EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 3, 2021, 923-941
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Some New Results of Fixed Point in Fuzzy Soft
G-Metric Spaces

Abdelhamied Farrag Sayed1,∗, Abdullah Alahmari2

1 Department of Mathematics, Al-Lith University College, Umm Al-Qura University,
P.O. Box 112, Al-Lith 21961, Makkah Al Mukarramah, Kingdom of Saudi Arabia
2 Department of Mathematics, College of Applied Sciences, Umm Al-Qura University,
Kingdom of Saudi Arabia.

Abstract. The main goal of the present paper is to study and prove some results of fixed points
for mappings satisfying different conditions in fuzzy soft G-metric spaces.

2020 Mathematics Subject Classifications: 03E72, 54A40, 54E40, 54E50, 47H10

Key Words and Phrases: Fuzzy set, Fuzzy soft set, Fuzzy soft G-metric space, Fixed point

1. Introduction

Most of the real life problems have various uncertainties and that classical mathematics
may not be able to model properly. Among these uncertainties, there are two types of
mathematical tools for dealing with such problems: fuzzy set theory [32] and theory of soft
sets due to Molodstov [19], both of which aid in the solution of problems in various fields.
Maji et al. [23] introduced the fuzzy soft set, which is a hybrid of fuzzy and soft sets.
Roy and Maji [22] presented some results on the use of fuzzy soft sets in decision-making
problems. The applications of fuzzy soft sets have been extensively studied (see, e.g.,
[1, 3, 4, 6, 12, 15, 16, 18, 31]). Beaulaa and Gunaseeli defined the fuzzy soft metric space
in [7], and Sayed and Alahmari [25] defined the notions of some mappings and proved
several fixed point theorems in fuzzy soft metric spaces. Mustafa and Sims [20] introduced
the concept of G-metric space to extend and generalize the concept of metric space. Since
then, a number of authors have investigated a variety of well-known results in G-metric
space (see, e.g., [5, 13, 21, 24, 26–30]). Güler et al. [11] proposed the idea of a soft G-
metric space based on a soft element and found some of its properties. Then they came up
with the concepts ”soft G-convergence” and ”soft continuity”. They also established that
fixed points in soft G-metric spaces exist and are unique. Güler and Yildirim presented

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.4028

Email addresses: dr.afsayed@hotmail.com, afssayed@uqu.edu.sa (A. F. Sayed),
aaahmari@uqu.edu.sa (A. Alahmari)

http://www.ejpam.com 923 © 2021 EJPAM All rights reserved.



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 924

soft G-Cauchy sequences and soft G-complete metric spaces in [10], and Shrivastava et al.
established fixed point results of mapping defined on soft G-metric spaces in [17]. Using
fuzzy soft elements, Sayed et al. proposed the concept of fuzzy soft G-metric space in [2].
They also investigated on fuzzy soft continuity and convergence in fuzzy soft G-metric
spaces. The main goal of the present paper is to study and prove some results of fixed
points for mappings satisfying different conditions in fuzzy soft G-metric spaces.

2. Preliminaries

Basic definitions of fuzzy soft sets and fuzzy soft G-metric spaces are presented in this
section.
Throughout this study, we will use the terms X to refer to an initial universe, E to refer
to the set of all parameters for X, and I = [0, 1] to refer to the initial universe.

Definition 1. [32] The membership function µA(x) of a fuzzy set A is a function µA :
X → [0, 1] So, every element x in X has membership degree: µA(x) ∈ [0, 1].
A is completely determined by the set of tuples: A = {(x, µA(x)) : x ∈ X}.

An empty fuzzy set, denoted by 0̃ is one whose membership value is 0, ∀x ∈ X. In
contrast to the above, when µA(x) = 1, then the fuzzy set A is the universal fuzzy set,
denoted by 1̃.

Definition 2. [8] A fuzzy soft set fA over X is a pair (f,A), where f is a mapping
f : A→ IX defined by

fA(e) =

{
0̃, if e /∈ A;
otherwise, if e ∈ A.

Definition 3. [9] A fuzzy soft set fA over X is said to be:

(a) null fuzzy soft set, denoted by φ̃, if for all e ∈ A, fA(e) = 0̃,

(b) absolute fuzzy soft set, denoted by Ẽ, if for all e ∈ A, fA(e) = 1̃.

The fuzzy soft real numbers were defined in [14], denoted by ˜̃r, ˜̃s, ˜̃t, ... etc, and ¯̄r, ¯̄s, ¯̄t will
be denoted in particular type of fuzzy soft real numbers such that ¯̄r(e) is a fuzzy number
for all e ∈ E.
Let A ⊆ E. R(A)∗ be a set of all nonnegative fuzzy soft real numbers and FSC(fA)
denotes a collection of all fuzzy soft points of a fuzzy soft set fA over X.

Definition 4. [2] Let E be a nonempty set of parameters, A ⊆ E and Ẽ be the absolute
fuzzy soft. A mapping G̃ : FSC(Ẽ)× FSC(Ẽ)× FSC(Ẽ)→ R(A)∗ is said to be a fuzzy
soft G-metric on Ẽ if G̃ satisfies the following conditions:
(FSG̃1) : G̃(fe1 , fe2 , fe3) = 0̃ if fe1 = fe2 = fe3;
(FSG̃2) : 0̃<̃G̃(fe1 , fe2 , fe3) for all fe1 , fe2∈̃FSC(Ẽ) with fe1 6= fe2;
(FSG̃3) : G̃(fe1 , fe1 , fe2)≤̃G̃(fe1 , fe2 , fe3) for all fe1 , fe2 , fe3∈̃FSC(Ẽ) with fe2 6= fe3;



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 925

(FSG̃4) : G̃(fe1 , fe2 , fe3) = G̃(fe1 , fe3 , fe2) = G̃(fe2 , fe3 , fe1) = ...;
(FSG̃5) : G̃(fe1 , fe2 , fe3)≤̃G̃(fe1 , fe, fe) + G̃(fe, fe2 , fe3) for all fe1 , fe2 , fe3 , fe∈̃FSC(Ẽ).
The fuzzy soft set Ẽ with a fuzzy soft G-metric G̃ on Ẽ is said to be a fuzzy soft G-metric
space and is denoted by (Ẽ, G̃).

Definition 5. [2] Let (Ẽ, G̃) be a fuzzy soft G-metric space and {fen} a sequence of fuzzy
soft elements in Ẽ. The sequence {fen} is said to be fuzzy soft G-convergent to fe in
Ẽ if for every ε̃≥̃0̃, chosen arbitrary, there exists a natural number N = N(ε̃) such that

0̃≤̃G̃(fen , fen , fe)<̃ε̃ whenever n ≥ N i.e n ≥ N ⇒ {fen}∈̃
˜̃B(fe,

˜̃t, ε̃). We denote this by
fen → fe as n→∞ or Limn→∞{fen} = fe.

Definition 6. [2] Let (Ẽ, G̃) be a fuzzy soft G-metric space and {fen} be sequence of fuzzy
soft elements in Ẽ. Then the sequence {fen} is said to be fuzzy soft G-Cauchy if for every
ε̃≥̃0̃, there exist δ̃>̃0̃ and a positive integer N = N(ε̃) such that G̃(fen , fem , fel)<̃ε̃ for all
n,m, l ≥ N ; that is G̃(fen , fem , fel)→ 0̃ as n,m, l→∞.

Definition 7. [2] A fuzzy soft G-metric space (Ẽ, G̃) is said to be fuzzy soft G-complete
if every fuzzy soft G-Cauchy sequence in (Ẽ, G̃) is fuzzy soft G-convergent in (Ẽ, G̃).

Definition 8. [2] Let (Ẽ, G̃), (
˜́
E,

˜́
G) be two fuzzy soft G-metric spaces. Then a function

T : Ẽ → ˜́
E is fuzzy soft G-continuous at a fuzzy soft element fe∈̃FSC(Ẽ) if and only if

for every ε̃>̃0̃, there exists δ̃≥̃0̃ such that fe1 , fe2∈̃FSC(Ẽ) and G̃(fe, fe1 , fe2)<̃δ̃ implies

that
˜́
G(Tfe, T fe1 , T fe2)<̃ε̃.

A function T is fuzzy soft G-continuous if and only if it is fuzzy soft G-continuous at all
fuzzy soft elements fe∈̃FSC(Ẽ)

3. Main Results

We start our main results in this paper with the following theorem (Theorem 4.2 in
[2])

Theorem 1. Let (Ẽ, G̃) be a fuzzy soft G-complete and T : (Ẽ, G̃)→ (Ẽ, G̃) be a mapping
that satisfies the following condition for all fe1 , fe2 , fe3∈̃FSC(Ẽ),

G̃(Tfe1 , Tfe2 , Tfe3)≤̃¯̄aG̃(fe1 , Tfe1 , Tfe1) + ¯̄bG̃(fe2 , Tfe2 , Tfe2)+

¯̄cG̃(fe3 , T fe3 , Tfe3) + ¯̄dG̃(fe1 , fe2 , fe3),
(1)

where 0̃≤̃¯̄a+ ¯̄b+ ¯̄c+ ¯̄d<̃1̃.
Then T has a unique fixed point, say fe, and T is a fuzzy soft G-continuous at fe.

We will show that this theorem fails to verify if the set of parameter is not finite. For
this, we will provide the following two examples:



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 926

Example 1. Let X = E = { 1
n : n ∈ N}. Consider the fuzzy soft G-metric space (Ẽ, G̃),

where

G̃(fe1 , fe2 , fe3) =
˜̃1

3
{|fe1 − fe2 |+ |fe2 − fe3 |+ |fe1 − fe3 |} ,∀fe1 , fe2 , fe3∈̃FSC(Ẽ).

Note that, (Ẽ, G̃) is a symmetric fuzzy soft G-metric(Proposition 3.3, [2]).
Now, we show that (Ẽ, G̃) is fuzzy soft G-complete. For this, suppose that {fen}n∈N is a
fuzzy soft G-Cauchy sequence of fuzzy soft elements in (Ẽ, G̃). Take the fuzzy soft real
number ε̃ such that ε̃(λ) = λ,∀λ ∈ E, that is ε̃( 1

h) = 1
h ,∀h ∈ N. Then, ∃k ∈ N such that

G̃(fen , fem , fel)<̃ε̃, ∀n,m, l ≥ k, which implies to G̃(fen , fem , fel)(
1
h) < ε̃( 1

h),∀h ∈ N.
Hence

˜̃1

3
(|fen − fem |+ |fem − fel |+ |fen − fel |) (

1

h
) <

1

h′
, ∀n,m, l ≥ k, h ∈ N

Which implies to

˜̃1

3
(|fen − fem |+ |fem − fel |+ |fen − fel |) ≤

1

h′
, ∀n,m, l ≥ k, h ∈ N

So, we obtain that the sequence {fen}n∈N is eventually constant, and hence fuzzy soft
G-convergent. Hence (Ẽ, G̃) is fuzzy soft G-complete.

Now suppose T : (Ẽ, G̃)→ (Ẽ, G̃) is defined as:

T (fe) =
˜̃1
8(fe),∀fe ∈ FSC(Ẽ)

Note that T satisfies the condition (1). For any fe1 , fe2 , fe3∈̃FSC(Ẽ) and η ∈ E, we have

G̃(Tfe1 , T fe2 , T fe3)(η) =

(
˜̃1
3

{
˜̃1
8 |fe1 − fe2 |+

˜̃1
8 |fe2 − fe3 |+

˜̃1
8 |fe1 − fe3 |

})
(η)

=
˜̃1
24 {|fe1 − fe2 |+ |fe2 − fe3 |+ |fe1 − fe3 |}

=
˜̃1
8G̃(fe1 , fe2 , fe3)(η)

≤̃˜̃1
8

{
G̃(fe1 , Tfe1 , Tfe1) + G̃(fe2 , Tfe2 , Tfe2) + G̃(fe3 , Tfe3 , T fe3)

}
(η).

Hence, we obtain

G̃(Tfe1 , T fe2 , T fe3)≤̃˜̃1
8G̃(fe1 , T fe1 , T fe1) +

˜̃1
8G̃(fe2 , Tfe2 , Tfe2) +

˜̃1
8G̃(fe3 , T fe3 , T fe3) +

˜̃1
8G̃(fe1 , fe2 , fe3).

All conditions of Theorem 1 are satisfied but T is a fixed point free map.

Example 2. Let X = E = { 1
n : n ∈ N}. Consider the fuzzy soft G-metric space (Ẽ, G̃),

where the fuzzy soft G-metric is defined as:

G̃(fe1 , fe2 , fe3) =



fe1 + fe2 + fe3 + 1̃, if fe1 6= fe2 6= fe3 6= 0̃;

fe1 + fe3 + 1̃, if fe1 = fe2 6= fe3 6= 0̃;

fe2 + fe3 + ˜̃2, if fe1 = 0̃, fe2 6= fe3 6= 0̃;

fe2 + ˜̃3, if fe1 = 0̃, fe2 = fe3 6= 0̃;

fe3 + ˜̃2, if fe1 = fe2 = 0̃, fe3 6= 0̃;

0̃, if fe1 = fe2 = fe3,



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 927

and extend the definition by symmetry in its arguments.
It is easy to show that (Ẽ, G̃) is a fuzzy soft G-metric which is not symmetric.
Now, we show that (Ẽ, G̃) is fuzzy soft G-complete. For this, suppose that {fen}n∈N is a
fuzzy soft G-Cauchy sequence of fuzzy soft elements in (Ẽ, G̃).
Take the fuzzy soft real number ε̃ such that ε̃(λ) = λ,∀λ ∈ E, that is ε̃( 1

h) = 1
h ,∀h ∈ N.

Then, ∃k ∈ N such that G̃(fen , fem , fel)<̃ε̃,∀n,m, l ≥ k, Which implies to
G̃(fen , fem , fel)(

1
h)<̃ε̃( 1

h),∀h ∈ N, that is G̃(fen , fem , fel)(
1
h)<̃ 1

h′ ,∀n,m, l ≥ k,∀h ∈ N.
This is possible only if the sequence {fen}n∈N is constant, and therefore it is fuzzy soft
G-convergent. We conclude that (Ẽ, G̃) is fuzzy soft G-complete.

Now suppose T : (Ẽ, G̃)→ (Ẽ, G̃) is defined as:

T (fe) =
˜̃1
8(fe),∀fe ∈ FSC(Ẽ)

Note that T satisfies the condition (1). For any fe1 , fe2 , fe3∈̃FSC(Ẽ) and η ∈ E,
we have

G̃(Tfe1 , T fe2 , T fe3)(η) =
˜̃1
8G̃(fe1 , fe2 , fe3)(η)

≤̃˜̃1
8{G̃(fe1 , Tfe1 , Tfe1) + G̃(fe2 , Tfe2 , T fe2) + G̃(fe3 , T fe3 , T fe3) +

G̃(fe1 , fe2 , fe3)}(η).
Hence, we obtain

G̃(Tfe1 , T fe2 , T fe3)≤̃˜̃1
8G̃(fe1 , T fe1 , T fe1) +

˜̃1
8G̃(fe2 , Tfe2 , Tfe2) +

˜̃1
8G̃(fe3 , T fe3 , T fe3) +

˜̃1
8G̃(fe1 , fe2 , fe3).

All conditions of Theorem 1 are satisfied but T can be seen it has no fixed point.

Next, we see that keeping the set of parameters finite, one can obtain the following
new results.

Theorem 2. Suppose (Ẽ, G̃) is a fuzzy soft G-metric space and T : (Ẽ, G̃)→ (Ẽ, G̃) is a
mapping satisfying the following condition:

¯̄aG̃(Tfe1 , T fe2 , Tfe3)≤̃¯̄b


G̃(fe1 , T fe2 , Tfe2)

+G̃(fe1 , T fe3 , T fe3)

+G̃(fe3 , T fe1 , T fe1)

+ ¯̄cG̃(fe1 , fe2 , fe3) (2)

∀fe1 , fe2 , fe3∈̃FSC(Ẽ) and 0̃≤̃¯̄a, ¯̄b, ¯̄c<̃1̃ with ˜̃3¯̄b+ ¯̄c<̃¯̄a.
Then T has an unique fixed point, say fe, and at fe, T is fuzzy soft G-continuous.

Proof. Assume that fe0∈̃FSC(Ẽ) is an arbitrary fuzzy soft element and define the
sequence {gen}n∈N as follows: Tge0 = ge1 , T ge1 = ge2 , T ge2 = ge3 , ..., T gen = gen+1 .
Consider that gen 6= gen+1 .
Substituting fe1 = gen , fe2 = gen+1 and fe3 = gen+1 in (2), we obtain

¯̄aG̃(Tgen , T gen+1 , T gen+1)≤̃¯̄b


G̃(gen , T gen+1 , T gen+1)

+G̃(gen+1 , T gen+1 , T gen+1)

+G̃(gen+1 , T gen , T gen)

+ ¯̄cG̃(gen , gen+1 , gen+1)



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 928

⇒ ¯̄aG̃(gen+1 , gen+2 , gen+2)≤̃¯̄b


G̃(gen , gen+2 , gen+2)

+G̃(gen+1 , gen+2 , gen+2)

+G̃(gen+1 , gen+1 , gen+1)

+ ¯̄cG̃(gen , gen+1 , gen+1)

⇒ ¯̄aG̃(gen+1 , gen+2 , gen+2)≤̃
¯̄b+¯̄c

(¯̄a−˜̃2¯̄b)
G̃(gen , gen+1 , gen+1)

⇒ ¯̄aG̃(gen+1 , gen+2 , gen+2)≤̃¯̄kG̃(gen , gen+1 , gen+1), where ¯̄k =
¯̄b+¯̄c

(¯̄a−˜̃2¯̄b)
<̃1̃.

On continuing this process (n+ 1) times; we obtain
¯̄aG̃(gen+1 , gen+2 , gen+2)≤̃¯̄k(n+1)G̃(ge0 , ge1 , ge1).

Similarly, we will conclude that
¯̄aG̃(gen , gen+1 , gen+1)≤̃¯̄knG̃(ge0 , ge1 , ge1).

Next, we show that {gen}n∈N is a fuzzy soft G-Cauchy sequence.
Then for all n,m ∈ N, n < m, we have
G̃(gen , gem , gem)≤̃G̃(gen , gen+1 , gen+1) + G̃(gen+1 , gen+2 , gen+2) + ...+ G̃(gem−1 , gem , gem)

≤̃(¯̄kn + ¯̄k(n+1) + ...+ ¯̄kmG̃(ge0 , ge1 , ge1)

≤̃
¯̄kn

1̃−¯̄k
G̃(ge0 , ge1 , ge1).

Hence, {gen}n∈N is a fuzzy soft G-Cauchy sequence.
Since (Ẽ, G̃) is fuzzy soft G-complete, there exists fe∈̃FSC(Ẽ) such that {gen}n∈N fuzzy
soft G-converges to fe.
Next, we will show that fe is a fixed point of T .
For this, we take fe1 = gen and fe2 = fe3 = fe in (3.1), then

¯̄aG̃(Tgen , T fe, T fe)≤̃¯̄b


G̃(gen , T fe, T fe)

+G̃(fe, Tfe, Tfe)

+G̃(fe, T gen , T gen)

+ ¯̄cG̃(gen , fe, fe)

⇒ ¯̄aG̃(fe, T fe, T fe)≤̃¯̄b


G̃(fe, T fe, T fe)

+G̃(fe, T fe, T fe)

+G̃(fe, fe, fe)

+ ¯̄cG̃(fe, fe, fe).

⇒ G̃(fe, Tfe, Tfe)≤̃
˜̃2¯̄b
¯̄a G̃((fe, Tfe, Tfe).

This is a contradiction, so Tfe = fe i.e. fe is a fixed point of T
Now, to prove uniqueness, assume that fe and ge are two fixed points of T . Then by
inequality (2), we have

¯̄aG̃(Tfe, T ge, T ge)≤̃¯̄b


G̃(fe, T ge, T ge)

+G̃(ge, T ge, T ge)

+G̃(ge, T fe, T fe)

+ ¯̄cG̃(fe, ge, ge)

⇒ ¯̄aG̃(fe, ge, ge)≤̃¯̄b


G̃(fe, ge, ge)

+G̃(ge, ge, ge)

+G̃(ge, fe, fe)

+ ¯̄cG̃(fe, ge, ge).

So, we deduct that (¯̄a− ¯̄b− ¯̄c)G̃(fe, ge, ge)≤̃¯̄bG̃(ge, fe, fe).

⇒ G̃(fe, ge, ge)≤̃
¯̄b

(¯̄a−¯̄b−¯̄c)
G̃(ge, fe, fe) and by use of the same argument, we will find

G̃(ge, fe, fe)≤̃
¯̄b

(¯̄a−¯̄b−¯̄c)
G̃(fe, ge, ge).



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 929

Therefore, we get G̃(fe, ge, ge)≤̃
(

¯̄b

(¯̄a−¯̄b−¯̄c)

)2
G̃(fe, ge, ge), since ˜̃3¯̄b + ¯̄c<̃¯̄a, this is a contra-

diction implies that fe = ge.
To show that T is fuzzy soft G-continuous at fe.
Let {hen}n∈N be a sequence of fuzzy soft elements in Ẽ such that {hen}n∈N → fe, then we
can deduce that using inequality (2), we have

¯̄aG̃(Tfe, Then , Then)≤̃¯̄b


G̃(fe, Then , Then)

+G̃(hen , Then , Then)

+G̃(hen , T fe, T fe)

+ ¯̄cG̃(fe, hen , hen)

⇒ ¯̄aG̃(fe, Then , Then)≤̃¯̄b


G̃(fe, Then , Then)

+G̃(hen , Then , Then)

+G̃(hen , fe, fe)

+ ¯̄cG̃(fe, hen , hen)

Taking the limit as n→∞ from which, we see that

(ā− ˜̃2¯̄b)G̃(fe, Then , Then)→ 0̃ and so, by proposition (4.1) in [2] we have that the sequence
{Then}n∈N is fuzzy soft G-convergent to Tfe = fe, therefore proposition (4.4) in [2] implies
that T is fuzzy soft G-continuous at fe.

Theorem 3. Suppose (Ẽ, G̃) is a fuzzy soft G-metric space and T : (Ẽ, G̃)→ (Ẽ, G̃) is a
mapping satisfying the following condition:

¯̄aG̃(Tfe1 , T fe2 , T fe3) + ¯̄b min


G̃(Tfe1 , T fe2 , T fe3),

G̃(fe1 , T fe1 , T fe1),

G̃(fe2 , T fe2 , T fe2),

G̃(fe3 , T fe3 , T fe3)

 ≤̃¯̄cG̃(fe1 , fe2 , fe3) (3)

∀fe1 , fe2 , fe3∈̃FSC(Ẽ) and 0̃≤̃¯̄a, ¯̄b, ¯̄c<̃1̃ with ¯̄c− ¯̄b<̃¯̄a.
Then T has an unique fixed point, say fe, and at fe, T is fuzzy soft G-continuous.

Proof. Assume that fe0∈̃FSC(Ẽ) is an arbitrary fuzzy soft element and define the
sequence {gen}n∈N as follows: Tge0 = ge1 , T ge1 = ge2 , T ge2 = ge3 , ..., T gen = gen+1 .
Consider that gen 6= gen+1 .
Substituting fe1 = gen , fe2 = gen+1 and fe3 = gen+1 in (3), we obtain

¯̄aG̃(Tgen , T gen+1 , T gen+1) + ¯̄b min


G̃(Tgen , T gen+1 , T gen+1),

G̃(gen , T gen , T gen),

G̃(gen+1 , T gen+1 , T gen+1),

G̃(gen+1 , T gen+1 , T gen+1)

 ≤̃¯̄cG̃(gen , gen+1gen+2),

⇒ ¯̄aG̃(gen+1 , gen+2 , gen+2) + ¯̄b min


G̃(gen+1 , gen+2 , gen+2),

G̃(gen , gen+1 , gen+1),

G̃(gen+1 , gen+2 , T gen+2),

G̃(gen+1 , gen+2 , gen+2)

 ≤̃¯̄cG̃(gen , gen+1 , gen+1)

⇒ ¯̄aG̃(gen+1 , gen+2 , gen+2) + ¯̄b min

{
G̃(gen+1 , gen+2 , gen+2),

G̃(gen , gen+1 , T gen+1)

}
≤̃¯̄cG̃(gen , gen+1 , gen+1)



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 930

We now have two cases:

Case (1): If min

{
G̃(gen+1 , gen+2 , , gen+2),

G̃(gen , gen+1 , T gen+1)

}
= G̃(gen+1 , gen+2 , gen+2), then

¯̄aG̃(gen+1 , gen+2 , gen+2) + ¯̄bG̃(gen+1 , gen+2 , gen+2)≤̃¯̄cG̃(gen , gen+1 , gen+1)

⇒ G̃(gen+1 , gen+2 , gen+2)≤̃ ¯̄c
¯̄a+¯̄b

G̃(gen , gen+1 , gen+1)

Case (2): If min

{
G̃(gen+1 , gen+2 , , gen+2),

G̃(gen , gen+1 , T gen+1)

}
= G̃(gen , gen+1 , gen+1), then

¯̄aG̃(gen+1 , gen+2 , gen+2) + ¯̄bG̃(gen , gen+1 , gen+1)≤̃¯̄cG̃(gen , gen+1 , gen+1)

⇒ G̃(gen+1 , gen+2 , gen+2)≤̃ ¯̄c−¯̄b
¯̄a G̃(gen , gen+1 , gen+1)

From (1) and (2), we have
G̃(gen+1 , gen+2 , gen+2)≤̃¯̄kG̃(gen , gen+1 , gen+1)

Similarly, we will conclude that
G̃(gen , gen+1 , gen+1)≤̃¯̄kG̃(gen−1 , gen , gen) and so

G̃(gen , gen+1 , gen+1)≤̃¯̄knG̃(ge0 , ge1 , ge1)
Next, we show that {gen}n∈N is a fuzzy soft G-Cauchy sequence.
Then for all n,m ∈ N, n < m, we have
¯̄aG̃(gen , gem , gem)≤̃G̃(gen , gen+1 , gen+1) + G̃(gen+1 , gen+2 , gen+2) + ...+ G̃(gem−1 , gem , gem)

≤̃(¯̄kn + ¯̄k(n+1) + ...+ ¯̄k(m−1))G̃(ge0 , ge1 , ge1)

≤̃
¯̄kn

1̃−¯̄k
G̃(ge0 , ge1 , ge2).

Hence, {gen}n∈N is a fuzzy soft G-Cauchy sequence.
Since (Ẽ, G̃) is fuzzy soft G-complete, there exists fe∈̃FSC(Ẽ) such that {gen}n∈N fuzzy
soft G-converges to fe.
Next, we will show that fe is a fixed point of T .
For this, we take fe1 = gen and fe2 = fe3 = fe in (3), then

¯̄aG̃(Tgen , Tfe, Tfe) + ¯̄b min


G̃(Tgen , Tfe, T fe),

G̃(gen , T gen , T gen),

G̃(fe, T fe, Tfe),

G̃(fe, T fe, Tfe)

 ≤̃¯̄cG̃(gen , fe, fe)

As n→∞, we have

¯̄aG̃(fe, T fe, T fe) + ¯̄b min


G̃(fe, T fe, T fe),

G̃(fe, T fe, T fe),

G̃(fe, T fe, T fe),

G̃(fe, Tfe, T fe)

 ≤̃¯̄cG̃(fe, fe, fe)

⇒ ¯̄aG̃(fe, T fe, T fe)≤̃0̃, since 0̃≤̃¯̄a<̃1̃
This is a contradiction, so Tfe = fe i.e. fe is a fixed point of T .
Now, to prove uniqueness, assume that fe and ge are two fixed points of T . Then by
inequality (3), we have

¯̄aG̃(Tfe, T ge, T ge) + ¯̄b min


G̃(Tfe, T ge, T ge),

G̃(fe, Tfe, T ge),

G̃(ge, T ge, T ge),

G̃(ge, T ge, T ge)

 ≤̃¯̄cG̃(fe, ge, ge)



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 931

⇒ G̃(fe, ge, ge)≤̃ ¯̄c
¯̄aG̃(fe, ge, ge), this is a contradiction implies that fe = ge. To show that

T is fuzzy soft G-continuous at fe.
Let {hen}n∈N be a sequence of fuzzy soft elements in Ẽ such that {hen}n∈N → fe, then we
can deduce that using inequality (3), we have

¯̄aG̃(Tfe, Then , Then) + ¯̄b min


G̃(Tfe, Then , Then),

G̃(fe, Tfe, Tfe),

G̃(hen , Then , Then),

G̃(hen , Then , Then)

 ≤̃¯̄cG̃(fe, hen , hen)

⇒ ¯̄aG̃(fe, Then , Then) + ¯̄b min


G̃(fe, Then , Then),

G̃(fe, fe, fe),

G̃(hen , Then , Then),

G̃(hen , Then , Then)

 ≤̃¯̄cG̃(fe, hen , hen)

Taking the limit as n→∞ from which, we see that
G̃(fe, Then , Then) → 0̃ and so, by proposition (4.1) in [2] we have that the sequence
{Then}n∈N is fuzzy soft G-convergent to Tfe = fe, therefore proposition (4.4) in [2] implies
that T is fuzzy soft G-continuous at fe.

Theorem 4. Suppose (Ẽ, G̃) is a fuzzy soft G-metric space and T : (Ẽ, G̃)→ (Ẽ, G̃) is a
mapping satisfying the following condition:

¯̄aG̃(Tfe1 , Tfe2 , T fe3) + ¯̄b


min

{
G̃(Tfe1 , T fe2 , T fe2) · G̃(fe1 , Tfe1 , T fe1),

G̃(fe1 , fe2 , T fe3) · G̃(fe2 , Tfe2 , Tfe2)

}
min

{
G̃(Tfe1 , T fe2 , T fe3) · G̃(fe1 , Tfe1 , T fe1),

G̃(fe1 , fe2 , fe3) · G̃(fe2 , T fe2 , T fe2)

}


≤̃¯̄cG̃(fe1 , fe2 , fe2)

(4)

∀fe1 , fe2 , fe3∈̃FSC(Ẽ) and 0̃≤̃¯̄a, ¯̄b, ¯̄c<̃1̃ with ¯̄c− ¯̄b<̃¯̄a.
Then T has an unique fixed point, say fe, and at fe, T is fuzzy soft G-continuous.

Proof. Assume that fe0∈̃FSC(Ẽ) is an arbitrary fuzzy soft element and define the
sequence {gen}n∈N as follows: Tge0 = ge1 , T ge1 = ge2 , T ge2 = ge3 , ..., T gen = gen+1 .
Consider that gen 6= gen+1 .
Substituting fe1 = gen , fe2 = gen+1 and fe3 = gen+1 in (4), we obtain

¯̄aG̃(Tgen , T gen+1 , T gen+1) + ¯̄b


min

{
G̃(Tgen , T gen+1 , T gen+1) · G̃(gen , T gen , T gen),

G̃(gen , gen+1 , gen+1) · G̃(gen+1 , T gen+1 , T gen+1)

}
min

{
G̃(Tgen , T gen+1 , T gen+1), G̃(gen , T gen , T gen),

G̃(gen , gen+1 , gen+1), G̃(gen+1 , T gen+1 , T gen+1)

}


≤̃¯̄cG̃(gen , gen+1 , gen+1)



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 932

¯̄aG̃(gen+1 , gen+2 , gen+2) + ¯̄b


min

{
G̃(gen+1 , gen+2 , gen+2) · G̃(gen , gen+1 , gen+1),

G̃(gen , gen+1 , gen+1) · G̃(gen+1 , gen+2 , gen+2)

}
min

{
G̃(gen+1 , gen+2 , gen+2), G̃(gen , gen+1 , gen+1),

G̃(gen , gen+1 , gen+1), G̃(gen+1 , gen+2 , gen+2)

}


≤̃¯̄cG̃(gen , gen+1 , gen+1)

(5)

⇒ ¯̄aG̃(gen+1 , gen+2 , gen+2) + ¯̄b


min

 G̃(gen+1 , gen+2 , gen+2) · G̃(gen , gen+1 , gen+1),

G̃(gen , gen+1 , gen+1) · G̃(gen+1 , gen+2 , gen+2)


min

{
G̃(gen+1 , gen+2 , gen+2), G̃(gen , gen+1 , gen+1)

}


≤̃¯̄cG̃(gen , gen+1 , gen+1)
We now have four cases:

Case (1): If


min

 G̃(gen+1 , gen+2 , gen+2) · G̃(gen , gen+1 , gen+1),

G̃(gen , gen+1 , gen+1) · G̃(gen+1 , gen+2 , gen+2)


min

{
G̃(gen+1 , gen+2 , gen+2), G̃(gen , gen+1 , gen+1)

}
=

[{
G̃(gen+1 , gen+2 , gen+2) · G̃(gen , gen+1 , gen+1)

}
G̃(gen , gen+1 , gen+1)

]
= G̃(gen+1 , gen+2 , gen+2)

Then, we have
¯̄aG̃(gen+1 , gen+2 , gen+2) + ¯̄bG̃(gen+1 , gen+2 , gen+2)≤̃¯̄cG̃(gen , gen+1 , gen+1)

⇒ G̃(gen+1 , gen+2 , gen+2)≤̃ ¯̄c
¯̄a+¯̄b

G̃(gen , gen+1 , gen+1)

Case (2): If


min

 G̃(gen+1 , gen+2 , gen+2) · G̃(gen , gen+1 , gen+1),

G̃(gen , gen+1 , gen+1) · G̃(gen+1 , gen+2 , gen+2)


min

{
G̃(gen+1 , gen+2 , gen+2), G̃(gen , gen+1 , gen+1)

}
=

[{
G̃(gen+1 , gen+2 , gen+2) · G̃(gen , gen+1 , gen+1)

}
G̃(gen+1 , gen+2 , gen+2)

]
= G̃(gen , gen+1 , gen+1)

Then, we have
¯̄aG̃(gen+1 , gen+2 , gen+2) + ¯̄bG̃(gen , gen+1 , gen+1)≤̃¯̄cG̃(gen , gen+1 , gen+1)

⇒ G̃(gen+1 , gen+2 , gen+2)≤̃ ¯̄c−¯̄b
¯̄a G̃(gen , gen+1 , gen+1)

Case (3): If


min

 G̃(gen+1 , gen+2 , gen+2) · G̃(gen , gen+1 , gen+1),

G̃(gen , gen+1 , gen+1) · G̃(gen+1 , gen+2 , gen+2)


min

{
G̃(gen+1 , gen+2 , gen+2), G̃(gen , gen+1 , gen+1)

}
=

[{
G̃(gen , gen+1 , gen+1) · G̃(gen+1 , gen+2 , gen+2)

}
G̃(gen , gen+1 , gen+1)

]
= G̃(gen+1 , gen+2 , gen+2)

Then, we have
¯̄aG̃(gen+1 , gen+2 , gen+2) + ¯̄bG̃(gen+1 , gen+2 , gen+2)≤̃¯̄cG̃(gen , gen+1 , gen+1)

⇒ G̃(gen+1 , gen+2 , gen+2)≤̃ ¯̄c
¯̄a+¯̄b

G̃(gen , gen+1 , gen+1)



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 933

Case (4): If


min

 G̃(gen+1 , gen+2 , gen+2) · G̃(gen , gen+1 , gen+1),

G̃(gen , gen+1 , gen+1) · G̃(gen+1 , gen+2 , gen+2)


min

{
G̃(gen+1 , gen+2 , gen+2), G̃(gen , gen+1 , gen+1)

}
=

[{
G̃(gen , gen+1 , gen+1) · G̃(gen+1 , gen+2 , gen+2)

}
G̃(gen+1 , gen+2 , gen+2)

]
= G̃(gen , gen+1 , gen+1)

Then, we have
¯̄aG̃(gen+1 , gen+2 , gen+2) + ¯̄bG̃(gen , gen+1 , gen+1)≤̃¯̄cG̃(gen , gen+1 , gen+1)

⇒ G̃(gen+1 , gen+2 , gen+2)≤̃ ¯̄c−¯̄b
¯̄a G̃(gen , gen+1 , gen+1)

From Cases (1), (2), (3) and (4), we have
G̃(gen+1 , gen+2 , gen+2)≤̃¯̄kG̃(gen , gen+1 , gen+1)

On continuing this process (n+ 1) times, we have
G̃(gen+1 , gen+2 , gen+2)≤̃¯̄kn+1G̃(ge0 , ge1 , ge1)

Similarly, we will conclude that
G̃(gen , gen+1 , gen+1)≤̃¯̄knG̃(ge0 , ge1 , ge1)

Next, we show that {gen}n∈N is a fuzzy soft G-Cauchy sequence.
Then for all n,m ∈ N, n < m, we have
¯̄aG̃(gen , gem , gem)≤̃G̃(gen , gen+1 , gen+1) + G̃(gen+1 , gen+2 , gen+2) + ...+ G̃(gem−1 , gem , gem)

≤̃(¯̄kn + ¯̄k(n+1) + ...+ ¯̄k(m−1))G̃(ge0 , ge1 , ge1)

≤̃
¯̄kn

1̃−¯̄k
G̃(ge0 , ge1 , ge2).

Hence, {gen}n∈N is a fuzzy soft G-Cauchy sequence.
Since (Ẽ, G̃) is fuzzy soft G-complete, there exists fe∈̃FSC(Ẽ) such that {gen}n∈N fuzzy
soft G-converges to fe.
Next, we will show that fe is a fixed point of T .
For this, we take fe1 = gen and fe2 = fe3 = fe in the inequality (4), then from the inequal-
ity (5), we have

¯̄aG̃(Tgen , T fe, T fe)+
¯̄b


min

 G̃(Tgen , T fe, T fe) · G̃(gen , T gen , T gen),

G̃(gen , fe, fe) · G̃(fe, T fe, T fe)


min

 G̃(Tgen , Tfe, Tfe), G̃(gen , T gen , T gen),

G̃(gen , fe, fe), G̃(fe, T fe, T fe)



 ≤̃¯̄cG̃(gen , fe, fe)

As n→∞, we have

¯̄aG̃(fe, Tfe, Tfe) + ¯̄b


min

 G̃(fe, Tfe, Tfe) · G̃(fe, fe, fe),

G̃(fe, fe, fe) · G̃(fe, Tfe, Tfe)


min

 G̃(fe, T fe, T fe), G̃(fe, T fe, T fe),

G̃(fe, fe, fe), G̃(fe, Tfe, T fe)



 ≤̃¯̄cG̃(fe, fe, fe)

⇒ ¯̄aG̃(fe, T fe, T fe)≤̃0̃, since 0̃≤̃¯̄a<̃1̃.
This is a contradiction, so Tfe = fe i.e. fe is a fixed point of T .
Now, to prove uniqueness, assume that fe and ge are two fixed points of T . Then by
inequality (5), we have



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 934

¯̄aG̃(Tfe, T ge, T ge) + ¯̄b


min

 G̃(Tfe, T ge, T ge) · G̃(ge, T ge, T ge),

G̃(fe, ge, ge) · G̃(ge, T ge, T ge)


min

 G̃(Tfe, T ge, T ge) · G̃(fe, T fe, T fe),

G̃(fe, ge, ge) · G̃(ge, T ge, T ge)



 ≤̃¯̄cG̃(fe, ge, ge)

⇒ ¯̄aG̃(fe, ge, ge) + ¯̄b


min

 G̃(fe, ge, ge) · G̃(ge, ge, ge),

G̃(fe, ge, ge) · G̃(ge, ge, ge)


min

 G̃(fe, ge, ge) · G̃(fe, fe, fe),

G̃(fe, ge, ge) · G̃(ge, ge, ge)



 ≤̃¯̄cG̃(fe, ge, ge)

⇒ G̃(fe, ge, ge)≤̃ ¯̄c
¯̄aG̃(fe, ge, ge), since 0̃≤̃ ¯̄c

¯̄a<̃1̃.
This is a contradiction, implies that fe = ge.
To show that T is fuzzy soft G-continuous at fe.
Let {hen}n∈N be a sequence of fuzzy soft elements in Ẽ such that {hen}n∈N → fe, then we
can deduce that using inequality (5), we have

¯̄aG̃(Tfe, Then , Then) + ¯̄b


min

 G̃(Tfe, Then , Then) · G̃(fe, Tfe, T fe),

G̃(fe, hen , hen) · G̃(fe, T fe, T fe)


min

 G̃(Tfe, Then , Then) · G̃(fe, T fe, T fe),

G̃(fe, hen , hen) · G̃(fe, Tfe, Tfe)



 ≤̃
¯̄cG̃(fe, hen , hen)

⇒ ¯̄aG̃(fe, Then , Then) + ¯̄b


min

 G̃(fe, Then , Then) · G̃(fe, fe, fe),

G̃(fe, hen , hen) · G̃(fe, fe, fe)


min

 G̃(fe, Then , Then) · G̃(fe, fe, fe),

G̃(fe, hen , hen) · G̃(fe, fe, fe)



 ≤̃¯̄cG̃(fe, hen , hen)

Taking the limit as n→∞ from which, we see that
G̃(fe, Then , Then) → 0̃ and so, by proposition (4.1) in [2] we have that the sequence
{Then}n∈N is fuzzy soft G-convergent to Tfe = fe, therefore proposition (4.4) in [2] im-
plies that T is fuzzy soft G-continuous at fe.

Theorem 5. Suppose (Ẽ, G̃) is a fuzzy soft G-metric space and T : (Ẽ, G̃)→ (Ẽ, G̃) is a
mapping satisfying the following condition:

min


[
G̃(fe1 , T fe1 , T fe1) + G̃(fe1 , T fe2 , T fe2)

]
,

G̃(Tfe1 , Tfe2 , Tfe3),[
G̃(fe2 , T fe2 , T fe2) + G̃(fe2 , T fe1 , T fe1)

]
 ≤̃¯̄aG̃(fe1 , fe2 , fe3) (6)

∀fe1 , fe2 , fe3∈̃FSC(Ẽ) and 0̃≤̃¯̄a<̃1̃.
Then T has an unique fixed point, say fe, and at fe, T is fuzzy soft G-continuous.

Proof. Assume that fe0∈̃FSC(Ẽ) is an arbitrary fuzzy soft element and define the
sequence {gen}n∈N as follows: Tge0 = ge1 , T ge1 = ge2 , T ge2 = ge3 , ..., T gen = gen+1 .
Consider that gen 6= gen+1 .
Substituting fe1 = gen , fe2 = gen+1 and fe3 = gen+1 in (6), we obtain



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 935

min


[
G̃(gen , T gen , T gen) + G̃(gen , T gen+1 , T gen+1)

]
,

G̃(Tgen , T gen+1 , T gen+1),[
G̃(gen+1 , T gen+1 , T gen+1) + G̃(gen+1 , T gen , T gen)

]
 ≤̃¯̄aG̃(gen , gen+1 , gen+1)

⇒ min


[
G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)

]
,

G̃(gen+1 , gen+2 , gen+2),[
G̃(gen+1 , gen+2 , gen+2) + G̃(gen+1 , gen+1 , gen+1)

]
 ≤̃¯̄aG̃(gen , gen+1 , gen+1)

⇒ min

{ [
G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)

]
,

G̃(gen+1 , gen+2 , gen+2)

}
≤̃¯̄aG̃(gen , gen+1 , gen+1) (7)

We now have two cases:

Case (1): If min

{ [
G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)

]
,

G̃(gen+1 , gen+2 , gen+2)

}
= G̃(gen+1 , gen+2 , gen+2)

Then, (7) is reduced to

G̃(gen+1 , gen+2 , gen+2)≤̃¯̄aG̃(gen , gen+1 , gen+1)

Case (2): If min

{ [
G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)

]
,

G̃(gen+1 , gen+2 , gen+2)

}
=[

G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)
]

Then, (7) is reduced to

G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)≤̃¯̄aG̃(gen , gen+1 , gen+1)

G̃(gen , gen+1 , gen+1) +
[
G̃(gen+1 , gen+2 , gen+2)− G̃(gen , gen+1 , gen+1)

]
≤̃¯̄aG̃(gen , gen+1 , gen+1)

⇒ G̃(gen+1 , gen+2 , gen+2)≤̃¯̄aG̃(gen , gen+1 , gen+1)

On continuing this process (n+ 1) times; we obtain

G̃(gen+1 , gen+2 , gen+2)≤̃¯̄an+1G̃(ge0 , ge1 , ge1)

Similarly, we will conclude that

G̃(gen , gen+1 , gen+1)≤̃¯̄anG̃(ge0 , ge1 , ge1)

Next, we show that {gen}n∈N is a fuzzy soft G-Cauchy sequence.
Then for all n,m ∈ N, n < m, we have
G̃(gen , gem , gem)≤̃G̃(gen , gen+1 , gen+1) + G̃(gen+1 , gen+2 , gen+2) + ...+ G̃(gem−1 , gem , gem)

≤̃(¯̄an + ¯̄a(n+1) + ...+ ¯̄amG̃(ge0 , ge1 , ge1)
≤̃ ¯̄an

1̃−¯̄a
G̃(ge0 , ge1 , ge1).



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 936

Hence, {gen}n∈N is a fuzzy soft G-Cauchy sequence.
Since (Ẽ, G̃) is fuzzy soft G-complete, there exists fe∈̃FSC(Ẽ) such that {gen}n∈N fuzzy
soft G-converges to fe.
Next, we will show that fe is a fixed point of T .
For this, we take fe1 = gen and fe2 = fe3 = fe in (6), then

min


[
G̃(gen , T gen , T gen) + G̃(gen , Tfe, T fe)

]
,

G̃(Tgen , T fe, T fe),[
G̃(fe, Tfe, T fe) + G̃(fe, T gen , T gen)

]
 ≤̃¯̄aG̃(gen , fe, fe)

⇒ min


[
G̃(fe, fe, fe) + G̃(fe, Tfe, T fe)

]
,

G̃(fe, T fe, T fe),[
G̃(fe, T fe, T fe) + G̃(fe, fe, fe)

]
 ≤̃¯̄aG̃(fe, fe, fe)

⇒ G̃(fe, Tfe, Tfe)≤̃0̃
This is a contradiction, so Tfe = fe i.e. fe is a fixed point of T .
Now, to prove uniqueness, assume that fe and ge are two fixed points of T . Then by
inequality (6), we have

min


[
G̃(fe, fe, fe) + G̃(fe, ge, ge)

]
,

G̃(fe, ge, ge),[
G̃(ge, ge, ge) + G̃(ge, fe, fe)

]
 ≤̃¯̄aG̃(fe, fe, fe)

⇒ G̃(fe, ge, ge)≤̃¯̄aG̃(fe, ge, ge), this is a contradiction implies that fe = ge.
To show that T is fuzzy soft G-continuous at fe.
Let {hen}n∈N be a sequence of fuzzy soft elements in Ẽ such that {hen}n∈N → fe, then we
can deduce that using inequality (6), we have

min


[
G̃(fe, Tfe, T fe) + G̃(fe, Then , Then)

]
,

G̃(Tfe, Then , Then),[
G̃(hen , Then , Then) + G̃(hen , T fe, T fe)

]
 ≤̃¯̄aG̃(fe, hen , hen)

⇒ min


[
G̃(fe, fe, fe) + G̃(fe, Then , Then)

]
,

G̃(fe, Then , Then),[
G̃(hen , Then , Then) + G̃(hen , fe, fe)

]
 ≤̃¯̄aG̃(fe, hen , hen)

Taking the limit as n→∞ from which, we see that
G̃(fe, Then , Then) → 0̃ and so, by proposition (4.1) in [2] we have that the sequence
{Then}n∈N is fuzzy soft G-convergent to Tfe = fe, therefore proposition (4.4) in [2] implies
that T is fuzzy soft G-continuous at fe.



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 937

Theorem 6. Suppose (Ẽ, G̃) is a fuzzy soft G-metric space and T : (Ẽ, G̃)→ (Ẽ, G̃) is a
mapping satisfying the following condition:

G̃(Tfe1 , Tfe2 , Tfe3)≤̃¯̄a max


[
G̃(fe1 , T fe1 , Tfe1) + G̃(fe1 , Tfe2 , Tfe2)

]
,

G̃(fe1 , fe2 , fe3),[
G̃(fe2 , T fe2 , Tfe2) + G̃(fe2 , Tfe1 , Tfe1)

]
 (8)

∀fe1 , fe2 , fe3∈̃FSC(Ẽ) and 0̃≤̃¯̄a<̃1̃.
Then T has an unique fixed point, say fe, and at fe, T is fuzzy soft G-continuous.

Proof. Assume that fe0∈̃FSC(Ẽ) is an arbitrary fuzzy soft element and define the
sequence {gen}n∈N as follows: Tge0 = ge1 , T ge1 = ge2 , T ge2 = ge3 , ..., T gen = gen+1 .
Consider that gen 6= gen+1 .
Substituting fe1 = gen , fe2 = gen+1 and fe3 = gen+1 in (8), we obtain

G̃(Tgen , T gen+1 , T gen+1)≤̃¯̄a max


[
G̃(gen , T gen , T gen) + G̃(gen , T gen+1 , T gen+1)

]
,

G̃(gen , gen+1 , gen+1),[
G̃(gen+1 , T gen+1 , T gen+1) + G̃(gen+1 , T gen , T gen)

]


⇒ G̃(gen+1 , gen+2 , gen+2)≤̃¯̄a max


[
G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)

]
,

G̃(gen , gen+1 , gen+1),[
G̃(gen+1 , gen+2 , gen+2) + G̃(gen+1 , gen+1 , gen+1)

]


⇒ G̃(gen+1 , gen+2 , gen+2)≤̃¯̄a max

{ [
G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)

]
,

G̃(gen , gen+1 , gen+1), G̃(gen+1 , gen+2 , gen+2)

}
(9)

We now have three cases:

Case (1): If max

{ [
G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)

]
,

G̃(gen , gen+1 , gen+1), G̃(gen+1 , gen+2 , gen+2)

}
=[

G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)
]

Then, (9) is reduced to

G̃(gen+1 , gen+2 , gen+2)≤̃¯̄a
[
G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)

]
⇒ G̃(gen+1 , gen+2 , gen+2)≤̃ ¯̄a

1̃−˜̃a
G̃(gen , gen+1 , gen+1)

Case (2): If max

{ [
G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)

]
,

G̃(gen , gen+1 , gen+1), G̃(gen+1 , gen+2 , gen+2)

}
= G̃(gen , gen+1 , gen+1)

Then, (9) is reduced to
⇒ G̃(gen+1 , gen+2 , gen+2)≤̃¯̄aG̃(gen , gen+1 , gen+1)

Case (3): Ifmax

{ [
G̃(gen , gen+1 , gen+1) + G̃(gen , gen+2 , gen+2)

]
,

G̃(gen , gen+1 , gen+1), G̃(gen+1 , gen+2 , gen+2)

}
= G̃(gen+1 , gen+2 , gen+2)



A. F. Sayed, A. Alahmari / Eur. J. Pure Appl. Math, 14 (3) (2021), 923-941 938

Then, (9) is reduced to
G̃(gen+1 , gen+2 , gen+2)≤̃¯̄aG̃(gen+1 , gen+2 , gen+2), which is a contradiction.

From cases (1), (2) and (3); we have
G̃(gen+1 , gen+2 , gen+2)≤̃¯̄aG̃(gen , gen+1 , gen+1)

On continuing this process (n+ 1) times; we have
G̃(gen+1 , gen+2 , gen+2)≤̃¯̄an+1G̃(ge0 , ge1 , ge1)

Similarly, we will conclude that
G̃(gen , gen+1 , gen)≤̃¯̄aG̃(gen , gen+1 , gen+1)

Next, we show that {gen}n∈N is a fuzzy soft G-Cauchy sequence.
Then for all n,m ∈ N, n < m, we have
G̃(gen , gem , gem)≤̃G̃(gen , gen+1 , gen+1) + G̃(gen+1 , gen+2 , gen+2) + ...+ G̃(gem−1 , gem , gem)

≤̃(¯̄an + ¯̄a(n+1) + ...+ ¯̄amG̃(ge0 , ge1 , ge1)
≤̃ ¯̄an

1̃−¯̄a
G̃(ge0 , ge1 , ge1).

Hence, {gen}n∈N is a fuzzy soft G-Cauchy sequence.
Since (Ẽ, G̃) is fuzzy soft G-complete, there exists fe∈̃FSC(Ẽ) such that {gen}n∈N fuzzy
soft G-converges to fe.
Next, we will show that fe is a fixed point of T .
For this, we take fe1 = gen and fe2 = fe3 = fe in (8), then

G̃(Tgen , Tfe, Tfe)≤̃¯̄a max


[
G̃(gen , T gen , T gen) + G̃(gen , Tfe, Tfe)

]
,

G̃(gen , fe, fe),[
G̃(fe, Tfe, Tfe) + G̃(fe, T gen , T gen)

]


⇒ G̃(fe, T fe, T fe)≤̃¯̄a max


[
G̃(fe, fe, fe) + G̃(fe, Tfe, T fe)

]
,

G̃(fe, fe, fe),[
G̃(fe, T fe, T fe) + G̃(fe, fe, fe)

]


⇒ G̃(fe, T fe, T fe)≤̃¯̄aG̃(fe, T fe, T fe)
⇒ (1̃− ¯̄a)G̃(fe, T fe, T fe)≤̃0̃
This is a contradiction, so Tfe = fe i.e. fe is a fixed point of T .
Now, to prove uniqueness, assume that fe and ge are two fixed points of T . Then by
inequality (8), we have

G̃(Tfe, T ge, T ge)≤̃¯̄a max


[
G̃(fe, Tfe, Tfe) + G̃(fe, T ge, T ge)

]
,

G̃(fe, ge, ge),[
G̃(ge, T ge, T ge) + G̃(ge, T fe, T fe)

]


⇒ G̃(fe, ge, ge)≤̃¯̄a max


[
G̃(fe, fe, fe) + G̃(fe, ge, ge)

]
,

G̃(fe, ge, ge),[
G̃(ge, ge, ge) + G̃(ge, fe, fe)

]


⇒ G̃(fe, ge, ge)≤̃¯̄a max
{
G̃(fe, ge, ge), G̃(ge, fe, fe)

}
We now have two cases:
Case (I): If max

{
G̃(fe, ge, ge), G̃(ge, fe, fe)

}
= G̃(fe, ge, ge), then we get



REFERENCES 939

G̃(fe, ge, ge)≤̃¯̄aG̃(fe, ge, ge)
This is a contradiction implies that fe = ge
Case (II): If max

{
G̃(fe, ge, ge), G̃(ge, fe, fe)

}
= G̃(ge, fe, fe), then we get

G̃(fe, ge, ge)≤̃¯̄aG̃(ge, fe, fe)
So, we deduct that G̃(fe, ge, ge)≤̃¯̄aG̃(ge, fe, fe).
By repeated use of the same argument, we will find G̃(ge, fe, fe)≤̃¯̄aG̃(fe, ge, ge).
Therefore, we get G̃(fe, ge, ge)≤̃¯̄a2G̃(ge, fe, fe).

Since ¯̄a<̃(
˜̃

1/2), this is a contradiction implies that fe = ge
To show that T is fuzzy soft G-continuous at fe.
Let {hen}n∈N be a sequence of fuzzy soft elements in Ẽ such that {hen}n∈N → fe, then by
using inequality (8), we can deduce that

G̃(fe, Then , Then) = G̃(Tfe, Then , Then)≤̃

¯̄a max


[
G̃(fe, Tfe, Tfe) + G̃(fe, Then , Then)

]
,

G̃(fe, hen , hen),[
G̃(hen , Then , Then) + G̃(hen , T fe, T fe)

]


⇒ G̃(fe, Then , Then)≤̃¯̄aG̃(fe, Then , Then)
⇒ (1̃− ¯̄a)G̃(fe, Then , Then)≤̃0̃
Taking the limit as n→∞ from which, we see that
G̃(fe, Then , Then) → 0̃ and so, by proposition (4.1) in [2] we have that the sequence
{Then}n∈N is fuzzy soft G-convergent to Tfe = fe, therefore proposition (4.4) in [2] im-
plies that T is fuzzy soft G-continuous at fe.

4. Conclusion

In this paper, some new results of fixed points for mappings satisfying different condi-
tions in fuzzy soft G-metric spaces were presented and proved. We’ll hope to improve the
search performance even more in the future for some more important results in this space.

Acknowledgements

The authors are very grateful to the editor and the reviewers for their valuable sug-
gestions.

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