EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6931
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fixed Point Results in Complex-Valued Neutrosophic
Metric Spaces with Application to Integral Equations

Syed Muhammad Umair Ud-din1, Umar Ishtiaq2, Hafiz Fukhar Ud-din1,
Ibrahim Alraddadi3,∗, Dragan Pamucar4

1 Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100,
Pakistan
2 Office of Research Innovation and Commercialization, University of Management and
Technology, Lahore 54770, Pakistan
3 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah,
Saudi Arabia
4 Széchenyi István University, Győr, Hungary

Abstract. This work presents the notion of complex-valued neutrosophic metric spaces (CVN-
MSs) and provides a fresh mathematical framework extending conventional fuzzy and intuitionistic
fuzzy metric spaces (IFMSs). This new method is especially appropriate for studying complicated
mathematical structures since it helps to better depict uncertainty and imprecision by including
neutrosophic sets. We determine the existence and uniqueness of fixed points under several con-
tractive mappings in this newly defined metric space. Our results extend the classical Banach
contraction ideas and modify them for the neutrosophic environment. We demonstrate several
fixed-point theorems, expanding the applicability of current fixed-point results in non-classical
metric spaces. We use our results to solve Fredholm integral equations and show the efficiency of
CVNMSs in tackling real-world mathematical problems by illustrating the pragmatic relevance of
our results. Furthermore, comprehensive cases are included to show the relevance of our findings.
This work also extends the fixed-point theory in neutrosophic environments and provides fresh
directions for investigation in integral equations and contractive mappings.

2020 Mathematics Subject Classifications: 47H10, 54H25

Key Words and Phrases: Fixed point, neutrosophic sets, existence and uniqueness, metric
spaces, contraction mappings

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6931

Email addresses: m.umairuddin@iub.edu.pk (S. M. U. Ud-din),
hfdin@yahoo.com (H. F. Ud-din), umarishtiaq@umt.edu.pk (U. Ishtiaq),
ialraddadi@iu.edu.sa (I. Alraddadi), pamucar.dragan@sze.hu (D. Pamucar)

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



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 2 of 38

1. Introduction

A neutrosophic metric space expands on the traditional notion of metric spaces by in-
cluding the concept of neutrosophy, which deals with uncertainty and indeterminacy. The
distances between points in a typical metric space are well defined and accurate; however,
in a neutrosophic metric space, the distance between points is specified by three distinct
functions that indicate the degree of truth, indeterminacy, and falsity. This approach
enables more sophisticated modeling of complicated, uncertain, or ambiguous processes in
which information may be inadequate or inconsistent. Essentially, it provides a mathe-
matical structure for analyzing environments where traditional metrics fall short, allowing
more adaptable and resilient approaches to problems in fields such as fuzzy logic, deci-
sion making, and applied research. Neutrosophic metric spaces allow for a more in-depth
examination of phenomena that classical approaches cannot fully represent.

In mathematical analysis, fixed-point theory is a powerful tool with wide-ranging appli-
cations. The most applicable fixed point theorem in metric spaces is the Banach contrac-
tion mapping theorem, which was introduced in [1]. It has been generalized into several
versions of fixed-point theorems and has been advanced through numerous methodolo-
gies. Mathematical models based on classical set theory cannot always capture ambiguous
circumstances in nature or real-world challenges. To overcome this challenge, Zadeh [2]
introduced fuzzy sets to represent an element’s membership by designating an element’s
inclusion in a set by allocating a value within the interval [0, 1]. Afterward, Atanassov [3]
Presented intuitionistic fuzzy sets, which facilitates the representation of level of ambiguity
when determining whether an element in a set is a member or not.

Kramosil and Michalek [4] proposed fuzzy metric spaces (FMSs), which extend prob-
abilistic metric spaces. Grabiec [5] was the first to research the ideology of fuzzy metric
fixed-point theory (FMFPT). By introducing G-completeness and G-Cauchy sequences,
he established a fuzzy counterpart to the Banach contraction principle on FMSs inspired
by [4]. In 1994, George and Veeramani [6] developed a Hausdorff topology for FMSs. Ad-
ditionally, they showed several fixed-point outcomes on the modified spaces and suggested
changes to Grabiec’s Cauchy sequence concept. In 2004, Park [7] proposed the framework
of IFMSs, which increased the scope of fuzzy metrics. Researchers are still delving deeper
into FMFPT. The study mostly goes in two different directions: expanding the scope of
FMSs (comprehensive analysis provided in [8–13]) and examining the existence of fixed
points for mappings adhering to various contractive conditions (for comprehensive infor-
mation, refer to [14–17]. Azam et al. [18] introduced complex-valued metric spaces to
metric fixed-point theory in 2011. Instead of using non-negative real numbers, they used
ordered complex numbers to provide fixed-point results for translations that meet logical
inequality criteria. Shukla et al. [19] used this notion in FMFPT. The authors defined
complex-valued fuzzy metric spaces (CVFMSs) and identified fixed-point transformations
that meet contractive conditions. Umar et al. [20] worked on Some Common Fixed-Point
Theorems in Neutrosophic metric spaces (NMSs). Current research focuses on analyzing
fixed-point translations using CVFMSs. Examples of relevant research include publica-
tions by [21, 22] and Humaira et al. [23–27], which provide practical applications. Kiri̧ sci



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 3 of 38

and Simsek [28] proposed NMSs for membership, nonmembership, and naturalness func-
tions. The authors in [29, 30] and Sowndrarajan et al. [31] demonstrated fixed point
findings in NMSs. Umar et al. [32] worked on fixed point results in orthogonal NMSs. See
[33, 34] for applications and research direction.

This paper introduces CVNMSs (CVNMSs) as a new type of fuzzy metric space. This
novel idea encompasses both CVFMSs by [19] and IFMSs by [7]. We give some fixed-point
outcomes for transformations with contractive constraints in freshly formed spaces. We
apply Banach’s fuzzy variation to intuitionistic fuzzy spaces, resulting in common fixed-
point outcomes in CVFMSs. Our findings are demonstrated through practical examples
and applications.

2. Preliminaries

This section summarizes key concepts in CVFMSs, as defined in previous work by
[19]. Throughout the paper, we denote the set of positive integers by M and the set
of nonnegative integers by M0. We represent any complex number z = c + id by (c, d).
Assume that S = {(c, d) : 0 ≤ c < ∞, 0 ≤ d < ∞} ⊂ C, where C is the set of complex
numbers. We express (0,0),(1,1) and (3,1) in C as ∅,ℑ′ and ℑ respectively. We denote the
closed unit complex interval by T = {(c, d) : 0 ≤ c ≤ 1, 0 ≤ d ≤ 1}, the open unit complex
interval by T0 = {(c, d) : 0 < c < 1, 0 < d < 1}, and S0 = {(c, d) : 0 < c < ∞, 0 < d < ∞}.

Assume that ⪯ is a partial order in C, e2 − e1 ∈ S if and only if e1 ⪯ e2, where
e1, e2 ∈ C. We write e1 ≺ e2 to express Re(e2) > Re(e1) and Im(e2) > Im(e1). It is
obvious that e1 ≺ e2 if and only if e2 − e1 ∈ S0. Assume that {en} is a sequence in C. If
for each n ∈ M, cn+1 ⪯ en or en ⪯ en+1 hold, then {en} is called a monotonic sequence
in relation to ⪯ .

Remark 1. [19] Let en ∈ S for every n ∈ M, all of the following propositions are true:

(1) If {en} is a monotonic sequence with respect to ⪯ and for some x́, ý ∈ S fulfil
x́ ⪯ en ⪯ ý for each n ∈ M, then there exists e ∈ S such that limn→∞ en = e.

(2) ⪯ creates a lattice structure on set of complex numbers C, but does not create a total
ordering on C.

(3) If for each k ∈ L satisfies x́ ⪯ k ⪯ ý for some x́, ý ∈ C, then infimum of L and
supremum of L are exists.

Remark 2. [19] Considering that the following criteria hold for each n ∈ M, en, e
′
n ∈ S,

(1) If for each n ∈ M, we have en ⪯ e′n ⪯ ℑ′ in addition to en → ℑ′ as n approaches to
infinity , it follows that e′n = ℑ′.

(2) If every element en in a sequence satisfies en ⪯ z, and if the sequence en has a limit
e ∈ S, then the limit e also satisfies e ⪯ z.

(3) If every element en in a sequence satisfies z ⪯ en, and if the sequence en has a limit
e ∈ S, then the limit e also satisfies z ⪯ e.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 4 of 38

Definition 1. [19] Assume that V is a set that is not empty. Complex fuzzy set (CFS) E
is defined as the mapping from V to the closed unit complex interval T.

Definition 2. [19] A binary operation ⋆ : T×T → T is called a complex-valued t-norm if
it meets the following conditions:

(1) ∅ ⋆ e = ∅,ℑ′ ⋆ e = e for every e ∈ T;

(2) ⋆ is associative and commutative

(3) e3 ⋆ e4 ⪰ e2 ⋆ e1 given that e3 ⪰ e1, e4 ⪰ e2 for each e1, e2, e3, e4 ∈ T.

Example 1. [19] Suppose that ei = (τi, ξi) ∈ T for i = 1, 2, binary operations ⋆x, ⋆y, ⋆z :
T× T → T are defined below:

(1) e1 ⋆x e2 = (τ1τ2, ξ1ξ2);

(2) e1 ⋆y e2 = (min{τ1, τ2},min{ξ1, ξ2});

(3) e1 ⋆z e2 = (max{τ1 + τ2 − 1, 0},max{ξ1 + ξ2 − 1, 0}).

Therefore, ⋆x, ⋆y, ⋆z are complex-valued t-norms.

Definition 3. [19] Let V ̸= 0, ⋆ be a continuous complex-valued t-norms and E be a CFS
defined on V2 × S0 whereby the criteria following hold:

(1) E(ϱo, ν, e) ≻ ∅;

(2) E(ϱo, ν, e) = ℑ′ for each e ∈ S0 if and only if ϱo = ν;

(3) E(ϱo, ν, e) = E(ν, ϱo, e);

(4) E(ϱo, ς, e+ e′) ⪰ E(ϱo, ν, e) ⋆ E(ν, ς, e′);

(5) E(ϱo, ν, ·) : S0 → T is continuous;

for every ϱo, ν, ς ∈ V and e, e′ ∈ S0.

Then E is called a complex-valued fuzzy metric on V, and (V, E , ⋆) is called a complex-
valued fuzzy metric space. A CVFM E characterizes the degree of nearness between two
points of the set V relative to a complex factor e ∈ S0.

Definition 4. A binary operation △ : T× T → T is called complex-valued t-conorm if it
meets the following conditions:

(1) e△∅ = e, e△ℑ′ = ℑ′ for every e ∈ T;

(2) △ is associative and commutative;

(3) e3 △ e4 ⪰ e1 △ e2 given that e3 ⪰ e1, e4 ⪰ e2 for each e1, e2, e3, e4 ∈ T.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 5 of 38

Example 2. Suppose that ei = (τi, ξi) ∈ T for i = 1, 2, binary operations △x,△y,△z :
T× T → T are defined as below:

(1) e1 △x e2 = (τ1 + τ2, ξ1 + ξ2)− (τ1τ2, ξ1ξ2);

(2) e1 △y e2 = (max{τ1, τ2},max{ξ1, ξ2});

(3) e1 △z e2 = (min{τ1 + τ2, 1},min{ξ1 + ξ2, 1}).

Therefore, △x,△y and △z are complex-valued triangular-conorms.

Remark 3. Both binary operations, triangular-norm, and triangular-conorm, are fre-
quently employed in fuzzy set theory, especially within the contexts of [0, 1] and lattices.
The former denotes the shared area between two fuzzy sets, sometimes expressed as a con-
junction in fuzzy logic. The t-conorm, which is the dual of the t-norm, is seen as the area
where two fuzzy sets meet, or, in other words, a disjunction in fuzzy logic. For additional
insights into the ideas of t-norm and t-conorm are encouraged to refer to [35] and [36].

Definition 5. Assume that V ̸= 0, ⋆ and △ are continuous complex-valued t-norm and
t-conorm, respectively, and E ,G are complex fuzzy sets (CFSs) defined on V2×S0 in which
the following conditions are satisfied:

(1) E(ϱo, ν, e) + G(ϱo, ν, e) ⪯ ℑ′;

(2) E(ϱo, ν, e) ≻ ∅;

(3) E(ϱo, ν, e) = ℑ′ for each e ∈ S0 if and only if ϱo = ν;

(4) E(ϱo, ν, e) = E(ν, ϱo, e);

(5) E(ϱo, ς, e+ e′) ⪰ E(ϱo, ν, e) ⋆ E(ν, ς, e′);

(6) E(ϱo, ν, ·) : S0 → T is continuous;

(7) G(ϱo, ν, e) ≺ ℑ′;

(8) G(ϱo, ν, e) = ∅ for each e ∈ S0 if and only if ϱo = ν;

(9) G(ϱo, ν, e) = G(ν, ϱo, e);

(10) G(ϱo, ς, e+ e′) ⪯ G(ϱo, ν, e)△G(ν, ς, e′);

(11) G(ϱo, ν, ·) : S0 → T is continuous;

for every ϱo, ν, ς ∈ V and e, e′ ∈ S0.

The pair (E ,G) is called complex-valued intuitionistic fuzzy metric on V and (V, E ,G, ⋆,△)
is called complex-valued intuitionistic fuzzy metric space. The pair (E ,G) denotes the de-
gree of nearness and the degree of non-nearness between two points of the set V with
respect to a complex parameter e ∈ S0.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 6 of 38

3. Main Results

In this section, we introduce the concept of complex-valued neutrosophic metric space
(CVNMS).

Definition 6. [28] Assume that V ̸= 0, and let ⋆ and △ represent continuous triangular
norm and continuous triangular conorm, respectively, and E ,G,H are the neutrosophic
sets defined on V× V× (0,∞) in which the following conditions are satisfied:

(1) E(ϱo, ν, e) + G(ϱo, ν, e) +H(ϱo, ν, e) ≤ 3;

(2) E(ϱo, ν, e) > 0;

(3) E(ϱo, ν, e) = 1 if and only if ϱo = ν;

(4) E(ϱo, ν, e) = E(ν, ϱo, e);

(5) E(ϱo, ς, e+ e′) ≥ E(ϱo, ν, e) ⋆ E(ν, ς, e′);

(6) E(ϱo, ν,△) is a non decreasing function of R+ and lime→∞ E(ϱo, ν, e) = 1;

(7) G(ϱo, ν, e) < 1;

(8) G(ϱo, ν, e) = 0 if and only if ϱo = ν;

(9) G(ϱo, ν, e) = G(ν, ϱo, e);

(10) G(ϱo, ς, e+ e′) ≤ G(ϱo, ν, e)△G(ν, ς, e′);

(11) G(ϱo, ν,△) is a non increasing function of R+ and lime→∞ G(ϱo, ν, e) = 0;

(12) H(ϱo, ν, e) < 1;

(13) H(ϱo, ν, e) = 0 if and only if ϱo = ν;

(14) H(ϱo, ν, e) = H(ν, ϱo, e);

(15) H(ϱo, ς, e+ e′) ⪯ H(ϱo, ν, c)△H(ν, ς, e′);

(16) H(ϱo, ν,△) is a non increasing function of R+ and lime→∞H(ϱo, ν, e) = 0;

(17) If e ≤ 0 then E(ϱo, ν, e) = 0,G(ϱo, ν, e) = 1 and H(ϱo, ν, e) = 1;

for every ϱo, ν, ς ∈ V and e, e′ ∈ (0,∞).

The triplet (E ,G,H) is called neutrosophic metric on V and (V, E ,G,H, ⋆,△) is called
CVNMS. The triplet (E ,G,H) indicates the closeness degree, the non-closeness degree,
and the neutralness degree between two points of the set V with respect to a parameter
e ∈ (0,∞).



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 7 of 38

Definition 7. Assume that V ̸= 0, ⋆ and △ represent continuous complex-valued triangu-
lar norm and continuous complex-valued triangular conorm, respectively, and E ,G,H are
CFSs defined on V2 × S0 in which the following conditions are satisfied:

(1) E(ϱo, ν, e) + G(ϱo, ν, e) +H(ϱo, ν, e) ⪯ ℑ;

(2) E(ϱo, ν, e) ≻ ∅;

(3) E(ϱo, ν, e) = ℑ for each e ∈ S0 if and only if ϱo = ν;

(4) E(ϱo, ν, e) = E(ν, ϱo, e);

(5) E(ϱo, ς, e+ e′) ⪰ E(ϱo, ν, e) ⋆ E(ν, ς, c′);

(6) E(ϱo, ν, ·) : S0 → T is continuous;

(7) G(ϱo, ν, e) ≺ ℑ;

(8) G(ϱo, ν, e) = ∅ for each e ∈ S0 if and only if ϱo = ν;

(9) G(ϱo, ν, e) = G(ν, ϱo, e);

(10) G(ϱo, ς, e+ e′) ⪯ G(ϱo, ν, e)△G(ν, ς, e′);

(11) G(ϱo, ν, ·) : S0 → T is continuous;

(12) H(ϱo, ν, e) ≺ ℑ;

(13) H(ϱo, ν, e) = ∅ for each e ∈ S0 if and only if ϱo = ν;

(14) H(ϱo, ν, e) = H(ν, ϱo, e);

(15) H(ϱo, ς, e+ e′) ⪯ H(ϱo, ν, e)△H(ν, ς, e′);

(16) H(ϱo, ν, ·) : S0 → T is continuous;

for every ϱo, ν, ς ∈ V and e, e′ ∈ S0.

Then the triplet (E ,G,H) is called complex-valued neutrosophic metric on V and
(V, E ,G,H, ⋆,△) is called CVNMS. The triplet (E ,G,H) characterizes the degree of near-
ness, the degree of non-nearness , and the neutralness degree between two points of the
set V relative to a complex parameter e ∈ S0.

Example 3. Consider the metric space (V, d). Two binary operations, ⋆y and △y, are
defined for ei = (τi, ξi) ∈ T, where i=1,2, as follows:

e1 ⋆y e2 = (min{τ1, τ2},min{ξ1, ξ2}) and e1 △y e2 = (max{τ1, τ2},max{ξ1, ξ2}).

The CFSs E ,G, and H are defined as follows:

E(ϱo, ν, e) = τ + ξ

τ + ξ + d(ϱo, ν)
ℑ, G(ϱo, ν, e) = d(ϱo, ν)

τ + ξ + d(ϱo, ν)
ℑ, H(ϱo, ν, e) =

d(ϱo, ν)

τ + ξ
ℑ

for each ϱo, ν ∈ V and e = (τ, ξ) ∈ S0. Consequently, (V, E ,G,H, ⋆y,△y) is a CVNMS.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 8 of 38

Lemma 1. Given that (V, E ,G,H, ⋆,△) is a CVNMS, E(ϱo, ν, ·) is non-decreasing, G(ϱo, ν, ·)
is non-increasing, and H(ϱo, ν, ·) is non-increasing, that is, for any e, e′ ∈ S0 with e ≺ e′, it
follows that E(ϱo, ν, e) ⪯ E(ϱo, ν, e′),G(ϱo, ν, e) ⪰ G(ϱo, ν, e′) and H(ϱo, ν, e) ⪰ H(ϱo, ν, e′)
for each ϱo, ν ∈ V.
Proof. Consider e, e′ ∈ S0 where e ≺ e′, this implies that e′ − e ∈ S0. By using condition
(5) from Definition 7, we obtain

E(ϱo, ν, e′) = E(ϱo, ν, e′ − e+ e)

⪰ E(ϱo, ϱo, e′ − e) ⋆ E(ϱo, ν, e)
= ℑ ⋆ E(ϱo, ν, e)
= E(ϱo, ν, e).

Hence E(ϱo, ν, e′) ⪰ E(ϱo, ν, e). Alternatively by using condition (10) from Definition 7, we
have

G(ϱo, ν, e′) = G(ϱo, ν, e′ − e+ e)

⪯ G(ϱo, ϱo, e′ − e)△ G(ϱo, ν, e)
= ∅△G(ϱo, ν, e)
= G(ϱo, ν, e).

Hence G(ϱo, ν, e′) ⪯ G(ϱo, ν, e). Similarly by using condition (15) from Definition 7, we
have

H(ϱo, ν, e′) = H(ϱo, ν, e′ − e+ e)

⪯ H(ϱo, ϱo, e′ − e)△H(ϱo, ν, e)

= ∅△H(ϱo, ν, e)

= H(ϱo, ν, e).

Hence H(ϱo, ν, e′) ⪯ H(ϱo, ν, e).

Definition 8. Let (V, E ,G,H, ⋆,△) be a CVNMS. A sequence {ϱon} in V converges to
ϱo ∈ V provided that all r ∈ T0 as well as e ∈ S0, for some n0 ∈ M if it meets the following
criteria:

E(ϱon, ϱo, e) ≻ ℑ− r, G(ϱon, ϱo, e) ≺ r and H(ϱon, ϱ
o, e) ≺ r for each n > n0.

Definition 9. Let (V, E ,G,H, ⋆,△) be a CVNMS. A cauchy sequence is a sequence {ϱon}
in V that satisfies the following criteria:

lim
n→∞

inf
m>n

E(ϱon, ϱom, e) = ℑ,

lim
n→∞

sup
m>n

G(ϱon, ϱom, e) = ∅,

lim
n→∞

sup
m>n

H(ϱon, ϱ
o
m, e) = ∅,

for each e ∈ S0.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 9 of 38

A CVNMS is said to be complete if every Cauchy sequence in V converges.
The examples below help to clarify the ideas covered in Definitions 8 and 9.

Example 4. Observe the CVNMS, denoted as (V, E ,G,H, ⋆y,△y) in example 3. Further-
more, set V = [4, 5] and define d as d(ϱo, ν) = |ϱo − ν| for all ϱo, ν ∈ V. Let the sequence
{ϱon} = {4 + 1

n} and ϱo = 4.
Now we verify that E(ϱon, ϱo, e) ≻ ℑ− s for each s = (s1, s2) ∈ T0 and e ∈ S0. For the real
part,

Re(E(ϱon, ϱo, e)−ℑ+ s) =
τ + ξ

τ + ξ + d(ϱon, ϱ
o)

− 1 + s1

=
τ + ξ

τ + ξ + |4 + 1
n − 4|

− 1 + s1

=
τ + ξ

τ + ξ + 1
n

− 1 + s1.

As n → ∞, then Re(E(ϱon, ϱo, e) − ℑ + s) → s1. Therefore, for each s ∈ T0 together
with e ∈ S0, there is always an M1 ∈ M in which Re(E(ϱon, ϱo, e) − ℑ + s) > 0 holds for
all n > M1. The approach for determining the imaginary part is the same, leading to
Im(E(ϱon, ϱo, e)− ℑ + s) → s2 as n → ∞. Therefore, for every s ∈ T0 and e ∈ S0, always
there is a M2 ∈ M such that Im(E(ϱon, ϱo, e)) > 0 holds for all n > M2. Therefore, for
every s ∈ T0 and e ∈ S0, by taking n0 = max{M1,M2}, we establish E(ϱon, ϱo, e) > ℑ − s
foe each n > n0.
Now we verify that G(ϱon, ϱo, e) < s for every (s1, s2) ∈ T0 and e ∈ S0. For the real part,

Re(s− G(ϱon, ϱo, e)) = s1 −
d(ϱon, ϱ

o)

τ + ξ + d(ϱon, ϱ
o)

= s1 −
|4 + 1

n − 4|
τ + ξ + |4 + 1

n − 4|

= s1 −
1
n

τ + ξ + 1
n

.

As n → ∞, then Re(s − G(ϱon, ϱo, e)) → s1. Therefore, for each s ∈ T0 and e ∈ S0,
there is always an M1 ∈ M in which Re(G(ϱon, ϱo, e) − ℑ + s) > 0 holds for each n >
M1. The approach for determining the imaginary part is the same, this leads to Im(s −
G(ϱon, ϱo, e)) → s2 as n → ∞. Therefore, for each r ∈ T0 and e ∈ S0, there is always an
M2 ∈ M in which Im(s − G(ϱon, ϱo, e)) > 0 holds for all n > M2. Therefore, for every
s ∈ T0 and e ∈ S0, by taking n0 = max{M1,M2}, we establish G(ϱon, ϱo, e) < s for each
n > n0.

Now we verify that H(ϱon, ϱ
o, e) < s for every (s1, s2) ∈ T0 and e ∈ S0. For the real

part,

Re(s−H(ϱon, ϱ
o, e)) = s1 −

d(ϱon, ϱ
o)

τ + ξ



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 10 of 38

= s1 −
|4 + 1

n − 4|
τ + ξ

= s1 −
1
n

τ + ξ
.

As n → ∞, then Re(s − H(ϱon, ϱ
o, e)) → s1. Therefore, for each s ∈ T0 and e ∈ S0,

there is always an M1 ∈ M in which Re(H(ϱon, ϱ
o, e) − ℑ + s) > 0 holds for each n >

M1. The approach for determining the imaginary part is the same, this leads to Im(s −
H(ϱon, ϱ

o, e)) → s2 as n → ∞. Therefore, for each s ∈ T0 and e ∈ S0, there is always an
M2 ∈ M in which Im(s − H(ϱon, ϱ

o, e)) > 0 holds for all n > M2. Therefore, for every
s ∈ T0 and e ∈ S0, by taking n0 = max{M1,M2}, we establish H(ϱon, ϱ

o, e) < s for each
n > n0. Every requirement of Definition 8 is fulfilled. We may therefore say that {4 + 1

n}
converges to 4.

Example 5. Using the same conditions as the previous example, we will demonstrate that
{4 + 1

n} is a Cauchy sequence. For all e ∈ S for any n,m ∈ M where m > n,

E(ϱon, ϱom, e) =
τ + ξ

τ + ξ + d(ϱon, ϱ
o
m)

ℑ

=
τ + ξ

τ + ξ + |4 + 1
n − (4 + 1

m)|
ℑ

=
τ + ξ

τ + ξ + | 1n − 1
m |

ℑ,

G(ϱon, ϱom, e) =
d(ϱon, ϱ

o
m)

τ + ξ + d(ϱon, ϱ
o
m)

ℑ

=
|4 + 1

n − (4 + 1
m)|

τ + ξ + |4 + 1
n − (4 + 1

m)|
ℑ

=
| 1n − 1

m |
τ + ξ + | 1n − 1

m |
ℑ

And

H(ϱon, ϱ
o
m, e) =

d(ϱon, ϱ
o
m)

τ + ξ
ℑ

=
|4 + 1

n − (4 + 1
m)|

τ + ξ
ℑ

=
| 1n − 1

m |
τ + ξ

ℑ.

As m,n → ∞, we observe that E(ϱon, ϱom, e) → ℑ,G(ϱon, ϱom, e) → ∅ and H(ϱon, ϱ
o
m, e) →

∅, which leads to limn→∞ infm>n E(ϱon, ϱom, e) = ℑ, limn→∞ supm>n G(ϱon, ϱom, e) = ∅ and
limn→∞ supm>nH(ϱon, ϱ

o
m, e) = ∅. Hence show that 4 + 1

n is a Cauchy sequence.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 11 of 38

Lemma 2. Let (V, E ,G,H, ⋆,△) be a CVNMS. The sequence {ϱon} ∈ V converges to ϱo ∈ V
if and only if limn→∞ E(ϱon, ϱo, e) = ℑ, limn→∞ G(ϱon, ϱo, e) = ∅ and limn→∞H(ϱon, ϱ

o, e) =
∅ are satisfied for each e ∈ S0.
Proof. Assume that limn→∞ E(ϱon, ϱo, e) = ℑ, limn→∞ G(ϱon, ϱo, e) = ∅ and limn→∞H(ϱon, ϱ

o, e) =
∅ for every e ∈ S0. Assume that e be a fixed element from S0. It is feasible to identify a
real number ϵ > 0 such that z ≺ r for all z ∈ C and |z| < ϵ for any r ∈ T0. Considering
this particular ϵ, we may find a n0 ∈ M such that

|ℑ − E(ϱon, ϱo, e)| < ϵ, |G(ϱon, ϱo, e)| < ϵ and|H(ϱon, ϱ
o, e)| < ϵ for each n ∈ n0.

These two inequalities indicate that

ℑ− E(ϱon, ϱo, e) ≺ s

−E(ϱon, ϱo, e) ≺ s−ℑ

E(ϱon, ϱo, e) ≻ ℑ− s

as well as
G(ϱon, ϱo, e) ≺ s

also
H(ϱon, ϱ

o, e) ≺ s

for all n > n0 respectively. Consequently, {ϱon} is convergent to ϱo ∈ V. Conversely,
assume that e ∈ S0 is fixed and a real value ϵ > 0 is specified. Assume that a sequence
{ϱon} is converges to ϱo ∈ V that is, For each element s in the set T0 and every element no

in the set M, it is possible to select a value such that E(ϱon, ϱo, e) > ℑ − s,G(ϱon, ϱo, e) < s
and H(ϱon, ϱ

o, e) < s for each n > n0. A complex number s is selected from the set T0 such
that |s| < ϵ. Therefore

|ℑ − E(ϱon, ϱo, e)| < |s|, |G(ϱon, ϱo, e)| < |s| < ϵ and |H(ϱon, ϱ
o, e)| < |s| < ϵ for any n > n0.

Therefore, limn→∞ E(ϱon, ϱo, e) = ℑ, limn→∞ G(ϱon, ϱo, e) = ∅ and limn→∞H(ϱon, ϱ
o, e) =

∅ is satisfied for every e ∈ S0.

Lemma 3. Let (V, E ,G,H, ⋆,△) be a CVNMS. A sequence {ϱon} ∈ V is classified as a
cauchy sequence if and only if for every s ∈ T0 and e ∈ S0, one can find n0 ∈ M satisfying

E(ϱon, ϱo, e) ≻ ℑ− s, G(ϱon, ϱo, e) ≺ s and H(ϱon, ϱ
o, e) ≺ s for each n,m > n0.

Proof. Assume that {ϱon} is Cauchy sequence. Suppose e is a fixed element from S0,
Then, for each s ∈ T0, it is possible to identify n0 ∈ M that satisfies the conditions
ℑ − infm>n E(ϱon, ϱom, e) ≺ s, supm>n G(ϱon, ϱom, e) ≺ s, and supm>nH(ϱon, ϱ

o
m, e) ≺ s for

each n > n0. Here, we look at three situations. For the situation where m > n > n0, this
leads to ℑ−s ≺ infm>n E(ϱon, ϱom, e) ≺ E(ϱon, ϱom, e), G(ϱon, ϱom, e) ≺ supm>n G(ϱon, ϱom, e) ≺ s
and H(ϱon, ϱ

o
m, e) ≺ supm>nH(ϱon, ϱ

o
m, e) ≺. Now if m = n > n0, then ℑ − s ≺ ℑ =

E(ϱon, ϱom, e), G(ϱon, ϱom, e) = ∅ ≺ s and H(ϱon, ϱ
o
m, e) = ∅ ≺ s. Finally, but not least, given



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 12 of 38

the situation when n > m > n0, it follows that ℑ−s ≺ infn>m E(ϱom, ϱon, e) ⪯ E(ϱom, ϱon, e) =
E(ϱon, ϱom, e), G(ϱom, ϱon, e) = G(ϱon, ϱom, e) ⪯ supn>m G(ϱon, ϱom, e) ≺ s and H(ϱom, ϱon, e) =
H(ϱon, ϱ

o
m, e) ⪯ supn>mH(ϱon, ϱ

o
m, e) ≺ s. Thus, we deduce that E(ϱon, ϱom, e) ≻ ℑ−s, G(ϱon, ϱom, e) ≺

s as well as H(ϱon, ϱ
o
m, e) ≺ s for any n,m > n0. Conversely, let e ∈ S be fixed, and a

real number ϵ > 0 be given. Suppose that for each s ∈ T0, one can identify an n0 ∈ M in
which E(ϱon, ϱom, e) > ℑ − s, G(ϱon, ϱom, e) ≺ s and H(ϱon, ϱ

o
m, e) ≺ s for each n > m > n0.

Consequently
ℑ− 2s ≺ ℑ− s ⪯ inf

m>n
E(ϱon, ϱom, e)

sup
m>n

G(ϱon, ϱom, e) ⪯ s ≺ 2s

and
sup
m>n

H(ϱon, ϱ
o
m, e) ⪯ s ≺ 2s

for every n > n0. Select a complex number s ∈ T0 that fulfils |s| < ϵ
2 , then we get

|ℑ− inf
m>n

E(ϱon, ϱom, e)| < 2|s| < ϵ, | sup
m>n

G(ϱon, ϱom, e)| < 2|s| < ϵ and | sup
m>n

H(ϱon, ϱ
o
m, e)| < 2|s| < ϵ for each n > n0.

Thus, we have limn→∞ infm>n E(ϱon, ϱom, e) = ℑ, limn→∞ supm>n G(ϱon, ϱom, e) = ∅, and
limn→∞ supm>nH(ϱon, ϱ

o
m, e) = ∅, This demonstrates that the sequence {ϱo}n is Cauchy.

4. Fixed-point Results

We will now examine the existence and uniqueness of fixed points for self-mappings
that satisfy the specified contractive requirements in CVNMS. Let {en} be a sequence
from C. It is said that limn→∞ en = ∞ = (∞,∞) if for every c ∈ C, there is a n0 ∈ M
such that en ⪰ e for all n > n0

Theorem 1. Let (V, E ,G,H, ⋆,△) be a complete CVNMS with the characteristic that any
sequence {en} ∈ S0 fulfills limn→∞ en = ∞ implies

lim
n→∞

inf
ν∈V

E(ϱo, ν, en) = ℑ, lim
n→∞

sup
ν∈V

G(ϱo, ν, en) = ∅, lim
n→∞

sup
ν∈V

H(ϱo, ν, en) = ∅,

for any ϱo ∈ V. Consider a self-mapping h : V → V meets the following condition:

E(hϱo, hν, ke) ⪰ E(ϱo, ν, e), G(hϱo, hν, ke) ⪯ G(ϱo, ν, e) and H(hϱo, hν, ke) ⪯ H(ϱo, ν, e)
(1)

for each ϱo, ν ∈ V and e ∈ S0, where k ∈ (0, 1). Then, there is a unique fixed point of the
mapping h that is located in V.
Proof. Let ϱo ∈ V be an arbitrarily chosen point. In V, a sequence {ϱon} is defined by
ϱon = hϱon−1 for all n ∈ M. The existence of a n0 ∈ M such that ϱon0

= ϱon0−1 guarantees
that ϱon0

is a fixed point of h. To prove that the sequence {ϱon} is Cauchy, we see that
ϱon ̸= ϱon−1 for each n ∈ M.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 13 of 38

For every n ∈ M and a fixed e ∈ S0, we define

An := {E(ϱon, ϱom, e) : m > n} ⊂ T,

Bn := {G(ϱon, ϱom, e) : m > n} ⊂ T,

Cn := {H(ϱon, ϱ
o
m, e) : m > n} ⊂ T.

As ∅ < E(ϱon, ϱom, e) ⪯ ℑ for every n ∈ M and n < m, subsequent to the remarks 1, the
greatest lower bound of the set An, denoted as inf An = x́n, exist for every n ∈ M. In the
same way, since ∅ ⪯ G(ϱon, ϱom, e) ≺ ℑ for all n ∈ M such that n < m, subsequent to the
remarks 1, the least upper bound of the set Bn, denoted as supBn = ýn, exists for every
n ∈ M. and also, since ∅ ⪯ H(ϱon, ϱ

o
m, c) ≺ ℑ for all n ∈ M where n < m, subsequent to

the remarks 1, The supremum of the set Cn, denoted as supCn = ćn, exists for every n
belonging to the set M. For c ∈ S0 and n,m ∈ M where m > n, using equation (1), we
obtain

E(ϱon+1, ϱ
o
m+1, e) = E(hϱon+1, hϱ

o
m, e) ⪰ E

(
ϱon, ϱ

o
m,

e

k

)
(2)

G(ϱon+1, ϱ
o
m+1, e) = G(hϱon, hϱom, e) ⪯ G

(
ϱon, ϱ

o
m,

e

k

)
(3)

and
H(ϱon+1, ϱ

o
m+1, e) = H(hϱon, hϱ

o
m, e) ⪯ H

(
ϱon, ϱ

o
m,

e

k

)
(4)

Since k ∈ (0, 1), according to Lemma 1, this implies that

E
(
ϱon, ϱ

o
m,

e

k

)
⪰ E(ϱon, ϱom, e), G

(
ϱon, ϱ

o
m,

e

k

)
⪯ G(ϱon, ϱom, e) and H

(
ϱon, ϱ

o
m,

e

k

)
⪯ H(ϱon, ϱ

o
m, e).

This ultimately results in

E(ϱon+1, ϱ
o
m+1, e) ⪰ E(ϱon, ϱom, e)

G(ϱon+1, ϱ
o
m+1, e) ⪯ G(ϱon, ϱom, e)

and
H(ϱon+1, ϱ

o
m+1, e) ⪯ H(ϱon, ϱ

o
m, e)

for every n,m ∈ M and m > n.After verifying the inf(E), sup(G) and sup(H) above, we
may conclude that

∅ ⪯ x́n ⪯ x́n+1 ⪯ ℑ, ∅ ⪯ ýn+1 ⪯ ýn ⪯ ℑ, ∅ ⪯ ćn+1 ⪯ ćn ⪯ ℑ

for any n ∈ M. Thus {x́n}, {ýn} and {ćn} are monotonic sequences in S. According to
Remarks 1, there are complex numbers x́, ý, ć ∈ S satisfying limn→∞ x́n = x́, limn→∞ ýn =
ý and limn→∞ ćn = ć. By using equations (2), (3), and (4), we have

x́n+1 = inf
m>n

E(ϱon+1, ϱ
o
m+1, e) ⪰ inf

m>n
E
(
ϱon, ϱ

o
m,

e

k

)



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 14 of 38

ýn+1 = sup
m>n

G(ϱon+1, ϱ
o
m+1, e) ⪯ sup

m>n
G
(
ϱon, ϱ

o
m,

e

k

)
and

ćn+1 = sup
m>n

H(ϱon+1, ϱ
o
m+1, e) ⪯ sup

m>n
H

(
ϱon, ϱ

o
m,

e

k

)
for c ∈ S0 and n ∈ M. By successively applying equation (1) to the inequalities mentioned
above, we obtain

x́n+1 ⪰ inf
m>n

E(ϱon, ϱom,
e

k
)

⪰ inf
m>n

E(ϱon−1, ϱ
o
m−1,

e

k2
)

⪰ inf
m>n

E(ϱon−2, ϱ
o
m−2,

e

k3
)

.

.

.

⪰ inf
m>n

E(ϱo0, ϱom−n,
e

kn+1
)

ýn+1 ⪯ sup
m>n

G(ϱon, ϱom,
e

k
)

⪯ sup
m>n

G(ϱon−1, ϱ
o
m−1,

e

k2
)

⪯ sup
m>n

G(ϱon−2, ϱ
o
m−2,

e

k3
)

.

.

.

⪯ sup
m>n

G(ϱo0, ϱom−n,
e

kn+1
)

and

ćn+1 ⪯ sup
m>n

H(ϱon, ϱ
o
m,

e

k
)

⪯ sup
m>n

H(ϱon−1, ϱ
o
m−1,

e

k2
)

⪯ sup
m>n

H(ϱon−2, ϱ
o
m−2,

e

k3
)

.

.

.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 15 of 38

⪯ sup
m>n

H(ϱo0, ϱ
o
m−n,

e

kn+1
)

for any e ∈ S0 and n ∈ M. In addition, we obtain

x́n+1 ⪰ inf
m>n

E
(
ϱo0, ϱ

o
m−n,

e

kn+1

)
⪰ inf

ν∈V
E
(
ϱo0, ν,

e

kn+1

)
ýn+1 ⪯ sup

m>n
G
(
ϱo0, ϱ

o
m−n,

e

kn+1

)
⪯ sup

ν∈V
G
(
ϱo0, ν,

e

kn+1

)
and

ćn+1 ⪯ sup
m>n

H
(
ϱo0, ϱ

o
m−n,

e

kn+1

)
⪯ sup

ν∈V
H

(
ϱo0, ν,

e

kn+1

)
for each e ∈ S0 and n ∈ M. As n → ∞ on both sides of the given inequality, the hypothesis
yields

x́ = lim
n→∞

x́n+1 ⪰ lim
n→∞

inf
ν∈V

E
(
ϱo0, ν,

e

kn+1

)
= ℑ

ý = lim
n→∞

ýn+1 ⪯ lim
n→∞

sup
ν∈V

G
(
ϱo0, ν,

e

kn+1

)
= ∅

and
ć = lim

n→∞
ćn+1 ⪯ lim

n→∞
sup
ν∈V

H
(
ϱo0, ν,

e

kn+1

)
= ∅

Which imply x́ = ℑ, ý = ∅ and ć = ∅. Thus,

lim
n→∞

inf
m>n

E(ϱon+1, ϱ
o
m+1, e) = lim

n→∞
x́n = ℑ,

lim
n→∞

sup
m>n

G(ϱon+1, ϱ
o
m+1, e) = lim

n→∞
ýn = ∅,

lim
n→∞

sup
m>n

H(ϱon+1, ϱ
o
m+1, e) = lim

n→∞
ćn = ∅,

for all e ∈ S0 which show sequence {ϱon} is Cauchy. Given that (V, E ,G,H, ⋆,△) is com-
plete, 2 implies the existence of ϱo ∈ V satisfying

lim
n→∞

E(ϱon, ϱo, e) = ℑ, lim
n→∞

G(ϱon, ϱom, e) = ∅ and lim
n→∞

H(ϱon, ϱ
o
m, e) = ∅ for any e ∈ S0

(5)
As a result of equation (1), and conditions (5), (10), (15) of Definition 7 for any e ∈ S0,
lead us to the conclusion that

E(ϱo, hϱo, e) ⪰ E
(
ϱo, ϱon+1,

e

2

)
∗ E

(
ϱon+1, hϱ

o,
e

2

)
= E

(
ϱo, ϱon+1,

c

2

)
∗ E

(
hϱon, hϱ

o,
c

2

)
⪰ E

(
ϱo, ϱon+1,

e

2

)
∗ E

(
ϱon, ϱ

o,
e

2k

)
,

G(ϱo, hϱo, e) ⪯ G
(
ϱo, ϱon+1,

e

2

)
△G

(
ϱon+1, hϱ

o,
e

2

)



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 16 of 38

= G
(
ϱo, ϱon+1,

e

2

)
△G

(
hϱon, hϱ

o,
e

2

)
⪯ G

(
ϱo, ϱon+1,

e

2

)
△G

(
ϱon, ϱ

o,
e

2k

)
,

and

H(ϱo, hϱo, e) ⪯ H
(
ϱo, ϱon+1,

e

2

)
△H

(
ϱon+1, hϱ

o,
e

2

)
= H

(
ϱo, ϱon+1,

e

2

)
△H

(
hϱon, hϱ

o,
e

2

)
⪯ H

(
ϱo, ϱon+1,

e

2

)
△H

(
ϱon, ϱ

o,
e

2k

)
.

Now considering the limit as n → ∞ for both given inequalities, utilizing equation (5)
together with Remarks 2, for any e ∈ S0, it follows that

E(ϱo, hϱo, e) = ℑ, G(ϱo, hϱo, e) = ∅ and H(ϱo, hϱo, e) = ∅.

By conditions (3), (8), and (11) of Definition 7, it can be concluded that ϱo is equal to
hϱo, indicating that ϱo is a fixed point of h. To demonstrate uniqueness, let us assume that
z and ϱo are two distinct fixed points of h. This indicates that there are some elements
e′ ∈ S0 for which E(ϱo, z, e′) ̸= ℑ,G(ϱo, z, e′) ̸= ∅ and H(ϱo, z, e′) ̸= ∅. By iteratively
applying equation (1), for every n ∈ M we obtain

E(ϱo, z, e′) = E(hϱo, hz, e′) ⪰ E(ϱo, z, e
′

k
)

⪰ E(ϱo, z, e
′

k2
)

.

.

.

⪰ E(ϱo, z, e
′

kn
)

G(ϱo, z, e′) = G(hϱo, hz, e′) ⪯ G(ϱo, z, e
′

k
)

⪯ G(ϱo, z, e
′

k2
)

.

.

.

⪯ G(ϱo, z, e
′

kn
)



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 17 of 38

and

H(ϱo, z, e′) = ẃ(hϱo, hz, e′) ⪯ H(ϱo, z,
e′

k
)

⪯ H(ϱo, z,
e′

k2
)

.

.

.

⪯ H(ϱo, z,
e′

kn
).

Consequently, we derive that

E(ϱo, z, e′) ⪰ E(ϱo, z, e
′

kn
) ⪰ inf

ν∈V
E(ϱo, z, e

′

kn
)

G(ϱo, z, e′) ⪯ G(ϱo, z, e
′

kn
) ⪯ sup

ν∈V
G(ϱo, z, e

′

kn
)

and

H(ϱo, z, e′) ⪯ H(ϱo, z,
e′

kn
) ⪯ sup

ν∈V
H(ϱo, z,

e′

kn
).

As k ∈ (0, 1), it is obvious that limn→∞
e′

kn = ∞. So by assuming the limit as n approaches
infinity for the above inequalities, it implies

E(ϱo, z, e′) = ℑ, G(ϱo, z, e′) = ∅ and H(ϱo, z, e′) = ∅

which is a contradiction. Thus E(ϱo, z, e) = ℑ,G(ϱo, z, e) = ∅ and H(ϱo, z, e) = ∅ for
each e ∈ S0. The uniqueness of the fixed points of h is confirmed by the fact that ϱo = z,
as deduced from the conditions (3), (8), and (11) of Definitions 7.

Remark 4. The proof of Theorem 1 remains the same by substituting equation (1) with
the subsequently contractive condition of mapping h:

E(hϱo, hν, κ(e)e) ⪰ E(ϱo, ν, e), G(hϱo, hν, κ(e)e) ⪯ G(ϱo, ν, e) and H(hϱo, hν, κ(e)e) ⪯ H(ϱo, ν, e).

For each e ∈ S0 and ϱo, ν ∈ V, κ denotes a mapping from S0 to (0, 1).

Example 6. Assume that (V, d) is a metric space and V = [0, 1] combined with d(ϱo, ν) =
|ϱo − ν| for each ϱo, ν ∈ V. Define the complex-valued t-norm ∗ and the complex-valued
t-conorm △ by ẃ1 ∗ ẃ2 = (µ1µ2,κ1κ2) and ẃ1 △ ẃ2 = (max{µ1, µ2},max{κ1,κ2}) for all
ẃ1 = (µ1,κ1), ẃ2 = (µ2,κ2) ∈ T respectively. Define complex-valued fuzzy sets E, G, and
H as:

E(ϱo, ν, e) = τξ

τξ + d(ϱo, ν)
ℑ, G(ϱo, ν, e) = d(ϱo, ν)

τξ + d(ϱo, ν)
ℑ, H(ϱo, ν, e) =

d(ϱo, ν)

τξ
ℑ,



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 18 of 38

for each ϱo, ν ∈ V and e = (τ, ξ) ∈ S0. It is simple to prove that (V, E ,G,H, ⋆,△) is a
complete CVNMS induced by metric d. Let us consider a sequence {en} ∈ S0 where each
element en = (τn, ξn) for n ∈ M is arbitrary, Along with the fact that for all ν ∈ V,
0 ≤ d(ϱo, ν) ≤ 1, this means that

ℑ ⪰ inf
ν∈V

E(ϱo, ν, en)

= inf
ν∈V

τnξn
τnξn + d(ϱo, ν)

ℑ

=
τnξn

τnξn + supν∈V d(ϱo, ν)
ℑ

⪰ τnξn
τnξn + 1

ℑ.

As n approaches infinity, we have

ℑ ⪰ lim
n→∞

inf
ν∈V

E(ϱo, ν, en) ⪰ lim
n∈∞

τnξn
τnξn + 1

ℑ = ℑ

the given expression implies that limn→∞ infν∈V E(ϱo, ν, en) = ℑ. Furthermore, we obtain

∅ ⪯ sup
ν∈V

G(ϱo, ν, en)

= sup
ν∈V

d(ϱo, ν)

τnξn + d(ϱo, ν)
ℑ

=
supν∈V d(ϱo, ν)

τnξn + infν∈V d(ϱo, ν)
ℑ

⪯ 1

τnξn
ℑ.

As n approaches infinity, we have

∅ ⪯ lim
n→∞

sup
ν∈V

G(ϱo, ν, en) ⪯ lim
n∈∞

1

τnξn
ℑ = ∅

the given expression implies that limn→∞ supν∈V G(ϱo, ν, en) = ∅. In a similar way, we
have

∅ ⪯ sup
ν∈V

H(ϱo, ν, en)

= sup
ν∈V

d(ϱo, ν)

τnξn
ℑ

=
supν∈V d(ϱo, ν)

τnξn
ℑ

⪯ 1

τnξn
ℑ.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 19 of 38

As n approaches infinity, we have

∅ ⪯ lim
n→∞

sup
ν∈V

H(ϱo, ν, en) ⪯ lim
n∈∞

1

τnξn
ℑ = ∅

the given expression implies that limn→∞ supν∈VH(ϱo, ν, en) = ∅. Let K be a mapping
from V to V which defined as Kϱo = ϱo

2 for each ϱo ∈ V. If we choose a number k ∈
[1/2, 1) ⊂ (0, 1), then for all ϱo, ν ∈ V and e ∈ S0, fulfills equation (1). However, since
2k > 1, we have

E(Kϱo,Kν, ke) =
kτξ

kτξ + d(Kϱo,Kν)
ℑ

=
kτξ

kτξ + |ϱo2 − ν
2 |
ℑ

=
kτξ

kτξ + 1
2 |ϱo − ν|

ℑ

=
2kτξ

2kτξ + |ϱo − ν|
ℑ

⪰ τξ

τξ + |ϱo − ν|
ℑ

= E(ϱo, ν, e)

for any ϱo, ν ∈ V and e = (τ, ξ) ∈ S0.

G(Kϱo,Kν, ke) =
d(Kϱo,Kν)

kτξ + d(Kϱo,Kν)
ℑ

=
|ϱ

o

2 − ν
2 |

kτξ + |ϱo2 − ν
2 |
ℑ

=
1
2 |ϱ

o − ν|
kτξ + 1

2 |ϱo − ν|
ℑ

=
|ϱo − ν|

2kτξ + |ϱo − ν|
ℑ

⪯ |ϱo − ν|
τξ + |ϱo − ν|

ℑ

= G(ϱo, ν, e)

for any ϱo, ν ∈ V and e = (τ, ξ) ∈ S0.

H(Kϱo,Kν, ke) =
d(Kϱo,Kν)

kτξ
ℑ

=
|ϱ

o

2 − ν
2 |

kτξ
ℑ



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 20 of 38

=
1
2 |ϱ

o − ν|
kτξ

ℑ

=
|ϱo − ν|
2kτξ

ℑ

⪯ |ϱo − ν|
τξ

ℑ

= H(ϱo, ν, c)

for each e = (τ, ξ) ∈ S0 and ϱo, ν ∈ V. Consequently, all the conditions given in Theorem
1 are met. Hence in K, 0 is the only fixed point.

The following example demonstrates that Theorem 1 is not unnecessary.

Example 7. Suppose that V is equal to M. We define two binary operations, ⋆ and △, as
e1 ⋆ e2 = (τ1τ2, ξ1ξ2) and e1 △ e2 = (τ1 + τ2, ξ1 + ξ2)− (τ1τ2, ξ1ξ2) for any ei = (τi, ξi) ∈ T
for i=1,2. Consider the CFSs E, G, and H, which are defined as follows:

E(ϱo, ν, e) = min{ϱo, ν}
max{ϱo, ν}

ℑ, G(ϱo, ν, e) =
(
1− min{ϱo, ν}

max{ϱo, ν}

)
ℑ, H(ϱo, ν, e) =

(
1− 2min{ϱo, ν}

min{ϱo, ν}+max{ϱo, ν}

)
ℑ,

for each ϱo, ν ∈ V and e = (τ, ξ) ∈ S0. It is simple to prove that (V, E ,G,H, ⋆,△) is a
complete CVNMS. For any ϱo ∈ V, consider a mapping K : V → V represented by ϱo2+5.
Consider the sequence {en}, defined as en = (n, n) for all n ∈ M. The construction of en
ensures that there is no ambiguity in the expression limn→∞ = ∞. For every element ϱo

in the set V, with a fixed element ν ∈ V such that ϱo is not equal to ν, and for all n ∈ M,
we may prove that

inf
ν∈V

E(ϱo, ν, en) = inf
min{ϱo, ν}
max{ϱo, ν}

ℑ = ∅

sup
ν∈V

G(ϱo, ν, en) = sup

(
1− min{ϱo, ν}

max{ϱo, ν}

)
ℑ = ℑ

and

sup
ν∈V

H(ϱo, ν, en) = sup

(
1− 2min{ϱo, ν}

min{ϱo, ν}+max{ϱo, ν}

)
ℑ = ℑ.

Therefore, we get
lim
n→∞

inf
ν∈V

E(ϱo, ν, en) = ∅ ̸= ℑ

lim
n→∞

sup
ν∈V

G(ϱo, ν, en) = ℑ ̸= ∅

and
lim
n→∞

sup
ν∈V

H(ϱo, ν, en) = ℑ ̸= ∅

for each ϱo ∈ V. For every k ∈ (0, 1), ϱo, ν ∈ V and e ∈ S0, note that

E(Kϱo,Kν, ke) =
min{ϱo2 + 5, ν2 + 5}
max{ϱo2 + 5, ν2 + 5}

ℑ



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 21 of 38

⪰ min{ϱo, ν}
max{ϱo, ν}

ℑ

= E(ϱo, ν, e)

G(Kϱo,Kν, ke) =

(
1− min{ϱo2 + 5, ν2 + 5}

max{ϱo2 + 5, ν2 + 5}

)
ℑ

⪯
(
1− min{ϱo, ν}

max{ϱo, ν}

)
ℑ

= G(ϱo, ν, e)

and

H(Kϱo,Kν, ke) =

(
1− 2min{ϱo2 + 5, ν2 + 5}

min{ϱo2 + 5, ν2 + 5}+max{ϱo2 + 5, ν2 + 5}

)
ℑ

⪯
(
1− 2min{ϱo, ν}

min{ϱo, ν}+max{ϱo, ν}

)
ℑ

= H(ϱo, ν, e).

Equation (1) is satisfied by the mapping K, however, V has no fixed point.

In order to obtain the next result, Ω is defined as the set of all mappings φ : T → T,
where φ is continuous, φ(ℑ) = ℑ, φ(c) ≻ c for every e ∈ T0, and limn→ φn(e) = ℑ for
every e ∈ T0. Similarly, 𭟋 is defined as the set of all mappings σ : T → T in which σ is
continuous, σ(∅) = ∅, σ(e) ≺ e for every e ∈ T, and limn→ σn(e) = ∅ for every e ∈ T.

Theorem 2. Assume that the CVNMS (V, E ,G,H, ⋆,△) is complete. If the following
conditions are satisfied by a mapping K : V → V:

E(Kϱo,Kν, e) ⪰ φ(E(ϱo, ν, e)), G(Kϱo,Kν, e) ⪯ σ(G(ϱo, ν, e)) and H(Kϱo,Kν, e) ⪯ σ(H(ϱo, ν, e))
(6)

for each ϱo, ν ∈ V and e ∈ S0, with σ ∈ 𭟋 and φ ∈ Ω. Then the mapping K has a fixed
point in the set V.

Proof. Let ϱo0 ∈ V be Any given point. In V, a sequence {ϱon} is defined by ϱon = hϱon−1

for all n ∈ M. The existence of an element n0 ∈ M such that ϱon0
= ϱon0−1 makes sure

that ϱon0
is a fixed point in K. Now, we assume that ϱon is not equal to ϱon−1 for every

nM, and we prove that the sequence {ϱon} is Cauchy. For each n ∈ M and a given e ∈ S0,
let us define

An := {E(ϱon, ϱom, e) : m > n} ⊂ T,

Bn := {G(ϱon, ϱom, e) : m > n} ⊂ T,

Cn := {H(ϱon, ϱ
o
m, e) : m > n} ⊂ T.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 22 of 38

As ∅ ≺ E(ϱon, ϱom, e) ⪯ ℑ According to Remarks 1, the greatest lower bound of An, that
is, inf An = x́n, exists for any n ∈ N such that n < m. In the same way, since ∅ ⪯
G(ϱon, ϱom, e) ≺ ℑ for each n ∈ M such that n < m, According to Remarks 1, the least upper
bound Bn, that is, supBn = ýn exist for any n ∈ N. Also, since ∅ ⪯ H(ϱon, ϱ

o
m, e) ≺ ℑ for

each n ∈ M such that n < m, According to Remarks 1, the sup(Cn), that is, supCn = ćn
exist for any n ∈ N. According to equation (6), for each n,m ∈ M such that m > n, this
implies that

E(ϱon+1, ϱ
o
m+1, e) = E(Kϱon,Kϱom, e) ⪰ φ(E(ϱon, ϱom, e)) ≻ E(ϱon, ϱom, e) (7)

G(ϱon+1, ϱ
o
m+1, e) = G(Kϱon,Kϱom, e) ⪯ σ(G(ϱon, ϱom, e) ≺ G(ϱon, ϱom, e) (8)

and
H(ϱon+1, ϱ

o
m+1, e) = H(Kϱon,Kϱom, e) ⪯ σ(H(ϱon, ϱ

o
m, e) ≺ H(ϱon, ϱ

o
m, e). (9)

From this, we can conclude that

E(ϱon+1, ϱ
o
m+1, e) ≻ E(ϱon, ϱom, e)

G(ϱon+1, ϱ
o
m+1, e) ≺ G(ϱon, ϱom, e)

and
H(ϱon+1, ϱ

o
m+1, e) ≺ H(ϱon, ϱ

o
m, e)

for all n,m ∈ M such that m > n with e ∈ S0. By taking the inf(E), sup(G), and sup(H),
we obtain

∅ ⪯ x́n ⪯ x́n+1 ⪯ ℑ, ∅ ⪯ ýn+1 ⪯ ýn ⪯ ℑ, ∅ ⪯ ćn+1 ⪯ ćn ⪯ ℑ

for any n ∈ M. Thus {x́n}, {ýn} and {ćn} are monotonic sequences in S. According to
Remarks 1, there are complex numbers x́, ý, ć ∈ S satisfying limn→∞ x́n = x́, limn→∞ ýn =
ý and limn→∞ ćn = ć. By using equations (7), (8), and (9), and employing equation (6)
gradually, we have

E(ϱon+1, ϱ
o
m+1, e) ⪰ φ(E(ϱon, ϱom, e))

⪰ φ2(E(ϱon−1, ϱ
o
m−1, e))

.

.

.

⪰ φ(E(ϱo0, ϱom−n, e))

G(ϱon+1, ϱ
o
m+1, e) ⪯ σ(G(ϱon, ϱom, e))

⪯ σ2(G(ϱon−1, ϱ
o
m−1, e))



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 23 of 38

.

.

.

⪯ σ(G(ϱo0, ϱom−n, e))

and

H(ϱon+1, ϱ
o
m+1, e) ⪯ σ(H(ϱon, ϱ

o
m, e))

⪯ σ2(H(ϱon−1, ϱ
o
m−1, e))

.

.

.

⪯ σ(H(ϱo0, ϱ
o
m−n, e))

for every n ∈ M such that m > n and e ∈ S0. This implies that

x́n+1 ⪰ inf
m>n

φn(E(ϱo0, ϱom−n, e)

ýn+1 ⪯ sup
m>n

σn(G(ϱo0, ϱom−n, e)

and
ćn+1 ⪯ sup

m>n
σn(H(ϱo0, ϱ

o
m−n, e),

for every n ∈ M along with c ∈ S0. On both sides of the above inequality, as n approaches
infinity, we deduce that

x́ ⪰ lim
n→∞

inf
m>n

φn(E(ϱo0, ϱom−n, e))

= lim
n→∞

φn(E(ϱo0, ϱom−n, e))

= ℑ

ý ⪯ lim
n→∞

sup
m>n

σn(G(ϱo0, ϱom−n, e))

= lim
n→∞

σn(G(ϱo0, ϱom−n, e))

= ∅

and

ć ⪯ lim
n→∞

sup
m>n

σn(H(ϱo0, ϱ
o
m−n, e))



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 24 of 38

= lim
n→∞

σn(H(ϱo0, ϱ
o
m−n, e))

= ∅.

Hence, x́ = ℑ, ý = ∅ and ć = ∅. Thus,

lim
n→∞

inf
m>n

E(ϱon+1, ϱ
o
m+1, e) = lim

n→∞
x́n = ℑ,

lim
n→∞

sup
m>n

G(ϱon+1, ϱ
o
m+1, e) = lim

n→∞
ýn = ∅,

lim
n→∞

sup
m>n

H(ϱon+1, ϱ
o
m+1, e) = lim

n→∞
ýn = ∅,

for all e ∈ S0 which show sequence {ϱon} is Cauchy.
Given that (V, E ,G,H, ⋆,△) is complete, 2 implies the existence of ϱo ∈ V satisfying

lim
n→∞

E(ϱon, ϱo, e) = ℑ, lim
n→∞

G(ϱon, ϱom, e) = ∅ and lim
n→∞

H(ϱon, ϱ
o
m, e) = ∅ (10)

for any e ∈ S0.
As a result of the conditions (5), (10) and (15) of Definition 7 and equation (6), for

any e ∈ S0, we can conclude that

E(ϱo, hϱo, e) ⪰ E
(
ϱo, ϱon+1,

e

2

)
∗ E

(
ϱon+1,Kϱo,

e

2

)
= Ea

(
ϱo, ϱon+1,

e

2

)
∗ E

(
Kϱon,Kϱo,

e

2

)
⪰ E

(
ϱo, ϱon+1,

e

2

)
∗ φ

(
E
(
ϱon, ϱ

o,
e

2

))
≻ E

(
ϱo, ϱon+1,

e

2

)
∗ E

(
ϱon, ϱ

o,
e

2

)

G(ϱo,Kϱo, e) ⪯ G
(
ϱo, ϱon+1,

e

2

)
△G

(
ϱon+1,Kϱo,

e

2

)
= G

(
ϱo, ϱon+1,

e

2

)
△G

(
Kϱon,Kϱo,

e

2

)
⪯ G

(
ϱo, ϱon+1,

e

2

)
△ σ

(
G
(
ϱon, ϱ

o,
e

2

))
≺ G

(
ϱo, ϱon+1,

e

2

)
△G

(
ϱon, ϱ

o,
e

2

)
and

H(ϱo,Kϱo, e) ⪯ H
(
ϱo, ϱon+1,

e

2

)
△H

(
ϱon+1, hϱ

o,
e

2

)
= H

(
ϱo, ϱon+1,

e

2

)
△H

(
Kϱon,Kϱo,

e

2

)
⪯ H

(
ϱo, ϱon+1,

e

2

)
△ σ

(
H

(
ϱon, ϱ

o,
e

2

))
≺ H

(
ϱo, ϱon+1,

e

2

)
△H

(
ϱon, ϱ

o,
e

2

)
.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 25 of 38

Considering the preceding inequalities and applying limn→∞, by using equation (10), we
can conclude that

E(ϱo,Kϱo, e) = ℑ, G(ϱo,Kϱo, e) = ∅ and H(ϱo,Kϱo, e) = ∅

for any e ∈ S0. By applying the conditions (3), (8) and (13) of Definition 7, it can be
derived that ϱo = Kϱo, that is, ϱois a fixed point of K.

To prove the uniqueness of fixed point, suppose that z and ϱo are two distinct fixed
points of K. According to equation (6) for every c ∈ S0, it implies

E(ϱo, z, e) = E(Kϱo,Kz, e) ⪰ φ(E(ϱo, z, e)) ≻ E(ϱo, z, e)

G(ϱo, z, e) = G(Kϱo,Kz, e) ⪯ σ(G(ϱo, z, e)) ≺ G(ϱo, z, e)

and
H(ϱo, z, e) = H(Kϱo,Kz, e) ⪯ σ(H(ϱo, z, e)) ≺ H(ϱo, z, e)

It is a contradiction. Consequently, x = z, which shows that fixed point is unique.

5. Common Fixed-Point Results

This section investigates several standard fixed-point theorems for two mappings that
satisfy the given contraction condition on CVNMSs. It extends the concept of fuzzy
Banach contraction to these spaces.

Definition 10. Let (V, E ,G,H, ⋆,△) be a CVNMS. A neutrosophic Banach contraction
is defined as a pair of two mappings, I and J , both mapping from V to V, such that there
exists a real number k in the interval (0, 1) where

ℑ− E(Iϱo,J ν, e) ⪯ k(ℑ− E(Iϱo,J ν, e),

G(Iϱo,J ν, e) ⪯ kG(Iϱo,J ν, e), (11)

H(Iϱo,J ν, e) ⪯ kH(Iϱo,J ν, e)

holds for any ϱo, ν ∈ V and e ∈ S0.

Theorem 3. Suppose that (V, E ,G,H, ⋆,△) is a complete CVNMS and a pair of two
mappings I,J : V → V is a neutrosophic Banach contraction. Then both the mappings I
and J have a unique shared fixed point that belongs to V.
Proof. Take an arbitrarily selected point ϱo ∈ V. For any n ∈ M0, a sequence {ϱon} is
defined in V as

ϱo2n+1 = Iϱo2n,
ϱo2n+2 = J ϱo2n+1.

It is guaranteed that ϱon0
is a common fixed point of I, if there is a n0 ∈ M such that

ϱon0
= ϱon0+1. Using the equation (11), we also have

ℑ− E(ϱo2n+1, ϱ
o
2n+2, e) = ℑ− E(Iϱo2n,J ϱo2n+1, e)



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 26 of 38

⪯ k(ℑ− E(ϱo2n, ϱo2n+1, e))

= k(ℑ− ℑ)
= ∅

G(ϱo2n+1, ϱ
o
2n+2, e) = G(Iϱo2n,J ϱo2n+1, e)

⪯ kG(ϱo2n, ϱo2n+1, e)

= k(∅)

= ∅

and

H(ϱo2n+1, ϱ
o
2n+2, e) = H(Iϱo2n,J ϱo2n+1, e)

⪯ kH(ϱo2n, ϱ
o
2n+1, e)

= k(∅)

= ∅

for each e ∈ S0. Consequently, E(ϱo2n+1, ϱ
o
2n+2, e) = ℑ,G(ϱo2n+1, ϱ

o
2n+2, e) = ∅ and

H(ϱo2n+1, ϱ
o
2n+2, e) = ∅. By conditions (3), (8) and (13) of Definition 7, ϱo2n+1 =

ϱo2n+2 = J ϱo2n+1, this shows that ϱo2n+1 is a fixed point of J . We can infer that ϱo2n
is a common fixed point of I and J because ϱo2n = ϱo2n+1. Similarly, if there exists an
element n ∈ M0 such that ϱo2n+1 = ϱo2n+2, we can demonstrate using equation (11) that
ϱo2n+1 is a shared fixed point of I and J .

Suppose that ϱon ̸= ϱon+1 for every n ∈ M0. There are two scenarios that we will
examine. Suppose that n is an odd number in the first case. By substituting ϱo = ϱon−1

and ν = ϱon into equation (11), for every e ∈ S0, we obtain

ℑ− E(ϱon, ϱon+1, e) = ℑ− E(Iϱon−1,J ϱon, e)

⪯ k(ℑ− E(ϱon−1, ϱ
o
n, e))

≺ ℑ− E(ϱon−1, ϱ
o
n, e),

G(ϱon, ϱon+1, e) = G(Iϱon−1,J ϱon, e)

⪯ kG(ϱon−1, ϱ
o
n, e)

≺ G(ϱon−1, ϱ
o
n, e)

and

H(ϱon, ϱ
o
n+1, e) = H(Iϱon−1,J ϱon, e)

⪯ kH(ϱon−1, ϱ
o
n, e)

≺ H(ϱon−1, ϱ
o
n, e).



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 27 of 38

It follows that
E(ϱon, ϱon+1, e) ≻ E(ϱon−1, ϱ

o
n, e)

G(ϱon, ϱon+1, e) ≺ G(ϱon−1, ϱ
o
n, e)

and
H(ϱon, ϱ

o
n+1, e) ≺ H(ϱon−1, ϱ

o
n, e)

for any e ∈ S. For the second case, let us assume that n is an even number. By substituting
ϱo = ϱon and ν = ϱon−1 into equation (11), for every e ∈ S0, we obtain

ℑ− E(ϱon+1, ϱ
o
n, e) = ℑ− E(Iϱon,J ϱon−1, e)

⪯ k(ℑ− E(ϱon, ϱon−1, e))

≺ ℑ− E(ϱon, ϱon−1, e),

G(ϱon+1, ϱ
o
n, e) = G(Iϱon,J ϱon−1, e)

⪯ kG(ϱon, ϱon−1, e)

≺ G(ϱon, ϱon−1, e)

and

H(ϱon+1, ϱ
o
n, e) = H(Iϱon,J ϱon−1, e)

⪯ kH(ϱon, ϱ
o
n−1, e)

≺ H(ϱon, ϱ
o
n−1, e).

It follows that
E(ϱon, ϱon+1, e) ≻ E(ϱon−1, ϱ

o
n, e)

G(ϱon, ϱon+1, e) ≺ G(ϱon, ϱon−1, e)

and
H(ϱon, ϱ

o
n+1, e) ≺ H(ϱon, ϱ

o
n−1, e)

for any e ∈ S. Therefore, we conclude that

E(ϱon, ϱon+1, e) ≻ E(ϱon−1, ϱ
o
n, e), G(ϱon, ϱon+1, e) ≺ G(ϱon, ϱon−1, e), H(ϱon, ϱ

o
n+1, e) ≺ H(ϱon, ϱ

o
n−1, e)

for every n ∈ M0 and e ∈ S0. Denote E(ϱon, ϱon+1, e) = An,G(ϱon, ϱon+1, e) = Bn and
H(ϱon, ϱ

o
n+1, e) = Cn for each n ∈ M0. Since

ℑ ⪰ An ≻ An−1 ≻ ∅

∅ ⪯ Bn ≺ Bn−1 ≺ ℑ

∅ ⪯ Cn ≺ Cn−1 ≺ ℑ



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 28 of 38

for every n ∈ M0, It concludes that sequences {An}, {Bn} and {Cn} are monotonic in S.
By Remarks 1, one is possible to locate x́, ý, ć ∈ S satisfying

lim
n→∞

An = x́, lim
n→∞

Bn = ý, lim
n→∞

Cn = ć.

By using equation equation (11), for n ∈ M0 and c ∈ S0 we obtain

ℑ− E(ϱon, ϱon+1, e) ⪯ k(ℑ− E(ϱon−1, ϱ
o
n, e))

ℑ− An ⪯ k(ℑ− An−1)

G(ϱon, ϱon+1, e) ⪯ kG(ϱon−1, ϱ
o
n, e)

Bn ⪯ k(Bn−1)

and

H(ϱon, ϱ
o
n+1, e) ⪯ kH(ϱon−1, ϱ

o
n, e)

Cn ⪯ k(Cn−1).

As the value of n approaches infinity for inequalities, we get

ℑ− x́ ⪯ k(ℑ− x́)

ý ⪯ ký

and
ć ⪯ kć.

As k ∈ (0, 1), if x́ ≺ ℑ, ý ≻ ∅, and ć ≻ ∅, it is a contradiction. Therefore x́ = ℑ, ý = ∅
and ć = ∅ it indicates that

lim
n→∞

E(ϱon, ϱon+1, e) = ℑ,

lim
n→∞

G(ϱon, ϱon+1, e) = ∅,

lim
n→∞

H(ϱon, ϱ
o
n+1, e) = ∅

for every n ∈ M0 and e ∈ S0.
Now we will show that the {ϱon} is a Cauchy sequence. For every n ∈ M0 as well as

fixed e ∈ S0, consider

Dn = {E(ϱon, ϱom, e) : m > n} ⊆ T,
En = {G(ϱon, ϱom, e) : m > n} ⊆ T,
Fn = {H(ϱon, ϱ

o
m, e) : m > n} ⊆ T.

Since ∅ ≺ E(ϱon, ϱom, e) ⪯ ℑ,∅ ⪯ G(ϱom, ϱon, e) ≺ ℑ and ∅ ⪯ H(ϱom, ϱon, e) ≺ ℑ, by Remarks
1, The infimum of the CFS E(ϱon, ϱom, e), the supremum of the CFS G(ϱon, ϱom, e), and the



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 29 of 38

supremum of the CFS H(ϱon, ϱ
o
m, e) all exist. By iteratively applying the condition (5) of

Definition 7 for any positive integers m > n, we can obtain

E(ϱon, ϱom, e) ⪰ E
(
ϱon, ϱ

o
n+1,

e

m− n

)
⋆E

(
ϱon+1, ϱ

o
n+2,

e

m− n

)
⋆...⋆E

(
ϱom−1, ϱ

o
m,

e

m− n

)
.

Consequently,

lim
n→∞

inf
m>n

E(ϱon, ϱom, e) ⪰ ℑ ⋆ ℑ ⋆ ... ⋆ ℑ

= ℑ

which leads to
lim
n→∞

inf
m>n

E(ϱon, ϱom, e) = ℑ

for every e ∈ S0. Furthermore, By iteratively applying the condition (10) of Definition 7
for any positive integers m > n, we can obtain

G(ϱon, ϱom, e) ⪯ G
(
ϱon, ϱ

o
n+1,

c

m− n

)
△G

(
ϱon+1, ϱ

o
n+2,

e

m− n

)
△...△G

(
ϱom−1, ϱ

o
m,

e

m− n

)
.

Consequently,

lim
n→∞

sup
m>n

G(ϱon, ϱom, e) ⪯ ∅△∅△ ...△∅

= ∅

which leads to
lim
n→∞

sup
m>n

G(ϱon, ϱom, e) = ∅

for every e ∈ S0. Also By iteratively applying the condition (15) of Definition 7 for any
positive integers m > n, we can obtain

H(ϱon, ϱ
o
m, e) ⪯ H

(
ϱon, ϱ

o
n+1,

e

m− n

)
△H

(
ϱon+1, ϱ

o
n+2,

e

m− n

)
△...△H

(
ϱom−1, ϱ

o
m,

e

m− n

)
.

Consequently,

lim
n→∞

sup
m>n

H(ϱon, ϱ
o
m, e) ⪯ ∅△∅△ ...△∅

= ∅

which leads to
lim
n→∞

sup
m>n

H(ϱon, ϱ
o
m, e) = ∅

for every e ∈ S0. Hence, sequence {ϱon} is Cauchy.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 30 of 38

Given that (V, E ,G,H, ⋆,△) is complete, Lemma 2 implies the existence of u ∈ V
satisfying

lim
n→∞

E(ϱon, u, e) = ℑ, lim
n→∞

G(ϱon, u, e) = ∅ and lim
n→∞

H(ϱon, u, e) = ∅

for all e ∈ S0. For any n ∈ M0 and e ∈ S0, by equation (11) we yield

ℑ− E(Fu,Gϱo2n+1, e) ⪯ k(ℑ− E(u, ϱo2n+1, e))

≺ ℑ− E(u, ϱo2n+1, e)

G(Iu,J ϱo2n+1, e) ⪯ kG(u, ϱo2n+1, e)

≺ G(u, ϱo2n+1, e)

and

H(Iu,J ϱo2n+1, e) ⪯ kH(u, ϱo2n+1, e)

≺ H(u, ϱo2n+1, e).

This implies that
E(Iu,J ϱo2n+1, e) ≻ E(u, ϱo2n+1, e) (12)

G(Iu,J ϱo2n+1, e) ≺ G(u, ϱo2n+1, e) (13)

and
H(Iu,J ϱo2n+1, e) ≺ H(u, ϱo2n+1, e) (14)

for each n ∈ M0 and e ∈ S0. As a result of conditions (5), (10) and (15) of Definition 7,
equations (12), (13) and (14), for each n ∈ M0 and e ∈ S0, we may deduce that

E(u, Iu, e) ⪰ E
(
u, ϱo2n+2,

e

2

)
⋆ E

(
ϱo2n+2, Iu,

e

2

)
= E

(
u, ϱo2n+2,

e

2

)
⋆ E

(
J ϱo2n+1, Iu,

e

2

)
= E

(
u, ϱo2n+2,

e

2

)
⋆ E

(
Iu,J ϱo2n+1,

e

2

)
⪰ E

(
u, ϱo2n+2,

e

2

)
⋆ E

(
u, ϱo2n+1,

e

2

)

G(u, Iu, e) ⪯ G
(
u, ϱo2n+2,

e

2

)
△G

(
ϱo2n+2, Iu,

e

2

)
= G

(
u, ϱo2n+2,

e

2

)
△G

(
J ϱo2n+1, Iu,

e

2

)
= G

(
u, ϱo2n+2,

e

2

)
△G

(
Iu,J ϱo2n+1,

e

2

)
⪯ G

(
u, ϱo2n+2,

e

2

)
△G

(
u, ϱo2n+1,

e

2

)



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 31 of 38

and

H(u, Iu, e) ⪯ H
(
u, ϱo2n+2,

e

2

)
△H

(
ϱo2n+2, Iu,

e

2

)
= H

(
u, ϱo2n+2,

e

2

)
△H

(
J ϱo2n+1, Iu,

e

2

)
= H

(
u, ϱo2n+2,

e

2

)
△H

(
Iu,J ϱo2n+1,

e

2

)
⪯ H

(
u, ϱo2n+2,

e

2

)
△H

(
u, ϱo2n+1,

e

2

)
As the value of n approaches infinity for inequalities, we get

E(u, Iu, e) = ℑ, G(u, Iu, e) = ∅ and H(u, Iu, e) = ∅

for every e ∈ S0. Under conditions (3), (8) and (13) of Definition 7, it indicates that u is
equal to Iu. By applying the same methods as previously, one can establish that

E(u,J u, e) = ℑ, G(u,J u, e) = ∅ and H(u,J u, e) = ∅

for every e ∈ S0. By the conditions (3), (8) and (13) of Definition 7, imply that u is equal
to J u. Consequently, it follows that u = Iu = J u which shows that u is a is common
fixed point of both functions I and J .

In order to prove uniqueness, assume that u and v are two distinct fixed points of K,
It is possible to locate e ∈ S0 satisfying E(u, v, e) ̸= ℑ,G(u, v, e) ̸= ∅ and H(u, v, e) ̸= ∅
By equation (11),

ℑ− E(u, v, e) = ℑ− E(Iu,J v, e)

⪯ k(ℑ− E(u, v, e))
≺ ℑ− E(u, v, e)

G(u, v, e) = G(Iu,J v, e)

⪯ k(G(u, v, e))
≺ G(u, v, e)

and

H(u, v, e) = H(Iu,J v, e)

⪯ k(H(u, v, e))

≺ H(u, v, e)

which contradicts our assumption. Thus E(u, v, e) = ℑ,G(u, v, e) = ∅ and H(u, v, e) = ∅
for all e ∈ S0. By the conditions (3), (8) and (13) of Definition 7, we may deduce that u is
equal to v which show that the K has a unique fixed point.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 32 of 38

Corollary 1. Suppose that (V, E ,G,H, ⋆,△) is a complete CVNMS. A mapping I : V → V
satisfying

ℑ− E(Iϱo, Iν, e) ⪯ k(ℑ− E(ϱo, ν, e)),
G(Iϱo, Iν, e) ⪯ kG(ϱo, ν, e),
H(Iϱo, Iν, e) ⪯ kH(ϱo, ν, e)

for each ϱo, ν ∈ V and e ∈ S0, where k ∈ (0, 1). In that case, the mapping I has a single
fixed point in V.
Proof. The desired outcome may be obtained by replacing I = J into Theorem 3.

An example of the idea presented in Corollary 1 is shown below.

Example 8. Consider V = [0, 1]. Define two binary operations ⋆ and △ by e1 ⋆ e2 =
(τ1τ2, ξ1ξ2) and e1 △ e2 = (max(τ1, τ2),max(ξ1, ξ2)) for any ei = (τi, ξi) ∈ T where i=1,2.
Let CFSs E ,G and H be defined as follows:

E(ϱo, ν, e) =
(
τξ +min{ϱo, ν}
τξ +max(ϱo, ν)

)
ℑ, G(ϱo, ν, e) = 1−

(
τξ +min{ϱo, ν}
τξ +max(ϱo, ν)

)
ℑ,

H(ϱo, ν, e) =

(
max{ϱo, ν} −min{ϱo, ν}

τξ +max{ϱo, ν}

)
ℑ

for all ϱo, ν ∈ V and e = (τ, ξ) ∈ S0. It is not difficult to show that (V, E ,G,H, ⋆,△) is a
complete CVNMS.

Consider a mapping I : V → V expressed by ϱo

2 for all ϱo ∈ V. For any ϱo, ν ∈ V
satisfying ϱo ≤ ν, it is clear that Iϱo ≤ Iν. It follows that

E(Iϱo, Iν, e) =
(
τξ +min{Iϱo, Iν}
τξ +max(Iϱo, Iν)

)
ℑ,

=

(
τξ + Iϱo

τξ + Iν

)
ℑ,

⪰
(
τξ + ϱo

τξ + ν

)
ℑ,

= E(ϱo, ν, e).

If we choose any k ∈ (12 , 1), we have

ℑ− E(Iϱo, Iν, e) ⪯ k(ℑ− E(ϱo, ν, e))

for every ϱo, ν ∈ V and e = (τ, ξ) ∈ S0. Similarly, we can deduce that

G(Iϱo, Iν, e) ⪯ kG(ϱo, ν, e)
and

H(Iϱo, Iν, e) ⪯ kH(ϱo, ν, e)

for all ϱo, ν ∈ V and e = (τ, ξ) ∈ S0. Thus, all the requirements stated in Corollary 1 are
satisfied. Specifically, the number 0 is the only fixed point of I.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 33 of 38

Theorem 4. Let (V, E ,G,H, ⋆,△) be a CVNMS. If there is a commuting pair of self-
mappings, I,J : V → V satisfying

ℑ− E(Inϱo,J nν, e) ⪯ k(ℑ− E(ϱo, ν, e)),
G(Inϱo,J nν, e) ⪯ kG(ϱo, ν, e),
H(Inϱo,J nν, e) ⪯ kH(ϱo, ν, e)

for every ϱo, ν ∈ V, e ∈ S0 and n ∈ M, where k is any real number from (0, 1). Then there
is a single shared fixed point of mappings I and J inside V.
Proof. Both In and J n satisfy all the conditions stated in Theorem 3. Thus, they have
a shared fixed point µ in V, for instance, Inµ = J nµ = µ. According to the provided
information

InIµ = IInµ = Iµ,

it may be deduced that Iµ is a point fixed by In. Since both the mappings I and J commute,
We may express the equation as follows:

J nIµ = IJ nµ = Iµ

this demonstrates that Iµ is a point fixed by J n. Thus, Iµ acts as a common fixed point
of In and J n.

Similarly, According to the provided information

J nJ µ = JJ nµ = J µ,

it may be deduced that J µ is a point fixed by J n. Since both the mappings I and J
commute,We may express the equation as follows:

InJ µ = J Inµ = J µ

this demonstrates that J µ is a point fixed by In. Thus, J µ acts as a common fixed point
of In and J n.

Given that the common fixed point of In and J n is unique, it follows that µ = J µ =
Iµ. Consequently, µ as the common fixed point for both I and J . If I and J have any
common fixed point, that point will also be a fixed point of In and J n. The common fixed
point of I and J is uniquely defined for this purpose.

Corollary 2. Let (V, E ,G,H, ⋆,△) be a CVNMS. If there is a mapping I : V → V
satisfying

ℑ− E(Inϱo, Inν, e) ⪯ k(ℑ− E(ϱo, ν, e)),
G(Inϱo, Inν, e) ⪯ kG(ϱo, ν, e),
H(Inϱo, Inν, e) ⪯ kH(ϱo, ν, e)

for each ϱo, ν ∈ V, e ∈ S0 and n ∈ M, where 0 < k < 1. Then mapping I has a single
common fixed point within V.
Proof. The desired outcome may be obtained by replacing I = J in Theorem 4.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 34 of 38

6. Application to Fredholm integral equations of the second kind

In this section, we use Theorem 1 to show that Fredholm integral equations have a
unique solution. The collection of all continuous functions mapping the interval [0, 1] to R
is represented by the set C([0, 1],R). An illustration of a second-kind nonlinear Fredholm
integral equation is given below:

φ(q) = Q(q) + ć

∫ 1

0
ẃ(q, r)ϖ(r, φ(r))dr (15)

where Q denotes is a continuous real-valued function on [0, 1], ẃ(q, r) denotes the kernel
of the integral function,ϖ(r, φ(r)) denotes nonlinear and continuous function defined on
[0, 1]× R and φ(q) symbolizes the function we want to be identified.

Theorem 5. Assume that the set V = C([0, 1] × R). Assume that the following circum-
stances are fulfilled:

(1) A member x́ ∈ (0, 1) can be identified in the following:

|ϖ(r, φ(r))−ϖ(r, σ(r))| ≤ x́|φ(r)− σ(r)|

for any φ, σ ∈ V and r ∈ [0, 1];

(2)
∫ 1
0 ẃ(q, r)dr ≤ ý;

(3) ć2ý2x́2 ≤ k < 1.

As a result, the integral equation (15) possesses a unique solution inside the set V.
Proof. Consider a mapping I : V → V defined as

Iφ(q) = Q(q) + ć

∫ 1

0
ẃ(q, r)ϖ(r, φ(r))dr

for each φ(q) ∈ V and q ∈ [0, 1]. The complex-valued t-norm is defined as ⋆x, whereas the
complex-valued t-conorm is defined as △x. Moreover, E(ϱo, ν, e),G(ϱo, ν, e) and H(ϱo, ν, e)
defined by

E(φ(q), σ(q), e) = τ + ξ

τ + ξ + |φ(q)− σ(q)|2
ℑ,

G(φ(q), σ(q), e) = |φ(q)− σ(q)|2

τ + ξ + |φ(q)− σ(q)|2
ℑ,

H(φ(q), σ(q), e) =
|φ(q)− σ(q)|2

τ + ξ
ℑ

for each φ, σ ∈ V, e = (τ, ξ) > 0 and q ∈ [0, 1]. It is easily established that (V, E ,G,H, ⋆,△)
is a CVNMS.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 35 of 38

For each φ, σ ∈ V and q ∈ [0, 1], it follows that

|Iφ(q)− Iσ(q)|2 = |Q(q) + ć

∫ 1

0
ẃ(q, r)ϖ(r, φ(r))dr −Q(q)− ć

∫ 1

0
ẃ(q, r)ϖ(r, σ(r))dr|2

= ć2|
∫ 1

0
ẃ(q, r)ϖ(r, φ(r))dr −

∫ 1

0
ẃ(q, r)ϖ(r, σ(r))dr|2

≤ ć2
(∫ 1

0
ẃ(q, r)dr

)2

|ϖ(r, φ(r))−ϖ(r, σ(r))|2

≤ ć2ý2x́2|φ(r)− σ(r)|2

≤ k|φ(r)− σ(r)|2.

Now, for each φ, σ ∈ V and e ∈ S0, it leads to

E(Iφ(q), Iσ(q), ke) = k(τ + ξ)

k(τ + ξ) + |Iφ(q)− Iσ(q)|2
ℑ

⪰ k(τ + ξ)

k(τ + ξ) + k|φ(q)− σ(q)|2
ℑ

=
(τ + ξ)

(τ + ξ) + |φ(q)− σ(q)|2
ℑ

= E(φ(q), σ(q), e)

G(Iφ(q), Iσ(q), ke) = |Iφ(q)− Iσ(q)|2

k(τ + ξ) + |Iφ(q)− Iσ(q)|2
ℑ

=

(
1− k(τ + ξ)

k(τ + ξ) + |Iφ(q)− Iσ(q)|2

)
ℑ

⪯
(
1− k(τ + ξ)

k(τ + ξ) + k|φ(q)− σ(q)|2

)
ℑ

=

(
1− k(τ + ξ)

k(τ + ξ) + k|φ(q)− σ(q)|2

)
ℑ

=
|φ(q)− σ(q)|2

τ + ξ + |φ(q)− σ(q)|2
ℑ

= G(φ(q), σ(q), e),

and

H(Iφ(q), Iσ(q), ke) = |Iφ(q)− Iσ(q)|2

k(τ + ξ)
ℑ

⪯ |Iφ(q)− Iσ(q)|2

(τ + ξ)
ℑ

= H(φ(q), σ(q), e).

Consequently, every condition listed in Theorem 1 is satisfied, suggesting that there is only
one solution to the equation (15) exists in the set C([0, 1],R).



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 36 of 38

7. Conclusion

In this paper, we introduced the concept of CVNMSs as a generalization of CVFMSs,
complex-valued IFMSs, and NMSs. Further, we proved the Banach contraction theo-
rem and common fixed point theorems in the setting of CVNMSs. We provide several
non-trivial examples to demonstrate how the new strategy outperforms literature-based
methods. Furthermore, we find the existence and uniqueness of the solution of the integral
equation by applying the main result. Our findings broaden the scope of previous research
beyond fuzzy metric, intuitionistic fuzzy metric, and NMSs. This work is extendable in
the context of complex-valued neutrosophic b-metric spaces, complex-valued neutrosophic
controlled metric spaces, complex-valued neutrosophic partial metric spaces, and many
other structures.

Conflict of interest

The authors declare that they have no conflicts of interest.

Authors Contribution

All authors contributed equally in this manuscript.

References

[1] S. Banach. Sur les opérations dans les ensembles abstraits et leur application aux
équations intégrales. Fundamenta Mathematicae, 3(1):133–181, 1922.

[2] L. A. Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965.
[3] K. T. Atanassov. On Intuitionistic Fuzzy Sets Theory, volume 283 of Studies in

Fuzziness and Soft Computing. Springer, 2012.
[4] U. Saeed and M. Umair. A modified method for solving non-linear time and space

fractional partial differential equations. Engineering Computations, 36(7):2162–2178,
2019.

[5] M. Grabiec. Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 27(3):385–
389, 1988.

[6] A. George and P. Veeramani. On some results in fuzzy metric spaces. Fuzzy Sets and
Systems, 64(3):395–399, 1994.

[7] J. H. Park. Intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals, 22(5):1039–
1046, 2004.

[8] A. Bartwal, R. C. Dimri, and G. Prasad. Some fixed point theorems in fuzzy bipolar
metric spaces. Journal of Nonlinear Sciences and Applications, 13:196–204, 2020.

[9] R. Chugh and S. Kumar. Weakly compatible maps in generalized fuzzy metric spaces.
Journal of Analysis, 10:65–74, 2002.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 37 of 38

[10] F. Mehmood, R. Ali, and N. Hussain. Contractions in fuzzy rectangular b-metric
spaces with application. Journal of Intelligent & Fuzzy Systems, 37(1):1275–1285,
2019.

[11] S. Nădăban. Fuzzy b-metric spaces. International Journal of Computers Communi-
cations & Control, 11(2):273–281, 2016.

[12] R. Saadati, S. Sedghi, and N. Shobe. Modified intuitionistic fuzzy metric spaces and
some fixed point theorems. Chaos, Solitons & Fractals, 38(1):36–47, 2008.

[13] K. S. Wong, Z. Salleh, C. M. I. C. Taib, and I. Abdullah. Some fixed point results
on fuzzy extended rectangular b-metric spaces. In American Institute of Physics
Conference Series, volume 2746, page 060004, October 2023.

[14] M. S. Ashraf, R. Ali, and N. Hussain. Geraghty type contractions in fuzzy b-metric
spaces with application to integral equations. Filomat, 34(9):3083–3098, 2020.

[15] D. Gopal. Contributions to fixed point theory of fuzzy contractive mappings. Advances
in Metric Fixed Point Theory and Applications, pages 241–282, 2021.

[16] D. Gopal, P. Kumam, and M. Abbas. Background and Recent Developments of Metric
Fixed Point Theory. 2017.

[17] N. Saleem, B. Ali, M. Abbas, and Z. Raza. Fixed points of suzuki type generalized
multivalued mappings in fuzzy metric spaces with applications. Fixed Point Theory
and Applications, pages 1–18, 2015.

[18] A. Azam, B. Fisher, and M. Khan. Common fixed point theorems in complex-valued
metric spaces. Numerical Functional Analysis and Optimization, 32(3):243–253, 2011.

[19] S. Shukla, R. Rodriguez-Lopez, and M. Abbas. Fixed point results for contractive
mappings in complex-valued fuzzy metric spaces. Fixed Point Theory, 19(2):751–774,
2018.

[20] K. P. Patel and G. M. Deheri. Extension of some common fixed point theorems.
International Journal of Applied Physics and Mathematics, 3(5):329, 2013.

[21] I. Demir. Fixed point theorems in complex-valued fuzzy b-metric spaces with appli-
cation to integral equations. Miskolc Mathematical Notes, 22(1):153–171, 2021.

[22] S. T. Zubair, K. Gopalan, T. Abdeljawad, and N. Mlaiki. Novel fixed point technique
to coupled system of nonlinear implicit fractional differential equations in complex-
valued fuzzy rectangular b-metric spaces. AIMS Mathematics, 7(6):10867–10891,
2022.

[23] Humaira, H. A. Hammad, M. Sarwar, and M. De la Sen. Existence theorem for a
unique solution to a coupled system of impulsive fractional differential equations in
complex-valued fuzzy metric spaces. Advances in Difference Equations, 2021(1):242,
2021.

[24] M. Sarwar and T. Abdeljawad. Existence of unique solution to nonlinear mixed
volterra fredholm-hammerstein integral equations in complex-valued fuzzy metric
spaces. Journal of Intelligent & Fuzzy Systems, 40(3):4065–4074, 2021.

[25] Humaira, M. Sarwar, and T. Abdeljawad. Existence of solutions for nonlinear impul-
sive fractional differential equations via common fixed-point techniques in complex-
valued fuzzy metric spaces. Mathematical Problems in Engineering, 2020(1):7042715,
2020.



S. M. U. Ud-din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6931 38 of 38

[26] M. Sarwar, T. Abdeljawad, and N. Mlaiki. Fixed point results via least upper bound
property and its applications to fuzzy caputo fractional volterra–fredholm integro-
differential equations. Mathematics, 9(16):1969, 2021.

[27] U. Saeed and M. Umair. A modified method for solving non-linear time and space
fractional partial differential equations. Engineering Computations, 36(7):2162–2178,
2019.

[28] M. Kirişci and N. Şimşek. Neutrosophic metric spaces. Mathematical Sciences,
14(3):241–248, 2020.

[29] A. Asghar, A. Hussain, K. Ahmad, U. Ishtiaq, H. Al Sulami, and N. Hussain.
On neutrosophic 2-metric spaces with application. Journal of Function Spaces,
2023(1):9057107, 2023.

[30] M. Akram, U. Ishtiaq, K. Ahmad, T. A. Lazăr, V. L. Lazăr, and L. Guran. Some
generalized neutrosophic metric spaces and fixed point results with applications. Sym-
metry, 16(8):965, 2024.

[31] S. Sowndrarajan, M. Jeyaraman, and F. Smarandache. Fixed Point Results for Con-
traction Theorems in Neutrosophic Metric Spaces, volume 36. Infinite Study, 2020.

[32] U. Ishtiaq, K. Javed, F. Uddin, M. D. L. Sen, K. Ahmed, and M. U. Ali. Fixed point
results in orthogonal neutrosophic metric spaces. Complexity, 2021(1):2809657, 2021.

[33] M. E. M. Abdalla, A. Uzair, A. Ishtiaq, M. Tahir, and M. Kamran. Algebraic struc-
tures and practical implications of interval-valued fermatean neutrosophic super hy-
persoft sets in healthcare. Spectrum of Operational Research, 2(1):199–218, 2025.

[34] T. Fujita. Shadowed offset: Integrating offset and shadowed set frameworks for en-
hanced uncertainty modeling. Spectrum of Operational Research, 4(1):1–17, 2025.

[35] E. P. Klement, R. Mesiar, and E. Pap. Triangular norms. position paper ii: general
constructions and parameterized families. Fuzzy Sets and Systems, 145(3):411–438,
2004.

[36] O. Yazdanbakhsh and S. Dick. A systematic review of complex fuzzy sets and logic.
Fuzzy Sets and Systems, 338:1–22, 2018.