EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6486
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Contractive Operators Controlled by Simulation
Functions in Fuzzy Metric Spaces with Transitive

K-Closed Binary Relations

Abdelhamid Moussaoui1, Mirjana Pantović2, Stojan Radenović3,∗

1 Laboratory of Applied Mathematics and Scientific Computing, Faculty of Sciences and
Techniques, Sultan Moulay Slimane University, PO Box 523, Beni Mellal, 23000, Morocco
2 Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac,
Radoja Domanovića 12, 34000 Kragujevac, Serbia
3 Faculty of Mechanical Engineering, University of Belgrade, 16, Beograd 35, 11120, Serbia

Abstract. In this study, we establish a novel fuzzy functional contraction within fuzzy metric
spaces equipped with a binary relation, relying on the weaker concept of R-completeness rather
than the classical completeness of the entire space or its subspaces. The usual continuity require-
ment on the mapping is relaxed and replaced by either R-continuity or the P-self-closedness of the
relation’s restriction, employing a broad class of control functions S. The theoretical results are
illustrated with examples and an application to solving an integral equation governed by a given
binary relation, accompanied by several corollaries and derived consequences. This work extends
the theory of relation-theoretic fuzzy fixed points and provides a rigorous basis for further study of
coincidence and common fixed points, with potential applications to nonlinear operator equations
in uncertain settings.

2020 Mathematics Subject Classifications: 47H10, 54H25

Key Words and Phrases: Fixed point, R-completeness, Contractive mappings, Binary relation,
Fuzzy metric, Simulation function, R-continuity, P-self-closedness, Integral equation.

1. Introduction and Preliminaries

Fixed point theory occupies a central role in nonlinear functional analysis, providing a
rich toolkit for resolving diverse and intricate problems across many mathematical disci-
plines. Among its fundamental results is the Banach contraction principle, a cornerstone
of metric fixed point theory, which has been extensively generalized and applied to nu-
merous abstract metric frameworks. A recent advancement in this domain was introduced
by Khojasteh et al. [1], who enriched fixed point theory by incorporating a novel class

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6486

Email addresses: a.moussaoui@usms.ma (A. Moussaoui),
mirjana.pantovic@pmf.kg.ac.rs (M. Pantović), radens@beotel.rs (S. Radenović)

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



A. Moussaoui, M. Pantović, S. Radenović / Eur. J. Pure Appl. Math, 18 (3) (2025), 6486 2 of 18

of control functions known as simulation functions, thereby broadening the theoretical
horizon.

Another vibrant and rapidly growing facet of fixed point theory involves relation-
theoretic results. This area traces its origins to the pioneering work of Turinici [2], who
formulated the concept of order-theoretic fixed points. Building on this foundation, Ran
and Reurings [3] developed an order-theoretic version of the Banach contraction principle
in 2004, which they successfully applied to matrix equations, expanding the practical ap-
plicability of fixed point results. Later, Alam and Imdad [4] extended this framework by
introducing a relation-theoretic version of the Banach contraction principle, unifying sev-
eral classical order-theoretic theorems through arbitrary binary relations. This approach
has since catalyzed a proliferation of fixed point theorems employing diverse notions of
binary relations [5–10].

The introduction of fuzzy set theory by Zadeh in 1965 [11] marked a significant mile-
stone, offering a robust mathematical apparatus to address uncertainty and vagueness
beyond the scope of classical crisp sets. This innovation has become instrumental in mod-
eling complex and imprecise phenomena. The extension of probabilistic metric spaces into
the fuzzy context was pioneered by Kramosil and Michalek [12], who introduced fuzzy
metric spaces a concept subsequently refined by George and Veeramani [13] to define a
Hausdorff topology.

In the setting of fuzzy metric spaces, distance is conceptualized not as a fixed numerical
quantity but as a degree of proximity influenced by a parameter ℑ > 0. This parameter
captures practical factors such as travel time or energy expenditure that influence the
degree of proximity between two points. For instance, consider the journey by car from
Marrakech (u) to Casablanca (v), the degree of proximity between these two cities can be
modeled by parameters representing the duration of travel or the amount of fuel consumed
along the route. These factors reflect real-world considerations affecting how “close” two
locations are in a fuzzy metric sense, beyond mere physical distance. According to axiom
(M2), when the two points coincide (u = v), their degree of proximity attains the maximal
value of 1, reflecting the intuitive concept of perfect proximity in practical scenarios where
time, energy, or other resources play a critical role.

Recent years have witnessed growing interest in fixed point theory within fuzzy metric
spaces. Early contributions include the introduction of fuzzy contractive mappings by
Gregori and Sapena [14], which established foundational fixed point theorems. This line
of research was further developed by Mihet [15] with ψ-contractive mappings, and by
Wardowski [16] who introduced H-contractive mappings. More recently, Melliani and
Moussaoui [17] advanced the field by adapting the simulation function methodology to
fuzzy metric spaces, proposing the notion of FZ-contractions. An extensive corpus of
literature on various contraction types in fuzzy metric spaces has since emerged, including
important works such as [3, 5, 18–32].

To establish our main results, we begin by recalling essential concepts from the theory
of binary relations, including definitions and key properties that will be used throughout
the work.

Definition 1. A binary relation R on a non-empty set E is defined as a subset of E × E.



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For any s, t ∈ E, if (s, t) ∈ R, we say that s is related to t via R, often denoted sRt.
Furthermore, if either (s, t) ∈ R or (t, s) ∈ R, this is indicated by [s, t] ∈ R.

Note that the Cartesian product E × E defines the universal relation on E , while the
empty set ∅ represents the empty relation.

A binary relation R on a nonempty set E is said to have the following properties: The
relation R is reflexive if for every s ∈ E , we have sRs. It is transitive if whenever sRt and
tRr, it follows that sRr for all s, t, r ∈ E . The relation is antisymmetric if for all s, t ∈ E ,
the conditions sRt and tRs imply s = t. Additionally, R is complete if for every s, t ∈ E ,
the pair [s, t] belongs to R.

Finally, given a self-mapping K : E → E , the relation R is said to be K-closed if
whenever (s, t) ∈ R, it follows that (Ks,Kt) ∈ R for all s, t ∈ E .

Definition 2. [4] Let E be a nonempty set and R a binary relation on E. A sequence
{sq} ⊆ E is called R-preserving if (sq, sq+1) ∈ R for every q ∈ N.

Definition 3. [33] A continuous function ⋏ : [0, 1]× [0, 1] → [0, 1] is called a t-norm if it
is commutative, associative, and satisfies:

(i) For every ℘1 ∈ [0, 1], ℘1 ⋏ 1 = ℘1,

(ii) For all ℘1, ℘2, ℘3, ℘4 ∈ [0, 1], if ℘1 ≤ ℘3 and ℘2 ≤ ℘4, then ℘1 ⋏ ℘2 ≤ ℘3 ⋏ ℘4.

Example 1. Some standard continuous t-norms are:

(1) Minimum t-norm: ℘1 ⋏m ℘2 = min{℘1, ℘2},

(2)  Lukasiewicz t-norm: ℘1 ⋏L ℘2 = max{℘1 + ℘2 − 1, 0},

(3) Product t-norm: ℘1 ⋏p ℘2 = ℘1 · ℘2,

for all ℘1, ℘2 ∈ [0, 1].

Definition 4. [13] Let E be a non-empty set, ⋏ a continuous t-norm, and P : E2 ×
(0,+∞) → [0, 1] a fuzzy set. The triple (E ,P,⋏) is called a fuzzy metric space if for all
s, t, r ∈ E and ℑ, κ > 0, the following conditions hold:

(M1) P(s, t,ℑ) > 0,

(M2) P(s, t,ℑ) = 1 if and only if s = t,

(M3) P(s, t,ℑ) = P(t, s,ℑ),

(M4) P(s, t,ℑ)⋏P(t, r, κ) ≤ P(s, r,ℑ+ κ),

(M5) The function P(s, t, ·) : (0,+∞) → [0, 1] is continuous,

Lemma 1. [19] For all s, t ∈ E, the function P(s, t, ·) is nondecreasing on (0,+∞).

Definition 5. [13] Let (E ,P,⋏) be a fuzzy metric space.



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(i) A sequence {sq} ⊆ E is said to converge to s ∈ E if and only if

lim
q→+∞

P(sq, s,ℑ) = 1 for all ℑ > 0.

(ii) A sequence {sq} ⊆ E is called a Cauchy sequence if for each α ∈ (0, 1) and ℑ > 0,
there exists q0 ∈ N such that

P(sq, sp,ℑ) > 1− α for all q, p ≥ q0.

(iii) The fuzzy metric space (E ,P,⋏) is said to be complete if every Cauchy sequence in
E converges.

The notion of fuzzy contractive mappings was initially formulated by Gregori and
Sapena [14] as follows:

Definition 6. [14] Let (E ,P,⋏) be a fuzzy metric space. A mapping K : E → E is fuzzy
contractive if there exists a constant ℏ ∈ (0, 1) such that for all s, t ∈ E and ℑ > 0,

1

P(K(s),K(t),ℑ)
− 1 ≤ ℏ

(
1

P(s, t,ℑ)
− 1

)
.

Define Ψ as the family of continuous non-decreasing functions ψ : (0, 1] → (0, 1] satis-
fying ψ(t) > t for all t ∈ (0, 1).

Definition 7. [15] A mapping K : E → E is fuzzy ψ-contractive if

P(Ks,Kt,ℑ) ≥ ψ
(
P(s, t,ℑ)

)
for all s, t ∈ E , ℑ > 0.

Example 2. [15] For each ℏ ∈ (0, 1), the function ψℏ : (0, 1] → (0, 1] given by

ψℏ(s) =
s

s+ ℏ(1− s)

belongs to Ψ. Every ψℏ-fuzzy contractive mapping is also a fuzzy contractive mapping as
defined by Gregori and Sapena [14].

Let H denote the set of strictly decreasing functions η : (0, 1] → [0,+∞) that map the
interval (0, 1] onto the entire range [0,+∞).

Definition 8. [16] A mapping K : E → E is fuzzy H-contractive with respect to η ∈ H if
there exists a constant ℏ ∈ (0, 1) such that

η
(
P(Ks,Kt,ℑ)

)
≤ ℏ · η

(
P(s, t,ℑ)

)
for all s, t ∈ E and ℑ > 0.

Definition 9. [17] (see also [23, 24]) A function S : (0, 1] × (0, 1] → R is termed an
FZ-simulation function if it satisfies:



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(S1) S(1, 1) = 0,

(S2) For all s, t ∈ (0, 1), it holds that S(s, t) < 1
t −

1
s ,

(S3) For sequences {sq} and {tq} in (0, 1] with limq→+∞ sq = limq→+∞ tq < 1, we have

lim
q→+∞

supS(sq, tq) < 0.

The collection of all such functions is denoted FZ.

Definition 10. [17] (see also [23, 24]) Let (E ,P,⋏) be a fuzzy metric space, K : E → E
a self-map, and S ∈ FZ. The mapping K is an FZ-contraction with respect to S if for
all s, t ∈ E and ℑ > 0,

S
(
P(Ks,Kt,ℑ),P(s, t,ℑ)

)
≥ 0.

Example 3. ([24]) Consider the following functions Si : (0, 1]× (0, 1] → R, i = 1, 2, 3:

(i) S1(s, t) = ℏ
(
1
t − 1

)
− 1

s + 1, for ℏ ∈ (0, 1) and all s, t ∈ (0, 1],

(ii) S2(s, t) =
1

ψ(t) −
1
s , where ψ ∈ Ψ, for all s, t ∈ (0, 1],

(iii) S3(s, t) =
1

η−1(ℏ·η(t)) −
1
s , where ℏ ∈ (0, 1), η ∈ H, for all s, t ∈ (0, 1].

Each Si is an FZ-simulation function.

Definition 11. [34] A binary relation R on E is said to be P-self-closed if for any R-
preserving sequence {sq} ⊆ E with

lim
q→+∞

P(sq, s,ℑ) = 1 for all ℑ > 0,

there exists a subsequence {sqk} of {sq} such that (sqk , s) ∈ R.

Definition 12. [10] Let (E ,P,⋏) be a fuzzy metric space and R a binary relation on E.
A sequence {sq} ⊆ E is called an R-Cauchy sequence if it is R-preserving and for every
ε ∈ (0, 1) and ℑ > 0, there exists q0 ∈ N such that

P(sq+p, sq,ℑ) > 1− ε for all q ≥ q0, p ∈ N.

Remark 1. [10] For any binary relation R, every standard Cauchy sequence is an R-
Cauchy sequence. The notions coincide when R is the universal relation on E.

Definition 13. [10] A fuzzy metric space (E ,P,⋏) endowed with a binary relation R is
said to be R-complete if every R-Cauchy sequence converges in E.

Remark 2. [10] Every complete fuzzy metric space is R-complete for any binary relation
R.

Definition 14. [35] Let E be a nonempty set and R a binary relation on E. For s, t ∈ E,
a path of length ℓ from s to t in R is a finite sequence {Γ0,Γ1, . . . ,Γℓ} ⊆ E such that:



A. Moussaoui, M. Pantović, S. Radenović / Eur. J. Pure Appl. Math, 18 (3) (2025), 6486 6 of 18

(L1) Γ0 = s and Γℓ = t,

(L2) (Γj ,Γj+1) ∈ R for all j = 0, 1, . . . , ℓ− 1.

Note that a path of length ℓ contains ℓ+ 1 elements.

Let E be a nonempty set, U ⊆ E , and K : E → E be a self-mapping. We denote:

E(K,R) := {s ∈ E : (s,Ks) ∈ R},

P(s, t,R) := the set of all paths in R connecting s to t,

and
R|U := R∩ (U × U),

which is the restriction of R to U , naturally induced by R.

Definition 15. Let (E ,P,⋏) be a fuzzy metric space, and R a binary relation on E. A
mapping K : E → E is said to be R-continuous at s ∈ E if for every R-preserving sequence
{sq} with

lim
q→+∞

P(sq, s,ℑ) = 1 for all ℑ > 0,

it follows that
lim

q→+∞
P(Ksq,Ks,ℑ) = 1 for all ℑ > 0.

If K is R-continuous at every s ∈ E, then K is R-continuous.

Remark 3. Every continuous mapping is R-continuous for any binary relation R. In
particular, if R is the universal relation on E, then R-continuity coincides with usual
continuity.

To support our main theorems, we use the following lemma.

Lemma 2. [10] Let K : E → E be a self-mapping, and let R be a binary relation on E
that is transitive and K-closed. Assume there exists s0 ∈ E such that s0RKs0. Define the
sequence {sq} recursively by

sq = Ksq−1, q ∈ N.

Then, for all p, q ∈ N with q < p, the relation

spRsq

holds.



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2. Main Results

In this section, we present the principal results. We begin by establishing a funda-
mental proposition that articulates a fuzzy functional contraction condition within fuzzy
metric spaces endowed with a binary relation. This condition is framed through a con-
trol function S belonging to the family FZ, which governs the contractive behavior of the
mapping K with respect to the fuzzy metricP. The proposition introduces complementary
hypotheses that will be needed throughout the rest of the paper.

Proposition 1. Let (E ,P,⋏) be a fuzzy metric space, R a binary relation on E, and let
K : E → E be a mapping such that for every pair of elements s, t ∈ E with (s, t) ∈ R and
every positive real number ℑ, the following inequality holds with respect to the function S
in the family FZ:

S
(
P(Ks,Kt,ℑ),P(s, t,ℑ)

)
≥ 0. (1)

Then the following two conditions complement each other:

(H1) For all s, t ∈ E such that (s, t) ∈ R, S
(
P(Ks,Kt,ℑ),P(s, t,ℑ)

)
≥ 0,

(H2) For all s, t ∈ E such that [s, t] ∈ R, S
(
P(Ks,Kt,ℑ),P(s, t,ℑ)

)
≥ 0.

Proof. The implication (H2) ⇒ (H1) is trivial. Conversely, assuming that (H1) holds,
let s, t ∈ E such that [s, t] ∈ R. In this case, (H2) follows directly from (H1). If, however,
(t, s) ∈ R, then by using the symmetry of the fuzzy metric P and the assumption (H1),
we deduce the following:

ξ(P(Ks,Kt,ℑ),P(t, s,ℑ)) = ξ(P(Kt,Ks,ℑ),P(s, t,ℑ)) ≥ 0.

Thus, we have shown that (H1) ⇒ (H2).

Theorem 1. Let (E ,P,⋏) be a fuzzy metric space equipped with a binary relation R, and
let K : E → E be a self-mapping. Suppose the following conditions hold:

(i) There exists a subset U ⊆ E such that K(E) ⊆ U ⊆ E, and (U ,P,⋏) is R-complete.

(ii) E(K,R) ̸= ∅.

(iii) R is K-closed and transitive.

(iv) For every pair of elements s, t ∈ E with (s, t) ∈ R and every positive real number ℑ,
the following inequality holds with respect to the function S in the family FZ:

S
(
P(Ks,Kt,ℑ),P(s, t,ℑ)

)
≥ 0.

(v) Either R|U is P-self-closed, or K is R-continuous.

Then, K has a fixed point.



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Proof. The implication from (H2) to (H1) is straightforward. On the other hand,
assuming that (H1) holds, let s, t ∈ E with [s, t] ∈ R. In this scenario, (H2) follows
immediately from (H1). However, if (t, s) ∈ R, applying the fuzzy metric symmetry P
and (H1), we conclude:

S
(
P(Kt,Ks,ℑ),P(t, s,ℑ)

)
= S

(
P(Ks,Kt,ℑ),P(s, t,ℑ)

)
≥ 0,

where P(s, t,ℑ) = P(t, s,ℑ). This shows that (H1) ⇒ (H2).
Since E(K,R) ̸= ∅, let s0 be an arbitrary element such that s0 ∈ E(K,R). Define the

sequence {sq} by sq+1 = Ksq for all q ∈ N. Since R is K-closed and (s0,Ks0) ∈ R, we
have the following relations:

(s0,Ks0), (Ks0,K2s0), (K2s0,K3s0), . . . , (Kqs0,Kq+1s0) ∈ R.

Thus, the sequence {sq} preserves the relation R, as shown by:

(sq, sq+1) ∈ R.

Given that K satisfies the contraction inequality (1) with respect to S ∈ FZ, we have:

S
(
P(Ksq−1,Ksq,ℑ),P(sq−1, sq,ℑ)

)
≥ 0.

Thus, we deduce the following:

0 ≤ S
(
P(Ksq−1,Ksq,ℑ),P(sq−1, sq,ℑ)

)
= S

(
P(sq, sq+1,ℑ),P(sq−1, sq,ℑ)

)
<

1

P(sq−1, sq,ℑ)
− 1

P(sq, sq+1,ℑ)
.

This implies thatP(sq−1, sq,ℑ) < P(sq, sq+1,ℑ), which means that the sequence {P(sq−1, sq,ℑ)}
is non-decreasing and bounded above by 1. Hence, there exists a limit m(ℑ) ≤ 1 such
that:

lim
q→+∞

P(sq, sq−1,ℑ) = m(ℑ) for all ℑ > 0.

To prove that m(ℑ) = 1, assume the contrary, that is, m(ℑ0) < 1 for some ℑ0 > 0.
Applying the contraction condition and (S1), we obtain

0 ≤ lim sup
q→+∞

S
(
P(Ksq−1,Ksq,ℑ0), P(sq−1, sq,ℑ0)

)
< 0.

We reach a contradiction, which implies that m(ℑ) = 1. Therefore, we have:

lim
q→+∞

P(sq, sq+1,ℑ) = 1 for all ℑ > 0. (2)

Next, we show that {sq} is a Cauchy sequence. Assume the opposite, that is, that {sq}
is not Cauchy. Then, there exist ζ ∈ (0, 1), ℑ0 > 0 and two subsequences {sqi} and {spi}
such that for sufficiently large i, we have:

P(spi , sqi ,ℑ0) ≤ 1− ζ.



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Taking into consideration Lemma 1, we get

P

(
spi , sqi ,

ℑ0

2

)
≤ 1− ζ. (3)

By selecting pi as the smallest index satisfying the above inequality, we obtain

P

(
spi , sqi−1,

ℑ0

2

)
> 1− ζ. (4)

In view of Lemma 2 and condition (1), we have

0 ≤ S
(
P(Kspi−1,Ksqi−1,ℑ0),P(spi−1, sqi−1,ℑ0)

)
= S

(
P(spi , sqi ,ℑ0),P(spi−1, sqi−1,ℑ0)

)
the last equation together with property (S2), we obtain

P(spi−1, sqi−1,ℑ0) < P(spi , sqi ,ℑ0). (5)

By (3), (4), and (M4), we have

1− ϵ ≥ P(spi , sqi ,ℑ0)

> P(spi−1, sqi−1,ℑ0)

≥ P

(
spi−1, spi ,

ℑ0

2

)
∗P

(
spi , sqi−1,

ℑ0

2

)
> P

(
sqi−1, sqi ,

ℑ0

2

)
∗ (1− ζ).

Taking limit as i→ +∞ and using (2), we obtain

lim
i→+∞

P(spi , sqi ,ℑ0) = lim
i→+∞

P(spi−1, sqi−1,ℑ0) = 1− ζ. (6)

Based on the preceding analysis, we identify the two sequences

{Ti} = {P(spi−1, sqi−1,ℑ0)}, {Si} = {P(spi , sqi ,ℑ0)}.

From the previous discussion, we consider the sequences

{Ti} = {P(spi−1, sqi−1,ℑ0)}, {Si} = {P(spi , sqi ,ℑ0)},

both converging to the same limit 1−ζ < 1. Moreover, noting that K fulfills the contraction
condition (1) with respect to S ∈ FZ, and property (S3), we get

0 ≤ lim
i→+∞

supS
(
P(spi−1, sqi−1,ℑ0),P(spi , sqi ,ℑ0)

)
< 0.

This leads to a contradiction, and therefore {sq} must be a Cauchy sequence. Since
{sq} ⊆ K(E) ⊆ U , it follows that {sq} is an R-preserving Cauchy sequence in U . Given



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that U is R-complete, the sequence converges to some point s̃ ∈ U . If K is R-continuous,
we obtain s̃ = limq→+∞ sq+1 = limq→+∞Ksq = Ks̃. Thus, s̃ is a fixed point of K.

Next, suppose that the binary relation R is P-self-closed. Since the sequence {sq}
preserves R and converges to s̃, there exists a subsequence {sqi} such that [sqi , s̃] ∈ R|U
for all i, and by Proposition 1, [sqi , s̃] ∈ R and sqi → s̃. Without losing generality, we
assume that sqi ̸= s̃ for every i ∈ N. Now making use contraction inequality (1), we obtain

0 ≤ S
(
P(Ksqi ,Ks̃,ℑ),P(sqi , s̃,ℑ)

)
Finally, we will demonstrate that s̃ is a fixed point of K. Suppose that Ks̃ ̸= s̃. Then, we
have P(s̃,Ks̃,ℑ) < 1. Now, by (S2) and (S3), we obtain

0 ≤ lim
i→+∞

supS
(
P(sqi+1,Ks̃,ℑ),P(sqi , s̃,ℑ)

)
= lim

i→+∞
supS

(
P(Ksqi ,Ks̃,ℑ),P(sqi , s̃,ℑ)

)
≤ lim

i→+∞
sup

( 1

P(sqi , s̃,ℑ)
− 1

P(Ksqi ,Ks̃,ℑ)

)
= 1− 1

P(s̃,Ks̃,ℑ)
.

Consequentlly,
0 ≤ lim

i→+∞
supS

(
P(sqi+1,Ks̃,ℑ),P(sqi , s̃,ℑ)

)
< 0.

This leads to a contradiction, which necessitates that P(s̃,Ks̃,ℑ) = 1. As a result, we
conclude that Ks̃ = s̃, indicating that s̃ is a fixed point of the mapping K.

Theorem 2. Under the assumptions of Theorem 1, if for every pair of points s, t ∈ E the
set P(s, t,R) is nonempty, then self-mapping K possesses a unique fixed point.

Proof. We proceed by contradiction. Suppose that there exist two distinct fixed points,
s and s̃, of the mapping K. Since P(s, s̃,R) is nonempty, there exists a finite sequence
{Γ0,Γ1, . . . ,Γq} in E with length q, satisfying:

Γ0 = s, Γq = s̃, and (Γj ,Γj+1) ∈ R for j = 0, 1, . . . , q − 1.

Due to the transitivity of R, it follows that (Γ0,Γq) ∈ R. As K satisfies the contraction
inequality (1) with respect to S ∈ FZ, we have:

0 ≤ S
(
P(KΓ0,KΓq,ℑ),P(Γ0,Γq,ℑ)

)
<

1

P(Γ0,Γq,ℑ)
− 1

P(KΓ0,KΓq,ℑ)

=
1

P(s, s̃,ℑ)
− 1

P(s, s̃,ℑ)
.

Thus, P(s, s̃,ℑ) < P(s, s̃,ℑ). This inequality leads to a contradiction. Therefore, the fixed
point of K is unique.



A. Moussaoui, M. Pantović, S. Radenović / Eur. J. Pure Appl. Math, 18 (3) (2025), 6486 11 of 18

Corollary 1. Let (E ,P,⋏) be a fuzzy metric space endowed with a binary relation R, and
let K : E → E be a self-mapping. Suppose the following conditions are satisfied:

(i) (E ,P,⋏) is complete;

(ii) E(K,R) ̸= ∅;

(iii) R is K-closed and transitive;

(iv) For every pair s, t ∈ E with (s, t) ∈ R and each ℑ > 0, the following inequality holds
for some S ∈ FZ:

S
(
P(Ks,Kt,ℑ),P(s, t,ℑ)

)
≥ 0;

(v) Either R is P-self-closed or K is R-continuous.

Then, the mapping K has a fixed point.

Proof. The result follows directly by taking U = E in Theorem 1.

Example 4. Consider the metric space (E ,P), where E = R is equipped with the fuzzy
metric

P(s, t,ℑ) = ℑ
ℑ+ |s− t|

,

for all s, t ∈ E and ℑ > 0. This space is complete. Define the binary relation R on E by

R = {(s, t) ∈ E2 : (s− t)(s+ t+ 11) = 0}.

Define the mapping K : E → E by

K(s) = s2 + 11s+ 16.

The mapping K is continuous. For any s, t ∈ E with (s, t) ∈ R, it follows that

K(s) = K(t), hence (K(s),K(t)) ∈ R,

which means R is K-closed. Consider the point −8 ∈ E. Then K(−8) = (−8)2+11(−8)+
16 = −8, therefore, (−8,K(−8)) ∈ R, confirming E(K,R) ̸= ∅.

Define the function S : (0, 1]× (0, 1] → R by

S(r, b) =
7

9

(
1

b
− 1

)
−
(
1

r
− 1

)
.

For any r, b ∈ E with (r, b) ∈ R,

S(P(Ks,Kt,ℑ),P(s, t,ℑ)) = 7

9

(
1

P(s, t,ℑ)
− 1

)
−
(

1

P(Ks,Kt,ℑ)
− 1

)
.

=
7

9

| s− t |
ℑ

− | Ks−Kt |
ℑ

=
7

9

| s− t |
ℑ

− | (s2 + 11s+ 16)− (t2 + 11t+ 16) |
ℑ

=
7

9

| s− t |
ℑ

≥ 0.



A. Moussaoui, M. Pantović, S. Radenović / Eur. J. Pure Appl. Math, 18 (3) (2025), 6486 12 of 18

Hence,
S(P(Ks,Kt,ℑ),P(s, t,ℑ)) ≥ 0,

confirming that K satisfies the contraction inequality (1) with respect to S.
By Theorem 1, K has at least one fixed point in E.

Corollary 2. Let (E ,P,⋏) be a fuzzy metric space endowed with a binary relation R, and
let K : E → E be a self-mapping. Suppose the following conditions are satisfied:

(i) For all s, t ∈ E with (s, t) ∈ R and for each ℑ > 0, the inequality

ψ
(
P(s, t,ℑ)

)
≤ P(Ks,Kt,ℑ)

holds for ψ ∈ Ψ;

(ii) There exists a subset U ⊆ E such that K(E) ⊆ U and (U ,P,⋏) is R-complete;

(iii) E(K,R) ̸= ∅;

(iv) The relation R is K-closed and transitive;

(v) Either R|U is P-self-closed or K is R-continuous.

Then, the mapping K has a fixed point.

Proof. Define S : (0, 1]× (0, 1] → R by

S(r, b) =
1

ψ(b)
− 1

r
, for all r, b ∈ (0, 1],

where η ∈ H. The result then follows by applying Theorem 1.

Corollary 3. Let (E ,P,⋏) be a fuzzy metric space endowed with a binary relation R, and
let K : E → E be a self-mapping. Suppose the following conditions are satisfied:

(i) For all s, t ∈ E with (s, t) ∈ R and for each ℑ > 0, the inequality

η
(
P(Ks,Kt,ℑ)

)
≤ ℏ · η

(
P(s, t,ℑ)

)
holds for some ℏ ∈ (0, 1) and a function η ∈ H;

(ii) There exists a subset U ⊆ E such that K(E) ⊆ U and (U ,P,⋏) is R-complete;

(iii) E(K,R) ̸= ∅;

(iv) The relation R is K-closed and transitive;

(v) Either R|U is P-self-closed or K is R-continuous.

Then, the mapping K has a fixed point.



A. Moussaoui, M. Pantović, S. Radenović / Eur. J. Pure Appl. Math, 18 (3) (2025), 6486 13 of 18

Proof. Define S : (0, 1]× (0, 1] → R by

S(r, b) =
1

η−1(ℏ · η(b))
− 1

r
, for all r, b ∈ (0, 1],

where η ∈ H. The result then follows by applying Theorem 1.

Corollary 4. If the R-completeness of the subset U is replaced by the completeness of
E, and the R-continuity of K is replaced by standard continuity, then the conclusion of
Theorem 1 remains valid.

Proof. The assertion follows immediately from Remark 2 and Remark 3.

Corollary 5. Let (E ,P,⋏) be a fuzzy metric space equipped with a binary relation R, and
let K : E → E be a self-mapping. Suppose the following conditions hold:

(i) For all s, t ∈ E with (s, t) ∈ R and for every ℑ > 0,

1

P(Ks,Kt,ℑ)
− 1 ≤ ℏ

(
1

P(s, t,ℑ)
− 1

)
,

for some constant ℏ ∈ (0, 1);

(ii) There exists a subset U ⊆ E such that K(E) ⊆ U and (U ,P,⋏) is R-complete;

(iii) E(K,R) ̸= ∅;

(iv) The relation R is both K-closed and transitive;

(v) Either R|U is P-self-closed or K is R-continuous.

Then the mapping K admits a fixed point.

Proof. The result follows by defining S : (0, 1]× (0, 1] → R as

S(r, b) = ℏ
(
1

b
− 1

)
− 1

r
+ 1 for all r, b ∈ (0, 1],

and applying Theorem 1.

Application

We solve an integral equation under a given binary relation by applying Theorem 1.
Consider the integral equation

δ(ℓ) = r(ℓ) + β

∫ v

u
h(e, ℓ) g

(
e, δ(e)

)
de, ℓ ∈ J := [u, v], (7)

where δ is an arbitrary function defined on the interval J = [u, v], r : J → R, h : J×J → R,
g : J × R → R, β > 0 is a positive parameter, and E := C(J,R) denotes the space of all
continuous functions from J to R.

Define Θ as the class of all mappings θ : [0,+∞) → [0,+∞) satisfying:



A. Moussaoui, M. Pantović, S. Radenović / Eur. J. Pure Appl. Math, 18 (3) (2025), 6486 14 of 18

(i) θ is non-decreasing;

(ii) For every z ≥ 0, θ(z) ≤ z.

Consider the space E = C(J,R) of continuous functions on J , equipped with the supremum
metric

d(δ, z) = sup
ℓ∈J

|δ(ℓ)− z(ℓ)|.

Define the fuzzy metric function P : E × E × (0,+∞) → [0, 1] as

P(δ, z,ℑ) = ℑ
ℑ+ d(δ, z)

, ℑ > 0.

With the operation ⋏ on [0, 1] defined by

℘1 ⋏ ℘2 = ℘1 × ℘2, ℘1, ℘2 ∈ [0, 1],

the triple (E ,P,⋏) forms a complete fuzzy metric space. We are now ready to present the
main result of this section.

Theorem 3. Assume the following assumptions for equation 7:

S1⟩ supℓ∈J
∫ v
u |h(e, ℓ)| de ≤ 2

3β , for all e, ℓ ∈ J,

S1⟩ |g(e, δ1(e))− g(e, δ2(e))| ≤ θ(|δ1(e)− δ2(e)|), for all δ1, δ2 ∈ R.

Then equation 7 admits at least one solution in E.

Proof. Consider the operator K : E → E defined by

Kδ(ℓ) = r(ℓ) + β

∫ b

a
h(e, ℓ) g(e, δ(e)) de, ℓ ∈ J := [a, b],

where E = C(J,R) and the binary relation

R = {(δ, z) ∈ E × E : δ(ℓ) ≤ z(ℓ) for all ℓ ∈ J}

is defined. Equipped with the metric function P : E × E × (0,+∞) → [0, 1], P(δ, z,ℑ) =
ℑ

ℑ+d(δ,z) , for all ℑ > 0, where d(δ, z) = supℓ∈J |δ(ℓ) − z(ℓ)|, the space (E ,P,⋏) is R-

complete. Next, Suppose a sequence {δq} in E is R-preserving and converges to δ with

respect to P, that is δq
P−→ δ. Therefore, the sequence {δq} satisfies the monotonic chain

δ0(ℓ) ≤ δ1(ℓ) ≤ δ2(ℓ) ≤ · · · ≤ δq(ℓ) ≤ δq+1(ℓ) ≤ · · · , ℓ ∈ J.

This convergence entails δn(ℓ) ≤ δ(ℓ) for all q ∈ N, implying the existence of a subsequence
{δni} such that [δni , δ] ∈ R, q ∈ N, and thus R is P-self-closed.



A. Moussaoui, M. Pantović, S. Radenović / Eur. J. Pure Appl. Math, 18 (3) (2025), 6486 15 of 18

If δ, z ∈ E satisfy δ(ℓ) ≤ z(ℓ) for every ℓ and h(ℓ, e) ≥ 0, then K preserves the order

(Kδ)(ℓ) = r(ℓ) +

∫ v

u
h(e, ℓ) g(e, δ(e)) de ≤ r(ℓ) +

∫ v

u
h(e, ℓ) g(e, z(e)) de = (Kz)(ℓ).

Therefore,
(Kδ)(ℓ) ≤ (Kz)(ℓ).

Hence, (Kδ,Kz) ∈ R, thereby establishing that R is K-closed.
Next, consider the pair (δ, z) ∈ R, we have

1

P(Ks,Kt,ℑ)
− 1 =

d(Kδ,Kz)
ℑ

=
supℓ∈J |(Kδ)(ℓ)− (Kz)(ℓ)|

ℑ

≤ β sup
ℓ∈J

∫ v

u
|h(e, ℓ)| de

∫ v

u
|g(e, δ(e))− g(e, z(e))| de

≤ 2

3ℑ

∫ v

u
|g(e, δ(e))− g(e, z(e))| de

≤ 2

3ℑ

∫ v

u
θ(|δ(e)− z(e)|) de

≤ 2

3ℑ
θ(d(δ, z))

≤ 2

3

d(δ, z)

ℑ
=

2

3

( 1

P(δ, z,ℑ)
− 1

)
.

Consequently, the operator K satisfies the contractive condition (iv) in Theorem 1 by
employing the control function S(ℓ, e) ∈ FZ defined as

S(ℓ, e) =
2

3

(
1

ℓ
− 1

)
−
(
1

e
− 1

)
.

Let γ ∈ E be such that γ(ℓ) ≤ (Kγ)(ℓ), which implies (γ,Kγ) ∈ R, so the set E(K,R)
is nonempty. Now, Define ϖ = max{δ, z}. Then δ(ℓ) ≤ ϖ(ℓ) and z(ℓ) ≤ ϖ(ℓ), which
shows

(δ,ϖ) ∈ R and (z,ϖ) ∈ R.

Hence, the triple {δ,ϖ, z} forms a path in R connecting δ and z. All assumptions of
Theorem 2 are thus satisfied, concluding that K has a unique fixed point that solves the
integral equation.



A. Moussaoui, M. Pantović, S. Radenović / Eur. J. Pure Appl. Math, 18 (3) (2025), 6486 16 of 18

3. Conclusion

This study introduced a novel fuzzy contraction inequality essential for establishing
the existence and uniqueness of fixed points in fuzzy metric spaces endowed with a binary
relation. Notably, unlike many classical approaches, we did not require the completeness
of the entire space or any of its subspaces. Instead, our results rely on the relatively weaker
concept of R-completeness within subspaces of the whole space. Moreover, we relaxed the
traditional continuity assumption on the involved mapping, replacing it with the more
generalized notions of R-continuity or the P-self-closedness of the restriction of R to a
subset U .

Our framework, leveraging versatile control functionsS, enriches the theory of relation-
theoretic and fuzzy fixed points. These findings also pave the way for future research
on coincidence and common fixed points within relation-theoretic fuzzy metric spaces,
highlighting the broad potential and applicability of the approach.

Future work may extend this framework to broader contraction classes, including
vector-valued and multi-dimensional fuzzy metric spaces. The results provide a solid ba-
sis for analyzing nonlinear operator equations and related problems involving uncertainty
modeled by fuzzy metrics and relations.

In conclusion, this study unifies relational structures and fuzzy metric theory to estab-
lish robust fixed point results, opening pathways for further research on coincidence and
common fixed points in abstract metric and relational frameworks.

Declarations:

Competing interests

The author reports no conflicts of interest

Funding

This work was supported by the Serbian Ministry of Science, Technological Develop-
ment and Innovation (Agreement No. 451-03-137/2025-03/200122).

Authors’ contributions

All authors read and approved the final manuscript.

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