tx),p(y,Ty), 12 [p(x,Ty) + p(Tx,y)]}.
Then T has a unique fixed point.

Corollary 2.18. ([4], Theorem 2.2) Let (X,p) be a complete partial metric space and T : X →
CBp(X) be a multi-valued mapping. Assume that there exist r ∈ [0, 1) and s ≥ 1 such that, for all
x,y ∈ X

p(y,Tx) ≤ sp(y,x) ⇒ Hp(Tx,Ty) ≤ rMp(x,y),

where Mp(x,y) = max{p(x,y),p(x,Tx),p(y,Ty), 12 [p(x,Ty) + p(Tx,y)]}.
Then T is an MWP operator.

Proof. It suffice to take ϕ(t) = st and ψ(t) = rt in Theorem 2.16. �

Corollary 2.19. Let (X,p) be a complete partial metric space and T : X → CBp(X) be a multi-valued
mapping. Assume that there exist r ∈ [0, 1) and s ≥ 1 such that, for all x,y ∈ X

p(y,Tx) ≤ sp(y,x) ⇒ Hp(Tx,Ty) ≤ r max{p(x,y),p(x,Tx),p(y,Ty)}.

Then T is an MWP operator.

Corollary 2.20. Let (X,p) be a complete partial metric space and T : X → CBp(X) be a multi-valued
mapping. Assume that there exist r ∈ [0, 1) and s ≥ 1 such that, for all x,y ∈ X

p(y,Tx) ≤ sp(y,x) ⇒ Hp(Tx,Ty) ≤
r

3
{p(x,y) + p(x,Tx) + p(y,Ty)}.

Then T is an MWP operator.

If T is a single-valued mapping, we deduce the following results.

Corollary 2.21. Let (X,p) be a complete partial metric space and T : X → X be a mapping. Assume
that there exist ϕ ∈ Φ and ψ ∈ Ψ such that, for all x,y ∈ X

p(y,Tx) ≤ ϕ(p(y,x)) ⇒ p(Tx,Ty) ≤ ψ(Mp(x,y)),

where Mp(x,y) = max{p(x,y),p(x,Tx),p(y,Ty), 12 [p(x,Ty) + p(Tx,y)]}.
Then T has a unique fixed point.



ON MULTI-VALUED WEAKLY PICARD OPERATORS 181

Proof. The existence follows immediately also from Theorem 2.16. Thus, we need to prove uniqueness
of fixed point. We assume that there exist x,y ∈ X such that x = Tx and y = Ty with x 6= y.

Since p(y,Tx) = p(y,x) ≤ ϕ(p(y,x)), then by (2.22), we get

0 < p(x,y) = p(Tx,Ty) ≤ ψ(max{p(x,y),p(x,Tx),p(y,Ty),
1

2
[p(x,Ty) + p(Tx,y)]})

= ψ(max{p(x,y),p(x,x),p(y,y),p(x,y)}) = ψ(p(x,y))
< p(x,y)

which is a contradiction. Hence x = y, so the uniqueness of the fixed point of T.
�

Corollary 2.22. Let (X,p) be a complete partial metric space and T : X → X be a mapping. Assume
that there exist r ∈ [0, 1) and s ≥ 1 such that, for all x,y ∈ X

p(y,Tx) ≤ sp(y,x) ⇒ p(Tx,Ty) ≤ rMp(x,y),

where Mp(x,y) = max{p(x,y),p(x,Tx),p(y,Ty), 12 [p(x,Ty) + p(Tx,y)]}.
Then T has a unique fixed point.

References

[1] IA. Rus, A. Petrusel and A. Sintamarian, Data dependence of the fixed points set of multi-valued weakly Picard

operators, Stud. Univ. Babe?s-Bolyai, Math. 46 (2001), 111-121.

[2] IA. Rus, A. Petrusel and A. Sintamarian, Data dependence of the fixed points set of some multi-valued weakly
Picard operators, Nonlinear Anal. 52 (2003), 1947-1959.

[3] T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Am. Math.
Soc. 136 (2008), 1861-1869.

[4] M. Jleli, H. K. Nashine, B. Samet and C. Vetro, On multi-valued weakly Picard operators in partial Hausdorff

metric spaces, Fixed Point Theory Appl. 20015 (2015), Art. ID 52.
[5] I. Altun and H. Simsek, Some fixed point theorems on dualistic partial metric spaces, J. Adv. Math. Stud. 1 (2008),

1C8.

[6] E. Karapinar, H. Aydi, A. Felhi and S. Sahmim, Hausdorff Metric-Like, generalized Nadler’s Fixed Point Theorem
on Metric-Like Spaces and application, Miskolc Math. Notes (in press).

[7] C.T. Aage and J.N. Salunke, The results on fixed points in dislocated and dislocated quasi-metric space, Appl.

Math. Sci, 2(59) (2008), 2941-2948.
[8] S.G. Matthews, Partial metric topology, Proc. 8th Summer Conference on General Topology and Applications,

Annals of the New York Academi of Sciences, 728 (1994), 183-197.

[9] H. Aydi, M. Abbas and C. Vetro, Partial Hausdorff metric and Nadlers fixed point theorem on partial metric spaces,
Topol. Appl. 159 (2012), 3234-3242.

[10] R.D. Daheriya, R. Jain and M. Ughade, Some fixed point theorem for expansive type mapping in dislocated metric
space, ISRN Math. Anal. 2012 (2012), Art. ID 376832.

[11] A. Isufati, Fixed point theorems in dislocated quasi-metric space, Appl. Math. Sci. 4(5) (2010), 217-233.

[12] Amini A. Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory Appl. 2012 (2012),
Art. ID 204.

[13] R. George, Cyclic contractions and fixed points in dislocated metric spaces, Int. J. Math. Anal. 7(9) (2013), 403-411.

[14] G.E. Hardy and T.D. Rogers, A generalization of a fixed point theorem of Reich, Canadian Mathematical Bulletin,
16 (1973), 201C206.

[15] E. Karapınar and P. Salimi, Dislocated metric space to metric spaces with some fixed point theorems, Fixed Point

Theory Appl. 2013 (2013), Art. ID 222.
[16] P.S. Kumari, W. Kumar and I.R. Sarma, Common fixed point theorems on weakly compatible maps on dislocated

metric spaces, Math. Sci. 6 (2012), 71.

[17] PS. Kumari, Some fixed point theorems in generalized dislocated metric spaces, Math. Theory Model. 1(4) (2011),
16-22.

[18] S.B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488.
[19] I.R. Sarma and P.S. Kumari, On dislocated metric spaces, Int. J. Math. Arch. 3(1) (2012), 72-77.
[20] R. Shrivastava, ZK. Ansari and M. Sharma, Some results on fixed points in dislocated and dislocated quasi-metric

spaces, J. Adv. Stud. Topol. 3(1) (2012), 25-31.
[21] M. Shrivastava, K. Qureshi and A.D. Singh, A fixed point theorem for continuous mapping in dislocated quasi-metric

spaces, Int. J. Theor. Appl. Sci. 4(1) (2012), 39-40.

[22] K. Zoto, Some new results in dislocated and dislocated quasi-metric spaces. Appl. Math. Sci. 6(71) (2012), 3519-3526.

1Department of Mathematics, College of Sciences, King Faisal University, Hafouf, Saudi Arabia

2Department of Mathematics, Preparatory Engineering Institute, Bizerte, Carthage University, Tunisia



182 FELHI

∗Corresponding author: afelhi@kfu.edu.sa