J. Nig. Soc. Phys. Sci. 1 (2019) 88–94

Journal of the
Nigerian Society

of Physical
Sciences

Original Research

Common Fixed Point Theorems for Multivalued Generalized
F-Suzuki-Contraction Mappings in Complete Strong b−Metric

Spaces

Yusuf Ibrahim∗

Department of Mathematics, Sa’adatu Rimi College of Education, Kumbotso Kano, Nigeria

Abstract

This paper introduces a new version of multivalued generalized F-Suzuki-Contraction mapping and then establish some new common fixed point
theorems for these new multivalued generalized F-Suzuki-Contraction Mappings in complete strong b−Metric Spaces.

Keywords: Common Fixed Point Problem, Multivalued Generalized F-Suzuki-Contraction Mapping, Complete Strong b−metric Space.

Article History :
Received: 04 April 2019
Received in revised form: 01 September 2019
Accepted for publication: 02 September 2019
Published: 28 September 2019

c©2019 Journal of the Nigerian Society of Physical Sciences. All Rights Reserved.
Communicated by: B. J. Falaye

1. Introduction
Let X be a nonempty set and s ≥ 1 be a given real number. A
mapping d : X × X → R∗ is said to be a b-metric if for all
x, y, z ∈ X the following conditions are satisfied:

1. d(x, y) = 0 if and only if x = y;
2. d(x, y) = d(y, x);
3. d(x, z) ≤ s[d(x, y) + d(y, z)].

The pair (X, d) is called a b-metric space with constant s.
A strong b−metric is a semimetric space (X, d) if there exists
s ≥ 1 for which d satisfies the following triangular inequality.

d(x, y) ≤ d(x, z) + sd(z, y), f or each x, y, z ∈ X. (1)

In 1922, a mathematician Banach [1] proved a very important
result regarding a contraction mapping, known as the Banach
contraction principle, which states that every self-mapping T
defined on a complete metric space (X, d) satisfying

∗Corresponding Author Tel. No: +2348062814778
Email address: danustazz@gmail.com (Yusuf Ibrahim )

∀x, y ∈ X, d(T x, T y) ≤ λd(x, y), where λ ∈ (0, 1)

has a unique fixed point and for every x0 ∈ X a sequence
{Tn x0}∞n=1converges to the fixed point. Subsequently, in 1962,
Edelstein [2] proved the following version of the Banach con-
traction principle. Let (X, d) be a compact metric space and let
T : X → X be a self-mapping. Assume that for all x, y ∈ X with
x , y,

d(x, T x) < d(x, y) =⇒ d(T x, T y) < d(x, y).

Then T has a unique fixed point in X. In 2012, Wardowski [3]
introduced a new type of contractions called F-contraction and
proved a new fixed point theorem concerning F-contractions.

Let (X, d) be a metric space. A mapping T : X → X is said to
be an F-contraction if there exists τ > 0 such that

∀x, y ∈ X, d(T x, T y) > 0 =⇒ τ + F(d(T x, T y)) ≤ F(d(x, y)),

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Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 88–94 89

where F : R+ → R is a mapping satisfying the following con-
ditions:

F1 F is strictly increasing, i.e. for all x, y ∈ R+ such that
x < y, F(x) < F(y);

F2 For each sequence {αn}∞n=1 of positive numbers, limn→∞
αn =

0 if and only if lim
n→∞

F(αn) = −∞;

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

αk F(α) = 0.

We denote by ζ, the set of all functions satisfying the conditions
(F1) − (F3). Wardowski [3] then stated a modified version of
the Banach contraction principle as follows. Let (X, d) be a
complete metric space and let T : X → X be an F-contraction.
Then T has a unique fixed point x∗ ∈ X and for every x ∈ X
the sequence {Tn x}∞n=1 converges to x

∗. In 2014, Hossein, P. and
Poom, K. [15] defined the F-Suzuki contraction as follows and
gave another version of theorem. Let (X, d) be a metric space.
A mapping T : X → X is said to be an F-Suzuki-contraction if
there exists τ > 0 such that for all x, y ∈ X with T x , T y

d(x, T x) < d(x, y) =⇒ τ + F(d(T x, T y)) ≤ F(d(x, y)),

where F : R+ → R is a mapping satisfying the following con-
ditions:

F1 F is strictly increasing, i.e. for all x, y ∈ R+ such that
x < y, F(x) < F(y);

F2 For each sequence {αn}∞n=1 of positive numbers, limn→∞
αn =

0 if and only if lim
n→∞

F(αn) = −∞;

F3 F is continuous on (0,∞)

We denote by ζ, the set of all functions satisfying the conditions
(F1) − (F3).

Let T be a self-mapping of a complete metric space X into itself.
Suppose F ∈ ζ and there exists τ > 0 such that

∀x, y ∈ X, d(T x, T y) > 0 =⇒ τ + F(d(T x, T y)) ≤ F(d(x, y)).

Then T has a unique fixed point x∗ ∈ X and for every x0 ∈ X
the sequence {Tn x0}∞n=1 converges to x

∗.

Following this direction of research (see examples, [4, 5, 6, 7,
8, 9, 10, 16, 17]), in this paper, fixed point results of Piri and
Kumam [11], Ahmad et al. [9], Suzuki [18] and Suzuki [19] are
extended by introducing common fixed point problem for mul-
tivalued generalized F-Suzuki-contraction mappings in strong
b-metric spaces.

Definition 1.1. (Hardy and Rogers [14])
(1) There exist non-negative constants a, satisfying

∑5
i=1 ai <

1 such that, for each x, y ∈ X, d( f (x), f (y)) < a1d(x, y) +
a2d(x, f (x)) + a3d(y, f (y)) + a4d(x, f (y)) + a5d(y, f (x)).

(2) There exist monotonically decreasing functions ai(t) :
(0,∞) → [0, 1) satisfying

∑5
i=1 ai(t) < 1 such that, for

each x, y ∈ X, x , y, d( f (x), f (y)) < a1(d(x, y))d(x, f (x))
+ a2(d(x, y))d(y, f (y)) + a3(d(x, y))d(x, f (y))
+ a4(d(x, y))d(y, f (x)) + a5(d(x, y))d(x, y).

(3) For each x, y ∈ X, x , y,
d( f (x), f (y)) < max{d(x, y), d(x, f (x)),
d(y, f (y)), d(x, f (y)), d(y, f (x))}.

Lemma 1.1. [13] From definition 1.1, (1) =⇒ (2) =⇒ (3).

Denote by C B(X), the collection of all nonempty closed and
bounded subsets of X and let H be the Hausdorff metric with
respect to the metric d; that is,

H(A, B) = max{sup
a∈A

d(a, B), sup
b∈B

d(b, A)}

for all A, B ∈ C B(X), where d(a, B) = inf
b∈B

d(a, b) is the distance

from the point a to the subset B.

2. Main Results
Definition 2.1. Let 0 be the family of all functions F : R+ → R
such that:

(F1) F is strictly increasing, i.e. for all x, y ∈ R+ such that
x < y, F(x) < F(y);

(F2) for each sequence {αn}∞n=1 of positive numbers, limn→∞
αn =

0 if and only if lim
n→∞

F(αn) = −∞;

(F3) F is continuous on (0,∞).

Definition 2.2. Let Ψ be the family of all functions ψ : [0,∞) →
[0,∞) such that ψ is continuous and ψ(t) = 0 iff t = 0.

Definition 2.3. Let (X, d) be a strong b−metric space. Map-
pings T, S : X → C B(X) are said to be multivalued generalized
F-Suzuki-Contraction on (X, d) if there exists F ∈ 0 and ψ ∈ Ψ
such that, ∀x, y ∈ X, x , y,

1
1 + s

d(x, T x) < d(x, y) and
1

1 + s
d(y, S y) < d(y, S T x)

⇒ ψ(Nφ(x, y)) + F(s4 H(T x, S y)) ≤ F(Nφ(x, y)) in which

Nφ(x, y) = φ1(d(x, y))(d(x, y)) + φ2(d(x, y))(d(y, S T x))

+ φ3(d(x, y))
(

(d(y, T x)) + d(x, S y)
2s

)
+ φ4(d(x, y))

(
(d(x, S T x)) + H(S T x, S y)

2s

)
+ φ5(d(x, y))(H(S T x, S y) + H(S T x, T x))
+ φ6(d(x, y))(H(S T x, S y) + d(T x, x))
+ φ7(d(x, y))(d(T x, y)) + d(y, S y)) (2)

for which φ : R+ → [0, 1), with
∑7

i=1 φi(d(x, y)) < 1, is mono-
tonically decreasing function.

Comsidering the definition S T x := {S y ⊆ C B(X) : ∀y ∈ T x},
we have the following result.

Theorem 2.1. Let (X, d) be a complete strong b−metric space
and let T, S : X → C B(X) be multivalued generalized F-
Suzuki-Contraction mappings. Then T and S has a common

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Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 88–94 90

fixed point x∗ ∈ X and for every x ∈ X the sequence {T n x}∞n and
{S n x}∞n converge to x

∗.

Proof Let x0 = x ∈ X. Let xn+1 ∈ T xn and xn+2 ∈ S xn+1 ∀n ∈
N. If there exists n ∈ N such that d(xn, T xn) = d(xn+1, S xn+1) =
0 then xn+1 = xn = x becomes a fixed point of T and S , re-
spectively, therefore the proof is complete. Now, suppose that
d(xn, T xn) > 0 and d(xn+1, S xn+1) > 0 ∀n ∈ N then the proof
will be divided in to two steps.
Step one. We show that {xn}∞n=1 is a Cauchy sequence.
Let

d(xn, T xn) > 0 and d(xn+1, S xn+1) > 0 ∀n ∈ N. (3)

therefore, we have that

1
s + 1

d(xn, T xn) < d(xn, T xn) and

1
s + 1

d(xn+1, S xn+1) < d(xn+1, S xn+1) ∀n ∈ N. (4)

By Definition 2.3, we get

F(H(T xn, S xn+1)) ≤ F(Nφ(xn, xn+1)) −ψ(Nφ(xn, xn+1)).

Since that

Nφ(xn, xn+1)
= φ1(d(xn, xn+1))(d(xn, xn+1)) + φ2(d(xn, xn+1))(d(xn+1, xn+2))

+ φ3(d(xn, xn+1))
(

d(xn, xn+2)
2s

)
+ φ4(d(xn, xn+1))

(
(d(xn, xn+2))

2s

)
+ φ5(d(xn, xn+1))(d(xn+2, xn+1)) + φ6(d(xn, xn+1))(d(xn, xn+1))
+ φ7(d(xn, xn+1))(d(xn+2, xn+1)
≤ φ1(d(xn, xn+1))(d(xn, xn+1)) + φ2(d(xn, xn+1))(d(xn+1, xn+2))

+ φ3(d(xn, xn+1))
(

d(xn, xn+1) + sd(xn+1, xn+2)
2s

)
+ φ4(d(xn, xn+1))

(
d(xn, xn+1) + sd(xn+1, xn+2)

2s

)
+ φ5(d(xn, xn+1))(d(xn+2, xn+1)) + φ6(d(xn, xn+1))(d(xn, xn+1))
+ φ7(d(xn, xn+1))(d(xn+2, xn+1)
≤ φ1(d(xn, xn+1))(d(xn, xn+1)) + φ2(d(xn, xn+1))(d(xn+1, xn+2))

+ φ3(d(xn, xn+1))
(

s[d(xn, xn+1) + d(xn+1, xn+2)]
2s

)
+ φ4(d(xn, xn+1))

(
s[d(xn, xn+1) + d(xn+1, xn+2)]

2s

)
+ φ5(d(xn, xn+1))(d(xn+2, xn+1)) + φ6(d(xn, xn+1))(d(xn, xn+1))
+ φ7(d(xn, xn+1))(d(xn+2, xn+1)
≤ φ1(d(xn, xn+1))(d(xn, xn+1)) + φ2(d(xn, xn+1))(d(xn+1, xn+2))
+ φ3(d(xn, xn+1))(d(xn, xn+1)) + φ3(d(xn, xn+1))(d(xn+2, xn+1)
+ φ4(d(xn, xn+1))(d(xn, xn+1)) + φ4(d(xn, xn+1))(d(xn+2, xn+1)
+ φ5(d(xn, xn+1))(d(xn+2, xn+1)) + φ6(d(xn, xn+1))(d(xn, xn+1))
+ φ7(d(xn, xn+1))(d(xn+2, xn+1)
= [φ1(d(xn, xn+1)) + φ3(d(xn, xn+1)) + φ4(d(xn, xn+1))

+ φ6(d(xn, xn+1))](d(xn, xn+1))
+ [φ2(d(xn, xn+1)) + φ3(d(xn, xn+1)) + φ4(d(xn, xn+1))
+ φ5(d(xn, xn+1))(d(xn+2, xn+1)) + φ7(d(xn, xn+1))](d(xn+2, xn+1)
= φ′(d(xn, xn+1))(d(xn, xn+1)) + φ

′′(d(xn, xn+1))(d(xn+2, xn+1))
(5)

then by (5) and definition 2.3, we get

F(d(xn+1, xn+2))
≤ F(φ′(d(xn, xn+1))(d(xn, xn+1)) + φ

′′(d(xn, xn+1))(d(xn+2, xn+1)))
−ψ(φ′(d(xn, xn+1))(d(xn, xn+1)) + φ

′′(d(xn, xn+1))(d(xn+2, xn+1))).
(6)

On contrary, if d(xn+1, xn+2) > d(xn, xn+1), then

φ′(d(xn, xn+1))(d(xn, xn+1))
+φ′′(d(xn, xn+1))(d(xn+2, xn+1)) < d(xn+1, xn+2)

and therefore (6) becomes

F(d(xn+1, xn+2)) ≤ F(d(xn+1, xn+2)) −ψ(d(xn+1, xn+2)).

But, from (3) and the fact that ψ(d(xn+1, xn+2)) > 0, this is a
contradiction. Thus, we conclude that

F(d(xn+1, xn+2)) ≤ F(d(xn, xn+1)) −ψ(d(xn, xn+1))
< F(d(xn, xn+1)). (7)

By (7) and Definition 2.1(F1), we have that

d(xn+1, xn+2) < d(xn, xn+1) < d(xn−1, xn) ∀n ∈ N. (8)

Therefore {d(xn, xn+1)} is a nonnegative decreasing sequence of
real numbers. Thus there exists γ ≥ 0 such that lim

n→∞
d(xn, xn+1) =

γ. From (7) as n →∞, we have that

F(γ) ≤ F(γ) −ψ(γ).

This implies that ψ(γ) = 0 and thus γ = 0. Consequently we
arrive at

lim
n→∞

d(xn, T xn) = lim
n→∞

d(xn, xn+1) = 0. (9)

Now, we claim that {xn}∞n=1 is a Cauchy sequence. On contrary,
we assume that there exists � > 0 and n, m ∈ N such that, for all
n ≥ n� and n� < n < m,

d(xn, xm) ≥ � and d(xn−1, xm) < �. (10)

It implies that

� ≤ d(xn, xm) ≤ d(xn, xn−1) + sd(xn−1, xm)
< d(xn, xn−1) + s�. (11)

By (11) and (9), we have that

� ≤ limsup
n→∞

d(xn, xm) < s�. (12)

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Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 88–94 91

By triangle inequality, we have that

� ≤ d(xn, xm) ≤ d(xn, xm+1) + sd(xm+1, xm)
≤ d(xn, xm) + 2sd(xm+1, xm). (13)

By (9),(10), (12) and (13), we have that

� ≤ limsup
n→∞

d(xn, xm+1) < s�. (14)

Similarly, we have that

� ≤ d(xn, xm) ≤ d(xn, xn+1) + sd(xn+1, xm)

≤ sd(xn, xm) + (s
2

+ 1)d(xn, xn+1). (15)

By (9),(10), (12) and (15), we have that

� ≤ limsup
n→∞

d(xn, xn+1) < s�. (16)

Observe that

d(xn, xm+1) ≤ d(xn, xn+1) + sd(xn+1, xm+1)
≤ d(xn, xn+1) + s[d(xn+1, xm) + sd(xm+1, xm)]
≤ d(xn, xn+1) + s[d(xn, xn+1) + sd(xn, xm)
+ sd(xm+1, xm)]. (17)

By (17), we have that

�

s
≤ limsup

n→∞
d(xn+1, xm+1) < s

2�. (18)

By (9)and (10), we select n� > 0 ∈ N such that

1
s + 1

d(xn, T xn) <
1

s + 1
� < � ≤ d(xn, xm) ∀n ≥ n(�)

⇔
1

s + 1
d(xn, T xn) <

1
s + 1

� < d(xn, xm)

∀n ≥ n(�)

and
1

s + 1
d(xn+1, S xn+1) <

1
s + 1

� <
�

s
≤ d(xn+1, xm+1) ∀n ≥ n�

⇔
1

s + 1
d(xn+1, S xn+1) <

1
s + 1

�

< d(xn+1, xm+1) ∀n ≥ n�

It follows that from Definition 2.3, we have, for every n ≥ n�

F(H(xn+1, xm+1)) ≤ F(Nφ(xn, xm)) −ψ(Nφ(xn, xm)). (19)

Since that

d(xn, xm) ≤ Nφ(xn, xm)

= φ1(d(xn, xm))(d(xn, xm)) + φ2(d(xn, xm))(d(xn+2, xm))

+ φ3(d(xn, xm))
(

d(xn+1, xm) + d(xn, xm+1)
2s

)
+ φ4(d(xn, xm))

(
(d(xn+2, xn) + d(xn+2, xm+1))

2s

)
+ φ5(d(xn, xm))(d(xn+2, xm+1) + d(xn+2, xn+1))

+ φ6(d(xn, xm))(d(xn+2, xm+1) + d(xn, xn+1))

+ φ7(d(xn, xm))(d(xm, xn+1 + d(xm, xm+1)))

≤ φ1(d(xn, xm))(d(xn, xm)) + φ2(d(xn, xm))(d(xn+2, xn+1) + sd(xn+1, xm))

+ φ3(d(xn, xm))
(

d(xn+1, xm) + d(xn, xm+1)
2s

)
+ φ4(d(xn, xm))

(
(d(xn+2, xn+1) + sd(xn+1, xn) + d(xn+2, xn+1)) + sd(xn+1, xm+1))

2s

)
+ φ5(d(xn, xm))(d(xn+2, xn+1) + sd(xn+1, xm+1) + d(xn+2, xn+1))

+ φ6(d(xn, xm))(d(xn+2, xn+1) + sd(xn+1, xm+1) + d(xn, xn+1))

+ φ7(d(xn, xm))(d(xm, xn+1) + d(xm, xm+1))). (20)

By (12), (14), (16), (18) and (20), we have that

limsup
n→∞

d(xn, xm) ≤ limsup
n→∞

Nφ(xn, xm) < φ1(�)(s�) + φ2(�)(s
2�)

+ φ3(�)(�) + φ4(�)(
s2�
2

) + φ5(�)(s
3�) + φ6(�)(s

3�) + φ7(�)(s�)

≤ max{s�, s2�,�,
s�
2
, s3�, s�}

= s3�

and therefore

� ≤ limsup
n→∞

Nφ(xn, xm) < s
3�. (21)

Similarly

� ≤ limin f
n→∞

Nφ(xn, xm) < s
3�. (22)

By (19), (21) and (22), we have that

F(s3�) = F(s4
�

s
) ≤ F(s4limsup

n→∞
d(xn+1, xm+1))

≤ F(limsup
n→∞

Nφ(xn, xm)) −ψ(limsup
n→∞

Nφ(xn, xm))

≤ F(s3�) −ψ(�). (23)

By (23) and the fact that � > 0, this is a contradiction. Hence
{xn} is a Cauchy sequence in X. By completeness of (X, d),
{xn}∞n=1 and {xn+1}

∞
n=1 converge to some point x

∗ ∈ X, that is,

lim
n→∞

d(xn, x
∗) = 0 and lim

n→∞
d(xn+1, x

∗) = 0. (24)

There exists increasing sequences {nk}, {n + 1k} ⊂ N such that
xnk ∈ T x

∗ and xn+1k ∈ S x
∗ for all k ∈ N. Since T x∗ and S x∗ are

closed and

lim
n→∞

d(xnk, x
∗) = 0 and lim

n→∞
d(xn+1k, x

∗) = 0,

we get x∗ ∈ T x∗ and x∗ ∈ S x∗.
Step two. We show that x∗ is a common fixed point of T and S .
It suffices to show that

1
1 + s

d(xn, T xn) < d(xn, x
∗) and

1
1 + s

d(xn+1, S xn+1) < d(xn+1, x
∗),

f or every n ∈ N, (25)

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Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 88–94 92

implies

F(d(T x∗, x∗)) ≤ F(Nφ(x
∗, T x∗)) −ψ(Nφ(x

∗, T x∗))

and

F(d(S x∗, x∗)) ≤ F(Nφ(S x
∗, x∗)) −ψ(Nφ(S x

∗, x∗)),

respectively.
On contrary, suppose there exists m ∈ N such that

1
1 + s

d(xm, T xm) ≥ d(xm, x
∗) or

1
1 + s

d(xm+1, S xm+1) ≥ d(xm+1, x
∗).

(26)

By (26), we have that

(s + 1)d(xm, x
∗) ≤ d(xm, T xm) ≤ d(xm, x

∗) + sd(T xm, x
∗)

or

(s+1)d(xm+1, x
∗) ≤ d(xm+1, S xm+1) ≤ d(xm+1, x

∗)+sd(S xm+1, x
∗),

and therefore

d(xm, x
∗) ≤ d(T xm, x

∗) = d(xm+1, x
∗) and

d(xm+1, x
∗) ≤ d(S xm+1, x

∗) = d(xm+2, x
∗). (27)

By (8), (26) and (27), this is a contradiction. Hence, (25) holds,
and therefore

F(d(xn+1, x
∗)) = F(H(T xn, S x

∗))
≤ F(Nφ(xn, x

∗)) −ψ(Nφ(xn, x
∗)), (28)

and

F(d(xn+2, x
∗)) = F(H(S xn+1, T x

∗))
≤ F(Nφ(xn+1, x

∗)) −ψ(Nφ(xn+1, x
∗)). (29)

Since that

d(x∗, T x∗) ≤ Nφ(xn, x
∗)

= φ1(d(xn, x
∗))(d(xn, x

∗)) + φ2(d(xn, x
∗))(d(xn+2, x

∗))

+ φ3(d(xn, x
∗))

(
d(xn+1, x∗) + d(xn, S x∗)

2s

)
+ φ4(d(xn, x

∗))
(

d(xn, S x∗) + d(S x∗, xn+2)
2s

)
+ φ5(d(xn, x

∗))(d(S x∗, xn+2) + d(xn+1, S x
∗))

+ φ6(d(xn, x
∗))(d(xn, xn+1) + d(xn+2, T x

∗))
+ φ7(d(xn, x

∗))(d(T x∗, x∗) + d(x∗, xn+1))
≤ max{(d(xn, x

∗), d(xn+2, x
∗),

d(xn+1, x∗) + d(xn, S x∗)
2s

,

d(xn, S x∗) + sd(S x∗, xn+2) + d(S x∗, xn+2)
2s

,

d(S x∗, xn+2) + d(xn+1, S x
∗), d(xn, xn+1) + d(xn+2, T x

∗),

d(T x∗, x∗) + d(x∗, xn+1)} (30)

and

d(x∗, S x∗) ≤ Nφ(xn+1, x
∗)

= φ1(d(xn+1, x
∗))(d(xn+1, x

∗)) + φ2(d(xn+1, x
∗))(d(x∗, xn+3))

+ φ3(xn+1, x
∗))

(
d(xn+2, x∗) + d(xn+1, x∗)

2s

)
+ φ4(d(xn+1, x

∗))
(

d(xn+1, S x∗) + d(S x∗, xn+3)
2s

)
+ φ5(d(xn+1, x

∗))(d(xn+3, S x
∗) + d(xn+2, S x

∗))
+ φ6(d(xn+1, x

∗))(d(xn+1, xn+2) + d(xn+3, S x
∗))

+ φ7(d(xn+1, x
∗))(d(S x∗, x∗) + d(x∗, xn+2))

≤ max{d(xn+1, x
∗), d(x∗, xn+3),

d(xn+2, x∗) + d(xn+1, x∗)
2s

,

d(xn+1, xn+2) + sd(xn+2, S x∗) + d(S x∗, xn+3)
2s

,

d(xn+3, S x
∗) + d(xn+2, S x

∗), d(xn+1, xn+2) + d(xn+3, S x
∗),

d(S x∗, x∗) + d(x∗, xn+2)}. (31)

By (24) and (30), we have that

lim
n→∞

Nφ(xn, x
∗) = d(T x∗, x∗).

By (24) and (31), we have that

lim
n→∞

Nφ(xn+1, x
∗) = d(S x∗, x∗).

By (28)and (29) and by the continuity of F and ψ, we have that

F(d(x∗, T x∗)) ≤ F(Nφ(x
∗, T x∗)) −ψ(Nφ(x

∗, T x∗)),

and

F(d(x∗, S x∗)) ≤ F(Nφ(x
∗, S x∗)) −ψ(Nφ(x

∗, S x∗)).

Hence, since T x∗ and S x∗ are closed then we have x∗ ∈ T x∗

and x∗ ∈ S x∗, that is, x∗ is a fixed point of T and S .

In Theorem 2.1, when T = S = U, then we have the following
result.

Corollary 2.1.1. Let (X, d) be a complete strong b−metric space
and let U : X → C B(X) be a multivalued generalized F-Suzuki-
Contraction mapping. Then U has a fixed point x∗ ∈ X and for
every x ∈ X the sequence {U n x}∞n=1 converges to x

∗.

In Corollary 2.1.1, when U is a single-valued then we have an-
other new result as follows.

Corollary 2.1.2. Let (X, d) be a complete strong b−metric space
and let U : X → X be a single-valued generalized F-Suzuki-
Contraction mapping. Then U has a fixed point x∗ ∈ X and for
every x ∈ X the sequence {U n x}∞n=1 converges to x

∗.

In Theorem 2.1, when T and S are two single-valued then the

92



Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 88–94 93

following result holds.

Corollary 2.1.3. Let (X, d) be a complete strong b−metric space
and let T, S : X → X be two single-valued generalized F-
Suzuki-Contraction mappings. Then T and S have a common
fixed point x∗ ∈ X and for every x ∈ X the sequence {T n x}∞n=1
and {S n x}∞n=1 converge to x

∗.

In Theorem 2.1, when (X, d) is a complete b−metric space then
the following new result holds.

Corollary 2.1.4. Let (X, d) be a complete b−metric space and
let T, S : X → X be two single-valued generalized F-Suzuki-
Contraction mappings. Then T and S have a common fixed
point x∗ ∈ X and for every x ∈ X the sequence {T n x}∞n=1 and
{S n x}∞n=1 converge to x

∗.

In corollary 2.1.4, when T = S = U, then we have the following
result.

Corollary 2.1.5. Let (X, d) be a complete b−metric space and
let U : X → C B(X) be a multivalued generalized F-Suzuki-
Contraction mapping. Then U has a fixed point x∗ ∈ X and for
every x ∈ X the sequence {U n x}∞n=1 converges to x

∗.

Corollary 2.1.6. Let (X, d) be a complete strong b−metric space
and let U : X → C B(X) be a multivalued generalized F-
Suzuki-Contraction mapping such that there exists F ∈ 0 and
ψ ∈ Ψ, ∀x, y ∈ X, x , y, 1s+1 d(x, U x) < d(x, y) ⇒ ψ(N(x, y)) +
F(s4d(U x, Uy)) ≤ F(N(x, y)) in which

N(x, y) = max{d(x, y), d(y, U 2 x),

(d(y, U x)) + d(x, Uy)
2s

,
(d(x, Uy)) + d(U 2 x, Uy)

2s
,

d(U 2 x, Uy) + d(Uy, U x), d(U 2 x, Uy) + d(U x, x),
d(U x, y)) + d(y, Uy)}. (32)

Then U has a fixed point x∗ ∈ X and for every x ∈ X the se-
quence {U n x}∞n=1 converges to x

∗.

Proof from Lemma 1.1, since (2) ⇒ (32) then by the corollary
2.1.1 the result follows immediately.

Corollary 2.1.7. Let (X, d) be a complete strong b−metric space
and let U : X → X be a single-valued generalized F-Suzuki-
Contraction mapping such that there exists F ∈ 0 and ψ ∈
Ψ, ∀x, y ∈ X, x , y, 1s+1 d(x, U x) < d(x, y) ⇒ ψ(N(x, y)) +
F(s4d(U x, Uy)) ≤ F(N(x, y)) in which

N(x, y) = max{d(x, y), d(y, U 2 x),

(d(y, U x)) + d(x, Uy)
2s

,
(d(x, Uy)) + d(U 2 x, Uy)

2s
,

d(U 2 x, Uy) + d(Uy, U x), d(U 2 x, Uy) + d(U x, x),
d(U x, y)) + d(y, Uy)}. (33)

Then U has a fixed point x∗ ∈ X and for every x ∈ X the se-
quence {U n x}∞n=1 converges to x

∗.

Proof from Lemma 1.1, since (2) ⇒ (33) then by the corollary
2.1.2 the result holds.

Corollary 2.1.8. Let (X, d) be a complete strong b−metric space
and let T, S : X → X be two single-valued generalized F-
Suzuki-Contraction mappings such that there exists F ∈ 0 and
ψ ∈ Ψ, ∀x, y ∈ X, x , y, 1s+1 d(x, T x) < d(x, y) and

1
s+1 d(y, S x) <

d(y, S T x) ⇒ψ(N(x, y))+F(s4 H(T x, S y)) ≤ F(N(x, y)) in which

N(x, y) = max{d(x, y), d(y, S T x),
(d(y, T x)) + d(x, S y)

2s
,

(d(x, S y)) + d(S T x, S y)
2s

,

d(S T x, S y) + d(S y, T x), d(S T x, S y) + d(T x, x), d(T x, y)) + d(y, S y)}.
(34)

Then T and S have a common fixed point x∗ ∈ X and for every
x ∈ X the sequence {T n x}∞n=1 and {S

n x}∞n=1 converge to x
∗.

Proof from Lemma 1.1, since (2) ⇒ (34) then by the corollary
2.1.4 the result holds.

Corollary 2.1.9. Let (X, d) be a complete b−metric space and
let U : X → C B(X) be a multivalued generalized F-Suzuki-
Contraction mapping such that there exists F ∈ 0 and ψ ∈
Ψ, ∀x, y ∈ X, x , y, 12s d(x, U x) < d(x, y) ⇒ ψ(N(x, y)) +
F(s6d(U x, Uy)) ≤ F(N(x, y)) in which

N(x, y) = max{d(x, y), d(y, U 2 x),

(d(y, U x)) + d(x, Uy)
2s

,
(d(x, Uy)) + d(U 2 x, Uy)

2s
,

d(U 2 x, Uy) + d(Uy, U x), d(U 2 x, Uy) + d(U x, x), d(U x, y)) + d(y, Uy)}.
(35)

Then U has a fixed point x∗ ∈ X and for every x ∈ X the se-
quence {U n x}∞n=1 converges to x

∗.

Proof from Lemma 1.1, since (2) ⇒ (35) then by the corollary
2.1.5 the result holds.

3. Example
Let X = [0, 1]. T, S : [0, 1] → C B([0, 1]) be defined by T x =
[0, x2 ] and S y = [0,

y
2 ] such that S T x = [0,

x
8 ] for all x ∈ [0, 1].

Let d be the usual metric on X. Taking F(t) = t10 and let x < y,
then ∀x, y ∈ [0, 1] d(x, y) > 0 and d(y, S T x) = |y− x8 | > |y−

y
8 | =

7
8 y >

y
4 . Now, for s = 1, we have that

1
2 d(x, T x) = 0 < d(x, y)

and 12 d(y, S y) =
y
4 < d(y, S T x). Without lose of generality, let

φ1(d(x, y)) = φ2(d(x, y)) = φ3(d(x, y)) =
1
5 ; and φ4(d(x, y)) =

φ5(d(x, y)) = φ6(d(x, y)) = φ7(d(x, y)) =
1

102 . Therefore, we
have that

F(H(T x, S y)) = ln (H(T x, S y)) + H(T x, S y)

=
1

10

∣∣∣∣∣ y2 − x4
∣∣∣∣∣ = 110

∣∣∣∣∣y − y2 − x4
∣∣∣∣∣

≤
1

10

(∣∣∣∣∣y − x4
∣∣∣∣∣ +

∣∣∣∣∣x − y2
∣∣∣∣∣)

=
1

10


∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣

2

 + 110

∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣

2


93



Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 88–94 94

≤
1
10


∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣

2

 + 110

∣∣∣y − x4 ∣∣∣ + ∣∣∣x − x8 ∣∣∣ + ∣∣∣ x8 − y2 ∣∣∣

2


=

1
10


∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣

2

 + 110

∣∣∣ x

8 −
y
2

∣∣∣ + ∣∣∣x − x8 ∣∣∣
2


+

1
10

(∣∣∣∣∣ y2 − x8
∣∣∣∣∣) ≤ 110


∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣

2


+

1
10


∣∣∣ x

8 −
y
2

∣∣∣ + ∣∣∣x − x8 ∣∣∣
2

 + 110
(∣∣∣∣∣ y2 − x8

∣∣∣∣∣ +
∣∣∣∣∣ x8 − x4

∣∣∣∣∣)
=

1
5


∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣

2

 + 15

∣∣∣ x

8 −
y
2

∣∣∣ + ∣∣∣x − x8 ∣∣∣
2


+

1
10

(∣∣∣∣∣ y2 − x8
∣∣∣∣∣ +

∣∣∣∣∣ x8 − x4
∣∣∣∣∣) + 1102 (|x − y|) + 1102

(∣∣∣∣∣y − x8
∣∣∣∣∣)

+
1

102

(∣∣∣∣∣ x8 − y2
∣∣∣∣∣ +

∣∣∣∣∣ x4 − x
∣∣∣∣∣) + 1102

(∣∣∣∣∣ x4 − y
∣∣∣∣∣ +

∣∣∣∣∣y − y2
∣∣∣∣∣)

−
1

102

[
(|x − y|) +

(∣∣∣∣∣y − x8
∣∣∣∣∣) + (

∣∣∣∣∣ x8 − y2
∣∣∣∣∣ +

∣∣∣∣∣ x4 − x
∣∣∣∣∣)]

+

(∣∣∣∣∣ x4 − y
∣∣∣∣∣ +

∣∣∣∣∣y − y2
∣∣∣∣∣) − 110

(∣∣∣∣∣ y2 − x8
∣∣∣∣∣ +

∣∣∣∣∣ x8 − x4
∣∣∣∣∣)

−
1

10


∣∣∣y − x4 ∣∣∣ + ∣∣∣x − y2 ∣∣∣

2

 − 110

∣∣∣ x

8 −
y
2

∣∣∣ + ∣∣∣x − x8 ∣∣∣
2

 .
= φ1(d(x, y))(d(x, y)) + φ2(d(x, y))(d(y, S T x))

+ φ3(d(x, y))
(

(d(y, T x)) + d(x, S y)
2s

)
+ φ4(d(x, y))(

(d(x, S T x)) + d(S T x, S y)
2s

)
+ φ5(d(x, y))(d(S T x, S y) + d(S T x, T x))
+ φ6(d(x, y))(d(S T x, S y) + d(T x, x))
+ φ7(d(x, y))(d(T x, y)) + d(y, S y)) −ψ(Nφ(x, y)).

4. Conclusion
Fixed point results of Piri and Kumam [11], Ahmad et al. [9],
Suzuki [18] and Suzuki [19] are extended by introducing com-
mon fixed point problem for multivalued generalized F-Suzuki-
contraction mappings in strong b-metric spaces. In specific,
Corollary 2.1.1 and corollary 2.1.2 generalize and extend the
work of Ahmad et al. [9] and Kumam and Hossein [5], respec-
tively.

Acknowledgments
I thank the referees for the positive enlightening comments and
suggestions, which have greatly helped me in making improve-

ments to this paper.

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