International Journal of Analysis and Applications

Volume 17, Number 5 (2019), 734-751

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-734

FIXED POINT THEOREMS FOR GENERALIZED F-CONTRACTIONS AND

GENERALIZED F-SUZUKI-CONTRACTIONS IN COMPLETE DISLOCATED

Sb-METRIC SPACES

HAMID MEHRAVARAN, MAHNAZ KHANEHGIR∗ AND REZA ALLAHYARI

Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran

∗Corresponding author: khanehgir@mshdiau.ac.ir

Abstract. In this paper, first we describe the notion of dislocated Sb-metric space and then we introduce

the new notions of generalized F-contraction and generalized F-Suzuki-contraction in the setup of dislocated

Sb-metric spaces. We establish some fixed point theorems involving these contractions in complete dislocated

Sb-metric spaces. We also furnish some examples to verify the effectiveness and applicability of our results.

1. Introduction and Preliminaries

Bakhtin [1] and Czerwik [2] introduced b-metric spaces and proved the contraction principle in this frame-

work. In recent times, many authors obtained fixed point results for single-valued or set-valued functions,

in the setting of b-metric spaces.

In 2012, Sedghi et al. [11] introduced the concept of S-metric space by modifying D-metric and G-metric

spaces and proved some fixed point theorems for a self-mapping on a complete S-metric space. After that

Özgür and TaŞ studied some generalizations of the Banach contraction principle on S-metric spaces in [8].

They also obtained some fixed point theorems for the Rhoades’ contractive condition on S-metric spaces [7].

Sedghi et al. [10] introduced the concept of Sb-metric space as a generalization of S-metric space and proved

some coupled common fixed point theorems in Sb-metric space. Kishore et al. [4] proved some fixed point

Received 2019-03-11; accepted 2019-07-24; published 2019-09-02.

2010 Mathematics Subject Classification. 47H09, 47H10.

Key words and phrases. Dislocated metric space; Fixed point; Generalized F-contraction; Generalized F-Suzuki-contraction;

Sb-Metric space.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

734

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-734


Int. J. Anal. Appl. 17 (5) (2019) 735

theorems for generalized contractive conditions in partially ordered complete Sb-metric spaces and gave some

applications to integral equations and homotopy theory.

On the other hand, Wardowski [12] introduced a new contraction, the so-called F-contraction, and ob-

tained a fixed point result as a generalization of the Banach contraction principle. Thereafter, Dung and

Hang [3] studied the notion of a generalized F-contraction and established certain fixed point theorems for

such mappings. Recently, Piri and Kumam [6] extended the fixed point results of [12] by introducing a

generalized F-Suzuki-contraction in b-metric spaces.

Motivated by the aforementioned works, in this paper, we first introduce the notion of dislocated Sb- metric

space and then we describe some fixed point results of [3], [6] by introducing generalized F-contractions and

generalized F-Suzuki-contractions in dislocated Sb-metric spaces. We begin with some basic well-known

definitions and results which will be used further on.

Throughout this paper R, R+, N denote the set of all real numbers, the set of all nonnegative real numbers

and the set of all positive integers, respectively.

Definition 1.1. [11] Let X be a nonempty set. An S-metric on X is a function S : X3 → R+ that satisfies

the following conditions:

(S1) 0 < S(x,y,z) for each x,y,z ∈ X with x 6= y 6= z 6= x,

(S2) S(x,y,z) = 0 if and only if x = y = z,

(S3) S(x,y,z) ≤ S(x,x,a) + S(y,y,a) + S(z,z,a) for each x,y,z,a ∈ X.

Then the pair (X,S) is called an S-metric space.

Definition 1.2. [10] Let X be a nonempty set and b ≥ 1 be a given real number. Suppose that a mapping

Sb : X
3 → R+ satisfies:

(Sb1) 0 < Sb(x,y,z) for all x,y,z ∈ X with x 6= y 6= z 6= x,

(Sb2) Sb(x,y,z) = 0 if and only if x = y = z,

(Sb3) Sb(x,y,z) ≤ b
(
Sb(x,x,a) + Sb(y,y,a) + Sb(z,z,a)

)
for all x,y,z,a ∈ X.

Then Sb is called an Sb-metric on X and the pair (X,Sb) is called an Sb-metric space.

Definition 1.3. [10] If (X,Sb) is an Sb-metric space, a sequence {xn} in X is said to be:

(1) Cauchy sequence if, for each ε > 0, there exists n0 ∈ N such that Sb(xn,xn,xm)

< ε for all m,n ≥ n0.

(2) convergent to a point x ∈ X if, for each ε > 0, there exists a positive integer n0 such that

Sb(xn,xn,x) < ε or Sb(x,x,xn) < ε for all n ≥ n0, and we denote by lim
n→∞

xn = x.

Definition 1.4. [10] An Sb-metric space (X,Sb) is called complete if every Cauchy sequence is convergent

in X.



Int. J. Anal. Appl. 17 (5) (2019) 736

Example 1.1. [9] Let X = R. Define Sb : X3 → R+ by Sb(x,y,z) = |x− z| + |y − z| for all x,y,z ∈ X.

Then (X,Sb) is a complete Sb-metric space with b = 2.

Definition 1.5. Let (X,Sb) be an Sb-metric space. Then Sb is called symmetric if

Sb(x,x,y) = Sb(y,y,x) (1.1)

for all x,y ∈ X.

It is easy to see that the symmetry condition (1.1) is automatically satisfied by an S-metric [11].

We conclude this section recalling the following fixed point theorems of Dung and Hang [3] and Piri and

Kumam [6]. For this, we need some preliminaries.

Definition 1.6. [12] Let F be the family of all functions F : (0, +∞) → R such that:

(F1) F is strictly increasing, that is for all α,β ∈ (0, +∞) such that α < β, F(α) < F(β),

(F2) for each sequence {αn} of positive numbers, lim
n→+∞

αn = 0 if and only if lim
n→+∞

F(αn) = −∞,

(F3) there exists k ∈ (0, 1) such that lim
α→0+

αkF(α) = 0.

In 2014, Piri and Kumam [5] described a large class of functions by replacing the condition (F3) in the

above definition with the following one:

(F3′) F is continuous on (0, +∞).

They denote by F the family of all functions F : (0, +∞) → R which satisfy conditions (F1), (F2), and

(F3′).

Example 1.2. (see [5], [13]) The following functions F : (0, +∞) → R are the elements of F.

(1) F(α) = − 1√
α
,

(2) F(α) = −1
α

+ α,

(3) F(α) = 1
1−eα ,

(4) F(α) = ln α,

(5) F(α) = ln α + α.

Definition 1.7. [3] Let (X,d) be a metric space. A mapping T : X → X is said to be a generalized
F -contraction on (X,d) if there exist F ∈F and τ > 0 such that, for all x,y ∈ X,

d(Tx,Ty) > 0 ⇒ τ + F
(
d(Tx,Ty)

)
≤ F

(
N(x,y)

)
,

in which

N(x,y) = max
{
d(x,y),d(x,Tx),d(y,Ty),

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

2
,
d(T2x,x) + d(T2x,Ty)

2
,d(T2x,Tx),d(T2x,y),d(T2x,Ty)

}
.



Int. J. Anal. Appl. 17 (5) (2019) 737

Theorem 1.1. [3] Let (X,d) be a complete metric space and let T : X → X be a generalized F -contraction

mapping. If T or F is continuous, then T has a unique fixed point x∗ ∈ X and for every x ∈ X the sequence

{Tnx} converges to x∗.

We use FG to denote the set of all functions F : (0, +∞) → R which satisfy conditions (F1) and (F3′)

and Ψ to denote the set of all functions ψ : R+ → R+ such that ψ is continuous and ψ(t) = 0 if and only if

t = 0 (see [6]).

Definition 1.8. [6] Let (X,d) be a b-metric space. A self-mapping T : X → X is said to be a generalized
F -Suzuki-contraction if there exists F ∈ FG such that, for all x,y ∈ X with x 6= y,

1

2s
d(x,Tx) < d(x,y) ⇒ F

(
s5d(Tx,Ty)

)
≤ F

(
MT (x,y)

)
−ψ

(
MT (x,y)

)
,

in which ψ ∈ Ψ and

MT (x,y) = max
{
d(x,y),d(T2x,y),

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

2s
,
d(T2x,x) + d(T2x,Ty)

2s
,

d(T2x,Ty) + d(T2x,Tx),d(T2x,Ty) + d(Tx,x),d(Tx,y) + d(y,Ty)
}
.

Theorem 1.2. [6] Let (X,d) be a complete b-metric space and T : X → X be a generalized F -Suzuki-

contraction.Then T has a unique fixed point x∗ ∈ X and for every x ∈ X the sequence {Tnx} converges to

x∗.

2. Main results

In this section, we first introduce the concept of dislocated Sb-metric space and then we demonstrate some

fixed point results for generalized F-contractions and generalized F-Suzuki-contractions in such spaces. Our

results are remarkable for two reasons: first dislocated Sb- metric is more general, second the contractivity

condition involves auxiliary functions form a wider class.

Definition 2.1. Let X be a nonempty set and b ≥ 1 be a given real number. A mapping Sb : X3 → R+ is a

dislocated Sb-metric if, for all x,y,z,a ∈ X, the following conditions are satisfied:

(dSb1) Sb(x,y,z) = 0 implies x = y = z,

(dSb2) Sb(x,y,z) ≤ b
(
Sb(x,x,a) + Sb(y,y,a) + Sb(z,z,a)

)
.

A dislocated Sb-metric space is a pair (X,Sb) such that X is a nonempty set and Sb is a dislocated Sb-metric

on X. In the case that b = 1, Sb is denoted by S and it is called dislocated S-metric, and the pair (X,S) is

called dislocated S-metric space.

Definition 2.2. Let (X,Sb) be a dislocated Sb-metric space, {xn} be any sequence in X and x ∈ X. Then:

(i) The sequence {xn} is said to be a Cauchy sequence in (X,Sb) if, for each ε > 0, there exists n0 ∈ N

such that Sb(xn,xn,xm) < ε for each m,n ≥ n0.



Int. J. Anal. Appl. 17 (5) (2019) 738

(ii) The sequence {xn} is said to be convergent to x if, for each ε > 0, there exists a positive integer n0

such that Sb(x,x,xn) < ε for all n ≥ n0 and we denote it by lim
n→∞

xn = x.

(iii) (X,Sb) is said to be complete if every Cauchy sequence is convergent.

The following example shows that a dislocated Sb-metric need not be a dislocated S-metric.

Example 2.1. Let X = R+, then the mapping Sb : X3 → R+ defined by

Sb(x,y,z) = x + y + 4z is a complete dislocated Sb-metric on X with b = 2. However, it is not a dislocated

S-metric space. Indeed, we have

4 = Sb(0, 0, 1) � 2Sb(0, 0, 0) + Sb(1, 1, 0) = 2.

Definition 2.3. Suppose that (X,Sb) is a dislocated Sb-metric space. A mapping T : X → X is said to be
a generalized F -contraction on (X,Sb) if there exist F ∈ F and τ > 0 such that for all x,y ∈ X,

Sb(Tx,Tx,Ty) > 0 ⇒ τ + F
(
b2Sb(Tx,Tx,Ty)

)
≤ F

(
N(x,y)

)
, (2.1)

where

N(x,y) = max
{
Sb(x,x,y),Sb(Tx,Tx,Ty),

Sb(y,y,Tx)

10b8
,
Sb(x,x,Ty)

10b9
,
Sb(y,y,T

2x)

10b4

}
.

Our first main result is the following.

Theorem 2.1. Let (X,Sb) be a complete dislocated Sb-metric space and T : X → X be a generalized
F -contraction mapping satisfying the following condition:

max
{Sb(y,y,Ty)

5b7
+
Sb(Tx,Tx,Ty)

10b7
,
Sb(x,x,Ty)

10b9
,
Sb(y,y,T

2x)

10b4

}
≤ Sb(Tx,Tx,Ty)

for all x,y ∈ X.

Then T has a unique fixed point v ∈ X.

Proof. Let x0 be an arbitrary point in X and let {xn} be the Picard sequence of T based on x0, that
is, xn+1 = Txn for n = 0, 1, 2, . . . . If there exists n0 ∈ N such that Sb(xn0,xn0,xn0+1 ) = 0, then xn0
is a fixed point of T and the existence part of the proof is finished. On the contrary case, assume that

Sb(xn,xn,xn+1) > 0 for all n ∈ N∪{0}. Applying the contractivity condition (2.1), we get

F
(
b2Sb(Txn−1,Txn−1,Txn)

)
≤ F

(
N(xn−1,xn)

)
− τ. (2.2)

Using the definition of N(x,y) and the property (dSb2), we obtain that

max
{
Sb(xn−1,xn−1,xn),Sb(xn,xn,xn+1)

}
≤ N(xn−1,xn) (2.3)

= max
{
Sb(xn−1,xn−1,xn),Sb(xn,xn,Txn),

Sb(xn,xn,Txn−1)

10b8
,

Sb(xn−1,xn−1,Txn)

10b9
,
Sb(xn,xn,Txn)

10b4

}
≤ max{Sb(xn−1,xn−1,xn),Sb(xn,xn,Txn),

3Sb(xn,xn,xn+1)

10b7
,

Sb(xn−1,xn−1,Txn)

10b9
,
Sb(xn,xn,Txn)

10b4

}
= max

{
Sb(xn−1,xn−1,xn),Sb(xn,xn,xn+1)

}
.



Int. J. Anal. Appl. 17 (5) (2019) 739

Then N(xn−1,xn) = max
{
Sb(xn−1,xn−1,xn),Sb(xn,xn,xn+1)

}
and so (2.2), becomes

F
(
b2Sb(Txn−1,Txn−1,Txn)

)
≤ F

(
max

{
Sb(xn−1,xn−1,xn),Sb(xn,xn,xn+1)

})
− τ.

If we assume that

max
{
Sb(xn−1,xn−1,xn),Sb(Txn−1,Txn−1,Txn)

}
= Sb(Txn−1,Txn−1,Txn)

for some n, then we have

F
(
b2Sb(Txn−1,Txn−1,Txn)

)
≤ F

(
Sb(Txn−1,Txn−1,Txn)

)
− τ

< F
(
Sb(Txn−1,Txn−1,Txn)

)
.

Using condition (F1) we conclude that Sb(xn,xn,xn+1) < Sb(xn,xn,xn+1), which is a contradiction. There-

fore, for each n ∈ N we have

max
{
Sb(xn−1,xn−1,xn),Sb(xn,xn,xn+1)

}
= Sb(xn−1,xn−1,xn).

Applying again (2.2) and condition (F1), we deduce that

Sb(xn,xn,xn+1) < Sb(xn−1,xn−1,xn)

for each n. Thus
{
Sb(xn,xn,xn+1)

}
is a nonnegative decreasing sequence of real numbers. Then there exists

A ≥ 0 such that

lim
n→+∞

Sb(xn,xn,xn+1) = inf
n∈N

Sb(xn,xn,xn+1) = A.

We claim that A = 0. To support the claim, let it be untrue and A > 0. Then, for any ε > 0, it is possible

to find a positive integer m so that

Sb(xm,xm,Txm) < A + ε.

So, from (F1), we get

F
(
Sb(xm,xm,Txm)

)
< F(A + ε). (2.4)

It follows from (2.1) that

τ + F
(
b2Sb(Txm,Txm,T

2xm)
)
≤ F

(
N(xm,Txm)). (2.5)

By a similar argument as (2.3), it yields that

N(xm,Txm) = max
{
Sb(xm,xm,Txm),Sb(Txm,Txm,T

2xm)
}
.

Hence (2.5), becomes

F
(
b2Sb(Txm,Txm,T

2xm)
)
≤ F

(
max

{
Sb(xm,xm,Txm),Sb(Txm,Txm,T

2xm)
})
− τ. (2.6)

Now if, max{Sb(xm,xm,Txm),Sb(Txm,Txm,T2xm)} = Sb(Txm,Txm,T2xm) for some m, then (2.6) gives us a

contradiction. Thus, we infer that

max
{
Sb(xm,xm,Txm),Sb(Txm,Txm,T

2xm)
}

= Sb(xm,xm,Txm),



Int. J. Anal. Appl. 17 (5) (2019) 740

and therefore, we have

F
(
b2Sb(Txm,Txm,T

2xm)
)
≤ F

(
Sb(xm,xm,Txm)

)
− τ.

It implies that

F
(
b2Sb(T

2xm,T
2xm,T

3xm)
)
≤ F

(
Sb(Txm,Txm,T

2xm)
)
− τ

≤ F
(
b2Sb(Txm,Txm,T

2xm)
)
− τ

≤ F
(
Sb(xm,xm,Txm)

)
− 2τ.

Continuing the above process and taking (2.4) into account, we deduce that

F
(
b2Sb(T

nxm,T
nxm,T

n+1xm)
)
≤ F

(
Sb(T

n−1xm,T
n−1xm,T

nxm)
)
− τ

≤ F
(
b2Sb(T

n−1xm,T
n−1xm,T

nxm)
)
− τ

≤ F
(
Sb(T

n−2xm,T
n−2xm,T

n−1xm)
)
− 2τ

.

.

.

≤ F
(
Sb(xm,xm,Txm)

)
−nτ

< F(A + ε) −nτ,

and by passing to the limit as n → +∞ we obtain

lim
n→+∞

F
(
b2Sb(T

nxm,T
nxm.T

n+1xm)
)

= −∞.

This fact together with the condition (F2) implies that

lim
n→+∞

Sb(T
nxm,T

nxm,T
n+1xm) = 0.

Thus Sb(T
nxm,T

nxm,T
n+1xm) < A for n sufficiently large, which is a contradiction with the definition of

A. Then,

lim
n→+∞

Sb(xn,xn,xn+1) = 0. (2.7)

Next, we intend to show that the sequence {xn} is a Cauchy sequence in X. Arguing by contradiction, we

assume that there exist ε > 0, and subsequences {xq(n)} and {xp(n)} of {xn} with n < q(n) < p(n) such that

Sb(xq(n),xq(n),xp(n)) ≥ ε (2.8)

for each n ∈ N. Further, corresponding to q(n), we can choose p(n) in such a way that it is the smallest

integer with q(n) < p(n) satisfying the above inequality, then

Sb(xq(n),xq(n),xp(n)−1) < ε (2.9)

for all n ∈ N.
In the light of (2.8) and the condition (2.1), we conclude that

F
(
b2Sb(Txq(n)−1,Txq(n)−1,Txp(n)−1)

)
≤ F

(
N(xq(n)−1,xp(n)−1)

)
− τ. (2.10)



Int. J. Anal. Appl. 17 (5) (2019) 741

By our hypothesis and in view of (dSb2), we get

max
{
Sb(xq(n)−1,xq(n)−1,xp(n)−1),Sb(Txq(n)−1,Txq(n)−1,Txp(n)−1)

}
≤ N(xq(n)−1,xp(n)−1)

= max
{
Sb(xq(n)−1,xq(n)−1,xp(n)−1),Sb(Txq(n)−1,Txq(n)−1,Txp(n)−1),

Sb(xp(n)−1,xp(n)−1,Txq(n)−1)

10b8
,
Sb(xq(n)−1,xq(n)−1,TxP(n)−1)

10b9
,

Sb(xp(n)−1,xp(n)−1,Txq(n))

10b4

}
≤ max

{
Sb(xq(n)−1,xq(n)−1,xp(n)−1),Sb(Txq(n)−1,Txq(n)−1,Txp(n)−1),

Sb(xp(n)−1,xp(n)−1,Txp(n)−1)

5b7
+
Sb(Txq(n)−1,Txq(n)−1,Txp(n)−1)

10b7
,

Sb(xq(n)−1,xq(n)−1,TxP(n)−1)

10b9
,
Sb(xp(n)−1,xp(n)−1,Txq(n))

10b4

}
≤ max

{
Sb(xq(n)−1,xq(n)−1,xp(n)−1),Sb(Txq(n)−1,Txq(n)−1,Txp(n)−1)

}
.

It enforces that

N(xq(n)−1,xp(n)−1) = max
{
Sb(xq(n)−1,xq(n)−1,xp(n)−1),Sb(Txq(n)−1,Txq(n)−1,Txp(n)−1)

}
.

Suppose that the maximum on the right-hand side is equal to Sb(Txq(n)−1,Txq(n)−1,Txp(n)−1) for some n,

then from relation (2.10) together with the condition (F1) we get

Sb(Txq(n)−1,Txq(n)−1,Txp(n)−1) < Sb(Txq(n)−1,Txq(n)−1,Txp(n)−1)

which is a contradiction. Thus, we find that

max
{
xq(n)−1,xq(n)−1,xp(n)−1),Sb(xq(n),xq(n),xp(n))

}
= Sb(xq(n)−1,xq(n)−1,xp(n)−1)

for all n. Accordingly, (2.10) becomes

F
(
b2Sb(Txq(n)−1,Txq(n)−1,Txp(n)−1)

)
≤ F

(
Sb(xq(n)−1,xq(n)−1,xp(n)−1)

)
− τ (2.11)

and so using (F1) we get

Sb
(
xq(n),xq(n),xp(n)

)
< Sb(xq(n)−1,xq(n)−1,xp(n)−1). (2.12)

Regarding to (2.8), (2.12) and employing (dSb2) we observe that

ε ≤ Sb(xq(n),xq(n),xp(n))

< Sb(xq(n)−1,xq(n)−1,xp(n)−1)

≤ 2bSb(xq(n)−1,xq(n)−1,xq(n)) + bSb(xp(n)−1,xp(n)−1,xq(n))

≤ 2bSb(xq(n)−1,xq(n)−1,xq(n)) + 2b
2Sb(xp(n)−1,xp(n)−1,xp(n)−1)

+b2Sb(xq(n),xq(n),xp(n)−1)

≤ 2bSb(xq(n)−1,xq(n)−1,xq(n)) + 6b
3Sb(xp(n)−1,xp(n)−1,xp(n))

+b2Sb(xq(n),xq(n),xp(n)−1).

Combining this result with (2.7) and (2.9) we get

ε ≤ lim sup
n→+∞

Sb(xq(n),xq(n),xp(n)) ≤ lim sup
n→+∞

Sb(xq(n)−1,xq(n)−1,xp(n)−1) ≤ b
2ε. (2.13)



Int. J. Anal. Appl. 17 (5) (2019) 742

In view of (2.13) and (2.11) and applying the conditions (F1) and (F3′), we have

F(b2ε) ≤ F
(
b2 lim sup
n→+∞

Sb(xq((n),xq(n),xp(n))
)

≤ F
(

lim sup
n→+∞

Sb(xq(n)−1,xq(n)−1,xp(n)−1)
)
− τ

≤ F(b2ε) − τ.

It is a contradiction with τ > 0, and therefore it follows that {xn} is a Cauchy sequence in X. By

completeness of (X,Sb), {xn} converges to some point v ∈ X. Then, for each ε > 0, there exists N1 ∈ N

such that

Sb(v,v,xn) < ε, (2.14)

for all n ≥ N1. We are going to show that v is a fixed point of T. For this aim, we consider two following

cases:

Case 1. If Sb(Tv,Tv,Txn) = 0 for some n ≥ N1, then from (dSb2) we find that

Sb(Tv,Tv,v) ≤ 2bSb(Tv,Tv,Txn) + bSb(v,v,Txn) ≤ bε.

Case 2. If Sb(Tv,Tv,Txn) > 0 for all n ≥ N1, then using (2.1), we get

F
(
b2Sb(Tv,Tv,Txn)

)
≤ F

(
N(v,xn)

)
− τ. (2.15)

From our assumptions, and using (dSb2), it follows that

max
{
Sb(v,v,xn),Sb(Tv,Tv,Txn)

}
≤ N(v,xn)

= max
{
Sb(v,v,xn),Sb(Tv,Tv,Txn),

Sb(xn,xn,Tv)

10b8
,
Sb(v,v,Txn)

10b9
,
Sb(xn,xn,T

2v)

10b4

}
≤ max

{
Sb(v,v,xn),Sb(Tv,Tv,Txn),

Sb(xn,xn,Txn)

5b7
+
Sb(Tv,Tv,Txn)

10b7
,

Sb(v,v,Txn)

10b9
,
Sb(xn,xn,T

2v)

10b4

}
= max

{
Sb(v,v,xn),Sb(Tv,Tv,Txn)

}
.

It enforces that N(v,xn) = max
{
Sb(v,v,xn),Sb(Tv,Tv,Txn)

}
. Now, if we assume that the maximum on the

right-hand side of this equality is equal to Sb(Tv,Tv,Txn), then by replacing it in (2.15), we obtain

Sb(Tv,Tv,Txn) < Sb(Tv,Tv,Txn) which is a contradiction. Consequently, for each n ∈ N we have

max
{
Sb(v,v,xn),Sb(Tv,Tv,Txn)

}
= Sb(v,v,xn).

Hence, (2.15) turns into

F
(
b2Sb(Tv,Tv,Txn)

)
≤ F

(
Sb(v,v,xn)

)
− τ

< F
(
Sb(v,v,xn)

)
.

Employing the condition (F1), we get

Sb(Tv,Tv,Txn) < Sb(v,v,xn). (2.16)



Int. J. Anal. Appl. 17 (5) (2019) 743

From (dSb2), (2.16) and (2.14), we deduce that

Sb(Tv,Tv,v) ≤ 2bSb(Tv,Tv,TxN1 ) + bSb(v,v,TxN1 ) < 3bε.

From the arbitrariness of ε in each case, it follows that Sb(Tv,Tv,v) = 0 which implies that Tv = v. Hence,

v is a fixed point of T.

Finally, we show that T has at most one fixed point. Indeed, if v1,v2 ∈ X are two fixed points of T, such
that v1 6= v2, then we obtain

F
(
b2Sb(Tv1,Tv1,Tv2)

)
≤ F(N(v1,v2)

)
− τ, (2.17)

From our hypothesis and by using (dSb2), it follows that

Sb(v1,v1,v2) ≤ N(v1,v2)

≤ max
{
Sb(v1,v1,v2),Sb(Tv1,Tv1,Tv2),

Sb(v2,v2,Tv2)

5b7
+
Sb(Tv1,Tv1,Tv2)

10b7
,

Sb(v1,v1,Tv2)

10b9
,
Sb(Tv2,Tv2,T

2v1)

10b4

}
= max

{
Sb(v1,v1,v2),Sb(Tv1,Tv1,Tv2)

}
= Sb(v1,v1,v2).

Then (2.17) becomes

F
(
b2Sb(v1,v1,v2)

)
≤ F(Sb(v1,v1,v2)

)
− τ.

It gives us a contradiction. Therefore, v1 = v2 and the fixed point is unique. �

Now we illustrate our result contained in Theorem 2.1 with help of two examples.

Example 2.2. Let (X,Sb) be as in Example 2.1 and let τ > 0 be an arbitrary fixed number. Define

the mapping T : X → X by T(x) = e−τ x
8

and take F(α) = ln α + α (α > 0). It is easily verified that

N(x,y) = Sb(x,x,y) = 2x + 4y. Assume that x or y is nonzero, then Sb(Tx,Tx,Ty) > 0 and we have

τ + F
(
b2Sb(Tx,Tx,Ty)

)
= τ + ln(e−τ (x + 2y)) + e−τ (x + 2y)

= ln(x + 2y) + e−τ (x + 2y)

≤ ln(2x + 4y) + 2x + 4y

= F
(
Sb(x,x,y)

)
= F

(
N(x,y)

)
.

Hence, T is a generalized F -contraction. On the other hand, if we assume that 0 < τ ≤ 0.0250587314, then
the following estimate holds:

max
{Sb(y,y,Ty)

5b7
+
Sb(Tx,Tx,Ty)

10b7
,
Sb(x,x,Ty)

10b9
,
Sb(y,y,T

2x)

10b4

}

= max
{2y + e−τ y

2

5 × 27
+
e−τ ( x

4
+
y
2

)

10 × 27
,

2x + e−τ
y
2

10 × 29
,

2y + e−2τ x
16

10 × 24
}

≤ e−τ (
x

4
+
y

2
) = Sb(Tx,Tx,Ty).

Thus all conditions of Theorem 2.1 hold and 0 is a unique fixed point of T.



Int. J. Anal. Appl. 17 (5) (2019) 744

Example 2.3. Let X = R, and Sb : X3 → R+ be a mapping defined by
Sb(x,y,z) =

x2

2
+ y

2

2
+ 2z2. Then (X,Sb) is a complete dislocated Sb-metric with b = 2. Define the mapping

T : X → X by T(x) = x
3

and take F(α) = ln α (α > 0). It is easily checked that N(x,y) = Sb(x,x,y) =

x2 + 2y2. Assume that x or y is nonzero, then Sb(Tx,Tx,Ty) > 0 and we have

τ + F
(
b2Sb(Tx,Tx,Ty)

)
≤ F

(
N(x,y)

)
⇔ ln(

9

4
) ≥ τ.

Also, we observe that

max
{Sb(y,y,Ty)

5b7
+
Sb(Tx,Tx,Ty)

10b7
,
Sb(x,x,Ty)

10b9
,
Sb(y,y,T

2x)

10b4

}
= max

{48y2 + 2x2
23040

,
18x2 + 4y2

92160
,

162y2 + 4x2

25920

}
≤

10240x2 + 20480y2

92160

= Sb(Tx,Tx,Ty)

for all x,y ∈ X. Now, if we assume that 0 < τ ≤ ln( 9
4
), then all the conditions of Theorem 2.1 hold and 0

is a unique fixed point of T .

Now, we describe the concept of generalized F-Suzuki-contraction in the framework of dislocated Sb-metric

spaces.

Definition 2.4. Let (X,Sb) be a dislocated Sb-metric space. A mapping T : X → X is said to be a
generalized F -Suzuki-contraction if there exists F ∈ F such that for all x,y ∈ X

1

2b
Sb(x,x,Tx) < Sb(x,x,y) ⇒ F

(
2b3Sb(Tx,Tx,Ty)

)
≤ F

(
MT (x,y)

)
−ψ

(
MT (x,y)

)
, (2.18)

where ψ ∈ Ψ and

MT (x,y) = max
{
Sb(x,x,y),

Sb(y,y,Ty)

10
,
Sb(x,x,Tx)

10
,Sb(Tx,Tx,Ty),

Sb(y,y,Tx)

18b
,
Sb(Tx,Tx,T

2x)

2

}
.

Our second main result is the following.

Theorem 2.2. Let (X,Sb) be a complete dislocated Sb-metric space and T : X → X be a generalized
F -Suzuki-contraction satisfying the following condition:

max
{Sb(y,y,Ty)

10
,
Sb(x,x,Tx)

10
,
Sb(y,y,Ty)

9
+
Sb(Tx,Tx,Ty)

18
,
Sb(Tx,Tx,T

2x)

2

}
≤ Sb(Tx,Tx,Ty)

for all x,y in X. Then T has a unique fixed point in X.

Proof. Let x0 be arbitrary. Define xn = Txn−1 for each n ∈ N. If there exists n ∈ N such that Sb(xn,xn,Txn) =
0, then xn = Txn and xn becomes a fixed point of T, which completes the proof. Therefore, we assume that

Sb(xn,xn,Txn) > 0 for all n ∈ N. Taking into account (2.18), we deduce

F
(
2b3Sb(Txn,Txn,Txn+1)

)
≤ F

(
MT (xn,xn+1)

)
−ψ

(
MT (xn,xn+1)

)
. (2.19)



Int. J. Anal. Appl. 17 (5) (2019) 745

Using (dSb2) we get

max
{
Sb(xn,xn,xn+1),Sb(xn+1,xn+1,Txn+1)

}
≤ MT (xn,xn+1)

≤ max
{
Sb(xn,xn,xn+1),

Sb(xn+1,xn+1,Txn+1)

10
,
Sb(xn,xn,Txn)

10
,

Sb(Txn,Txn,Txn+1),
Sb(Txn,Txn,Txn+1)

2
,
Sb(xn+1,xn+1,xn+2)

6

}
= max

{
Sb(xn,xn,xn+1),Sb(xn+1,xn+1,xn+2)

}
and combining it with the relation (2.19) we derive

F
(
2b3Sb(Txn,Txn,Txn+1)

)
≤ F

(
max

{
Sb(xn,xn,xn+1),Sb(xn+1,xn+1,xn+2

})
−ψ

(
max

{
Sb(xn,xn,xn+1),Sb(xn+1,xn+1,xn+2

})
. (2.20)

If max
{
Sb(xn,xn,xn+1),Sb(xn+1,xn+1,xn+2)

}
= Sb(xn+1,xn+1,xn+2), then (2.20) becomes

F
(
2b3Sb(Txn,Txn,Txn+1)

)
≤ F

(
Sb(xn+1,xn+1,xn+2)

)
−ψ

(
S(xn+1,xn+1,xn+2)

)
.

By the property of ψ and using condition (F1), we obtain

2b3Sb(Txn,Txn,Txn+1) < Sb(Txn,Txn,Txn+1),

which is a contradiction. Hence max
{
Sb(xn,xn,xn+1),Sb(xn+1,xn+1,xn+2)

}
= Sb(xn,xn,xn+1), then (2.20) be-

comes

F
(
2b3Sb(Txn,Txn,Txn+1)

)
≤ F

(
Sb(xn,xn,xn+1)

)
−ψ

(
S(xn,xn,xn+1)

)
. (2.21)

This together with condition (F1) implies that Sb(Txn,Txn,Txn+1) < Sb(xn,xn,xn+1) for each n ∈ N.
Then {Sb(xn,xn,xn+1)} is a nonnegative decreasing sequence of real numbers. Therefore, there exists A ≥ 0
such that lim

n→+∞
Sb(xn,xn,xn+1) = A.

Letting n → +∞ in (2.21) and using (F3′) and continuity of ψ, we get

F(2b3A) ≤ F(A) −ψ(A).

It gives us ψ(A) = 0. By property of ψ we deduce that A = 0. Consequently, we have

lim
n→+∞

Sb(xn,xn,xn+1) = 0. (2.22)

Next, we prove that {xn} is a Cauchy sequence in X. If it is not true, then there exist ε > 0 and increasing
sequences of natural numbers {p(n)} and {q(n)} such that

n < q(n) < p(n),

Sb(xq(n),xq(n),xp(n)) ≥ ε,

Sb(xq(n),xq(n),xp(n)−1) < ε (2.23)

for all n ∈ N.

Owing to (2.22), there exists N1 ∈ N such that

Sb(xq(n),xq(n),Txq(n)) < ε (2.24)



Int. J. Anal. Appl. 17 (5) (2019) 746

for all n ≥ N1. Hence, from (2.23) and (2.24) it follows that

1

2b
Sb(xq(n),xq(n),Txq(n)) <

1

2b
ε < Sb(xq(n),xq(n),xp(n))

for all n ≥ N1. By using (2.18) we obtain

F
(
2b3Sb(Txq(n),Txq(n),Txp(n))

)
≤ F

(
MT (xq(n),xp(n))

)
−ψ

(
MT (xq(n),xp(n))

)
. (2.25)

From our assumptions and regarding (dSb2), we get

max
{
Sb(xq(n),xq(n),xp(n)),Sb(Txq(n),Txq(n),Txp(n))

}
≤ MT (xq(n),xp(n))

≤ max
{
Sb(xq(n),xq(n),xp(n)),

Sb(xp(n),xp(n),xp(n)+1)

10
,Sb(Txq(n),Txq(n),Txp(n))

Sb(xq(n),xq(n),xq(n)+1)

10
,
Sb(xq(n)+1,xq(n)+1,xq(n)+2)

2
,

Sb(xp(n),xp(n),xp(n)+1)

9
+
Sb(xq(n)+1,xq(n)+1,xp(n)+1)

18

}
≤ max

{
Sb(xq(n),xq(n),xp(n)),Sb(xq(n)+1,xq(n)+1,xp(n)+1)

}
.

Then (2.25) becomes

F
(
2b3Sb(Txq(n),Txq(n),Txp(n))

)
≤ F

(
max

{
Sb(xq(n),xq(n),xp(n)),Sb(xq(n)+1,xq(n)+1,xp(n)+1)

})
− ψ

(
max

{
Sb(xq(n),xq(n),xp(n)),Sb(xq(n)+1,xq(n)+1,xp(n)+1)

})
.

If max
{
Sb(xq(n),xq(n),xp(n)),Sb(xq(n)+1,xq(n)+1,xp(n)+1)

}
= Sb(xq(n)+1,xq(n)+1,xp(n)+1) for some n, then we have

F
(
2b3Sb(Txq(n),Txq(n),Txp(n))

)
≤ F

(
Sb(xq(n)+1,xq(n)+1,xp(n)+1)

)
− ψ

(
Sb(xq(n)+1,xq(n)+1,xp(n)+1)

)
.

Obviously, Sb(xq(n)+1,xq(n)+1,xp(n)+1) > 0 and by the property of ψ and (F1), we get

Sb(Txq(n),Txq(n),Txp(n)) < Sb(xq(n)+1,xq(n)+1,xp(n)+1),

which is a contradiction. Duo to this fact, we find that

max
{
Sb(xq(n),xq(n),xp(n)),Sb(xq(n)+1,xq(n)+1,xp(n)+1)

}
= Sb(xq(n),xq(n),xp(n))

for all n. Therefore

F
(
2b3Sb(Txq(n),Txq(n),Txp(n))

)
≤ F

(
Sb(xq(n),xq(n),xp(n))

)
−ψ

(
Sb(xq(n),xq(n),xp(n))

)
, (2.26)

and by (F1), it follows that

Sb(xq(n)+1,xq(n)+1,xp(n)+1) < Sb(xq(n),xq(n),xp(n)).



Int. J. Anal. Appl. 17 (5) (2019) 747

In view of (2.23) and (dSb2), we infer that

ε ≤ Sb(xq(n),xq(n),xp(n)) ≤ 2bSb(xq(n),xq(n),xp(n)−1) + bSb(xp(n),xp(n),xp(n)−1)

≤ 2bSb(xq(n),xq(n),xp(n)−1) + 2b
2Sb(xp(n),xp(n),xp(n)) + b

2Sb(xp(n)−1,xp(n)−1,xp(n))

≤ 2bSb(xq(n),xq(n),xp(n)−1) + 6b
3Sb(xp(n),xp(n),xp(n)+1)

+b2Sb(xp(n)−1,xp(n)−1,xp(n)).

Taking the limit as n → +∞ in the above inequality and regarding (2.22) and (2.23), we deduce that

ε ≤ lim
n→+∞

Sb(xq(n),xq(n),xp(n)) ≤ 2bε. (2.27)

On the other hand, we have

ε ≤ Sb(xq(n),xq(n),xp(n)) ≤ 2bSb(xq(n),xq(n),xq(n)+1) + bSb(xp(n),xp(n),xq(n)+1)

≤ 2bSb(xq(n),xq(n),xq(n)+1) + 2b
2Sb(xp(n),xp(n),xp(n)+1)

+b2Sb(xq(n)+1,xq(n)+1,xp(n)+1).

Taking the limit supremum as n → +∞ in the above inequality. By using (2.22) we obtain

ε

b2
≤ lim sup

n→+∞
Sb(xq(n)+1,xq(n)+1,xp(n)+1). (2.28)

Taking the limit supremum as n → +∞ on each side of (2.26) and using conditions (2.27) and (2.28) together

with (F1) and (F3′), we deduce that

F(2bε) = F(2b3
ε

b2
) ≤ F

(
2b3 lim sup

n→+∞
Sb(xq(n)+1,Sb(xq(n)+1,xp(n)+1)

)
≤ F

(
lim sup
n→+∞

Sb(xq(n),Sb(xq(n),xp(n))
)

−ψ
(

lim inf
n→+∞

Sb(xq(n),Sb(xq(n),xp(n))
)

≤ F(2bε) −ψ(ε).

It enforces that ψ(ε) = 0, which leads to a contradiction. Therefore {xn} is a Cauchy sequence in X. Since

X is a complete dislocated Sb-metric space, it follows that there exists v ∈ X in which for each ε > 0, there

exists N2 ∈ N such that

Sb(v,v,xn) < ε (2.29)

for all n > N2. Now, we prove that v is a fixed point of T. To this end, we show that Sb(Tv,Tv,v) = 0.

We consider the following cases:

Case 1. If Sb(v,v,xn) = 0 for sufficiently large n, then v = xn. Thus, for sufficiently large n, we can

write

Sb(Tv,Tv,v) = Sb(Txn,Txn,v) ≤ 2bSb(xn+1,xn+1,xn+2) + bSb(v,v,xn+2).



Int. J. Anal. Appl. 17 (5) (2019) 748

Letting n → +∞ in the above inequality. From (2.22) and (2.29) we get Sb(Tv,Tv,v) = 0. Thus Tv = v

and v is a fixed point of T .

Case 2. If there exists n ≥ N2 such that Sb(v,v,xn) > 0 and Sb(Tv,Tv,Txn) = 0, then from (dSb2) we

have

Sb(Tv,Tv,v) ≤ 2bSb(Tv,Tv,Txn) + bSb(v,v,xn+1) ≤ bε,

which implies that Tv = v by virtue of the arbitrariness of ε.

Case 3. If Sb(v,v,xn) > 0 and Sb(Tv,Tv,Txn) > 0 for all n ≥ N2, then using (2.18) we obtain

F
(
2b3Sb(Tv,Tv,Txn)

)
≤ F

(
MT (v,xn)

)
−ψ

(
MT (v,xn)

)
. (2.30)

Thus, by using the hypothesis and taking into account (dSb2), it yields

max
{
Sb(v,v,xn),Sb(Tv,Tv,Txn)

}
≤ MT (v,xn)

≤ max
{
Sb(v,v,xn),Sb(Tv,Tv,Txn),

Sb(xn,xn,Txn)

10

Sb(v,v,Tv)

10
,
Sb(xn,xn,Txn)

9
+
Sb(Tv,Tv,Txn)

18
,
Sb(Tv,Tv,T

2v)

2

}
≤ max

{
Sb(v,v,xn),Sb(Tv,Tv,Txn)

}
.

Then (2.30) becomes

F
(
2b3Sb(Tv,Tv,Txn)

)
≤ F

(
max

{
Sb(v,v,xn),Sb(Tv,Tv,Txn)

})
− ψ

(
max

{
Sb(v,v,xn),Sb(Tv,Tv,Txn)

})
.

If max
{
Sb(v,v,xn),Sb(Tv,Tv,Txn)

}
= Sb(Tv,Tv,Txn), then we have

F
(
2b3Sb(Tv,Tv,Txn)

)
≤ F

(
Sb(Tv,Tv,Txn)

)
−ψ

(
Sb(Tv,Tv,Txn)

)
.

From this it follows that 2b3Sb(Tv,Tv,Txn) < Sb(Tv,Tv,Txn), which is a contradiction. Therefore,

max
{
Sb(v,v,xn),Sb(Tv,Tv,Txn)

}
= Sb(v,v,xn)

and (2.30) becomes

F
(
2b3Sb(Tv,Tv,Txn)

)
≤ F

(
Sb(v,v,xn)

)
−ψ

(
Sb(v,v,xn)

)
< F

(
Sb(v,v,xn)

)
.

Thus, from (F1) we get

Sb(Tv,Tv,Txn) < Sb(v,v,xn). (2.31)

Applying (2.29), (2.31) and (dSb2) we get

Sb(Tv,Tv,v) ≤ 2bSb(Tv,Tv,Txn) + bSb(v,v,xn+1) < 3bε

for sufficiently large n. It enforces that Tv = v by virtue of the arbitrariness of ε. Then v is a fixed point

of T.



Int. J. Anal. Appl. 17 (5) (2019) 749

Next, we show the uniqueness. Indeed, if v1, v2 are two fixed points of T such that v1 6= v2, then in view of
(2.18) we get

F
(
2b3Sb(Tv1,Tv1,Tv2)

)
≤ F

(
MT (v1,v2)

)
−ψ

(
MT (v1,v2)

)
. (2.32)

According to our assumptions and by using (dSb2), we find that

Sb(v1,v1,v2) ≤ MT (v1,v2)

≤ max
{
Sb(v1,v1,v2),Sb(Tv1,Tv1,Tv2),

Sb(v2,v2,Tv2)

10
,
Sb(v1,v1,Tv1)

10
,
Sb(v2,v2,Tv2)

9

+
Sb(Tv1,Tv1,Tv2)

18

}
≤ max

{
Sb(v1,v1,v2),Sb(Tv1,Tv1,Tv2)

}
= Sb(v1,v1,v2).

Then (2.32) becomes

F
(
2b3Sb(v1,v1,v2)

)
≤ F

(
Sb(v1,v1,v2)

)
−ψ

(
Sb(v1,v1,v2)

)
.

From this it follows that 2b3Sb(v1,v1,v2) < Sb(v1,v1,v2), which is a contradiction. Then v1 = v2 and so T

has a unique fixed point in X. �

Example 2.4. Let X = {−1, 0, 1}. Define the mapping Sb : X3 → R+ by

Sb(x,y,z) =




3
2
, 0 = x = y 6= z = 1 or −1 = x = y 6= z = 1

10
6
, 1 = x = y 6= z

0, x = y = z = −1 or 1

1
5
, otherwise

for all x,y,z ∈ X. It is easy to show that (X,Sb) is a complete dislocated Sb-metric space with b = 32. Put

F(α) = ln α (α > 0) and ψ(t) = t (t ≥ 0). Define T : X → X by

T(x) =




0, x = 1

−1, x = −1, 0.

Note that Sb(x,x,y) > 0 and Sb(T(x),T(x),T(y)) > 0 if and only if x ∈ {−1, 0}, y = 1 or x = 1, y ∈
{−1, 0}. Also, for each x,y ∈ X we have MT (x,y) = Sb(x,x,y) and we find that

F
(
2b3Sb(T(x),T(x),T(y))

)
≤ F

(
Sb(x,x,y)

)
−ψ

(
Sb(x,x,y)

)
⇔ ln

Sb(x,x,y)

2b3Sb(T(x),T(x),T(y))
≥ Sb(x,x,y).

Now, we consider two cases:



Int. J. Anal. Appl. 17 (5) (2019) 750

Case 2.1. Case 1. Let x ∈{−1, 0} and y = 1, then

Sb(x,x,y) =
3

2
,

Sb(T(x),T(x),T(y)) = Sb(−1,−1, 0) =
1

5
,

Sb(0, 0,T(0)) =
1

5
, Sb(−1,−1,T(−1)) = 0.

So, we have

ln
Sb(x,x,y)

2b3Sb(T(x),T(x),T(y))
= ln

3
2
54
40

= ln
120

108
= 3.0377 ≥ Sb(x,x,y) =

3

2
.

Case 2. Let x = 1 and y ∈{−1, 0}, then
Sb(x,x,y) =

10

6
,

Sb(T(x),T(x),T(y)) =
1

5
,

Sb(x,x,T(x)) =
10

6
.

So, we have

ln
Sb(x,x,y)

2b3Sb(T(x),T(x),T(y))
= ln

10
6
54
40

= ln
400

324
= 3.4369 ≥ Sb(x,x,y) =

10

6
.

On the other hand, for all x,y ∈ X we have

max
{Sb(y,y,T(y))

10
,
Sb(x,x,T(x))

10
,
Sb(y,y,T(y))

9
+
Sb(T(x),T(x),T(y))

18
,
Sb(T(x),T(x),T

2(x))

2

}
≤

1

5
= Sb(T(x),T(x),T(y)).

Hence, T is a generalized F -Suzuki-contraction which satisfies the assumption of Theorem 2.2 and so it has

a unique fixed point −1.

References

[1] I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal. Unianowsk Gos. Ped. Inst. 30

(1989), 26–37.

[2] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostra. 1 (1993), 5–11.

[3] N.V. Dung and V.L. Hang, A fixed point theorem for generalized F-contractions on complete metric spaces, Vietnam J.

Math. 43 (2015), 743–753.

[4] G.N.V. Kishore, K.P.R. Rao, D. Panthi, B. Srinuvasa Rao and S. Satyanaraya, Some applications via fixed point results

in partially ordered Sb-metric spaces, Fixed Point Theory Appl. 2017 (2017), 10.

[5] H. Piri and P. Kumam, Some fixed point theorems concerning F-contraction in complete metric spaces, Fixed Point Theory

Appl. 2014 (2014), 210.

[6] H. Piri and P. Kumam, Fixed point theorems for generalized F-Suzuki-contraction mappings in complete b-metric spaces,

Fixed Point Theory Appl. 2016 (2016), 90 .

[7] N.Y. Özgür and N. TaŞ, Some fixed point theorems on S-metric spaces, Mat. Vesnik 69 (1) (2017), 39-52.

[8] N.Y. Özgür, N. TaŞ and U. Celik, New fixed point-circle results on S-metric spaces, Bull. Math. Anal. Appl. 9 (2) (2017),

10–23.

[9] Y. Rohená,T. Došenović and S. Radenović, A note on the paper ”A Fixed point Theorems in Sb-Metric Spaces”, Filomal

31 (11) (2017), 3335–3346.



Int. J. Anal. Appl. 17 (5) (2019) 751

[10] Sh. Sedghi, A. Gholidahne, T. Došenovic, J. Esfahani and S. Radenovic, Common fixed point of four maps in Sb-metric

spaces, J. Linear Topol. Alg. 5 (2) (2016), 93–104.

[11] Sh. Sedghi, N. Shobe and A. Aliouche, A generalization of fixed point theorem in S-metric spaces, Mat. Vesn. 64 (2012),

258–266.

[12] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl.

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	1. Introduction and Preliminaries
	2. Main results
	References