EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3304-3335
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Intuitionistic Fuzzy Z-Contractions and Common Fixed
Points with Applications

Maliha Rashid1,, Naeem Saleem2,3,∗, Quratulain Mumtaz1,, Maggie Aphane3,,
Inayatur Rehman4

1 Department of Mathematics and Statistics,International Islamic University, Islamabad,
Pakistan
2 Department of Mathematics, University of Management and Technology, Lahore 54770,
Pakistan
3 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences
University, Ga-Rankuwa, Medunsa, Pretoria 0204, South Africa
4 Department of Mathematics and Sciences, College of Arts and Applied Sciences Dhofar
University, Salalah, Oman

Abstract. In the context of b-metric spaces, this paper introduces two concepts: admissible
hybrid intuitionistic fuzzy Z-contractions and pairwise admissible hybrid intuitionistic fuzzy Z-
contractions, and establishes criteria for intuitionistic fuzzy fixed points under such contractions. It
is demonstrated that a pair of set-valued maps possesses a common fixed point. Various illustrative
examples are provided to validate these results. Moreover, the significant implications of our main
theorem are explored and analyzed across different types of simulation functions. Furthermore,
we derive several fixed point results in the context of partially ordered b-metric spaces, offering
insights from an application-oriented perspective. These outcomes extend and generalize several
prior results documented in the literature.

2020 Mathematics Subject Classifications: 54H25, 47H10, 03E72, 54E35, 06A06

Key Words and Phrases: b-metric space, Common intuitionistic fuzzy fixed point, Intuitionistic
fuzzy set-valued map, Pairwise intuitionistic fuzzy Z-contraction, Simulation function

1. Introduction

Fixed point theory is a pivotal area of mathematical analysis with broad implications
across various disciplines, including functional analysis, topology, and applied mathemat-
ics. One of the foundational results in this field is the Banach Fixed Point Theorem, also
known as the Contraction Mapping Principle, a cornerstone in the study of fixed points

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5431

Email addresses: maliha.rashid@iiu.edu.pk (M. Rashid),
quratulain.msma749@iiu.edu.pk (Q. Mumtaz), naeem.saleem2@gmail.com (N. Saleem),
maggie.aphane@smu.ac.za (M. Aphane), irehman@du.edu.om (I. Rehman)

https://www.ejpam.com 3304 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3305

that plays a crucial role in proving the existence and uniqueness of solutions across a broad
spectrum of mathematical problems, including those in differential and integral equations.
The Banach Contraction Theorem [13], a seminal result in this field, has profoundly in-
fluenced subsequent research and applications.

Fixed point theory is instrumental in addressing a variety of complex issues in non-
linear analysis. Specifically, in operator equations such as Gx = 0, the existence of a
fixed point fx = x can be effectively established for self-mappings within the relevant
domain. Fixed point theory has evolved significantly, with numerous generalizations that
extend beyond metric spaces to more abstract settings such as topological vector spaces,
normed spaces, and ordered structures. These generalizations have been instrumental in
solving complex problems in economics, computer science, and engineering, where finding
equilibrium states or stable solutions is essential.

Fuzzy set theory, introduced by Lotfi A. Zadeh [2] in 1965, revolutionized classical set
theory by allowing partial membership of elements in a set. This approach is particularly
useful in modeling situations characterized by uncertainty, imprecision, or vagueness, as
it provides a more flexible and realistic representation of real-world phenomena. Fuzzy
sets have found extensive applications in various fields such as control systems, decision-
making, pattern recognition, and artificial intelligence. Building upon the foundation of
fuzzy sets, Krassimir Atanassov [23] introduced intuitionistic fuzzy sets (IFS) in 1986 as a
further extension to better handle situations where there is hesitation or lack of complete
information. In intuitionistic fuzzy sets, each element is described by two values: the
degree of membership. Additionally, there exists a hesitation margin, often referred to as
the degree of indeterminacy or uncertainty. This generalization has proven useful in many
applications, particularly in decision-making scenarios where incomplete or ambiguous
information exists. Notably, Gulzar [14] applied IFSs to group theory, while Kanwal and
Akbar [27] used them to define intuitionistic fuzzy mappings and identify common fixed
points [24, 26].

Incorporating fuzzy sets into traditional metric spaces has led to significant advance-
ments. Deng [8] introduced the concept of fuzzy pseudo-metric spaces, which extend
the classical metric framework to accommodate fuzzy topological and uniform structures.
Erceg [17] further explored the equivalence of pseudo-quasi-metrics to distance functions in
fuzzy set theory, proposing additional axioms for metrics and uniformity within this con-
text. These developments underscore the ongoing effort to apply mathematical structures
to fuzzy scenarios.

In mathematical analysis, the concept of metric spaces plays a fundamental role in un-
derstanding various aspects of distance and convergence. Extending the traditional notion
of a metric, b-metric spaces provide a more generalized framework that retains many essen-
tial properties while relaxing some strict conditions. The development of b-metric spaces
drew upon a long history of work on metric spaces and their generalizations. Several key
contributions paved the way for the formalization of b-metric spaces: This line of inquiry
was initiated by Bakhtin [5] and Bourbaki [7]. Czerwik [8] further contributed to this area
by introducing a concept that relaxes the constraints of the traditional triangle inequality
and formally defined b-MSs to enhance the Banach Fixed Point (FP) theorem. Since then,



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3306

there has been a proliferation of research papers examining single-valued and multi-valued
operators within b-MSs, often delving into FP theory or the variational principle (Refer
to [1, 9, 10, 12, 22]).

The introduction of γ-admissibility has played a key role in extending fixed point
theory to more complex spaces and functions. It has been particularly useful in the study
of non linear analysis, multi valued mappings, dynamic system and generalized metric
spaces. Several researchers have contributed to the development of fixed point theorems
involving beta admissibility. Samet et al. (2012)[25] explored fixed point theorems for
single and multi-valued mappings in spaces where γ-admissibility is assumed, contributing
to the generalization of classical fixed point results. Rhoades [23] worked on extending
fixed point theorems using γ-admissibility in partially ordered spaces, showing how this
concept could be applied to more structured settings. The concept of γ-admissibility serves
as a powerful generalization in fixed point theory, allowing for the study of mappings
that do not necessarily satisfy the strict contractive conditions required in classical fixed
point theorems. By introducing a function γ that controls the behavior of the mapping,
γ-admissibility provides the flexibility needed to extend fixed point results to broader
classes of spaces and functions. This has led to significant advances in nonlinear analysis,
dynamic systems, and the study of generalized metric spaces. The foundational work on γ-
admissibility continues to influence ongoing research in applied mathematics, offering new
tools for solving complex problems in fields like optimization, economics, and mathematical
modeling.

In a related vein, the exploration of fixed points arising from hybrid contractions
represents a burgeoning area of research within fixed point theory. Recent contributions
[18] have provided criteria for establishing the existence of intuitionistic fuzzy fixed points
(IFFPs) within b-metric spaces, focusing on admissible hybrid intuitionistic fuzzy (AHIF)
Z-contractions and HIF Z-contractions. This article introduces a modified form of an
admissible hybrid intuitionistic fuzzy Z-contraction within the framework of IFS-valued
mappings in extended b-metric spaces, providing sufficient conditions for intuitionistic
fuzzy fixed point (IFFP) results. Several special cases of the main result are discussed
through corollaries. The application of these findings pertains to the IFFP result in the
context of an orderes b-metric space. Each result is supported by examples to validate
the hypotheses. To the best of our knowledge, common fixed point theorems within the
framework of IFSs utilizing simulation functions have not yet been explored, making the
concepts presented here novel.

2. Preliminaries

The collections of natural, non-negative real and real numbers are indicated by N, R+,
and R, respectively.

Definition 1. [8] Assume a set X that is not empty and h ≥ 1 is a constant. Suppose
that the mapping δ : X ×X → R+ meets the criteria below for all σ, τ,ϖ ∈ X;

(i) δ(σ, τ) = 0 if and only if σ = τ ;



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3307

(ii) δ(σ, τ) = δ(τ, σ);

(iii) δ(σ, τ) ≤ h[δ(σ,ϖ) + δ(ϖ, τ)] for all σ, τ,ϖ ∈ X.

Then, the triplet (X, δ, h) is called a b-MS.

Example 2. Let X = lp(R) with 0 < p < 1, where lp(R) = {σ = {σn} ⊂ R :∑+∞n=1|σn|p < +∞}. Then δ(σ, τ) = (
∑+∞

n=1 |σn − yn|p)
1
p is a b-MS on X with h = 2

1
p .

Notice that (X, δ) is not a MS.

Definition 3. [6] Let (X, δ, h) be a b-MS. Then, the subset A of X is called:

(i) compact if and only if for every sequence of elements of A, there exists a subsequence
that is convergent to an element of A.

(ii) closed if and only if for every sequence {σn}n∈N of elements of A converging to an
element σ, we have σ ∈ A.

Definition 4. [19] A set A, containing one or more elements, in X is called proximal if,
for some σ ∈ X, there is k ∈ A such that δ(σ, k) = δ(σ,A).

Denote by N (X), CB(X), Pr(X) and Q(X), the families of nonempty subsets of X,
all nonempty closed and bounded subsets of X, all nonempty proximal subsets of X and
all nonempty compact subsets of X, respectively.
Let (X, δ) be a b-MS. For A,B ∈ Q(X), the function ℵ : Q(X)×Q(X) → R+, defined by,

ℵ(A,B) =
{

max{supσ∈A δ(σ,B), supx∈B δ(σ,A)}, if it exists
+∞, otherwise ,

is named as generalized Hausdorff distance on X and Q(X, δ, h) is Hausdorff b-M, where
δ(σ,B) = infτ∈B δ(σ, τ).

Lemma 5. [27] Let (X, δ, h) be a b-MS. For A,B ∈ Q(X) and σ, τ ∈ X, the specifications
listed below are true:

(i) δ(σ,B) ≤ ℵ(A,B) for every σ ∈ A.

(ii) δ(σ,B) ≤ δ(σ, b) for any b ∈ B.

(iii) δ(σ,A) ≤ h[δ(σ, τ) + δ(τ,A)].

(iv) δ(σ,A) = 0 ⇔ σ ∈ A.

(v) ℵ(A,B) = 0 ⇔ A = B.

(vi) ℵ(A,B) = ℵ(B,A).



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3308

Khojasteh et al. [17] introduced a family of auxiliary functions known as SFs (simula-
tion functions) to unify distinct types of contraction.

Definition 6. [17] A SF (simulation function) is an operator ℘ : R+ × R+ → R that
meets the requirements listed below:

(i) ℘(0, 0) = 0;

(ii) ℘(a, b) < b− a for all a, b > 0;

(iii) if {an}n≥1 and {bn}n≥1 are sequences in (0,+∞) such that limn→+∞ an = limn→+∞ bn >
0, then limn→+∞ sup℘(an, bn) < 0.

The collection of SFs is denoted by Z.

Example 7. Let o and ψ be two altering distance functions such that ψ(t) < t ≤ ϕ(t) for
all t > 0, also ϕ and ψ are continuous functions such that ϕ(t) = ψ(t) = 0 if and only if
t = 0. Then the mapping ℘(a, b) = ψ(b)− ϕ(a) for all a, b ∈ [0,+∞) is a SF.

To study even more details and examples of SF see [2, 11, 17]

Definition 8. [17] Let (X, δ) be a MS. A correspondence T : X → X is named as Z-
contraction with regard to ℘ ∈ Z, if

℘(δ(T σ, T τ), δ(σ, τ)) ≥ 0 for all σ, τ ∈ X

Remark 9. The description of the SF makes it evident that ℘(a, b) < 0 for all a ≥ b > 0.
Therefore, if T is a Z-contraction with respect to ℘ ∈ Z then δ(T σ, T τ) < δ(σ, τ) for
all distinct σ, τ ∈ X. As a result, it can be seen that every Z contraction mapping is
contractive and continuous.

Theorem 10. [17] Every Z-contraction on a complete MS has a unique FP. Let X rep-
resent a universal set. A function with a domain of X and values in [0, 1] = I is referred
to as a FS in X. If A is a fuzzy set in X, then the function value A(X) is called the grade
of membership of σ in A.

Definition 11. [4] Let X be a universal set. An IFS (intuitionistic fuzzy set) A is
described as:

A = {σ ∈ X; (µA(σ), νA(σ))}

with µA : X → [0, 1] and νA : X → [0, 1] denote the degree of membership and the degree of
nonmembership of each element σ to set A respectively, such that 0 ≤ µA(σ) + νA(σ) ≤ 1
for all σ ∈ X.



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3309

X

σ1
σ2
σ3
.

.

.

IFS(X)

X [0, 1]

Tσ1 =

µσ1 : σi λ
ασj

γ

β
νσ1 :

σi
σj

such that µσ1
+ νσ1

:

X [0, 1]

σi

σj

a

b

Figure 1: Pictorial representation of intuitionistic fuzzy set-valued mappings.

Example 12. Consider the following real-world example of an IFS representing the idea
of ”customer satisfaction” for a restaurant. We will design eight customer satisfaction
elements and give membership and non-membership values to every single factor.

Table 1: Intuitionistic Fuzzy Set Data

Factors Membership Non Membership

Food quality 0.8 0.1
Service speed 0.7 0.2
Cleanliness 0.6 0.2
Atmosphere 0.6 0.3

Prices 0.4 0.5
Menu variety 0.8 0.2

Staff friendliness 0.6 0.4
Noise level 0.5 0.3

Definition 13. Let L = {(⋉, β);⋉+ β ≤ 1, (⋉, β) ∈ (0, 1]× [0, 1)} and A is an IFS, then
(⋉, β)-cut set of A is defined as:

[A](⋉,β) = {σ ∈ X : µA(σ) ≥ ⋉ and νA(σ) ≤ β}

Let X be a nonempty set and Υ be a MS. A mapping T : X → IFS(X) is called IFS-valued
map. An IFS-valued map T is an IF subset of X with membership function µT (σ)(τ) and
nonmembership function νT (σ)(τ).

An IFS A in a metric linear space V is known to be an approximate quantity if and only
if [A](⋉,β) is compact and convex in V for each (⋉, β) ∈ (0, 1]×[0, 1) with supσ∈V µA(σ) = 1
and infσ∈V νA(σ) = 0. The compilation of all approximate quantities in V is denoted by



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3310

1 2 3 4 5 6 7 8

0.2

0.4

0.6

0.8
=M(A)

=N(A)

Figure 2: Scattered Plot for Membership and Non-Membership of an IFS A

W (V ).
If there is (⋉, β) ∈ (0, 1]×[0, 1) in such a way that [A](⋉,β), [B](⋉,β) ∈ Q(X), then establish

D(⋉,β)(A,B) = ℵ([A](⋉,β), [B](⋉,β))

δ+∞(A,B) = sup
(⋉,β)

D(⋉,β)(A,B).

Definition 14. [4] Let X be an arbitrary set, a point ς ∈ X is called an IFFP of an IFM
S : X → IFS(X), if there exists (⋉, β) ∈ (0, 1]× [0, 1) so that ς ∈ [Sς](⋉,β).
A point ς ∈ X is referred to as FP of an IFM S : X → IFS(X), if µ(Sς)(ς) ≥ µ(Sς)(σ)
and ν(Sς)(ς) ≤ ν(Sς)(σ) for all σ ∈ X.

Give the set of all FPs of T , the symbol Fix(T ).

Rus [24] introduced the idea of comparison function (CF), which a lot of authors have
thoroughly reviewed to expand more generic contraction type mappings.

Definition 15. [24] A function φ : R+ → R+ is known as CF (comparison function) if it
does not decrease and φn(t) → 0 as n→ +∞ for all t ≥ 0.

Definition 16. [24] A nondecreasing function φ : R+ → R+ is called:

(i) a c-CF if it fulfils the criteria that
∑+∞

n=0 φ
n(t) converges for all t > 0.

(ii) a b-CF if it meets the requirement that
∑+∞

n=0 S
nφn(t) converges for all t ∈ R+ where

(X, δ) be a b-MS with S ≥ 1.

Example 17. Let (X, δ) be a b-MS with coefficient h ≥ 1. Then φ(t) = ℘t; t ∈ R+ with
0 < ℘ < 1

h is a b-CF. The definition of b-CF becomes equivalent to c-CF when h = 1.

Indicate by Λb, the family of all functions φ : R+ × R+ fulfilling:



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3311

(i) φ is a b-CF;

(ii) φ(t) = 0 if and only if t = 0;

(iii) φ is continuous.

Lemma 18. [24] For a CF φ : R+ → R+, the characteristics, defined below, are true:

(i) each iterate φn, n ∈ N is also a CF;

(ii) φ(t) < t for all t > 0.

Lemma 19. [24] Let φ : R+ → R+ be a b-CF. Then, the series
∑+∞k=0h

kφk(t) con-
verges for each t ∈ R+.

3. Main Results

Within this part, we’ll discuss the idea of γ-admissibility of IFS-valued maps, which is
inspired by the idea of γ-admissibility raised by Samet et al. [25].

Definition 20. Let (X, δ) be a metric linear space. A mapping T : X → IFS(X) is called
an IF λ-contraction, if there is λ ∈ (0, 1) in a way that for each σ, τ ∈ X,
δ+∞(T (σ), T (τ)) ≤ λδ(σ, τ).

Definition 21. For a b-MS (X, δ, h), γ : X × X → R+ and IFSs S and T , the ordered
pair (S, T ) is known as γ-admissible if the criteria bellow are met:

(i) for each σ ∈ X and τ ∈ [Sσ](⋉(σ),β(σ)), where (⋉(σ), β(σ)) ∈ (0, 1] × [0, 1),
with γ(σ, τ) ≥ 1, we have γ(τ,ϖ) ≥ 1 for all ϖ ∈ [T τ ](⋉(τ),β(τ)) ̸= φ where
(⋉(τ), β(τ)) ∈ (0, 1]× [0, 1)

(ii) for each σ ∈ X and τ ∈ [T σ](⋉(σ),β(σ)), where (⋉(σ), β(σ)) ∈ (0, 1] × [0, 1)
with γ(σ, τ) ≥ 1, we have γ(τ,ϖ) ≥ 1 for all ϖ ∈ [Sτ ](⋉(τ),β(τ)) ̸= φ, where
(⋉(τ), β(τ)) ∈ (0, 1]× [0, 1).

If S = T , then T is called γ-admissible.

Definition 22. Let (X, δ, h) be a b-MS and S, T : X → IFS(X) are IFS-valued maps.
The ordered pair (S, T ) is said to be ℵ-continuous at ς ∈ X if for each sequence {σn}n≥1

in X,

(i) limn→+∞ δ(σn, ς) = 0 implies that limn→+∞ ℵ([Sσn](⋉(σn),β(σn)), [T ς](⋉(ς),β(ς))) = 0
where (⋉(σn), β(σn)), (⋉(ς), β(ς)) ∈ (0, 1]× [0, 1).

(ii) limn→+∞ δ(σn, ς) = 0 implies that limn→+∞ ℵ([T σn](⋉(σn),β(σn)), [Sς](⋉(ς),β(ς))) = 0
where (⋉(σn), β(σn)), (⋉(ς), β(ς)) ∈ (0, 1]× [0, 1).



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3312

Definition 23. Let (X, δ, h) be a b-MS and S and T are IFS-valued maps. The mappings
S and T are called pairwise AHIF Z-contraction with respect to ℘ ∈ Z, if there exists
(⋉, β) ∈ (0, 1]× [0, 1) a function γ : X ×X → R+ and a b-CF φ : R+ × R+ in a manner
that the following prerequisites are fulfilled:

(i) ℘(γ(σ, τ)ℵ([Sσ](⋉(σ),β(σ)), [T τ ](⋉(τ),β(τ))), φ(M
r
(S,T )(σ, τ))) ≥ 0

for all σ, τ ∈ X, where

M r
(S,T )(σ, τ) =

{
[A(σ, τ)]

1
r , for r > 0, σ, τ ∈ X,

B(σ, τ), for r = 0, σ, τ ∈ X.

A(σ, τ) = k1(δ(σ, τ))
r + k2(δ(σ, [Sσ](⋉(σ),β(σ))))

r + k3(δ(y, [T τ ](⋉(τ),β(τ))))
r

+k4

(
δ(τ, [T τ ](⋉(τ),β(τ)))(1 + δ(σ, [Sσ](⋉(σ),β(σ))))

1 + δ(σ, τ)

)r

+k5

(
δ(τ, [Sσ](⋉(σ),β(σ)))(1 + δ(σ, [T τ ](⋉(τ),β(τ))))

1 + δ(σ, τ)

)r

B(σ, τ) = (δ(σ, τ))k1 × (δ(σ, [Sσ](⋉(σ),β(σ))))
k2 × (δ(τ, [T τ ](⋉(τ),β(τ))))

k3

×
(
δ(τ, [T τ ](⋉(τ),β(τ)))(1 + δ(σ, [Sσ](⋉(σ),β(σ))))

1 + δ(σ, τ)

)k4

×
(
δ(σ, [T τ ](⋉(τ),β(τ))) + δ(τ, [Sσ](⋉(σ),β(σ)))

2h

)k5

(ii) ℘(γ(σ, τ)ℵ([T σ](⋉(σ),β(σ)), [Sτ ](⋉(τ),β(τ))), φ(M
r
(T ,S)(σ, τ))) ≥ 0

for all σ, τ ∈ X, where

M r
(T ,S)(σ, τ) =

{
[A(σ, τ)]

1
r , for r > 0, σ, τ ∈ X,

B(σ, τ), for r = 0, σ, τ ∈ X.

A(σ, τ) = k1(δ(σ, τ))
r + k2(δ(σ, [T σ](⋉(σ),β(σ))))

r + k3(δ(τ, [Sτ ](⋉(τ),β(τ))))
r

+k4

(
δ(τ, [Sτ ](⋉(τ),β(τ)))(1 + δ(σ, [T σ](⋉(σ),β(σ))))

1 + δ(σ, τ)

)r

+k5

(
δ(τ, [T σ](⋉(σ),β(τ)))(1 + δ(σ, [Sτ ](⋉(τ),β(τ))))

1 + δ(σ, τ)

)r

B(σ, τ) = (δ(σ, τ))k1 × (δ(σ, [T σ](⋉(σ),β(σ))))
k2 × (δ(τ, [Sτ ](⋉(τ),β(τ))))

k3

×
(
δ(τ, [Sτ ](⋉(τ),β(τ)))(1 + δ(σ, [T σ](⋉(σ),β(σ))))

1 + δ(σ, τ)

)k4



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3313

×
(
δ(σ, [Sτ ](⋉(τ),β(τ))) + δ(τ, [T σ](⋉(σ),β(σ)))

2h

)k5

where (⋉(σ), β(σ)), (⋉(τ), β(τ)) ∈ (0, 1] × [0, 1) with γ ≥ 0 and ki ≥ 0 such as∑5
i=1 ki = 1.

Remark 24.

(i) If the pair (S, T ) is ℵ-continuous, then (T , S) is also ℵ-continuous.

(ii) If (S, T ) is pairwise AHIF Z-contraction then so is (T , S).

Let IFSs(X) be a subset of IFS(X) defined by:

IFSs(X) = {A ∈ IFS(X) : [A](⋉,β) ∈ Q(X), (⋉, β) ∈ (0, 1]× [0, 1)}

Theorem 25. Let (X, δ, h) be a complete b-MS and S, T : X → IFS(X) be pairwise
AHIF Z-contraction pertaining to ℘ ∈ Z.
Furthermore, suppose that:

(i) (S, T ) is a γ-admissible pair;

(ii) there is σ0 ∈ X and

(a) σ1 ∈ [Sσ0](⋉(σ0),β(σ0)) such that γ(σ0, σ1) ≥ 1;

(b) σ1 ∈ [T σ0](⋉(σ0),β(σ0)) such that γ(σ0, σ1) ≥ 1;

(iii) (S, T ) is ℵ-continuous;

(iv) The sets [Sσ](⋉(σ),β(σ)) and [T σ](⋉(σ),β(σ)) are proximal for each σ ∈ X.

Then S and T have at least single common IFFP in X.

Proof. Using (ii)(a), we have (⋉(σ0), β(σ0)) ∈ (0, 1] × [0, 1), σ0 ∈ X and σ1 ∈
[Sσ0](⋉(σ0),β(σ0)) such that γ(σ0, σ1) ≥ 1. If σ1 = σ0 and T = S then, from condition
(i) in definition (23), we have

0 ≤ ℘(γ(σ, τ)ℵ([Sσ0](⋉(σ0),β(σ0)), [T σ1](⋉(σ1),β(σ1))), φ(M
r
(S,T )(σ0, σ1)))

< φ(M r
(S,T )(σ0, σ1))− γ(σ0, σ1)ℵ([Sσ0](⋉(σ0),β(σ0)), [T σ1](⋉(σ1),β(σ1)))

which is analogous to

γ(σ0, σ1)ℵ([Sσ0](⋉(σ0),β(σ0)), [T σ1](⋉(σ1),β(σ1))) ≤ φ(M r
(S,T )(σ0, σ1)) (1)

Then for r > 0, we get

M r
(S,T )(σ0, σ1) = [A(σ0, σ1)]

1
r



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3314

=

[
k1(δ(σ0, σ1))

r + k2(δ(σ0, [Sσ0](⋉(σ0),β(σ0))))
r + k3(δ(σ1, [T σ1](⋉(σ1),β(σ1))))

r

+k4

(
δ(σ1, [T σ1](⋉(σ1),β(σ1)))(1 + δ(σ0, [Sσ0](⋉(σ0),β(σ0))))

1 + δ(σ0, σ1)

)r

+k5

(
δ(σ1, [Sσ0](⋉(σ0),β(σ0)))(1 + δ(σ0, [T σ1](⋉(σ1),β(σ1))))

1 + δ(σ0, σ1)

)r
] 1

r

using proximality of S, σ1 ∈ [Sσ0](⋉(σ0),β(σ0))

M r
(S,T )(σ0, σ1) =

[
k1(δ(σ0, σ1))

r + k2(δ(σ0, σ1))
r + k3(δ(σ1, [T σ1](⋉(σ1),β(σ1))))

r

+k4

(
δ(σ1, [T σ1](⋉(σ1),β(σ1)))(1 + δ(σ0, σ1))

1 + δ(σ0, σ1)

)r

+k5

(
δ(σ1, σ1)(1 + δ(σ0, [T σ1](⋉(σ1),β(σ1))))

1 + δ(σ0, σ1)

)r
] 1

r

= [(k1 + k2)(δ(σ0, σ1))
r + (k3 + k4)(δ(σ1, [T σ1](⋉(σ1),β(σ1))))

r]
1
r

for σ1 = σ0 and T = S, we get

M r
(S,T )(σ0, σ1) = [(k1 + k2)(δ(σ0, σ0))

r + (k3 + k4)(δ(σ1, [Sσ0](⋉(σ1),β(σ1))))
r]

1
r

= 0

Similarly B(σ0, σ1) = 0. Hence (1) becomes,

γ(σ0, σ1)ℵ([Sσ0](⋉(σ0),β(σ0)), [T σ1](⋉(σ1),β(σ1))) ≤ φ(0) = 0

which implies that

ℵ([Sσ0](⋉(σ0),β(σ0)), [T σ1](⋉(σ1),β(σ1))) = 0

that is, σ1 ∈ [Sσ0](⋉(σ0),β(σ0)) = [T σ1](⋉(σ1),β(σ1))

which means σ1 is IFFP of T .
Since σ1 ∈ [Sσ0](⋉(σ0),β(σ0)) with γ(σ0, σ1) ≥ 1,so by (i), γ(σ1, ϖ) ≥ 1 for all ϖ ∈
[T σ1](⋉(σ1),β(σ1)).
As σ1 ∈ [T σ1](⋉(σ1),β(σ1)),we have γ(σ1, σ1) ≥ 1 and

0 ≤ ℘(γ(σ1, σ1)ℵ([Sσ1](⋉(σ1),β(σ1)), [T σ1](⋉(σ1),β(σ1))), φ(M
r
(S,T )(σ1, σ1)))

which is equivalent to

γ(σ1, σ1)ℵ([Sσ1](⋉(σ1),β(σ1)), [T σ1](⋉(σ1),β(σ1))) ≤ φ(M r
(S,T )(σ1, σ1)) (2)



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3315

Then for r > 0, we get

M r
(S,T )(σ1, σ1) = [A(σ1, σ1)]

1
r

=

[
k1(δ(σ1, σ1))

r + k2(δ(σ1, [Sσ1](⋉(σ1),β(σ1))))
r + k3(δ(σ1, [T σ1](⋉(σ1),β(σ1))))

r

+k4

(
δ(σ1, [T σ1](⋉(σ1),β(σ1)))(1 + δ(σ1, [Sσ1](⋉(σ1),β(σ1))))

1 + δ(σ1, σ1)

)r

+k5

(
δ(σ1, [Sσ1](⋉(σ1),β(σ1)))(1 + δ(σ1, [T σ1](⋉(σ1),β(σ1))))

1 + δ(σ1, σ1)

)r
] 1

r

Using the proximality of T , we get

M r
(S,T ) = [k1(0) + k2(δ(σ1, [Sσ1](⋉(σ1),β(σ1))))

r + k3(0) + k4(0) + k5(δ(σ1, [Sσ1](⋉(σ1),β(σ1))))
r]

1
r

= 0

Similarly B(σ1, σ1) = 0. Therefore (2) becomes,

γ(σ1, σ1)ℵ([Sσ1](⋉(σ1),β(σ1)), [T σ1](⋉(σ1),β(σ1))) ≤ φ(0) = 0

which yields

ℵ([Sσ1](⋉(σ1),β(σ1)), [T σ1](⋉(σ1),β(σ1))) = 0

implies that [Sσ1](⋉(σ1),β(σ1)) = [T σ1](⋉(σ1),β(σ1))

that is, σ1 is an IFFP of S, which yields

σ1 ∈ [T σ1](⋉(σ1),β(σ1)) ∩ [Sσ1](⋉(σ1),β(σ1))

Henceforth we presume that σ0 ̸= σ1 and S ̸= T so, σ1 /∈ [T σ1](⋉(σ1),β(σ1))∩[Sσ1](⋉(σ1),β(σ1))

which implies that σ1 /∈ [T σ1](⋉(σ1),β(σ1)). So that

δ(σ1, [T σ1](⋉(σ1),β(σ1))) > 0.

Since [T σ1](⋉(σ1),β(σ1)) ∈ Q(X) and σ1 ∈ [Sσ0](⋉(σ0),β(σ0)), there is σ2 ∈ [T σ1](⋉(σ1),β(σ1))

with σ1 ̸= σ2 such that

δ(σ1, σ2) ≤ ℵ([Sσ0](⋉(σ0),β(σ0)), [T σ1](⋉(σ1),β(σ1)))

≤ γ(σ0, σ1)ℵ([Sσ0](⋉(σ0),β(σ0)), [T σ1](⋉(σ1),β(σ1))) (3)

using (2) and (3), we get
δ(σ1, σ2) ≤ φ(M r

(S,T )(σ0, σ1))

Provided that (S, T ) is γ-admissible pair and σ2 ∈ [T σ1](⋉(σ1),β(σ1)), we have γ(σ1, σ2) ≥ 1.



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3316

If σ2 ∈ [Sσ2](⋉(σ2),β(σ2)), then taking σ1 = σ2 and T = S consistent with earlier
steps, we find directly that σ2 ∈ [Sσ2](⋉(σ2),β(σ2)) ∩ [T σ2](⋉(σ2),β(σ2)) so we suppose σ2 ̸=
[Sσ2](⋉(σ1),β(σ1)) > 0. So that

δ(σ2, [Sσ2](⋉(σ2),β(σ2))) > 0.

Since [T σ1](⋉(σ1),β(σ1)), [Sσ2](⋉(σ2),β(σ2)) ∈ Q(X) and σ2 ∈ [T σ1](⋉(σ1),β(σ1)) there exists a
point σ3 ∈ [Sσ2](⋉(σ2),β(σ2)) with σ2 ̸= σ3 such that

δ(σ2, σ3) ≤ ℵ([T σ1](⋉(σ1),β(σ1)), [Sσ2](⋉(σ2),β(σ2)))

≤ γ(σ1, σ2)ℵ([T σ1](⋉(σ1),β(σ1)), [Sσ2](⋉(σ2),β(σ2))) (4)

from condition (ii) in definition (23), we get

0 ≤ ℘(γ(σ1, σ2)ℵ([T σ1](⋉(σ1),β(σ1)), [Sσ2](⋉(σ2),β(σ2))), φ(M
r
(T ,S)(σ1, σ2)))

< φ(M r
(T ,S)(σ1, σ2))− γ(σ1, σ2)ℵ([T σ1](⋉(σ1),β(σ1)), [Sσ2](⋉(σ2),β(σ2)))

which can be written as

γ(σ1, σ2)ℵ([T σ1](⋉(σ1),β(σ1)), [Sσ2](⋉(σ2),β(σ2))) ≤ φ(M r
(T ,S)(σ1, σ2)) (5)

combining (4) and (5) yields,

δ(σ2, σ3) ≤ φ(M r
(T ,S)(σ1, σ2))

Likewise, we generate a sequence {σn}n≥1 with σ2n+1 ∈ [Sσ2n](⋉(σ2n),β(σ2n)),
σ2n+2 ∈ [T σ2n+1](⋉(σ2n+1),β(σ2n+1)), γ(σ2n, σ2n+1) ≥ 1 and γ(σ2n+1, σ2n+2) ≥ 1 such that

δ(σ2n+1, σ2n+2) ≤ φ(M r
(S,T )(σ2n, σ(2n+1))) (6)

δ(σ2n+2, σ2n+3) ≤ φ(M r
(T ,S)(σ2n+1, σ2n+2)) (7)

Now we investigate (6) and (7) under the criteria below,
Case 1: r > 0
In this case, from condition (i) in definition (23) using the proximality of T and S in (6)
we have

M r
(S,T )(σ2n, σ2n+1) = [A(σ2n, σ2n+1)]

1
r

=

[
k1(δ(σ2n, σ2n+1))

r + k2(δ(σ2n, [Sσ2n](⋉(σ2n),β(σ2n))))
r

+ k3(δ(σ2n+1, [T σ2n+1](⋉(σ2n+1),β(σ2n+1))))
r

+ k4

(
δ(σ2n+1, [T σ2n+1](⋉(σ2n+1),β(σ2n+1)))(1 + δ(σ2n, [Sσ2n](⋉(σ2n),β(σ2n))))

1 + δ(σ2n, σ2n+1)

)r

+ k5

(
δ(σ2n+1, [Sσ2n](⋉(σ2n),β(σ2n)))(1 + δ(σ2n, [T σ2n+1](⋉(σ2n+1),β(σ2n+1))))

1 + δ(σ2n, σ2n+1)

)r
] 1

r



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3317

=

[
k1(δ(σ2n, σ2n+1))

r + k2(δ(σ2n, σ2n+1))
r + k3(δ(σ2n+1, σ2n+2))

r

+ k4

(
δ(σ2n+1, σ2n+2)(1 + δ(σ2n, σ2n+1))

1 + δ(σ2n, σ2n+1)

)r

+ k5

(
δ(σ2n+1, σ2n+1)(1 + δ(σ2n, σ2n+2))

1 + δ(σ2n, σ2n+1)

)r
] 1

r

= [(k1 + k2)(δ(σ2n, σ2n+1))
r + (k3 + k4)(δ(σ2n+1, σ2n+2))

r]
1
r (8)

using (8) in (6), we get

δ(σ2n+1, σ2n+2) ≤ φ([(k1 + k2)(δ(σ2n, σ2n+1))
r + (k3 + k4)(δ(σ2n+1, σ2n+2))

r]
1
r ) (9)

Assume that δ(σ2n, σ2n+1) ≤ δ(σ2n+1, σ2n+2), then since φ is non-decreasing, we get

δ(σ2n+1, σ2n+2) ≤ φ([(k1 + k2)(δ(σ2n+1, σ2n+2))
r + (k3 + k4)(δ(σ2n+1, σ2n+2))

r]
1
r )

noting that k1 + k2 + k3 + k4 ≤ 1,

δ(σ2n+1, σ2n+2) ≤ φ([(δ(σ2n+1, σ2n+2))
r]

1
r )

= φ(δ(σ2n+1, σ2n+2))

< δ(σ2n+1, σ2n+2)

a contradiction. Consequently, we have

δ(σ2n+1, σ2n+2) ≤ δ(σ2n, σ2n+1)

so, (9) becomes

δ(σ2n+1, σ2n+2) ≤ φ(δ(σ2n, σ2n+1))

≤ φ2(δ(σ2n−1, σ2n))

...

≤ φ2n+1(δ(σ0, σ1))

δ(σ2n+1, σ2n+2) ≤ φ2n+1(δ(σ0, σ1))

Similarly, for (7) we are able to demonstrate that

δ(σ2n+2, σ2n+3) ≤ φ2n+2(δ(σ0, σ1))

from above two equations, we conclude that

δ(σn, σn+1) ≤ φn(δ(σ0, σ1)) (10)



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3318

Take m,n ∈ N where m > n, then

δ(σn, σm) ≤ hδ(σn, σn+1) + h2δ(σn+1, σn+2) + · · ·+ hm−nδ(σm−1, σm)

using (10), we get

δ(σn, σm) ≤ hφn(δ(σ0, σ1)) + h2φn+1(δ(σ0, σ1)) + · · ·+ hm−nφm−1(δ(σ0, σ1))

= hn−n+1φn(δ(σ0, σ1)) + hn−n+2φn+1(δ(σ0, σ1)) + · · ·+ hm−n+1−1φm−1(δ(σ0, σ1))

=
1

hn−1
[hnφn(δ(σ0, σ1)) + hn+1φn+1(δ(σ0, σ1)) + · · ·+ hm−1φm−1(δ(σ0, σ1))]

=
1

hn−1

m−1∑
i=n

hiφi(δ(σ0, σ1))

≤ 1

hn−1

+∞∑
i=0

hiφi(δ(σ0, σ1))

thus

δ(σn, σm) ≤ 1

hn−1

+∞∑
i=0

hiφi(δ(σ0, σ1)) (11)

Since φ is a b-CF, it follows that the series
∑+∞

i=0 h
iφi(δ(σ0, σ1)) is convergent. Setting,

Sk =
k∑

i=1

hiφi(δ(σ0, σ1))

then (11) becomes

δ(σn, σm) ≤ 1

hn−1
(Sm−1 − Sn−1) (12)

applying limit as n,m→ +∞ in (12) we attain δ(σn, σm) → 0, this indicates that {σn}n≥1

is a Cauchy sequence in X. Completeness of X demonstrates that there is ς ∈ X, so that

lim
n→+∞

δ(σn, ς) = 0.

Now we show that ς ∈ [T ς](⋉(ς),β(ς)) by using TI in X,

δ(ς, [T ς](⋉(ς),β(ς))) ≤ h(δ(ς, σ2n+1)) + δ(σ2n+1, [T ς](⋉(ς),β(ς)))

= h(δ(ς, σ2n+1)) + h(δ(σ2n+1)).

using δ(σ,B) ≤ ℵ(A,B) for σ ∈ A,

δ(ς, [T ς](⋉(ς)+β(ς))) ≤ hδ(ς, σ2n+1) + hℵ([Sσ2n](⋉(σ2n),β(σ2n)), [T ς](⋉(ς),β(ς))) (13)

Since (S, T ) pair is ℵ-continuous, by applying limit as n→ +∞ in (13), we get



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3319

δ(ς, [T ς](⋉(ς),β(ς))) = 0 this indicate that ς ∈ [T ς](⋉(ς),β(ς))

Analogously it can be shown that ς ∈ [Sς](⋉(ς),β(ς)).
Thus

ς ∈ [T ς](⋉(ς),β(ς)) ∩ [Sς](⋉(ς),β(ς))

that is, ς is a common IFFP of mappings S and T .
Case 2: r = 0
For r = 0, taking inequality (6) with condition (i), we have

M r
(S,T )(σ2n, σ2n+1) = B(σ2n, σ2n+1)

= (δ(σ2n, σ2n+1))
k1 × (δ(σ2n, [Sσ2n](⋉(σ2n),β(σ2n))))

k2

× (δ(σ2n+1, [T σ2n+1](⋉(σ2n+1),β(σ2n+1))))
k3

×
(
δ(σ2n+1, [T σ2n+1](⋉(σ2n+1),β(σ2n+1)))(1 + δ(σ2n, [Sσ2n](⋉(σ2n),β(σ2n))))

1 + δ(σ2n, σ2n+1)

)k4

×
(
δ(σ2n, [T σ2n+1](⋉(σ2n+1),β(σ2n+1))) + δ(σ2n+1, [Sσ2n](⋉(σ2n),β(σ2n)))

2h

)k5

= (δ(σ2n, σ2n+1))
k1 × (δ(σ2n, σ2n+1))

k2 × (δ(σ2n+1, σ2n+2))
k3

×
(
δ(σ2n, σ2n+2)(1 + δ(σ2n, σ2n+1))

1 + δ(σ2n, σ2n+1)

)k4

×
(
δ(σ2n, σ2n+2) + δ(σ2n+1, σ2n+1)

2h

)k5

≤ (δ(σ2n, σ2n+1))
k1+k2 × (δ(σ2n+1, σ2n+2)

k3+k4)

×
(
h(δ(σ2n, σ2n+1) + δ(σ2n+1, σ2n+2))

2h

)k5

= (δ(σ2n, σ2n+1))
k1+k2 × (δ(σ2n+1, σ2n+2))

k3+k4

×
(
δ(σ2n, σ2n+1) + δ(σ2n+1, σ2n+2)

2

)k5

(14)

It is widely acknowledged that, for any p, q, l > 0 we have(
p+ q

2

)l

≤ pl + ql

2
(15)

applying (15) to (14), we get

M r
(S,T )(σ2n, σ2n+1) ≤ (δ(σ2n, σ2n+1))

k1+k2 × (δ(σ2n+1, σ2n+2))
k3+k4

×
(
(δ(σ2n, σ2n+1))

k5

2
+

(δ(σ2n+1, σ2n+2))
k5

2

)
(16)



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3320

using (16) in (6), we get

δ(σ2n+1, σ2n+2) ≤ φ(δ(σ2n, σ2n+1))
k1+k2 × (δ(σ2n+1, σ2n+2))

k3+k4

×
(
(δ(σ2n, σ2n+1))

k5

2
+

(δ(σ2n+1, σ2n+2))
k5

2

)
(17)

Suppose δ(σ2n, σ2n+1) ≤ δ(σ2n+1, σ2n+2) and noting that φ is non-decreasing, (17) gives:

δ(σ2n+1, σ2n+2) ≤ φ((δ(σ2n+1, σ2n+2))
k1+k2+k3+k4+k5)

= φ(δ(σ2n+1, σ2n+2))

< δ(σ2n+1, σ2n+2)

a contradiction. So we have

δ(σ2n+1, σ2n+2) ≤ δ(σ2n, σ2n+1)

therefore (17) becomes

δ(σ2n+1, σ2n+2) ≤ φ(δ(σ2n, σ2n+1))

≤ φ2(δ(σ2n−1, σ2n))

...

≤ φ2n+1(δ(σ0, σ1))

δ(σ2n+1, σ2n+2) ≤ φ2n+1(δ(σ0, σ1)).

Similarly, using (7) we can show that

δ(σ2n+2, σ2n+3) ≤ φ2n+2(δ(σ0, σ1))

combining above two equations, we can write

δ(σn, σn+1) ≤ φn(δ(σ0, σ1)). (18)

Adhering to the same process as in case 1, this can be deduced from (18) that {σn}n≥1 is
a Cauchy sequence in X. Completeness of X reveals that there is an element ς ∈ X such
as,

lim
n→+∞

δ(σn, ς) = 0. (19)

Now, to illustrate that ς ∈ [T ς](⋉(ς),β(ς)), consider

δ(ς, [T ς](⋉(ς),β(ς))) ≤ h(δ(ς, σ2n+1) + δ(σ2n+1, [T ς](⋉(ς),β(ς))))

≤ h(δ(ς, σ2n+1) + ℵ([Sσ2n](⋉(σ2n),β(σ2n)), [T ς](⋉(ς),β(ς)))) (20)

using ℵ-continuity of (S, T ) pair, by allowing n → +∞ in (20) and considering (19), we
attain



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3321

δ(ς, [T ς](⋉(ς),β(ς))) = 0 which implies that ς ∈ [T ς](⋉(ς),β(ς)).
Correspondingly we can prove that δ(ς, [Sς](⋉(ς),β(ς))) = 0 which means that ς ∈ [Sς](⋉(ς),β(ς)).
Thus we have

ς ∈ [T ς](⋉(ς),β(ς)) ∩ [Sς](⋉(ς),β(ς))

that is, ς is common IFFP of S and T .

Example 26. Let X = [1,+∞) and δ(σ, τ) = |σ−τ |2 for all σ, τ ∈ X. Then (X, δ, h = 2)
is a complete b-MS. For each σ ∈ X, consider IFS-valued maps S, T : X → IFS(X) and
Sσ, T σ are IFSs such that µT σ : X → [0, 1], µSσ : X → [0, 1] are membership functions of
T σ and Sσ respectively and νT σ : X → [0, 1] and νSσ : X → [0, 1] are non-membership
functions of T σ and Sσ respectively with µT σ(t) + νT σ(t) ≤ 1 and µSσ(t) + νSσ(t) ≤ 1
for all t ∈ X.
If σ = 1:

µT1(t) =

{
1
8 , if t = 1
2
5 , if t ̸= 1

νT1(t) =

{
3
7 , if t = 1
4
7 , if t ̸= 1

µS1(t) =

{
1, if t = 1
1
5 , if t ̸= 1

νS1(t) =

{
0, if t = 1
4
9 , if t ̸= 1.

If σ ̸= 1;

µT σ(t) =


⋉, if 1 ≤ t ≤ 3σ
1− 2

3⋉, if 3σ < t ≤ 5σ
⋉
6 , if 5σ < t < +∞

νT σ(t) =


β
4 , if 1 ≤ t ≤ 2σ
0, if 2σ < t ≤ 5σ
β, if 5σ < t < +∞

µSσ(t) =


⋉, if 1 ≤ t ≤ 5σ
⋉
3 , if 5σ < t ≤ 9σ
2⋉
19 , if 9σ < t < +∞



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3322

νSσ(t) =


β2, if 1 ≤ t ≤ 7σ
β3

5 , if 7σ < t ≤ 11σ
β, if 11σ < t < +∞.

Let ⋉ = 3
8 and β = 1

2 . Then

[T σ](⋉,β) =

{
{1}, if σ = 1
[1, 5σ], if σ ̸= 1

and

[Sσ](⋉,β) =

{
{1}, if σ = 1
[1, 5σ], if σ ̸= 1.

Clearly Sσ, T σ ∈ IFS(X) for each σ ∈ X. Define the functions γ : X × X → R+ and
φ : R+ → R+ by

γ(σ, τ) =


6, if σ = τ = 1
1

430 , if σ, τ ∈ {4, 5}
0, elsewhere.

and φ(t) = t
4 for all t > 0. Let ℘(a, b) = 1

2b − a for all a, b ∈ R+. Obviously ℘ ∈ Z and
φ ∈ Λb

Now we verify conditions

℘(γ(σ, τ)ℵ([Sσ](⋉,β), [T τ ](⋉,β)), φ(M
r
(S,T )(σ, τ))) ≥ 0

and
℘(γ(σ, τ)ℵ([T σ](⋉,β), [Sτ ](⋉,β)), φ(M

r
(T ,S)(σ, τ))) ≥ 0

for r > 0 under the these cases;
Case 1
if σ = τ = 1, Then [Sσ](⋉,β) = {1} = [T τ ](⋉,β), this implies that

ℵ([Sσ](⋉,β), [T τ ](⋉,β)) = 0

so,
℘(6(0), φ(M r

(S,T )(σ, τ))) ≥ 0.

Similarly
℘(γ(σ, τ)ℵ([T σ](⋉,β), [Sτ ](⋉,β)), φ(M

r
(T ,S)(σ, τ))) ≥ 0.

Case 2
if σ, τ ∈ {4, 5} such that σ ̸= τ , let σ = 4 and τ = 5. Then [Sσ](⋉,β) = [1, 20] and
[T τ ](⋉,β) = [1, 25].

ℵ([Sσ](⋉,β), [T τ ](⋉,β)) = ℵ([1, 20], [1, 25])



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3323

= δ(20, 25) = 25

M r
(S,T )(4, 5) = [A(4, 5)]

1
r

=

[
k1(δ(4, 5))

r + k2(δ(4, [S4](⋉,β)))
r + k3(δ(5, [T 5](⋉,β)))

r

+k4

(
δ(5, [T 5](⋉,β))(1 + δ(4, [S4](⋉,β)))

1 + δ(4, 5)

)r

+k5

(
δ(5, [S4](⋉,β))(1 + δ(4, [T 5](⋉,β)))

1 + δ(4, 5)

)r
] 1

r

taking k1 = k2 =
1
2 and k3 = k4 = k5 = 0,

M r
(S,T )(4, 5) = [

1

2
(1)r +

1

2
(0)r]

1
r

=

[
1

2

] 1
r

→ 1 as r → +∞

Therefore,

℘

(
1

430
(25), φ(1)

)
=

1

2

(
1

4

)
− 25

430
≥ 0

Similarly,
℘(γ(σ, τ)ℵ([T σ](⋉,β), [Sτ ](⋉,β)), φ(M

r
(T ,S)(σ, τ))) ≥ 0

Case 3
If σ, τ ∈ X − {1, 4, 5}. Then γ(σ, τ) = 0

℘(0, φ(M r
(S,T )(σ, τ)) =

1

2
φ(M r

(S,T )(σ, τ))) ≥ 0

Similarly,
℘(0, φ(M r

(T ,S)(σ, τ))) ≥ 0.

Moreover, it is clear that the pair (S, T ) is γ-admissible, ℵ-continuous and [Sσ](⋉,β), [T σ](⋉,β)

are proximal for each σ ∈ X. As the conditions outlined in Theorem 2 are fulfilled, we see
that S and T have many common IFFPs.

Here is another example(non-trivial) for our main result,

Example 27. Let X = lp(R) with 0 < p < 1, where lp(R) = {σ = {σn} ⊂ R :∑+∞n=1|σn|p < +∞}. Then δ(σ, τ) = (
∑+∞

n=1 |σn − yn|p)
1
p is a complete b-MS on X

with h = 2
1
p , consider IFS-valued maps S, T : X → IFS(X) and Sσ, T σ are IFSs such

that µT σ : X → [0, 1], µSσ : X → [0, 1] are membership functions of T σ and Sσ



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3324

respectively and νT σ : X → [0, 1] and νSσ : X → [0, 1] are non-membership functions
of T σ and Sσ respectively with µT σ(t)+νT σ(t) ≤ 1 and µSσ(t)+νSσ(t) ≤ 1 for all t ∈ X.

µTσ(t) =


1
8 , if t contains finite number of zero
9
10 , if t contains infinite number of zero
1

100 , if t contains no zero

νTσ(t) =


1
5 , if t contains finite number of zero
1

120 , if t contains infinite number of zero
1
7 , if t contains no zero

µSσ(t) =


1
50 , if t contains no zero
2
5 , if t contains finite number of zero
7
10 , if t contains infinite number of zero

νSσ(t) =


1
9 , if t contains no zero
3
7 , if t contains finite number of zero
2
15 , if t contains infinite number of zero

Let ⋉ = 1
2 and β = 1

6 , then
[Tσ](⋉,β) = {t contains infinitely many zeros}

and

[Sσ](⋉,β) = {t contains infinitely many zeros} Clearly Sσ, T σ ∈ IFS(X) for each
σ ∈ X. Define the functions γ : X ×X → R+ and φ : R+ → R+ by

γ(σ, τ) =

{
7, if σ = y
10, elsewhere.

and φ(t) = t
4 for all t > 0. Let ℘(a, b) = b−a

10 for all a, b ∈ R+. Obviously ℘ ∈ Z and
φ ∈ Λb

. Also,

ℵ([Sσ](⋉,β), [Tσ](⋉,β) = max(

∞∑
i=1

|σi − yi|p)
1
p

Clearly,
℘(γ(σ, τ)ℵ([T σ](⋉,β), [Sτ ](⋉,β)), φ(M

r
(T ,S)(σ, τ))) ≥ 0

As it can be seen that the pair (S, T ) is γ-admissible, ℵ-continuous and [Sσ](⋉,β), [T σ](⋉,β)

are proximal for each σ ∈ X. Since the conditions in Theorem 2 are fulfilled, we see that
S and T have many common IFFPs.



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3325

0.5 1.0 1.5 2.0 2.5 3.0
ϱ

5

10

15

[Tϱ]��,β

Figure 3: Graph representing common fixed points of the given mappings in example 25.

4. Consequences

In this section, we demonstrate how our primary theorem can be used to obtain a few
intriguing fixed point conclusions, particularly when using different simulation function
variations. All of the results reported here are also new as far as we can tell.

Corollary 1. Let (X, δ, h) be a complete b-MS and T be an AHIF Z-contraction regarding
℘ ∈ Z. Let’s also consider that:

(i) T is a γ-admissible IFS-valued map;

(ii) there are σ0 ∈ X and σ1 ∈ [T σ0](⋉,β) such that γ(σ0, σ1) ≥ 1, where (⋉, β) ∈
(0, 1]× [0, 1);

(iii) T is ℵ-continuous;

(iv) [T σ](⋉,β) is proximal for every σ ∈ X.

Then, T has at least one IFFP in X.

Proof. If we take the mapping T =S in Theorem 2, then it can easily be seen that the
AHIF Z-contraction T has many IFFPs in X.

Corollary 2. Let (X, δ, h) be a complete b-MS and T be an IFS-valued map satisfying:

(i) T is γ-admissible IFS-valued map;

(ii) there exists σ0 ∈ X and σ ∈ [T σ0](⋉,β) such that γ(σ0, σ1) ≥ 1 where (⋉, β) ∈
(0, 1]× [0, 1);

(iii) T is ℵ-continuous;

(iv) [T σ](⋉,β) is proximal for each σ ∈ X.



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3326

Additionally, assume that there exists ℘ ∈ Z, φ ∈ Λb and an operator γ : X ×X → R+ in
a way that for all σ, τ ∈ X,

℘(γ(σ, τ)ℵ([T σ](⋉,β), [T τ ](⋉,β)), φ([A(σ, τ)]
1
r )) ≥ 0 (21)

Then, T has at least one IFFP in X.

Proof. Set ρ(τ, κ) = ϕ(κ)− τ for all τ, κ ∈ R+ in Theorem 25. Then, 21 follows easily.
Note that ϕ(κ) − τ ∈ Z. Consequently Theorem 25 can be applied to find u ∈ X such
that u ∈ [Tu](⋉,β).

Corollary 3. Let (X, δ) be a complete b-MS and T : X → IFS(X) be an IFS-valued map
satisfying the condition:

γ(σ, τ)ℵ([T σ](⋉,β), [T τ ](⋉,β)) ≤ φ(M r
T (σ, τ)) (22)

for all σ, τ ∈ X, where φ ∈ Λb and γ : X ×X → R+ is a function. Furthermore, it can be
assumed that:

(i) T is γ-admissible;

(ii) there are σ0 ∈ X and σ1 ∈ [T σ0](⋉,β) with γ(σ0, σ1) ≥ 1

(iii) T is ℵ-continuous;

(iv) [T σ](⋉,β) is proximal for each σ ∈ X;

Then there is ς ∈ X such that ς ∈ [T ς](⋉,β).

Proof. Take ℘ := ℘(a, b) = φ(b) − a for each a, b ∈ R+ in corollary (1). Then (22)
follows easily. Notice that φ(b) − a ∈ Z. Accordingly, by utilizing corollary (1), ς ∈ X
can be determined such that ς ∈ [T ς](⋉,β).

The ensuing generalization and fuzzification of Rhoades’ outcome [23] are as follows:

Corollary 4. Let (X, δ, h) be a complete b-MS and T : X → IFS(X) be an IFS-valued
map satisfying the following:

ℵ([T σ](⋉,β), [T τ ](⋉,β)) ≤ φ(M r
T (σ, τ))− φ2(M r

T (σ, τ)) (23)

for all σ, τ ∈ X, where φ : R+ → R+ is lower semicontinuous and φ−1(0) = {0}. Further
let us suppose that:

(i) T is ℵ-continuous;

(ii) [T σ](⋉,β) is proximal for each σ ∈ X.

Then, there exists ς ∈ X such that ς ∈ [T ς](⋉,β).



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3327

Proof. In corollary (1), take ℘ := ℘(a, b) = b−φ(b)−a for all a, b ∈ R+ and γ(σ, τ) = 1
for all σ, τ ∈ X. Then (23) is attained. Observe that b−φ(b)− a ∈ Z. Thus, by corollary
(1), T has an IFFP in X.

Corollary 5. Nadler’s type [16]) Let (X, δ, h) be a complete b-MS and T : X → IFS(X)
be an IFS-valued map satisfying:

ℵ([T σ](⋉,β), [T τ ](⋉,β)) ≤ λδ(σ, τ) (24)

for all σ, τ ∈ X, where λ ∈ (0, 1). Assume also that:

(i) T is ℵ-continuous;

(ii) [T σ](⋉,β) is proximal for each σ ∈ X.

Then T has an IFFP in X.

Proof. Consider γ(σ, τ) = 1, ℘ := ℘(a, b) = λb − a for all a, b ∈ R+ and insert
φ(a) = λa for all a ≥ 0, with λ ∈ (0, 1) in corollary (1). Then (24) is achievable. It
can be obsereved that λb − a ∈ Z. Corollary (1) yields that there is ς ∈ X such that
ς ∈ [T ς](⋉,β).

The subsequent corollary builds upon Heilpern’s initial metric FP theorem [15]:

Corollary 6. Let (X, δ, h) ba e complete b-MS and T : X →W (X) be an IFS-valued map
satisfying:

δ+∞(T σ, T τ) ≤ λδ(σ, τ) (25)

for all σ, τ ∈ X, where λ ∈ (0, 1). In addition, suppose that:

(i) T is ℵ-continuous;

(ii) T σ is proximal for each σ ∈ X.

Then, there is ς ∈ X such that {ς} ∈ X such as {ς} ⊂ T ς.

Proof. Since ℵ([T σ](⋉,β), [T τ ](⋉,β)) ≤ δ+∞(T σ, T τ) for all σ, τ ∈ X, by employing
Corollary 5, we can identify ς ∈ X so that {ς} ⊂ T ς = [T ς](⋉,β) ∈W (X).

The definition of single-valued γ-admissible map proposed by Samet in [25] is provided
here:

Definition 28. Let O : X → X and γ : X ×X → R+ be mappings. Then, O is named as
γ-admissible if for all x, τ ∈ X,

γ(σ, τ) ≥ 1 ⇒ γ(Oσ,Oτ) ≥ 1. (26)

Chifu and Karapinar came up with this corollary without relying on triangular γ-
orbital admissibility of O:



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3328

Corollary 7. Let (X, δ, h) be a complete b-MS and O : X → X be a γ-admissible single-
valued mapping fulfilling:

℘(γ(σ, τ)ℵ(Oσ,Oτ), φ(M r
O(σ, τ)) ≥ 0 (27)

for all σ, τ ∈ X, where φ ∈ Λb,

M r
O(σ, τ) =

{
[W (σ, τ)]

1
r , for r > 0, σ, τ ∈ X

D(σ, τ), for r = 0, σ, τ ∈ X,

W (σ, τ) = k1(δ(σ, τ))
r + k2(δ(σ,Oσ))

r + k3(δ(τ,Oτ))
r

+k4

(
δ(τ,Oτ)(1 + δ(σ,Oσ))

1 + δ(σ, τ)

)r

+ k5

(
δ(τ,Oσ)(1 + δ(σ,Oτ))

1 + δ(σ, τ)

)r

and

D(σ, τ) = (δ(σ, τ))k1 × (δ(σ,Oσ))k2 × (δ(τ,Oτ))k3

×
(
δ(τ,Oτ)(1 + δ(σ,Oσ))

1 + δ(σ, τ)

)k4

×
(
δ(σ,Oτ) + δ(τ,Oσ)

2h

)k5

with r ≥ 0 and ki ≥ 0 (i = 1, 5) such that
∑5

i=1 ki = 1. Then, there exists ς ∈ X such that
Oς = ς.

Proof. Let (⋉, β) ∈ (0, 1] × [0, 1) and for each σ ∈ X, consider an IFS-valued map T
and for σ ∈ X, T σ is an IFS such that µT σ : X → [0, 1] is membership function and
νT σ : X → [0, 1] is nonmembership function. We define these maps as

µT σ(a) =

{
⋉, if a = Oσ
0, if a ̸= Oσ,

and

νT σ(a) =

{
0, if a = Oσ
β, if a ̸= Oσ,

Then,
[T x](⋉,β) = {a ∈ X : µT σ(a) ≥ ⋉ and νT σ(a) ≤ β}.

Obviously, {Oσ} ∈ Q(X). Notice that in this case, ℵ([T σ](⋉,β), [T τ ](⋉,β)) = δ(Oσ,Oτ).
Therefore, corollary (1) can be applied to obtain ς ∈ X such that ς ∈ [T ς](⋉,β) = {Oς},
which further implies that ς = Oς.



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3329

5. Application

One of the most helpful subfields of fixed theory, FP theory in partially ordered sets
has many applications, including solving matrix equations and solving boundary value
issues. See [3, 20, 21] for a few articles that go in this direction. Here, we will utilize
our principal discovery to derive its counterpart within the framework of ordered b-MS.
Importantly, a b-MS can be equipped with a partial ordering. To elaborate, if (X,⪯) is
a partially ordered set, then (X, δ, h,⪯) is recognized as an ordered b-MS. As a result,we
define, σ, τ ∈ X are comparable if either σ ⪯ τ or τ ⪯ σ is fulfilled. Consider £,ℜ ⊆ X,
then £ ⪯ ℜ if for all l ∈ £, there is r ∈ ℜ with l ⪯ r.

Theorem 29. Let (X, δ, h,⪯) be a complete ordered b-MS and S, T are IFS-valued maps.
Consider that there is ℘ ∈ Z, φ ∈ Λb and an operator γ : X ×X → R+ such that

℘(γ(σ, τ))ℵ([Sσ](⋉(σ),β(σ)), [T τ ](⋉(τ),β(τ))), φ(M
r
(S,T )(σ, τ)) ≥ 0 (28)

for all σ, τ ∈ X with [Sσ](⋉(σ),β(σ)) ⪯ [T τ ](⋉(τ),β(τ)) and

℘(γ(σ, τ))ℵ([T σ](⋉(σ),β(σ)), [Sτ ](⋉(τ),β(τ))), φ(M
r
(S,T )(σ, τ)) ≥ 0 (29)

for all σ, τ ∈ X with [T σ](⋉(σ),β(σ)) ⪯ [Sτ ](⋉(τ),β(τ)). We further require that the following
conditions be satisfied:

(i) there is σ0 ∈ X and
a) σ1 ∈ [Sσ0](⋉(σ0),β(σ0)) such that [Sσ0](⋉(σ0),β(σ0)) ⪯ [T σ1](⋉(σ1),β(σ1));
b) σ1 ∈ [T σ0](⋉(σ0),β(σ0)) such that [T σ0](⋉(σ0),β(σ0)) ⪯ [Sσ1](⋉(σ1),β(σ1))

(ii) for each σ ∈ X and
a) τ ∈ [Sσ](⋉(σ),β(σ)) with [Sσ](⋉(σ),β(σ)) ⪯ [T τ ](⋉(τ),β(τ)),
we have [T τ ](⋉(τ),β(τ)) ⪯ [Sϖ](⋉(ϖ),β(ϖ)) for all ϖ ∈ [T τ ](⋉(τ),β(τ));
b) τ ∈ [T σ](⋉(σ),β(σ)) with [T σ](⋉(σ),β(σ)) ⪯ [Sτ ](⋉(τ),β(τ)), we have
[Sτ ](⋉(τ),β(τ)) ⪯ [T ϖ](⋉(ϖ),β(ϖ)) for all ϖ ∈ [Sτ ](⋉(τ),β(τ))

(iii) The pair (S, T ) is ℵ-continuous;

(iv) The sets [Sσ](⋉(σ),β(σ)) and [T σ](⋉(σ),β(σ)) are proximal for each σ ∈ X.

Then, S and T have at least one common IFFP.

Proof. Let the function γ : X ×X → R+ be defined by

γ(σ, τ) =

{
1, if [Sσ](⋉(σ),β(σ)) ⪯ [T τ ](⋉(τ),β(τ)) or [T σ](⋉(σ),β(σ)) ⪯ [Sτ ](⋉(τ),β(τ)),

0, otherwise.

To show that the pair (S, T ) is γ-admissible, take

(i) σ ∈ X and τ ∈ [Sσ](⋉(σ),β(σ)) with γ(σ, τ) ≥ 1 then, [Sσ](⋉(σ),β(σ)) ⪯ [T τ ](⋉(τ),β(τ))

and by hypothesis (ii)(a),we have [T τ ](⋉(τ),β(τ)) ⪯ [Sϖ](⋉(ϖ),β(ϖ)) for all ϖ ∈
[T τ ](⋉(τ),β(τ)). It follows that γ(τ,ϖ) ≥ 1 for all ϖ ∈ [T τ ](⋉(τ),β(τ)).



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3330

(ii) σ ∈ X and τ ∈ [T σ](⋉(σ),β(σ)) with γ(σ, τ) ≥ 1 then, [T σ](⋉(σ),β(σ)) ⪯ [Sτ ](⋉(τ),β(τ))

and by hypothesis (ii)(b),we have [Sτ ](⋉(τ),β(τ)) ⪯ [T ϖ](⋉(ϖ),β(ϖ)) for all ϖ ∈
[Sτ ](⋉(τ),β(τ)). It follows that γ(τ,ϖ) ≥ 1 for all ϖ ∈ [Sτ ](⋉(τ),β(τ)).

Moreover, by inequalities (28) and (29), we find that the pair (S, T ) is an AHIF Z-
contraction regarding ℘ ∈ Z. As a result, it can be seen that all the axioms of the
Theorem 2 are met. Thus S and T have at least one common IFFP in X.

Example 30. Let X = N be a partially ordered set such that a ⪯ b if and only if b|a for
all a, b ∈ X. Define δ(σ, τ) = |σ − τ |3 for all σ, τ ∈ X. Then (X, δ, h = 4) is a complete
b-MS but not a MS. For each σ ∈ X consider two IFS-valued maps S, T : X → IFS(X)
and for σ ∈ X, T σ, Sσ are IFSs such that µSσ, µT σ : X → [0, 1] are membership functions
and νSσ, νT σ : X → [0, 1] are non-membership functions of S and T respectively with
µSσ(a) + νSσ(a) ≤ 1 and µT σ(a) + νT σ(a) ≤ 1 for all a ∈ X. We define these maps as:
Case 1:
If σ is even;

µT σ(a) =


⋉, if a = σ

2
⋉
3 , if a = σ
0, elsewhere

νT σ(a) =


β
5 , if a = σ

2
β
2 , if a = σ
β, elsewhere

µSσ(a) =


⋉, if a = σ

2
1− ⋉

3 , if a = σ
0, elsewhere

νSσ(a) =


β
6 , if a = σ

2
β
3 , if a = σ
β, elsewhere.

Case 2:
If σ is odd;

µT σ(a) =


1−⋉, if a = σ + 1
⋉, if a = σ
⋉3, elsewhere

νT σ(a) =


0, if a = σ + 1
β2, if a = σ
β4, elsewhere



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3331

µSσ(a) =


⋉
2 , if a = σ + 1
⋉, if a = σ
0, elsewhere

νSσ(a) =


β, if a = σ + 1
β2, if a = σ
β3, elsewhere.

Let ⋉ = 3
5 and β = 1

5 . Then

[T σ](⋉,β) =

{
{σ
2 }, if σ is even

{σ}, if σ is odd

and

[Sσ](⋉,β) =

{
{σ
2 }, if σ is even

{σ}, if σ is odd.

Clearly Sσ, T σ ∈ IFS(X) for each σ ∈ X. Define the functions γ : X × X → R+ and
φ : R+ → R+ by

γ(σ, τ) =


6, if σ = τ = 1
1

150 , if σ, τ ∈ {2, 3} such that σ ̸= τ
0, otherwise.

and φ(t) = t
4 for all t > 0. Let ℘(a, b) = 1

2b − a for all a, b ∈ R+. Obviously ℘ ∈ Z and
φ ∈ Λb

Now we verify conditions

℘(γ(σ, τ)ℵ([Sσ](⋉,β), [T τ ](⋉,β)), φ(M
r
(S,T )(σ, τ))) ≥ 0

and
℘(γ(σ, τ)ℵ([T σ](⋉,β), [Sτ ](⋉,β)), φ(M

r
(T ,S)(σ, τ))) ≥ 0

for r > 0 under the following cases;
Case 1:
If σ = τ = 1, then
[Sσ](⋉,β) = [T τ ](⋉,β) = {1} this implies that

ℵ([Sσ](⋉,β), [T τ ](⋉,β)) = 0

so,
℘(6(0), φ(M r

(S,T )(σ, τ))) ≥ 0

Similarly
℘(γ(σ, τ)ℵ([T σ](⋉,β), [Sτ ](⋉,β)), φ(M

r
(T ,S)(σ, τ))) ≥ 0



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3332

Case 2:
If σ, τ ∈ {2, 3} such that σ ̸= τ , Let σ = 2 and τ = 3. Then [Sσ](⋉,β) = {1} and
[T τ ](⋉,β) = {3} which implies that

ℵ([Sσ](⋉,β), [T τ ](⋉,β)) = ℵ({1}, {3})
= δ(1, 3) = 8

M r
(S,T )(2, 3) = [A(2, 3)]

1
r

=

[
k1(δ(2, 3))

r + k2(δ(2, [S2](⋉,β)))
r + k3(δ(3, [T 3](⋉,β)))

r

+k4

(
δ(3, [T 3](⋉,β))(1 + δ(2, [S2](⋉,β)))

1 + δ(2, 3)

)r

+k5

(
δ(3, [S2](⋉,β))(1 + δ(2, [T 3](⋉,β)))

1 + δ(2, 3)

)r
] 1

r

taking k1 = k2 =
1
2 and k3 = k4 = k5 = 0,

M r
(S,T )(2, 3) = [

1

2
(1)r +

1

2
(1)r]

1
r

= {1}
1
r = 1

Therefore,

℘

(
1

150
(8), φ(1)

)
≥ 0

Similarly,
℘(γ(σ, τ)ℵ([T σ](⋉,β), [Sτ ](⋉,β)), φ(M

r
(T ,S)(σ, τ))) ≥ 0

Case 3
If σ, τ ∈ X − {1, 2, 3}, then γ(σ, τ) = 0 which implies that

℘(0, φ(M r
(S,T )(σ, τ)) =

1

2
φ(M r

(S,T )(σ, τ))) ≥ 0

Similarly,
℘(0, φ(M r

(T ,S)(σ, τ))) ≥ 0.

Moreover, it is clear that the pair (S, T ) is γ-admissible, ℵ-continuous and the sets [Sσ](⋉,β), [T σ](⋉,β)

are proximal for each σ ∈ X. Hence S and T have at least one common IFFP in X.



N. Saleem et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3304-3335 3333

6. Conclusion

In this paper, we introduce the concept of Admissible Hybrid Intuitionistic Fuzzy
(AHIF) Z-contractions within the framework of complete b-metric spaces (b-MSs), a sig-
nificant innovation that aims to extend and generalize existing results in intuitionistic fuzzy
fixed point (IFFP) theory. By developing new definitions and constructing a theorem that
ensures the existence of at least one common IFFP, our work advances the understanding
of fixed points in the context of b-metric spaces, a structure that has gained considerable
attention due to its flexibility in handling generalized distance measures.

The importance of our work lies in its ability to unify and expand upon prior theorems,
which were limited in scope to more restrictive spaces and less generalized contractions.
Through the introduction of AHIF Z-contractions, we establish a broader framework for
studying common fixed points in intuitionistic fuzzy settings. Our results are not only
theoretical but also practical, as demonstrated by the example that showcases the appli-
cation of these findings in ordered b-metric spaces. This example illustrates the relevance
of our work for real-world problems where common IFFPs play a crucial role, further
highlighting the utility of our contributions.

Moreover, our work builds upon foundational fixed point theorems and intuitionistic
fuzzy set theory by providing new pathways for exploration within the framework of b-
metric spaces. The construction of AHIF Z-contractions and the corresponding fixed point
results represent a significant extension of existing literature. These contributions refine
and broaden the applicability of IFFP results, offering new perspectives for researchers
interested in the interplay between b-metric spaces and intuitionistic fuzzy systems.

The introduction of the AHIF Z-contraction, along with its practical implications, sig-
nals a meaningful advancement in fixed point theory, particularly within the context of
intuitionistic fuzzy systems in b-metric spaces. By demonstrating the existence of com-
mon fixed points in an ordered b-metric space, we not only validate our theoretical results
but also provide a robust example that may inspire further applications across various
domains. This research contributes to the growing body of work on generalized metric
spaces and intuitionistic fuzzy systems, and we anticipate that it will foster additional
exploration and innovation in these areas, leading to new developments in both theory
and application.
Open Problem: Discuss the limitations and potential challenges that may arise when
extending the proven results to Suzuki-type fuzzy weak ϕ-contraction, and propose future
research directions to address these issues.

Author Contributions: Conceptualization, M.R., N.S. and Q.M.; Formal analysis,
N.S., M.A. and M.R.; Investigation, N.S., Q.M. and M.A.; Writing original draft prepa-
ration, Q.M., M.A. and I.R.; Writing,review and editing, N.S., I.R. and Q.M. All authors
have read and agreed to the published version of the manuscript.



REFERENCES 3334

Funding: This research received no external funding.
Data Availability Statement: The data used to support the findings of this study are
available from the corresponding author upon request.
Conflict of Interest: The authors declare no conflict of interest.

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