Title


Science and Technology Indonesia
e-ISSN:2580-4391 p-ISSN:2580-4405
Vol. 8, No. 2, April 2023

Research Paper

Best Proximity Point Results in Fuzzy Normed Spaces
Raghad Ibrahaim Sabri1*, Buthainah Abd Al Hassan Ahmed1
1Department of Mathematics, College of Science, University of Baghdad, Baghdad, 10081, Iraq
*Corresponding author: raghad.i.sabri@uotechnology.edu.iq

AbstractFixed point (briefly FP ) theory is a potent tool for resolving several actual problems since many problems may be simplified to the
FP problem. The idea of Banach contraction mapping is a foundational theorem in FP theory. This idea has wide applications inseveral fields; hence, it has been developed in numerous ways. Nevertheless, all of these results are reliant on the existence anduniqueness of a FP on some suitable space. Because the FP problem could not have a solution in the case of nonself-mappings,the idea of the best proximity point (briefly Bpp) is offered to approach the best solution. This paper investigates the existenceand uniqueness of the Bpp of nonself-mappings in fuzzy normed space(briefly FNspace) to arrive at the best solution. Followingthe introduction of the definition of the Bpp, the existence, and uniqueness of the Bpp are shown in a FN space for diverse fuzzyproximal contractions such as 𝔅�̃� - fuzzy proximal contractive mapping and 𝔅h 𝔍h - fuzzy proximal contractive mapping.
KeywordsFuzzy Normed Space, Fuzzy Proximal Contractive, Best Proximity Point, Fuzzy Banach Space, Cauchy Sequence

Received: 18 December 2022, Accepted: 23 March 2023
https://doi.org/10.26554/sti.2023.8.2.298-304

1. INTRODUCTION

One of the most important branches of modern mathematics
is functional analysis. It is crucial in the theory of differential
equations, particularly partial differential equations, represen-
tation theory, and probability, as well as in the study of nu-
merous properties of different spaces such as normed space,
Hilbert space, Banach space, and others (Sabri and Ahmed,
2023; Nemah, 2017; Eiman and A.Mustafa, 2016; Zeana and
K.Assma, 2016; Dakheel and Ahmed, 2021).

Numerous applications in mathematics and allied fields,
such as inverse problems, are made possible by the well-known
Banach FP theorem (Banach, 1922) , which concerns the pres-
ence and uniqueness of FP of self-mappings defined on a com-
plete metric space (see (Lin et al., 2018; Zhang and Hofmann,
2020)). The Banach FP theorem has drawn many academics
to expand the reach of metric FP theory because of its wide va-
riety of applications (see (Abbas et al., 2012; Latif et al., 2015;
Mustafa et al., 2014)).

On the other hand, if U and V are both nonempty subsets
of (L; d) (where (L; d) is a metric space), consequently in the
situation of a mapping T U→ V, it is possible that there is
not a point u in U such that u=Tu: where d is the distance
between U and V. In these kinds of instances, it is preferable
to locate an element u in U such that the distance between u
and T u is as low as possible. If there is such an element u in

U, consequently it represents the Bpp of T.
Zadeh developed and examined the concept of a fuzzy set

in his groundbreaking research in Zadeh (1978) . The explo-
ration of fuzzy sets has resulted in the fuzzification of several
distinct mathematical ideas. It has potential applications in
many different domains. Kramosil and Michálek (1975) ini-
tially presented the idea of fuzzy metric spaces. Then the
concept of fuzzy metric spaces was modified by George and
Veeramani (1994) . Numerous articles have been published
on fuzzy metric spaces (Hussain et al., 2020; Paknazar, 2018;
Gregori et al., 2020; Zainab and Kider, 2021 Sabri, 2021; Li
and Zhang, 2023). FN space was subsequently introduced in
different methods by a large number of other mathematicians.
FN space has been the topic of a considerable number of pub-
lications; for instance, see (Miheţ and Zaharia, 2014; Sabri
and Ahmed, 2022; Kider and Kadhum, 2019; Sharma and
Hazarika, 2020; Sabre, 2012; Konwar and Debnath, 2023;
Sabri and Ahmed, 2023; Raghad, 2021).

Examining the presence and uniqueness of the Bpp in a
FN space is the goal of this work, offering an approach to ex-
pand and fuzzify results in normed spaces. To that end, three
theorems are presented that demonstrate the presence and
uniqueness of the Bpp under various circumstances. In ad-
dition, an example is provided to demonstrate the use of the
main theorem.

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Sabri et. al. Science and Technology Indonesia, 8 (2023) 298-304

2. EXPERIMENTAL SECTION

2.1 Preliminaries
This section defines the terminology and outcomes which is
going to be utilized throughout the paper. Materials :

2.1.1 Definition 3.1 Nadaban and Dzitac (2014)
Let D represent the vector space over the field R. A triplet
(D, FN , ⊗) is termed a FN space where ⊗ is a t-norm and FN
represent a fuzzy set on D×R satisfies the requirements below
for every p,q∈D:

(FN1)FN (p, 0) = 0,
(FN2)FN (p, 𝜏) = 1∀ > 0 if and only p = 0

(FN3)FN (rp, 𝜏) = FN
(
p,

𝜏

|r |

)
, ∀r ∈ R, where r

≠ 0 and 𝜏 ≥ 0
(FN4)FN (p, 𝜏) ⊗ FN (q, s) ≤ FN (p + q, 𝜏 + s)∀𝜏 ≥ 0
(FN5)FN (p, .)is the continuous for each p ∈ D and lim

𝜏→∞

FN (p, 𝜏) = 1

2.1.2 Definition 3.2 Bag and Samanta (2003)
Consider (D,FN ,⊗) be a FN space. then

1. A sequence pn is called a convergent if limn→∞ FN (pn-
p, 𝜏) = 1;∀ > 0 and p ∈D.

2. A sequence pn is called Cauchy if limn→∞ FN (pn+1- pn,
𝜏) = 1;∀ > 0 and j = 1,2,...

2.1.3 Definition 3.3 Bag and Samanta (2003)
A FN space (D, FN ,⊗) is called complete if every Cauchy
sequence in D is convergent in D. In a FN space (D,FN ,⊗),
Sabri and Ahmed (2022) presented the notion of fuzzy dis-
tance. Consider Ũ and Ṽ be subsets of (D,FM,⊗) which are
nonempty and Ũ ◦ (𝜏), Ṽ ◦ (𝜏) denoted by the following sets :

Ũ ◦ (𝜏) =
{
P ∈ U : FN (p − q, 𝜏) = Na(Ũ ,Ṽ , 𝜏) for some

q ∈ ṽ}
Ṽ ◦ (𝜏) =

{
q ∈ Ṽ : FN (p − q, 𝜏) = Na(Ũ ,Ṽ , 𝜏) for some

p ∈ Ũ
}

WhereNa(Ũ ,Ṽ , 𝜏) = sup
{
FN (p − q, 𝜏) : p ∈ Ũ , q ∈ Ṽ

}
3. RESULTS AND DISCUSSION

In this section, the definition 𝔅�̃�- fuzzy proximal contractive
mapping and 𝔅h 𝔍h- fuzzy proximal contractive mapping is
presented, then our main results are proved. In a fuzzy metric
space, Guria et al. (2019) proposed the notion of Bpp. In the
following, the notion of the Bpp in the framework of FN space
is introduced.

3.1 Definition 4.1
Let (D, FN , ⊗) be a FN space and Ũ, Ṽ are nonempty subsets
of D . An element P* ∈ Ṽ is called the Bpp of a mapping
T:Ũ→ Ṽ if FN (P*-Tp*, 𝜏)= Na(Ũ, Ṽ , 𝜏) for all 𝜏 > 0

Next, the definition of 𝔅�̃�- fuzzy proximal contractive
mapping is presented. Consider 𝜓 represents the collection of
all functions �̃�:[0,1] → [0,1], having the properties below:

1. �̃� is decreasing
2. �̃� is continuous
3. �̃� (`)= 0 if and only if ` = 1

3.2 Definition 4.2
Assume that (D, FN , ⊗) is a FN space and let Ũ, Ṽ subsets of
D which are nonempty.Let T Ũ → Ṽ be a mapping. Then T
𝔅�̃� - fuzzy proximal contractive mapping where �̃� ∈ 𝜓 if for
all p,q,u,v ∈ Ũ we have,

FN (u − Tp, 𝜏) = Na(Ũ ,Ṽ , 𝜏)
FN (u − Tq, 𝜏) = Na(Ũ ,Ṽ , 𝜏)

 ⇒�̃�(FN (u − v, 𝜏))
≤ 𝜗(𝜏)𝔅�̃�(p, q,
𝜏) (1)

where 𝜗:(0,∞) →(0,1) is a function and 𝔅�̃� (p,q,𝜏) = max
�̃�(FN (p-q,𝜏)) �̃�(FN (p-u,𝜏)) ⊗ �̃�(FN (p-u,𝜏)), �̃�(FN (q-u,𝜏))

3.3 Theorem 4.3
Assume that (D, FN , ⊗) be a fuzzy Banach space (brefily FB
space) where ⊗ is min t-norm and let Ũ, Ṽ subsets of D (where
Ũ and Ṽ closed). suppose Ũ ◦ (𝜏) is non-empty and T : Ũ →
Ṽ is nonself mapping fulfilling the following requirements:

1. T (Ũ◦(𝜏)) ⊆ ṽ◦ (𝜏), ∀ 𝜏 > 0
2. T is 𝔅�̃� - fuzzy proximal contractive mapping
3. if a sequence qn is in Ṽ ◦ (𝜏) and p ∈ Ũ such that FN

(p-qn,𝜏)= Nd (Ũ,Ṽ ,𝜏) as n → ∞ p ∈ Ũ ◦ (𝜏), ∀ 𝜏 > 0
Then T possesses a unique Bpp.
proof: Consider p◦ in Ũ ◦ (𝜏) since T (Ũ ◦ (𝜏)) ⊆ Ṽ (𝜏),

there exists p1 ∈ Ũ◦(𝜏) such that FN (p1-T◦,𝜏)=Na (Ũ,Ṽ ,𝜏) ∀
> 0

The process is repeated, and we get a sequence pn in Ũ◦(𝜏)
fulfilling

FN (Pn − Tp−1, 𝜏) = Na(Ũ ,Ṽ , 𝜏)
FN (Pn+1 − Tpn, 𝜏) = Na(Ũ ,Ṽ , 𝜏)

(2)

If for any n◦ ∈ N, Pn+1=Pn, then according ton(2), is a Bpp
of T. Consequently, suppose Pn+1 ≠ Pn for each n ∈ N. Now
for each 𝜏>0 and n ∈ N ∪ 0 define Ln (𝜏) = FN (Pn-Pn+1, 𝜏).
From (1) we get

�̃�(Ln(𝜏)) = �̃�(FN (Pn−Pn+1, 𝜏)) ≤ 𝜗(𝜏)𝔅�̃�(Pn−1−Pn, 𝜏) (3)

Where
𝔅�̃� (Pn−1-Pn′ , 𝜏) = max �̃� (FN (Pn−1-Pn′ , 𝜏)), �̃� (FN (Pn−1-Pn′ ,

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Sabri et. al. Science and Technology Indonesia, 8 (2023) 298-304

𝜏) ⊗ �̃� FN (Pn-Pn′ , 𝜏)), �̃� (FN (Pn-Pn+1, 𝜏)
= max {�̃� (FN (Pn−1-Pn′ , 𝜏)),�̃� (FN (Pn−1-Pn′ , 𝜏) �̃� (FN (Pn-
Pn+1, 𝜏) }
= max {�̃� (FN (Pn−1-Pn′ , 𝜏)), �̃� (FN (Pn-Pn+1, 𝜏)) }

If max {�̃� (FN (Pn−1-Pn′ , 𝜏)), �̃� (FN (Pn-Pn+1, 𝜏))}= �̃� (FN
(Pn-Pn+1, 𝜏)) then �̃� (Ln)(𝜏)) ≤ 𝜗 (𝜏) �̃�(Ln)(𝜏) < �̃�(Ln)(𝜏))

but this a contradiction since 0<𝜗(𝜏)<1. Hence

�̃�(Ln)(𝜏)) ≤ 𝜗(𝜏)�̃�(Ln−1)(𝜏) < �̃�(Ln−1)(𝜏))

Therefore Ln (𝜏) is increasing, so there is L(𝜏) ∈ (0,1] with
limn−∞ Ln(𝜏) = L(𝜏) ∀𝜏> 0. Now, it will be established that
L(𝜏)=1; ∀𝜏> 0 Suppose there is 𝜏◦ > 0 with 0 <L(𝜏◦) < 1. Then

�̃�(L)(𝜏◦)) = lim
n−∞

�̃�(L)(𝜏◦))

≤ 𝜗(𝜏◦) lim
n−∞

�̃�(Ln−1)(𝜏◦))

≤ 𝜗(𝜏◦) lim
n−∞

�̃�(L)(𝜏◦))

≤ �̃�(L)(𝜏◦))

a contradiction. Thus, L(𝜏)=1 and conclude that

lim
n−∞

Ln = 1 (4)

After that, in order to demonstrate that pn is a Cauchy
sequence. Consider pn is not Cauchy. Consequently there is ϶
∈(0,1) such that ∀k∈N, there are m(k),n(k) ∈N with mk > nk ≥
k and

FN (Pmk − Pnk, 𝜏◦) ≤ 1− ϶ where𝜏◦ > 0 (5)

Suppose m(k) is the smallest number that is larger than
n(k), and meets the Equation (5)

FN (Pmk−1 − Pnk, 𝜏◦) > 1− ϶

which indicates

1− ϶ ≥ FN (Pmk − Pnk, 𝜏◦)
≥ FN (Pmk − Pmk−1, 𝜏◦) ⊗ FN (Pmk−1 − Pnk, 𝜏◦)
> FN (Pmk − Pmk−1, 𝜏◦) ⊗ 1− ϶

As a result, we find

lim
k−∞

FN (Pmk − Pnk, 𝜏◦) = 1− ϶ (6)

Now from

FN (Pmk−1 − Pnk, 𝜏◦) ≥ FN (Pmk+1 − Pmk, 𝜏◦) ⊗ FN
(Pmk − Pnk, 𝜏◦) ⊗ FN (Pnk − Pnk+1, 𝜏◦)
Taking limit as k → ∞, we arrive
lim
k−∞

FN (Pm(k)+1 − Pn(k)+1, 𝜏◦) ≥ 1− ϶ (7)

Now, combining the results of (4) and (6), we yield

FN (Pmk − Pnk, 𝜏◦) ≥ FN (Pmk − Pmk+1, 𝜏◦) ⊗ FN
(Pmk+1 − Pnk+1, 𝜏◦) ⊗ FN (Pnk+1 − Pnk, 𝜏◦)
Thus, it follows

lim
k−∞

FN (Pmk+1 − Pnk+1, 𝜏◦) = 1− ϶

Additionally

FN (Pmk − Pnk, 𝜏◦) ≥ FN (Pmk − Pmk+1, 𝜏◦) ⊗ FN
(Pmk+1 − Pnk+1, 𝜏◦) ⊗ FN (Pnk+1 − Pnk, 𝜏◦)

Indicates

lim
k−∞

FN (Pmk − Pnk+1, 𝜏◦) ≥ 1− ϶

Likewise,

lim
k−∞

FN (Pnk − Pmk+1, 𝜏◦) ≥ 1− ϶

Now

FN (Pmk+1 − Tpmk, 𝜏) = Na(Ũ ,Ṽ , 𝜏)
FN (Pnk+1 − Tpnk, 𝜏) = Na(Ũ ,Ṽ , 𝜏) (8)

Indicates

�̃�(FN (Pmk+1 − Pnk+1, 𝜏) ≤ 𝜗(𝜏◦)𝔅�̃�(Pmk, Pnk, 𝜏◦)
≤ 𝜗(𝜏◦)max

{
�̃�(FN (Pmk − Pnk, 𝜏◦)�̃�(FN (Pmk, Pmk+1,

𝜏◦) ⊗ (FN (Pnk − Pmk+1, 𝜏)�̃�(FN (Pnk, Pnk+1, 𝜏◦))
}

As k goes to ∞ in above, we obtain
�̃�(1− ϶) ≤ 𝜗(𝜏◦)max

{
�̃�(1− ϶),

�̃�(1 ⊗ (1− ϶)), �̃�(1)
}
= 𝜗(𝜏◦)�̃�(1− ϶)

if �̃� (1-϶) = 0 then ϶=0 but this contradiction.

If ˜𝜓(1− ϶) > 0then�̃�(1− ϶) ≤ 𝜗(𝜏◦)�̃�(1− ϶) < �̃�(1− ϶)

A contradiction since 0<𝜗(𝜏◦)<1 therefore pn is a Cauchy.
Because (D,FN ),⊗) is complete then pn converges to p* ∈ D,

lim
n−∞

FN (Pn − P∗, 𝜏) = 1 (9)

Furthermore,

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Sabri et. al. Science and Technology Indonesia, 8 (2023) 298-304

Na(Ũ ,Ṽ , 𝜏) = (FN (Pn+1 − TPn, 𝜏)(by Equation 3)
≥ FN (Pn+1 − P∗, 𝜏) ⊗ FN (P∗ − TPn, 𝜏)
(applying condition(FN4))
≥ FN (Pn+1 − P∗, 𝜏) ⊗ FN (P∗ − Pn+1, 𝜏) ⊗ TPn, 𝜏)
(applying(FN4))
= FN (Pn+1 − P∗, 𝜏) ⊗ FN (P∗ − Pn+1, 𝜏) ⊗ Na(Ũ ,Ṽ , 𝜏)

which indicates

Na(Ũ ,Ṽ , 𝜏) ≥ FN (Pn+1 − P∗, 𝜏) ⊗ FN (P∗ − TPn, 𝜏)
≥ FN (Pn+1 − P∗, 𝜏) ⊗ FN (P∗ − Pn+1, 𝜏) ⊗ Na(Ũ ,Ṽ , 𝜏)

Using limit as n→∞ in the preceding inequality,
the following result is obtained:

Na(Ũ ,Ṽ , 𝜏) ≥ 1 ⊗ FN (P∗ − TPn, 𝜏)
≥ 1 ⊗ 1 ⊗ Na(Ũ ,Ṽ , 𝜏)

that is,

lim
n−∞

FN (P∗ − TPn, 𝜏) = Na(Ũ ,Ṽ , 𝜏) (10)

Now, to demonstrate that T has a Bpp. Note that (c) and (9)
imply p*∈ Ũ◦ (𝜏) and hence TP*(Ũ◦ (𝜏)). T(Ũ◦(𝜏)) ⊆ Ṽ◦ (𝜏)
assures the existence of ` ∈ Ũ◦ (𝜏) for which

FN ( ` − TP∗, 𝜏) = Na(Ũ ,Ṽ , 𝜏) (11)

We claim that `=P*. Contrary to this, suppose that `≠P*
By (1), (2), and (11), obtain �̃� (FN (`-Pn+1, 𝜏)) ≤ 𝜗 (𝜏) 𝔅�̃�
(Pn-P∗, 𝜏)

≤ 𝜗(𝜏)max
{
�̃�FN (Pn − P∗, 𝜏)�̃�(FN (Pn − `, 𝜏)⊗

FN (P∗ − `, 𝜏))�̃�(FN (P∗ − Pn+1, 𝜏))
}

Employing limit as n approaches to ∞ in the above equation,
one obtain

�̃� FN ( ` − P∗, 𝜏) ≤ 𝜗(𝜏)max
{
�̃�(FN (P∗ − P∗, 𝜏)

�̃�FN (P∗ − `, 𝜏)) ⊗ (FN (P∗ − `, 𝜏)), �̃�(FN (P∗ − P∗, 𝜏)
}

≤ 𝜗(𝜏)�̃�(FN (P∗ − `, 𝜏))
< �̃�(FN (P∗ − `, 𝜏))

a contradiction 0<𝜗(𝜏)<1. Therefore, `=P*and as a result
FN (p∗-`,𝜏)=Na (Ũ, Ṽ , 𝜏) thus p∗ is the Bpp of T. If 𝛼 is

another Bpp of T with 𝛼≠p∗, then 0< (FN (P∗-𝛼, 𝜏))<1 and
FN (P∗TP∗, 𝜏) = Na(Ũ, Ṽ , 𝜏) and FN (𝛼T𝛼, 𝜏) = Na(Ũ, Ṽ , 𝜏)

�̃� FN (𝛼 − P∗, 𝜏) ≤ 𝜗(𝜏)𝔅�̃�(𝛼 − P∗, 𝜏)
≤ 𝜗(𝜏)max

{
�̃�(FN (𝛼 − P∗, 𝜏)�̃�(FN (𝛼 − 𝛼, 𝜏)⊗

FN (P∗ − 𝛼, 𝜏))�̃�(FN (P∗ − P∗, 𝜏))
}

= 𝜗(𝜏)�̃�(FN (𝛼 − P∗, 𝜏)
< �̃�(FN (𝛼 − P∗, 𝜏)

a contradiction. Thus the Bpp is unique.

3.4 Example 4.4:
Let D = R. Suppose FN :D ×R→[0,1] is a fuzzy norm, defined
by: FN ( p,𝜏)=

𝜏
𝜏+p ; ∀p ∈ D 𝜏>0, where ||p||: R→[0,∞) with

||p||=|p|. Let Ũ= {1,2,3,4,5} and Ṽ = {6,7,8,9,10} So that Na(Ũ,
Ṽ , 𝜏) = supFN (p-q,𝜏): p ∈Ũ, q ∈Ũ = 𝜏𝜏+1 Define D : Ũ→Ṽ by

T(P) =
{

6 i f P = 5
P + 5 otherwise

We have Ũ◦𝜏=5 andṼ◦𝜏=6, T ⊆Ṽ◦𝜏 Since FN (`-Tp,𝜏)=
Na(Ũ, Ṽ , 𝜏) =

𝜏
𝜏+1 implies (u,p)=(5,5) or (u,p)=(5,1) Now, let

�̃� defined by �̃� (`)=1-` for each `∈[0,1]. Then from (1), we
have �̃�(FN (u-𝜗,𝜏) ≥ 𝜗 (𝜏) 𝔅�̃� (p,q,𝜏)

(FN (u − 𝜗, 𝜏) =
𝜏

𝜏 + ||u − v||
=

𝜏

𝜏 + |5 + 5|
= 1

which implies �̃�(FN (u-𝜗,𝜏)=1- (FN (u-𝜗,𝜏)=1-1=0 and this
shows that �̃�(FN (u-𝜗,𝜏) ≤𝜗(𝜏) 𝔅�̃�(p,q,𝜏)

hold for each p,q, u,v ∈ Ũ and for all 𝜏 >0 and 𝜗(𝜏) ∈(0,1)
.Therefore, each of the requirements of Theorem 4.3 are met,
and there exists a uniquep∗∈ Ũ such that FN (p∗-Tp∗,𝜏)= Na(Ũ,
Ṽ , 𝜏) for all 𝜏 >0. In this example p∗=5 is a unique Bpp. Now
if we assume that Ũ◦ (𝜏) is a nonempty closed set, then we may
reduce some requirements in Theorem 4.3 as shown below.

3.5 Theorem 4.5:
Assume that (D, FN , ⊗) is a FB space where ⊗ is min t-norm.
Suppose that Ũ◦(𝜏) is a closed subset of D and T Ũ →Ṽ is a
mapping meeting the following conditions:

1. T (Ũ◦(𝜏)) ⊆ ṽ◦ (𝜏) for each 𝜏 > 0
2. there is a function �̃� ∈ 𝜓 for which
3.

FN (u − Tp, 𝜏) = Na(Ũ ,Ṽ , 𝜏)
FN (u − Tq, 𝜏) = Na(Ũ ,Ṽ , 𝜏)

}
(12)

⇒ �̃� (FN (u-v,𝜏)) ≤ 𝜗 (𝜏) 𝔅�̃� (p,q, 𝜏)

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Sabri et. al. Science and Technology Indonesia, 8 (2023) 298-304

holds for each p,q,u,v ∈ Ũ, and 𝜏>0, where 𝜗:(0,∞) →(0,1)
is a function and 𝔅�̃�(p,q,𝜏)=max �̃�(FN (p,q,𝜏), �̃�(FN (p,q,𝜏) ⊗
(FN (p,q,𝜏), �̃�(FN (p,q,𝜏)))

Then T possesses a unique Bpp
Proof : Similar to the proof of Theorem 4.3, construct a

Cauchy sequence pn in Ũ◦ (𝜏). The sequence pn is convergent
to some p∗ in Ũ◦ (𝜏) because Ũ◦ (𝜏) is a closed set, and the
completeness of (D, FN , ⊗) guarantees this. The remainder of
the proof is identical to the proof given for Theorem 4.3. Now
the notion of 𝔅h 𝔍h- fuzzy proximal contractive mapping is
introduced as follows:

3.6 Definition 4.6:
Let (D, FN , ⊗) be a FN space and let Ũ,Ṽ nonempty subsets
of D. Assume that T:Ũ→Ṽ is a given mapping. Then T is
termed as 𝔅h 𝔍h- fuzzy proximal contractive mapping if for
each p,q,u,v ∈Ũ and 𝜏>0 we have,

FN (u − Tp, 𝜏) = Na(Ũ ,Ṽ , 𝜏)
FN (u − Tq, 𝜏) = Na(Ũ ,Ṽ , 𝜏)

}

⇒ (FN (u − v, 𝜏)) ≥ 𝔍h(p, q, 𝜏) + 𝔅h(p, q, 𝜏) (13)

where h:I→I with h(`)>0 for each ` ∈ (0,1] and 𝔅h (p,q,𝜏)=
min { h(p-q,𝜏), (FN (p-u,𝜏) ⊗ FN (p-u,𝜏), hFN (p-𝜗,𝜏))} 𝔍h (p-
q,𝜏) + {FN (p-q,𝜏),FN (p-u,𝜏)}

The next theorem employs a different contraction condi-
tion than the previous results.

3.7 Theorem 4.7:
Assume that (D, FN , ⊗) is a FB space where ⊗ is min t-norm
and let Ũ,Ṽ closed and nonempty subsets of D where Ũ◦𝜏
is non-empty. Let T Ũ→Ṽ be nonself mapping meeting the
following conditions:

1. T (Ũ◦(𝜏)) ⊆ ṽ◦ (𝜏)
2. T 𝔅h 𝔍h fuzzy proximal contractive mapping.
3. If sequence {qn} is in Ṽ◦ (𝜏) and p ∈ Ũ with FN (p-qn𝜏)=

Na(Ũ, Ṽ , 𝜏) as n → ∞ then p ∈ Ũ◦ (𝜏)
ThenT possesses a unique Bpp.
Proof : Consider p◦ in Ũ◦(𝜏. since T (Ũ◦(𝜏)) ⊆ Ṽ◦(𝜏,

there exists p1 ∈ Ũ◦(𝜏) such that FN (p1-TP◦𝜏) = Na(Ũ, Ṽ , 𝜏)
: for each 𝜏 > 0

The process is repeated, and we get a sequence pn in Ũ◦(𝜏)
fulfilling

FN (Pn1 − Tpn−1, 𝜏) = Na(Ũ ,Ṽ , 𝜏)
FN (Pnk+1 − Tpn, 𝜏) = Na(Ũ ,Ṽ , 𝜏) (14)

From (13) and (14) obtain, FN (Pn-Pn+1, 𝜏) ≥ 𝔍h (Pn−1,
pn,𝜏) 𝔅h (Pn−1, pn,𝜏)

≥ min{FN (Pn−1 − Pn, 𝜏)FN (Pn − Pn, 𝜏) + minh
{FN (Pn−1 − Pn, 𝜏)hFN (Pn−1 − Pn, 𝜏)
⊗ FN (Pn − Pn, 𝜏), FN (Pn − Pn−1, 𝜏)))} (15)

≥ FN (Pn−1-Pn, 𝜏)+min { h (FN (Pn−1-Pn,𝜏)), h (FN (Pn-
Pn+1,𝜏))}

which implies FN (Pn-Pn+1,𝜏) ≥ FN (Pn−1-Pn,𝜏)
That is {FN (Pn-Pn+1,𝜏)} is is increasing, so there is 𝔧(𝜏)

∈(0,1] with limn−∞ FN (Pn+1-Pn,𝜏) = 𝔧(𝜏) ∀ 𝜏 > 0
Now, it will be established that 𝔧(𝜏)=1 for each 𝜏 > 0. Sup-

pose there is 𝜏◦> 0 such that 0 <𝔧 (𝜏◦) < 1. Considering limit
as n goes to ∞ in (15), then

𝔧 (𝜏 ◦) ≥j(𝜏◦)+minh(𝔧(𝜏◦)),h(1)}
If min h(𝔧(𝜏◦ )),h(1) =h(𝔧(𝜏◦)) then obtain 𝔧(𝜏◦) ≤(𝔧(𝜏◦ ) +

h(𝔧(𝜏◦ )) implies thath (𝔧(𝜏◦ )=0 and this is a contradiction. If
minh𝔧(𝜏◦ )),h(1) =h(1) then obtain 𝔧(𝜏◦) ≥ 𝔧(𝜏◦)+h(1) implies
that h(1)=0 and this is a contradiction. This demonstrates that
for all 𝜏> 0, 𝔧(𝜏) = 1.

After that, to demonstrate pn is a Cauchy sequence. Con-
sider pn is not Cauchy. Then there is ϶ ∈ (0,1) such that ∀k∈N,
there are m(k),n(k) ∈N with mk > nk ≤k and

FN (Pmk − Pnk, 𝜏) ≤ 1− ϶ (16)

Suppose m(k) is the smallest number that is larger than
n(k), that meeting (16),

FN (Pmk − Pnk, 𝜏) > 1− ϶

In a manner analogous to the proof of Theorem 4.3, get

lim
n−∞

FN (Pmk − Pnk, 𝜏) = 1− ϶, lim
n−∞

FN (Pmk+1 − Pnk+1, 𝜏)

= 1− ϶

lim
n−∞

FN (Pmk − Pnk+1, 𝜏) ≥ 1− ϶, lim
n−∞

FN (Pnk − Pmk+1, 𝜏)

≥ 1− ϶

From (14) we get

FN (Pmk+1 −TPmk, 𝜏◦) = Na(Ũ ,Ṽ , 𝜏◦)and
FN (Pnk+ −TPnk, 𝜏◦) = Na(Ũ ,Ṽ , 𝜏◦)

Hence, (13) implies

FN (Pmk+1 − Pnk+1, 𝜏◦) ≥ 𝔍h(Pmk, Pnk, 𝜏◦) + 𝔅h(Pmk,
Pnk, 𝜏◦)
≥ min{FN (Pmk −TPnk, 𝜏◦), FN (Pnk −TPmk+1, 𝜏◦)}
+ min{h(FN (Pmk − Pnk, 𝜏◦))

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Sabri et. al. Science and Technology Indonesia, 8 (2023) 298-304

If k tends to ∞ in above inequality, one obtains
1-϶ ≥(1-϶)+min {h(1-϶) h(1)}
Thus

0≥ min {h(1-϶) h(1)}
then, either h(1-϶)=0 or h(1)=0 but this is in both cases a
contradiction, therefore pn is a Cauchy. Because (D,FN ,⊗) is
complete then pn converges to p* ∈ D,

lim
n−∞

FN (Pn − P∗, 𝜏) = 1f oreach𝜏 > 0 (17)

Now to demonstrate that T has Bpp. Like Theorem 4.3,
obtain P* ∈ Ũ◦(𝜏). As (Ũ◦(𝜏)) ⊆ Ṽ◦(𝜏) assures the existence
og ` ∈Ũ◦(𝜏)

FN ( ` −TP∗, 𝜏) = Na(Ũ ,Ṽ , 𝜏) (18)

We claim that `=p∗. Contrary to this, suppose that `≠p∗

By (14) and (18), obtain

FN (Pnk+1 − `, 𝜏) ≥ 𝔍h(Pn, P∗, 𝜏) + 𝔅h(Pn, P∗, 𝜏
≥ min{FN (Pn − P∗, 𝜏), FN (P∗ − Pn+1, 𝜏)}
+ min{h(FN (Pn − P∗, 𝜏)), h(FN (Pn − Pn+1, 𝜏)⊗
(FN (P∗ − Pn+1, 𝜏), h(FN (P∗ − `, 𝜏)

Using limit as n approaches to ∞ in above inequality, one
gets

FN (P∗ − `, 𝜏) ≥ 1 + min{h(1)FN (P∗, `, 𝜏))}
So1 ≥ FN (P∗ − `, 𝜏) ≥ 1, which impliesFN (P∗ − `, 𝜏)
= 1, that is` = P∗andFN (P∗ −TP∗, 𝜏) = Na(Ũ ,Ṽ , 𝜏)

To demonstrate the uniqueness, assume 𝛼 is another Bpp
of T such that 𝛼≠p∗, that is 0<

FN (P∗ − 𝛼, 𝜏) < 1for𝜏 > 0asFN (P∗ −TP∗, 𝜏) = Na
(Ũ ,Ṽ , 𝜏)andFN (𝛼 −T𝛼, 𝜏) = Na(Ũ ,Ṽ , 𝜏)

hence, from (13) we obtain

FN (P∗ − 𝛼, 𝜏) ≥ 𝔍h(P∗, 𝛼, 𝜏) + 𝔅h(P∗, 𝛼, 𝜏)
≥ min{FN (P∗, 𝛼, 𝜏), FN (𝛼, P∗𝜏)}
+ min{h(FN (P∗, 𝛼, 𝜏)), h(FN (P∗, P∗, 𝜏)⊗
(FN (𝛼, P∗, 𝜏), h(FN (𝛼, 𝛼, 𝜏)}
(FN (P∗, 𝛼, 𝜏) + min{h(FN (P∗, 𝛼, 𝜏))h(1)}

Therefore

(FN (P∗, 𝛼, 𝜏) ≥ (FN (P∗, 𝛼, 𝜏) + min{h(FN (P∗, 𝛼,
𝜏))h(1)} (19)

Which implies h FN (P∗,𝛼,𝜏)=0 or h which is in both
cases a contradiction as h(`)>0 for each ` ∈ (0,1]. Thus FN
(P∗,𝛼,𝜏)=1 and so p∗=𝛼.

4. CONCLUSION

The Bpp theorem for various non-self fuzzy contractive map-
ping types such as 𝔅h- fuzzy proximal contractive mapping
and 𝔅h 𝔍h- fuzzy proximal contractive mapping in a FN space
is established in this study. An example is presented to show
the significance of the results that were obtained. The general-
izations of these kinds of contraction mappings and studies of
their applications in the FN space needs more investigation in
upcoming work.

5. ACKNOWLEDGMENT

I would like to convey my profound appreciation to the Re-
viewers of this work and the publishers of the "Science and
Technology Indonesia journal" for presenting me with this
splendid opportunity.

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© 2023 The Authors. Page 304 of 304


	INTRODUCTION
	EXPERIMENTAL SECTION
	Preliminaries
	Definition 3.1 15
	Definition 3.2 2
	Definition 3.3 2 


	RESULTS AND DISCUSSION
	Definition 4.1
	Definition 4.2 
	Theorem 4.3
	Example 4.4:
	Theorem 4.5:
	Definition 4.6: 
	Theorem 4.7:

	CONCLUSION
	ACKNOWLEDGMENT