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EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 1, 2019, 88-100
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Fixed point results in metric-like spaces via
σ-simulation functions

Habes Alsamir1,∗, Mohd Selmi Noorani1, Wasfi Shatanawi2,3, Hassen Aydi4,
Habibulla Akhadkulov5, Haitham Qawaqneh1, Kareem Alanazi6

1 School of Mathematical Sciences, Faculty of Science and Technology,
Universiti Kebangsaan Malaysia, 43600 UKM, Selangor Darul Ehsan, Malaysia
2 Department of Mathematics, Hashemite University, Zarqa 1315, Jordan
3 Department of Mathematics and General Courses Prince Sultan University, Riyadh,
Saudi Arabia
4 Imam Abdulrahman Bin Faisal University, Department of Mathematics,
College of Education of Jubail, P.O: 12020, Industrial Jubail 31961. Saudi Arabia
5 School of Quantitative Sciences, University Utara Malaysia, CAS 06010, UUM Sintok,
Kedah Darul Aman, Malaysia
6 Mathematics Department, Science and Arts College, Aljouf University, Saudi Arabia

Abstract. The purpose of this paper is to establish some fixed point results for (α, β)-admissible
Z-contraction mappings in complete metric-like spaces. Our results generalize and extend several
known results on literature. Two illustrated examples are also presented.

2010 Mathematics Subject Classifications: 47H10, 54H25

Key Words and Phrases: Fixed point, metric-like space, simulation function, (α, β)-admissible
mapping

1. Introduction and preliminaries

Fixed point theory is an essential tool to resolve many equations appeared in applied
science such as Biology, Physics, Economics, Engineering and Game Theory. Banach
contraction principle [12] is considered the most important tool in fixed point theory. It
was extended in several directions. For more details, see [13, 17, 18, 20–25]. Going in this
direction, Harandi [16] reintroduced the concept of metric-like spaces.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i1.3331

Email addresses: h.alsamer@gmail.com (H. Alsamir), msn@ukm.my (M.S. Noorani),
swasfi@hu.edu.jo (W. Shatanawi), wshatanawi@psu.edu.sa (W. Shatanawi),
hmaydi@iau.edu.sa (H. Aydi), habibulla@uum.edu.my (H. Akhadkulov),
haitham.math77@gmail.com (H. Qawaqneh), sunshine-w@hotmail.com (K. Alanazi)

http://www.ejpam.com 88 c© 2019 EJPAM All rights reserved.



H. Alsamir et al. / Eur. J. Pure Appl. Math, 12 (1) (2019), 88-100 89

Definition 1. [16] Let X is a nonempty set. A function σ : X ×X → [0,∞) is said to
be a metric-like space (or a dislocated metric) on X if for any x,w, y ∈ X, the following
conditions hold:

(σ1) σ(x, y) = 0 implies that x = y;

(σ2) σ(x, y) = σ(y, x);

(σ3) σ(x, y) ≤ σ(x, z) + σ(z, y).

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

It is clear that every metric space and partial metric space is a metric-like space, but
the converse is not true.

Example 1. Let X = {0, 1} and

σ(x, y) =


2, if x = y = 0;

1, otherwise.

Then (X,σ) is a metric-like space. It is neither a partial metric space (σ(0, 0) 6≤ σ(0, 1)),
nor a metric space (σ(0, 0) = 2 6= 0).

Following [16], we have the following topological concepts. Each metric-like σ on X
generates a topology τσ on X whose base is the family of open σ-balls

Bσ(x, ε) = {y ∈ X :| σ(x, y)− σ(x, x) |< ε}, for all x ∈ X and ε > 0.

Now, let (X,σ) be a metric-like space. The mapping T : X → X is said σ-continuous at
x ∈ X if for all ε > 0, there exists δ > 0 such that T (Bσ(x, δ)) ⊆ Bσ(Tx, ε). Consequently,
if T : X → X is σ-continuous, then if limn→∞ xn = x, we have limn→∞ Txn = Tx. A
sequence {xn}n=0

∞ of elements of X is called σ-Cauchy if the limit limn,m→∞ σ(xn, ym)
exists and is a finite number. The metric-like space (X,σ) is called complete if for each
σ-Cauchy sequence {Xn}n∞, there is some y ∈ Y such that

lim
n→∞

σ(xn, x) = σ(x, x) = lim
n,m→∞

σ(xn, xm).

A subset A of a metric-like space (X,σ) is bounded if there is a point b ∈ X and a positive
constant K such that σ(a, b) ≤ K for all a ∈ A.

Remark 1. Let X = {0, 1} be endowed with σ(x, y) = 1 for each x, y ∈ X. Take xn = 1
for each n ∈ N. Using the convergence definition, it is is easy to see that xn → 0 and
xn → 1. In metric-like spaces, the limit of a convergent sequence is not necessarily unique.

The following lemma is known and useful for the rest of paper.



H. Alsamir et al. / Eur. J. Pure Appl. Math, 12 (1) (2019), 88-100 90

Lemma 1. [5, 16] Let (X,σ) be a metric-like space. Let {xn} be a sequence in X such
that xn → x where x ∈ X and σ(x, y) = 0. Then for all y ∈ X, we have limn→∞ σ(xn, y) =
σ(x, y).

In literature, there are several (common) fixed point works in the setting of metric-like
spaces. For instance, see [6, 8, 10].
On the one hand, Samet [26] presented the concept of α-admissible mappings and proved
some fixed point theorems in metric spaces. Recently, Chandok [14] introduced the notion
of (α, β)-admissible mappings and obtained some fixed point theorems.

Definition 2. [14] Let X be a nonempty set, f : X → X and α, β : X × X → R+.
We say that f is an (α, β)-admissible mapping if α(x, y) ≥ 1 and β(x, y) ≥ 1 imply that
α(fx, fy) ≥ 1 and β(fx, fy) ≥ 1 for all x, y ∈ X.

For other results using different concepts of α-admissible mappings, see [1, 2, 7, 9, 11,
15, 27–29]. On the other hand, Khojasteh et al. [19] introduced a new class of mappings
called simulation functions. They [19] proved several fixed point theorems and showed
that many results in the literature are simple consequences of their obtained results.

Definition 3. [19] A function ζ : [0,∞)× [0,∞)→ R is called a simulation function if ζ
satisfies the following conditions:

(ζ1) ζ(0, 0) = 0;

(ζ2) ζ(t, s) < s− t for all t, s > 0;

(ζ3) if {tn} and {sn} are sequences in (0,∞) such that limn→∞ tn = limn→∞ sn = ` ∈
(0,∞), then

lim
n→∞

sup ζ(tn, sn) < 0.

In [19], the following unique fixed point theorem is established.

Theorem 1. [19] Let (X, d) be a metric space and f : X → X be a Z-contraction with
respect to a simulation function ζ, that is,

ζ(d(fx, fy), d(x, y)) ≥ 0, forall x, y ∈ X.

Then T has a unique fixed point.

It is worth mentioning that the Banach contraction is an example of Z-contractions
by defining ζ : [0,∞)× [0,∞)→ R via

ζ(t, s) = γs− t, ∀ s, t ∈ [0,∞),

where γ ∈ [0, 1).

Argoubi et al. [4] modified Definition 3 as follows.



H. Alsamir et al. / Eur. J. Pure Appl. Math, 12 (1) (2019), 88-100 91

Definition 4. [4] A simulation function is a function ζ : [0,∞)× [0,∞)→ R that satisfies
the following conditions:

(i) ζ(t, s) < s− t for all t, s > 0;

(ii) if {tn} and {sn} are sequences in (0,∞) such that limn→∞ tn = limn→∞ sn = ` ∈
(0,∞), then

lim
n→∞

sup ζ(tn, sn) < 0.

It is clear that any simulation function in the sense of Khojasteh et al. (Definition 3)
is also a simulation function in the sense of Argoubi et al. (Definition 4). The converse is
not true. For more details, see [4].

Example 2. [4] Define a function ζ : [0,∞)× [0,∞)→ R by

ζ(t, s) =

{
1 if (s, t) = (0, 0),

λs− t otherwise,

where λ ∈ (0, 1). Then ζ is a simulation function in the sense of Argoubi et al.

In the following, some other examples of simulation functions in the sense of Definition
3 (see [3, 19, 31]).

(i) ζ(t, s) = cs− t for all t, s ∈ [0,∞) where c ∈ [0, 1).

(ii) ζ(t, s) = s − φ(s) − t for all t, s ∈ [0,∞), where φ : R+ → R+ is a lower semi-
continuous function such that φ(t) = 0 if and only if t = 0.

In this paper, we introduce the concept of (α, β)-admissible Z-contractions with respect
to ζ. We also establish the existence of fixed points for this class of mappings in metric-like
spaces. Our work generalizes and extends some theorems in the literature. Two illustrated
examples are given to support the obtained results.

2. Main results

First, we introduce the following.

Definition 5. Let (X,σ) be a metric-like space. Given f : X → X and α, β : X×X → R+.
Such f is said an (α, β)-admissible Z-contraction with respect to ζ if

ζ(α(x, y)β(x, y)σ(fx, fy), σ(x, y)) ≥ 0 (1)

for all x, y ∈ X, where ζ is a simulation function in the sense of Definition 3.

Now, we introduce our main theorem.



H. Alsamir et al. / Eur. J. Pure Appl. Math, 12 (1) (2019), 88-100 92

Theorem 2. Let (X,σ) be a complete metric-like space and let f be a self-mapping on X
satisfying the following conditions:

(i) f is (α, β)-admissible;

(ii) there exists x0 ∈ X such that α(x0, fx0) ≥ 1 and β(x0, fx0) ≥ 1;

(iii) f is an (α, β)-admissible Z-contraction on (X,σ);

(iv) f is σ−continuous.

Then f has a unique fixed point u ∈ X with σ(u, u) = 0.

Proof.
By (2), there exists x0 ∈ X such that α(x0, fx0) ≥ 1 and β(x0, fx0) ≥ 1. Define

the sequence {xn} by xn+1 = fxn for all n = 0, 1, 2, · · · . If xn = xn+1 for some n, then
xn = xn+1 = fxn. So xn is a fixed point of f, and the proof is completed. From now on,
assume that xn 6= xn+1 for all n ∈ N ∪ {0}. Since f is an (α, β)-admissible mapping, we
derive

α(x0, fx0) = α(x0, x1) ≥ 1⇒ α(fx0, fx1) = α(x1, x2) ≥ 1.

Continuing in this process, we get

α(xn, xn+1) ≥ 1, for all n ≥ 0. (2)

Similarly,
β(xn, xn+1) ≥ 1, for all n ≥ 0. (3)

From (1), (2) and (3), we have

0 ≤ ζ(α(xn, xn−1)β(xn, xn−1)σ(fxn, fxn−1), σ(xn, xn−1))

= ζ(α(xn, xn−1)β(xn, xn−1)σ(xn+1, xn), σ(xn, xn−1))

< σ(xn, xn−1)− α(xn, xn−1)β(xn, xn−1)σ(xn+1, xn). (4)

Consequently, we derive that

σ(xn+1, xn) ≤ α(xn, xn−1)β(xn, xn−1)σ(xn+1, xn) < σ(xn, xn−1) for all n ≥ 0. (5)

The sequence {σ(xn, xn−1)} is nondecreasing, so there exists r ≥ 0 such that limn→∞ σ(xn, xn−1) =
r. We prove that

lim
n→∞

σ(xn, xn−1) = 0. (6)

Suppose that r > 0. By (5), we derive that

lim
n→∞

α(xn, xn−1)β(xn, xn+1)σ(xn, xn−1) = r. (7)

Letting sn = α(xn, xn−1)β(xn, xn−1)σ(xn, xn+1) and sn = σ(xn, xn−1) and taking (ζ3) into
account, we have

0 ≤ lim sup
n→∞

ζ(α(xn, xn−1)β(xn, xn+1)σ(xn, xn−1)) < 0, (8)



H. Alsamir et al. / Eur. J. Pure Appl. Math, 12 (1) (2019), 88-100 93

which is a contradiction. Thus, r = 0.
Now, we will show that {xn} is a Cauchy sequence. Suppose on the contrary that {xn}
is not a Cauchy sequence. Then, there exists ε for which we can find subsequences {xnl

}
and {xml

} of {xn} with nl > ml > l such that for every l,

σ(xnl
, xml

) ≥ ε (9)

and nl is the smallest number such that (9) holds. From (9), we get

σ(xnl−1, xml
) < ε. (10)

Using the triangular inequality and (10),

ε ≤ σ(xnl
, xml

)

≤ σ(xnl
, xnl−1) + σ(xnl−1, xml

)

< σ(xnl
, xnl−1) + ε.

Letting n→∞ in the above inequality and using (6), we obtain

lim
n→∞

σ(xnl
, xml

) = ε. (11)

Also, from the triangular inequality, we have

| σ(xnl+1, xml
)− σ(xnl

, xml
) |≤ σ(xnl

, xnl+1).

On taking limit as l→∞ on both sides of above inequality and using (6) and (11), we get

lim
l→∞

σ(xnl+1, xml
) = ε. (12)

Similarly, it is easy to show that

lim
l→∞

σ(xnl+1, xml+1) = ε. (13)

Moreover, since f is an (α, β)-admissible mapping, we have

α(xnl
, xml

) ≥ 1 andβ(xnl
, xml

) ≥ 1. (14)

By the fact f is an (α, β)-admissible Z-contraction with respect to ζ , together with (11),
(14) and (ζ3), we get

0 ≤ lim sup
l→∞

ζ(α(xnl
, xml

)β(xnl
, xml

)σ(xnl+1, xml+1), σ(xnl
, xml

)) < 0,

which is a contradiction. Hence {xn} is a Cauchy sequence. Owing to the fact that (X,σ)
is a complete metric-like space, there exists some u ∈ X such that

lim
n→∞

σ(xn, u) = σ(u, u) = lim
n→∞

σ(xn, xm) = 0, (15)



H. Alsamir et al. / Eur. J. Pure Appl. Math, 12 (1) (2019), 88-100 94

which implies that σ(u, u) = 0. Moreover, the continuity of f implies that

lim
n→∞

σ(xn+1, fu) = σ(fxn, fu) = σ(fu, fu).

By Lemma 1 and (15), we obtain

lim
n→∞

σ(xn+1, fu) = σ(u, fu). (16)

Combining (15) and (16), we have σ(fu, fu) = σ(u, fu) , that is, fu = u.. To prove the
uniqueness of the fixed point, suppose that there exists w ∈ X such that fw = w and
w 6= u. Then

0 ≤ ζ(α(u,w)β(u,w)σ(fu, fw), σ(u,w)) < σ(u,w)− α(u,w)β(u,w)σ(fu, fw) ≤ 0,

which is a contradiction, so u = w.

Theorem 2 remains true if we drop the continuity hypothesis by the following property:

(H): If {xn} is a sequence in X such that α(xn, xn+1) ≥ 1 and β(xn, xn+1) ≥ 1 for
all n, then there exists a subsequence {xnl

} of {xn} such that α(xnl
, xnl+1) ≥ 1 and

β(xnl
, xnl+1) ≥ 1 for all l ∈ N and α(x, fx) ≥ 1 and β(x, fx) ≥ 1.

Theorem 3. Let (X,σ) be a complete metric-like space and let f be a self-mapping on X
satisfying the following conditions:

(i) f is (α, β)-admissible;

(ii) there exists x0 ∈ X such that α(x0, fx0) ≥ 1 and β(x0, fx0) ≥ 1;

(iii) f is an (α, β)-admissible Z-contraction on (X,σ);

(iv) (H) holds.

Then f has a unique fixed point u ∈ X with σ(u, u) = 0.
Proof. Following the proof of Theorem 2, we construct a sequence {xn} in X defined

by xn+1 = fxn, which converges to some u ∈ X. From definition (2) and (H), there exists
a subsequence {xnl

} of {xn} such that α(xnl
, xnl

) ≥ 1 and β(xnl
, xnl

) ≥ 1 for all l ∈ N.
Thus applying (1) for all l, we have

0 ≤ ζ(α(xnl
, u)β(xnl

, u)σ(fxn, fu), σ(xnl
, u))

= ζ(α(xnl
, u)β(xnl

, u)σ(xn+1, fu), σ(xnl
, u))

< σ(xnl
, u)− α(xnl

, u)β(xnl
, u)σ(xnl+1, fu) (17)

which is equivalent to

σ(xnl
+ 1, fu) = σ(fxnl

, fu) ≤ α(xnl
, u)β(xnl

, u)σ(fxn, fu) ≤ σ(xnl
, u). (18)

Letting l → ∞ in the above equality, we have σ(u, fu) = 0. Using similar arguments as
above, we can show that u is a fixed point of f. The uniqueness of the fixed point of f is
obtained by similar arguments as those given in the proof of Theorem 2.



H. Alsamir et al. / Eur. J. Pure Appl. Math, 12 (1) (2019), 88-100 95

3. Consequences

In this section, we apply Theorem 2 to obtain different results known in literature.
The first one is of Banach type.

Corollary 1. Let (X,σ) be a complete metric-like space and let f be a self-mapping on
X satisfying the following conditions:

(i) f is (α, β)-admissible;

(ii) there exists x0 ∈ X such that α(x0, fx0) ≥ 1 and β(x0, fx0) ≥ 1;

(iii)
α(x, y)β(x, y)σ(fx, fy) ≤ λσ(x, y),

for all x, y ∈ X and λ ∈ [0, 1);

(iv) f is σ-continuous.

Then f has a unique fixed point u ∈ X with σ(u, u) = 0.

Proof. Following the lines of Theorem 2, by taking as a σ-simulation function,

ζ(t, s) = λs− t.

Corollary 2. Let (X,σ) be a complete metric-like space and let f be a self-mapping on
X satisfying the following conditions:

(i) f is (α, β)-admissible;

(ii) there exists x0 ∈ X such that α(x0, fx0) ≥ 1 and β(x0, fx0) ≥ 1;

(iii) there exists a lower semi-continuous function ϕ : R+ → R+ with ϕ−1 = {0} such
that

α(x, y)β(x, y)σ(fx, fy) ≤ σ(x, y)− ϕ(σ(x, y))

for all x, y ∈ X;

(iv) f is σ-continuous.

Then f has a unique fixed point u ∈ X with σ(u, u) = 0.

Proof. It suffices to take
ζ(t, s) = s− ϕ(s)− t.

If we consider in Theorem 2, α(x, y) = β(x, y) = 1 for all x, y ∈ X, we have



H. Alsamir et al. / Eur. J. Pure Appl. Math, 12 (1) (2019), 88-100 96

Corollary 3. Let (X,σ) be a complete metric-like space and let f be a self-mapping on
X. Suppose that there exists a σ-simulation function ζ such that

ζ(σ(fx, fy), σ(x, y)) ≥ 0 (19)

for all x, y ∈ X. Then f has a unique fixed point u ∈ X with σ(u, u) = 0.

We present the following illustrated examples.

Example 3. Let X = [0,∞), σ(x, y) = (x+ y) for all x, y ∈ X and f : X → X be defined
by

fx =

{
1
4x if 0 ≤ x ≤ 1

4x otherwise.

Consider
ζ(s, t) = cs− t,

where 0 ≤ 1
4 < c < 1. Define α, β : X ×X → R+ as

α(x, y) =

{
4
3 if 0 ≤ x, y ≤ 1

0 otherwise,

β(x, y) =

{
3
2 if 0 ≤ x, y ≤ 1

0 otherwise.

We shall prove that Corollary 1 can be applied. Clearly, (X,σ) is a complete metric-like
space. Let x, y ∈ X such that α(x, y) ≥ 1 and β(x, y) ≥ 1. Since x, y ∈ [0, 1] and so
fx ∈ [0, 1], fy ∈ [0, 1] and α(fx, fy) = 1 and β(fx, fy) = 1. Hence f is (α, β)-admissible.
Condition (2) is satisfied with x0 = 1. Condition (4) is satisfied with xn = fnx1 = 1

n .
If 0 ≤ x ≤ 1, then α(x, y) = 4

3 and β(x, y) = 3
2 . We have

ζ(α(x, y)β(x, y)σ(fx, fy), σ(x, y)) = cσ(x, y)− α(x, y)β(x, y)σ(fx, fy)

=
3

4
(x+ y)− 2

1

4
(x+ y)

= (
3

4
− 1

2
)(x+ y)

=
1

4
(x+ y)

≥ 0.

If 0 ≤ x ≤ 1 and y > 1, then ζ(α(x, y)β(x, y)σ(fx, fy), σ(x, y)) ≥ 0 since α(x, y) =
β(x, y) = 0. Consequently, all assumptions of Corollary 1 are satisfied and hence f has a
unique fixed point, which is u = 0.

We also notice that (19) is not satisfied. In fact, for x = 1, y = 2, we get

σ(f1, f2) = (
33

4
)2 > 3 = σ(x, y).



H. Alsamir et al. / Eur. J. Pure Appl. Math, 12 (1) (2019), 88-100 97

Example 4. Consider X = {0, 1, 3} and define σ : X ×X → R+ as follows:
σ(0, 0) = 0, σ(1, 0) = σ(0, 1) = 1

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

3 , σ(1, 1) =
1
2 , σ(3, 3) = 7

2 .
Note that σ(3, 3) 6= 0, so (X,σ) is not a metric and σ(3, 3) > σ(0, 3), so (X,σ) is not
a partial metric. Clearly, (X,σ) is metric-like space. Let f : X → X be defined by
f0 = f1 = 0 and f3 = 1. Take α, β : X ×X → R+ given as

α(x, y) =

{
5
2 , if x ∈{0,1,3},
0, otherwise,

β(x, y) =

{
1, if x ∈{0,1,3},
0, otherwise.

Take ζ : X × X → R+ by ζ(t, s) = 1
2s − t. Let x, y ∈ X be such that α(x, y) ≥ 1 and

β(x, y) ≥ 1, then α(fx, fy) ≥ 1 and β(fx, fy) ≥ 1, that is, f is (α, β)-admissible. Now,
we consider the following cases:

(i) Case 1: x = 0 and y = 0. We have

ζ(α(0, 0)β(0, 0)σ(f0, f0), σ(0, 0)) = ζ(
5

2
.1.0, 0) = ζ(0, 0) = 0.

(ii) Case 2: x = 0 and y = 1. Here,

ζ(α(0, 1)β(0, 1)σ(f0, f1), σ(0, 1)) = ζ(
5

2
.1.0, 1) = ζ(0,

1

10
) =

1

20
> 0.

(iii) Case 3: x = 0 and y = 3. We have

ζ(α(0, 3)β(0, 3)σ(f0, f3), σ(0, 3)) = ζ(
5

2
.

1

10
,
1

2
) = ζ(

1

4
,
1

2
) = 0.

(iv) Case 4: x = 1 and y = 1. Here,

ζ(α(1, 1)β(1, 1)σ(f1, f1), σ(1, 1)) = ζ(
5

2
.1.0,

1

2
) = ζ(0,

1

2
) =

1

4
> 0.

(v) Case 5: x = 1 and y = 3. We have

ζ(α(1, 3)β(1, 3)σ(f1, f3), σ(1, 3)) = ζ(
5

2
.1.

1

10
,
2

3
) = ζ(

1

4
,
2

3
) =

1

12
> 0.

(vi) Case 6: x = 3 and y = 3. Here,

ζ(α(3, 3)β(3, 3)σ(f3, f3), σ(3, 3)) = ζ(
5

2
.1.

1

2
,
7

2
) = ζ(

5

4
,
7

2
) =

1

2
> 0.

Thus, f is an (α, β)-admissible Z-contraction with respect to ζ. Hence all conditions of
Theorem 2 are satisfied and f has a unique fixed point, which is, u = 0.



REFERENCES 98

Acknowledgements

The authors would like to acknowledge the grant: UKM Grant DIP-2017-011 and
Ministry of Education, Malaysia grant FRGS/1/2017/STG06/UKM/01/1 for financial
support.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the manuscript.

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