EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5785
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Quasi Contractions and Fixed Point Theorems in the
Context of Neutrosophic Fuzzy Metric Spaces

Anwar Bataihah1,∗, Ayman Hazaymeh1

1 Department of Mathematics, Faculty of Science, Jadara University, Irbid,Jordan

Abstract. Fixed point theory has garnered significant interest due to its applicability across
various scientific disciplines. In this manuscript, we present fixed point theorems pertaining to
neutrosophic fuzzy quasi-contractions, situated within the sophisticated framework of neutrosophic
fuzzy metric spaces. Furthermore, we establish several fixed point results pertinent to this field of
research.

2020 Mathematics Subject Classifications: 37C25, 55M20, 26E50, 46S40

Key Words and Phrases: Fixed point theory, Neutrosophic fuzzy metric spaces, Banach con-
traction principle, quasi-contractions, linear contractions

1. Introduction

The Banach fixed-point theorem [6], often referred to as the Banach contraction prin-
ciple, represents a cornerstone in mathematical analysis, especially within the realm of
metric spaces. This theorem guarantees both the existence and uniqueness of fixed points
for certain self-maps, provided specific conditions are met in metric spaces. Essentially,
the Banach fixed-point theorem offers a comprehensive framework for Picard’s method of
successive approximations and has found extensive application across various branches of
mathematics. Introduced by Stefan Banach (1892–1945) in 1922, this theorem has inspired
a multitude of mathematicians to explore numerous extensions and generalizations in di-
verse mathematical fields, as indicated by the references in [2, 8–10, 22, 23, 25, 29, 34–36].

In [30], the authors present the concept of α̌, η̌ proximal contractive mappings. Sub-
sequently, they establish a theorem regarding the best proximity point for this class of
mappings within the context of a fuzzy Banach space, demonstrating significant results
related to best proximity points for these contractions. In the work referenced as [33],
the authors established the existence of coincidence and best proximity points for fuzzy
(α−η)- and fuzzy (β−ψ)-generalized proximal contractions within the context of b-fuzzy

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5785

Email addresses: a.bataihah@jadara.edu.jo; anwerbataihah@gmail.com (A. Bataihah),
aymanha@jadara.edu.jo (A. Hazaymeh)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. Bataihah, A. Hazaymeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5785 2 of 15

metric spaces. In the work referenced as [31], the authors defined fuzzy normed spaces and
presented the concepts of the best proximity point and α̃−proximal admissibility within
this framework. Additionally, they introduced the concept of α̃, ψ̃-proximal contractive
mappings. A theorem regarding the best proximity point for these types of mappings
in fuzzy normed spaces was subsequently established. In reference [32], the authors es-
tablished the existence and uniqueness of the best proximity point for nonself-mappings
within a fuzzy normed space.

The notion of Fuzzy Sets (FSs), first introduced by Zadeh [39], has significantly influ-
enced a wide array of scientific fields since its emergence. While this framework is highly
pertinent to practical applications, it has not consistently offered satisfactory solutions to
various problems over the years. As a result, there has been a renewed interest in research
focused on resolving these challenges. In this context, Atanassov [4] proposed Intuitionistic
Fuzzy Sets (IFSs) as a strategy to tackle such issues. Furthermore, the Neutrosophic Set
(NS), created by Smarandache [37], serves as a sophisticated extension of traditional set
theory. Other important generalizations include interval-valued FS [38], interval-valued
IFS [5].

Neutrosophic sets exhibit a diverse range of applications across multiple fields. For
instance, Barbosa and Smarandache [7] proposed the Neutrosophic One-Round Zero-
Knowledge Proof protocol (N-1-R) ZKP, which enhances the One-Round (1-R) ZKP frame-
work by incorporating Neutrosophic numbers. Furthermore, the authors in [1] provide an
in-depth examination of effective and optimally appropriate solutions related to scalar
optimization problems. They also derive the Kuhn-Tucker conditions relevant to both ef-
ficiency and optimal efficiency. To gain a more comprehensive insight into the applications
of neutrosophic sets and their extensive uses, it is recommended to consult the literature
referenced in [3, 11, 14, 16–20] and associated sources.

2. Preliminary

In this framework, the interval ]0−, 1+[ is characterized as a non-standard unit interval.
Within this context, non-standard finite numbers are articulated as (1+) = 1 + ϵ, where
”1” represents the standard component and ϵ denotes the non-standard element. In a
similar manner, (0−) = 0− ϵ, with ”0” indicating the standard component and ’ϵ” as the
non-standard element. The numbers 0 and 1 can be interpreted as non-standard values
that are infinitesimally small yet less than 0 or infinitesimally small yet greater than
1, respectively, and these values are encompassed within the non-standard unit interval
]0−, 1 + [.

Definition 1. [39] In relation to a universal set U , a fuzzy set X is defined by the notation
X = {< ξ, µX (ξ) >: 0 ≤ µX (ξ) ≤ 1, ξ ∈ U}. In this context, µX : X → [0, 1], and µX (ξ)
represents the degree of membership of the element ξ within the fuzzy set X .

Definition 2. [37] A neutrosophic set V relative to a universal set U is defined as V =
{< ξ, (TN (ξ), IN (ξ), FN (ξ)) >: ξ ∈ U, TN (ξ), IN (ξ), FN (ξ) ∈ ]0−, 1 + [ }. In this context,
TN (ξ), IN (ξ), and FN (ξ) represent the membership degrees of truth, indeterminacy, and



A. Bataihah, A. Hazaymeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5785 3 of 15

falsity for an element ξ within the set V, respectively, while ]0−, 1 + [ signifies a non-
standard unit interval.

Definition 3. [13] A neutrosophic fuzzy set D within a universal set U is characterized
as follows: D = {< x, (µD(ξ), TD(ξ, µ), ID(ξ, µ), F (ξ, µ)) >: ξ ∈ U, µD(ξ) ∈ [0, 1],
TD(ξ, µ), ID(ξ, µ), F (ξ, µ) ∈]0−, 1 + [ }. In this framework, the membership degree µD(ξ)
is represented by three distinct components: the truth membership grade TD(ξ, µ), the
indeterminacy membership grade ID(ξ, µ), and the falsity membership grade F (ξ, µ). The
notation ]0−, 1 + [ signifies a nonstandard unit interval.

Triangular norms (TN), first proposed by Menger [26] (see also [24]), represent a cru-
cial concept in the realm of mathematical analysis. Menger’s pioneering methodology
utilized probability distributions to evaluate the distance between two elements in a given
space, thereby transcending the conventional dependence on numerical values. This ap-
proach enables the generalization of the triangle inequality within metric spaces via the
implementation of triangular norms. In contrast, triangular conorms (TC) act as dual
counterparts to t-norms. Both TN and TC play vital roles in fuzzy operations, particu-
larly in relation to intersections and unions.

In this manuscript, the notation R+ is used to represent the interval = [0,+∞), while
I denotes the unit interval [0, 1].

Definition 4. [26]
Consider an operation ⊙ : I×I → I. This operation is classified as continuous t-norm

(TN) if it meets the following criteria: for any elements ϖ,ϖ′, t, t′ ∈ I.

(i) ϖ ⊙ 1 = ϖ,

(ii) If ϖ ≤ ϖ′ and t ≤ t′, than ϖ ⊙ t ≤ ϖ′ ⊙ t′,

(iii) ⊙ is continuous,

(iv) ⊙ is commutative and associate.

Definition 5. [26]
Consider an operation ⊕ : I × I → I. This operation is classified as continuous

t-conorm (TC) if it meets the following criteria: for all elements ϖ,ϖ′, t, t′ ∈ I.

(i) ϖ ⊕ 0 = ϖ,

(ii) If ϖ ≤ ϖ′ and t ≤ t′, than ϖ ⊕ t ≤ ϖ′ ⊕ t′,

(iii) ⊕ is continuous,

(iv) ⊕ is commutative and associate.



A. Bataihah, A. Hazaymeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5785 4 of 15

Definition 6. [24] A 6-tuple (X ,H,K,L,⊙,⊕) is referred to as a Neutrophic Metric Space
(NMS) if the set X is a non-empty arbitrary collection, ⊙ signifies a continuous t-norm,
⊕ indicates a continuous t-conorm, and the elements H,K, and L are three fuzzy sets
established on the Cartesian product X 2 × (0,+∞). These components must satisfy the
following specific conditions for all elements ξ, ω, c ∈ X and for all positive real numbers
λ, ρ.

(i) 0 ≤ H(ξ, ω, λ) ≤ 1, 0 ≤ K(ξ, ω, λ) ≤ 1, 0 ≤ L(ξ, ω, λ) ≤ 1,

(ii) 0 ≤ H(ξ, ω, λ) +K(ξ, ω, λ) + L(ξ, ω, λ) ≤ 3,

(iii) H(ξ, ω, λ) = 1, for λ > 0 iff ξ = ω

(iv) H(ξ, ω, λ) = H(ω, ξ, λ), for λ > 0

(v) H(ξ, ω, λ)⊙H(ω, c, ρ) ≤ H(ξ, c, λ+ ρ)

(vi) H(ξ, ω, ·) : R+ → I is continuous

(vii) lim
λ→+∞

H(ξ, ω, λ) = 1

(viii) K(ξ, ω, λ) = 0 iff ξ = ω

(ix) K(ξ, ω, λ) = K(ω, ξ, λ),

(x) K(ξ, ω, λ)⊕K(ω, c, ρ) ≥ K(ξ, c, λ+ ρ),

(xi) K(ξ, ω, ·) : R+ → I is continuous

(xii) lim
λ→+∞

K(ξ, ω, λ) = 0

(xiii) L(ξ, ω, λ) = 0, for λ > 0 iff ξ = ω

(xiv) L(ξ, ω, λ) = L(ω, ξ, λ),

(xv) L(ξ, ω, λ)⊕ L(ω, c, ρ) ≥ S(ξ, c, λ+ ρ),

(xvi) L(ξ, ω, ·) : R → I is continuous

(xvii) lim
λ→+∞

L(ξ, ω, λ) = 0

(xviii) If λ ≤ 0, then H(ξ, ω, λ) = 0, K(ξ, ω, λ) = L(ξ, ω, λ) = 1

The functions H(ξ, ω, λ), K(ξ, ω, λ), and L(ξ, ω, λ) represent the degrees of nearness, neu-
tralness, and non-nearness between the elements ξ and ω in relation to the parameter λ,
respectively.

Recently, Ghosh et al. [15] presented the notion of neutrosophic fuzzy metric spaces
and examined various topological characteristics associated with this concept.



A. Bataihah, A. Hazaymeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5785 5 of 15

Definition 7. [15] A 7-tuple (X ,H,J ,K,L,⊙,⊕) is defined as a Neutrophic Fuzzy Metric
Space (NFMS) if X represents an arbitrary set, ⊙ denotes a continuous t-norm, ⊕ signifies
a continuous t-conorm, and the fuzzy sets H,J ,K, and L are defined on X 2 × (0,+∞).
These sets must satisfy specific conditions for all ξ, ω, c ∈ X and for λ, ρ > 0.

(i) 0 ≤ H(ξ, ω, λ) ≤ 1, 0 ≤ J (ξ, ω, λ) ≤ 1, 0 ≤ K(ξ, ω, λ) ≤ 1, 0 ≤ L(ξ, ω, λ) ≤ 1,

(ii) 0 ≤ H(ξ, ω, λ) + J (ξ, ω, λ) +K(ξ, ω, λ) + L(ξ, ω, λ) ≤ 4,

(iii) H(ξ, ω, λ) = 1,

(iv) H(ξ, ω, λ) = H(ω, ξ, λ),

(v) H(ξ, ω, λ)⊙H(ω, c, ρ) ≤ H(ξ, c, λ+ ρ), for ρ, λ > 0

(vi) H(ξ, ω, ·) : R → I is continuous

(vii) lim
λ→+∞

H(ξ, ω, λ) = 1

(viii) J (ξ, ω, λ) = 1, iff ξ = ω

(ix) J (ξ, ω, λ) = J (ω, ξ, λ), for λ > 0

(x) J (ξ, ω, λ)⊙ J (ω, c, ρ) ≤ J (ξ, c, λ+ ρ),

(xi) J (ξ, ω, ·) : R → I is continuous

(xii) lim
λ→+∞

J (ξ, ω, λ) = 1

(xiii) K(ξ, ω, λ) = 0, iff ξ = ω

(xiv) K(ξ, ω, λ) = K(ω, ξ, λ),

(xv) K(ξ, ω, λ)⊕K(ω, c, ρ) ≥ K(ξ, c, λ+ ρ),

(xvi) K(ξ, ω, ·) : R → I is continuous

(xvii) lim
λ→+∞

K(ξ, ω, λ) = 0

(xviii) L(ξ, ω, λ) = 0, iff ξ = ω

(xix) L(ξ, ω, λ) = L(ω, ξ, λ),

(xx) L(ξ, ω, λ)⊕ L(ω, c, ρ) ≥ L(ξ, c, λ+ ρ),

(xxi) L(ξ, ω, ·) : R → I is continuous

(xxii) lim
λ→+∞

L(ξ, ω, λ) = 0

(xxiii) If λ ≤ 0, then H(ξ, ω, λ) = 0, K(ξ, ω, λ) = L(ξ, ω, λ) = 1



A. Bataihah, A. Hazaymeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5785 6 of 15

In this context, H(ξ, ω, λ) indicates the certainty that the distance between ξ and ω is less
than λ. Meanwhile, J (ξ, ω, λ) signifies the level of proximity, K(ξ, ω, λ) represents the
degree of neutrality, and L(ξ, ω, λ) reflects the extent of non-proximity between ξ and ω in
relation to λ.

The convergence, Cauchyness, completeness are given as follows.

Definition 8. [15] Let (ξn) be a sequence in a NFMS (X ,H,J ,K,L,⊙,⊕). Then

(i) (ξn) converges to ξ ∈ X iff for a given ϵ ∈ (0, 1), λ > 0 there is n0 ∈ N such that for
each n ≥ n0

H(ξn, ξ, λ) > 1− ϵ, J (ξn, ξ, λ) > 1− ϵ, K(ξn, ξ, λ) < ϵ, L(ξn, ξ, λ) < ϵ

i.e.,
lim

n→+∞
H(ξn, ξ, λ) = 1, lim

n→+∞
J (ξn, ξ, λ) = 1,

lim
n→+∞

K(ξn, ξ, λ) = 0, lim
n→+∞

L(ξn, ξ, λ) = 0

(ii) (ξn) is called Cauchy iff for a given ϵ ∈ (0, 1), λ > 0 there is n0 ∈ N such that for
each n,m ≥ n0

H(ξn, ξm, λ) > 1− ϵ, J (ξn, ξm, λ) > 1− ϵ, K(ξn, ξm, λ) < ϵ, L(ξn, ξm, λ) < ϵ

i.e.,
lim

n.m→+∞
H(ξn, ξm, λ) = 1, lim

n.m→+∞
J (ξn, ξm, λ) = 1,

lim
n,m→+∞

K(ξn, ξm, λ) = 0, lim
n,m→+∞

L(ξn, ξm, λ) = 0

(iii) (X ,H,J ,K,L,⊙,⊕) is called complete if each Cauchy sequence is convergent to an
element in X .

Quasi contractions were first introduced by L.B. Ciric in 1974 within the framework
of metric spaces [12]. Ciric established both the existence and uniqueness of fixed points
for quasi contractions. Subsequently, numerous researchers have explored quasi contrac-
tions in various distance settings, as one can see the studies presented in [21, 27, 28] and
references therein.

3. Main Result

In the following sections, we will first present a valuable lemma that is integral to our
primary research. Subsequently, we will introduce our contractions within the framework
of neutrosophic fuzzy metric spaces. We will then demonstrate that these contractions have
a unique fixed point under certain conditions, and we will explore several consequential
results derived from our main findings.

We start with the following helpful lemma.



A. Bataihah, A. Hazaymeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5785 7 of 15

Lemma 1. Let (X ,H,J ,K,L,⊙,⊕) be a NFMS. Then

(i) H(ξ, ω, ·) : R → R is non-decreasing

(ii) J (ξ, ω, ·) : R → R is non-decreasing

(iii) K(ξ, ω, ·) : R → R is non-increasing

(iv) L(ξ, ω, ·) : R → R is non-increasing

Proof. (1) Let λ1, λ2 > 0, with λ1 > λ2. Then, there is δ > 0 such that λ1 = λ2 + δ.
From (5), we get

H(ξ, ω, λ1) = H(ξ, ω, λ2 + δ)
≥ H(ξ, ω, λ2)⊙H(ω, ω, δ)
= H(ξ, ω, λ2).

The proofs for (2),(3) and (4) are identical to that of (1).

Through this context we need the following notations.
If T : X 2 × R → R, and f is a self map on X , then

(i)
M 1

T (ξ, ω, λ) = max

{
1

T (ξ, ω, λ)
− 1,

1

T (ξ, fξ, λ)
− 1,

1

T (ω, fω, λ)
− 1,

1

T (ξ, fω, λ)
− 1,

1

T (fξ, ω, λ)
− 1

}
,

(ii) M 2
T (ξ, ω, λ) = max {T (ξ, ω, λ), T (ξ, fξ, λ), T (ω, fω, λ), T (ξ, fω, λ), T (fξ, ω, λ)} .

Additionally, we establish the following notation to define the concept of need:

(i) δ1T (ξ, f, λ) = max
i,j∈N

{
1

T (f iξ, f jξ, λ)
− 1

}
,

(ii) δ2T (ξ, f, λ) = max
i,j∈N

{
T (f iξ, f jξ, λ)

}
.

We will now provide the definition of a neutrosophic fuzzy quasi-contractions.

Definition 9. Let (X ,H,J ,K,L,⊙,⊕) represent a neutrosophic fuzzy metric space (NFMS).
A function f : X → X is defined as a neutrosophic fuzzy quasi-contraction if, for every pair
of elements ξ, ω ∈ X and for all λ > 0, the following condition holds for some k ∈ [0, 1):

1

H(fξ, fω, λ)
− 1 ≤ k M 1

H(ξ, ω, λ),

1

J (fξ, fω, λ)
− 1 ≤ k M 1

J (ξ, ω, λ),



A. Bataihah, A. Hazaymeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5785 8 of 15

K(fξ, fω, λ) ≤ k M 2
K(ξ, ω, λ),

and
L(fξ, fω, λ) ≤ k M 2

L(ξ, ω, λ).

Theorem 1. Let (X ,H,J ,K,L,⊙,⊕) be a complete NFMS, Suppose that f : X → X
is neutrosophic fuzzy quasi-contraction. Furthermore, consider that one of the following
statements holds true.

(*) f is continuous,

(**) The fuzzy sets H,J ,K, and L exhibit continuity with respect to their first two coor-
dinates.

Consequently, the function f possesses a unique fixed point.

Proof. Let ξ0 ∈ X represent any arbitrary element. We consider the sequence (ξn)
characterized by the relation ξn = fn(ξ0) for all n ≥ 0. By Definition 9 we have for all
i, j ∈ N

1
H(fn+iξ0,fn+jξ0,λ)

− 1 ≤ k max
{

1
H(fn+i−1ξ0,fn+j−1ξ0,λ)

− 1, 1
H(fn+i−1ξ0,fn+iξ0,λ)

− 1, 1
H(fn+j−1ξ0,fn+jξ0,λ)

− 1,
1

H(fn+i−1ξ0,fn+jξ0,λ)
− 1, 1

H(fn+j−1ξ0,fn+iξ0,λ)
− 1 } ,

1
J (fn+iξ0,fn+jξ0,λ)

− 1 ≤ k max
{

1
J (fn+i−1ξ0,fn+j−1ξ0,λ)

− 1, 1
J (fn+i−1ξ0,fn+iξ0,λ)

− 1, 1
J (fn+j−1ξ0,fn+jξ0,λ)

− 1,
1

J (fn+i−1ξ0,fn+jξ0,λ)
− 1, 1

J (fn+j−1ξ0,fn+iξ0,λ)
− 1 } ,

K(fn+iξ0, f
n+jξ0, λ) ≤ k max

{
K(fn+i−1ξ0, f

n+j−1ξ0, λ),K(fn+i−1ξ0, f
n+iξ0, λ),K(fn+j−1ξ0, f

n+jξ0, λ),
K(fn+i−1ξ0, f

n+jξ0, λ),K(fn+j−1ξ0, f
n+iξ0, λ) } ,

and

L(fn+iξ0, f
n+jξ0, λ) ≤ k max

{
L(fn+i−1ξ0, f

n+j−1ξ0, λ),L(fn+i−1ξ0, f
n+iξ0, λ),L(fn+j−1ξ0, f

n+jξ0, λ),
L(fn+i−1ξ0, f

n+jξ0, λ),L(fn+j−1ξ0, f
n+iξ0, λ) } .

Thus, we conclude

δH(f
nξ0, f, λ) ≤ k δ1H(f

n−1ξ0, f, λ),

δJ (f
nξ0, f, λ) ≤ k δ1J (f

n−1ξ0, f, λ),

σK(f
nξ0, f, λ) ≤ k δ2K(f

n−1ξ0, f, λ),

and
σL(f

nξ0, f, λ) ≤ k δ2L(f
n−1ξ0, f, λ).



A. Bataihah, A. Hazaymeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5785 9 of 15

Hence, we conclude that for each n ≥ 1

δH(f
nξ0, f, λ) ≤ kn δ1H(ξ0, f, λ),

δJ (f
nξ0, f, λ) ≤ kn δ1J (ξ0, f, λ),

σK(f
nξ0, f, λ) ≤ kn δ2K(ξ0, f, λ),

and
σL(f

nξ0, f, λ) ≤ kn δ2L(ξ0, f, λ).

The above inequalities, gives

1

H(fnξ0, fn+mξ0, λ)
− 1 ≤ δH(f

nξ0, f, λ) ≤ kn δ1H(ξ0, f, λ),

1

J (fnξ0, fn+mξ0, λ)
− 1 ≤ δJ (f

nξ0, f, λ) ≤ kn δ1J (ξ0, f, λ),

K(fnξ0, f
n+mξ0, λ) ≤ σK(f

nξ0, f, λ) ≤ kn δ2K(ξ0, f, λ),

and
L(fnξ0, fn+mξ0, λ) ≤ σL(f

nξ0, f, λ) ≤ kn δ2L(ξ0, f, λ).

By considering the limit as both n and m tend to infinity, we obtain that

lim
n,m→+∞

H(fnξ0, f
n+mξ0, λ) = 1,

lim
n,m→+∞

J (fnξ0, f
n+mξ0, λ) = 1,

lim
n,m→+∞

K(fnξ0, f
n+mξ0, λ) = 0,

lim
n,m→+∞

L(fnξ0, fn+mξ0, λ) = 0,

Consequently, the sequence (fnξ0) qualifies as a Cauchy sequence, which implies the
existence of an element u ∈ X such that fnξ0 converges to u.
If (*) holds, (i.e., f is continuous), then fn+1ξ0 = ffnξ0 → fu and so, u = fu.
If (**) holds, then Definition 9 implies that

1
H(fu,fn+1ξ0,λ)

− 1 ≤ k max
{

1
H(u,fnξ0,λ)

− 1, 1
H(u,fu,λ) − 1, 1

H(fnξ0,fn+1ξ0,λ)
− 1,

1
H(u,fn+1ξ0,λ)

− 1, 1
H(fnξ0,fu,λ)

− 1 } ,

1
J (fu,fn+1ξ0,λ)

− 1 ≤ k max
{

1
J (u,fnξ0,λ)

− 1, 1
J (u,fu,λ) − 1, 1

J (fnξ0,fn+1ξ0,λ)
− 1,

1
J (u,fn+1ξ0,λ)

− 1, 1
J (fnξ0,fu,λ)

− 1 } ,



A. Bataihah, A. Hazaymeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5785 10 of 15

K(fu, fn+1ξ0, λ) ≤ k max
{
K(u, fnξ0, λ),K(u, fu, λ),K(fnξ0, f

n+1ξ0, λ),
K(u, fn+1ξ0, λ),K(fnξ0, fu, λ) } ,

and

L(fu, fn+1ξ0, λ) ≤ k max
{
L(u, fnξ0, λ),L(u, fu, λ),L(fnξ0, fn+1ξ0, λ),
L(u, fn+1ξ0, λ),L(fnξ0, fu, λ) } .

By evaluating the limit, we obtain

(1− k)

(
1

H(fu, u, λ)
− 1

)
≤ 0,

(1− k)

(
1

J (fu, u, λ)
− 1

)
≤ 0,

(1− k) K(fu, u, λ) ≤ 0,

(1− k) L(fu, u, λ) ≤ 0.

Hence u = fu.
Let y ∈ X such that y = fy. If u ̸= y, then according to Definition 9, it can be inferred

that.

1
H(u,y,λ) − 1 = 1

H(fu,fy,λ) − 1 ≤ k max
{

1
H(u,y,λ) − 1, 1

H(u,u,λ) − 1, 1
H(y,y,λ) − 1,

1
H(u,y,λ) − 1, 1

H(y,u,λ) − 1 }
= k

(
1

H(u,y,λ) − 1
)
,

1
J (u,y,λ) − 1 = 1

J (fu,fy,λ) − 1 ≤ k max
{

1
J (u,y,λ) − 1, 1

J (u,u,λ) − 1, 1
J (y,y,λ) − 1,

1
J (u,y,λ) − 1, 1

J (y,u,λ) − 1 }
= k

(
1

J (u,y,λ) − 1
)
,

K(u, y, λ) = K(fu, fy, λ) ≤ k max {K(u, y, λ),K(u, u, λ),K(y, y, λ),
K(u, y, λ),K(y, u, λ) }

= k K(y, u, λ),
and

L(u, y, λ) = L(fu, fy, λ) ≤ k max {L(u, y, λ),L(u, u, λ),L(y, y, λ),
L(u, y, λ),L(y, u, λ) }

= k L(y, u, λ).



A. Bataihah, A. Hazaymeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5785 11 of 15

So,

(1− k)

(
1

H(u, y, λ)
− 1

)
≤ 0,

(1− k)

(
1

J (u, y, λ)
− 1

)
≤ 0,

(1− k) K(u, y, λ) ≤ 0,

and
(1− k) L(u, y, λ) ≤ 0.

Which implies u = y.

Corollary 1. Let (X ,H,J ,K,L,⊙,⊕) be a complete NFMS, Suppose that there is k ∈
[0, 1) such that f : X → X satisfies the following for each ξ, ω ∈ X and each λ > 0, we
have:

1

H(fξ, fω, λ)
− 1 ≤ k

(
1

H(ξ, ω, λ)
− 1

)
,

1

J (fξ, fω, λ)
− 1 ≤ k

(
1

J (ξ, ω, λ)
− 1

)
,

K(fξ, fω, λ) ≤ k K(ξ, ω, λ),

and
L(fξ, fω, λ) ≤ k L(ξ, ω, λ).

Furthermore, consider that one of the following statements holds true.

(i) f is continuous,

(ii) The fuzzy sets H,J ,K, and L exhibit continuity with respect to their first two coor-
dinates.

Consequently, the function f has a unique fixed point.

Corollary 2. Let (X ,H,J ,K,L,⊙,⊕) be a complete NFMS, Suppose that there is k ∈
[0, 1) such that f : X → X satisfies the following for each ξ, ω ∈ X and each λ > 0, we
have:

1

H(fξ, fω, λ)
− 1 ≤ k

2

(
1

H(ξ, fξ, λ)
− 1 +

1

H(ω, fω, λ)
− 1

)
,

1

J (fξ, fω, λ)
− 1 ≤ k

2

(
1

J (ξ, fξ, λ)
− 1 +

1

J (ω, fω, λ)
− 1

)
,

K(fξ, fω, λ) ≤ k

2
(K(ξ, fξ, λ) +K(ω, fω, λ)),



A. Bataihah, A. Hazaymeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5785 12 of 15

and

L(fξ, fω, λ) ≤ k

2
L(ξ, fξ, λ) + L(ω, fω, λ)).

Furthermore, consider that one of the following statements holds true.

(i) f is continuous,

(ii) The fuzzy sets H,J ,K, and L exhibit continuity with respect to their first two coor-
dinates.

Consequently, the function f has a unique fixed point.

Corollary 3. Let (X ,H,J ,K,L,⊙,⊕) be a complete NFMS, Suppose that there is k ∈
[0, 1) such that f : X → X satisfies the following for each ξ, ω ∈ X and each λ > 0, we
have:

1

H(fξ, fω, λ)
− 1 ≤ k

2

(
1

H(ξ, fω, λ)
− 1 +

1

H(ξ, fω, λ)
− 1

)
,

1

J (fξ, fω, λ)
− 1 ≤ k

2

(
1

J (ξ, fω, λ)
− 1 +

1

J (ξ, fω, λ)
− 1

)
,

K(fξ, fω, λ) ≤ k

2
(K(ξ, fω, λ) +K(ω, fξ, λ)),

and

L(fξ, fω, λ) ≤ k

2
(L(ξ, fω, λ) + L(ω, fξ, λ)).

Furthermore, consider that one of the following statements holds true.

(i) f is continuous,

(ii) The fuzzy sets H,J ,K, and L exhibit continuity with respect to their first two coor-
dinates.

Consequently, the function f has a unique fixed point.

4. Conclusion

The theory of fixed points stands as a cornerstone in both applied and pure mathemat-
ics, boasting a diverse array of applications across numerous fields. In this exploration, we
present several fixed point theorems pertaining to neutrosophic fuzzy quasi-contractions,
elegantly situated within the sophisticated framework of neutrosophic fuzzy metric spaces.
Furthermore, we unveil a multitude of fixed point results that are pertinent to this in-
triguing domain of study. Future research could extend these findings to various distance
spaces, such as neutrosophic 2-metric spaces, and explore potential applications related to
our work.



A. Bataihah, A. Hazaymeh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5785 13 of 15

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