EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6256
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Iterative Approaches to Multiple Fixed Points in
Generalized Metric Spaces

Yassin Alzubaidi

Mathematics Department, Umm Al-Qura University, Makkah, Saudi Arabia

Abstract. We combine the concept of having multiple fixed points for a mapping with the ap-
proach of obtaining these points through iterative methods. As is well known, contraction self-
mappings in standard metric spaces yield unique fixed points that can be obtained iteratively. To
overcome this limitation, we conduct our study within the framework of generalized MP-metric
spaces, utilizing their properties and the broader concept of limits. This enables us to establish our
main result: the existence of multiple fixed points that can be iteratively obtained for generalized
contraction mappings satisfying specific conditions.

2020 Mathematics Subject Classifications: 47H10, 54E35

Key Words and Phrases: Fixed point, Iterative methods, Generalized contractor, Multiple fixed
points, Generalized metric, MP-metric

1. Introduction

Fixed-point theory in metric spaces is one of the most active research areas in mathe-
matics due to its broad applications in both pure and applied fields. It plays a crucial role
in theoretical research and practical disciplines such as physics, computing, and engineer-
ing [1], [2]. The origins of fixed-point theory trace back to the late 19th and early 20th
centuries, with pioneering contributions from scholars such as Poincaré, Lefschetz–Hopf,
and Leray–Schauder [3]. However, the foundation of fixed-point theory in metric spaces
was largely established in 1922 when Banach published his seminal paper [4], proving the
existence and uniqueness of fixed points for a special class of functions called contraction
mappings. Banach’s proof was more than just a demonstration of the theorem; it im-
plicitly introduced an iterative method—specifically, Picard iteration—to obtain the fixed
point. His groundbreaking result inspired numerous researchers, leading to extensive gen-
eralizations and applications of Banach’s theorem. Among the major directions of research
in fixed-point theory, two prominent areas stand out: the first is the study of iterative
methods to obtain fixed points, and the second is the quest for the existence of more than
one fixed point. In this paper, we aim to combine these two directions and integrate them.

DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6256

Email address: yazubaidi@uqu.edu.sa (Y. Alzubaidi)

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



Y. Alzubaidi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6256 2 of 11

In this article, the notion of an iterative fixed point refers to a fixed point that can
be obtained through iterative methods. An iterative fixed point is of dual importance;
not only is it a fixed point, but we also have established methods for locating or ap-
proximating it, making it particularly useful in practical applications. Consequently, it is
unsurprising that significant attention has been devoted to iterative methods in fixed point
theory, with various approaches being implemented within this framework. In addition to
the straightforward Picard iterative method, more complex iterative techniques—such as
the Krasnoselskij, Mann, and Ishikawa methods—have been developed and successfully
employed to obtain fixed points. ( see, for example, [5], [6], [7], [8] and [9] ).

On the other hand, the study of the existence of multiple fixed points began in the
1960s with the works of Browder and Kirk ([10], [11], [12], and [13]). Their research
demonstrated that non-expansive mappings can possess more than one fixed point, thus
establishing a foundation for examining the potential of multiple fixed points in specific
contexts. Subsequently, the explicit investigation of the existence of multiple fixed points
gained traction. For instance, results concerning multiple fixed points for monotone and
positive mappings are available in [14] and [15]. Moreover, further applications have
illustrated that the significance of multiple fixed points is partially derived from the fact
that, in many practical scenarios, the solution to a system corresponds to a fixed point of
a particular mapping. This underscores a side of the importance of studying the existence
of more than one fixed point.

However, in the standard metric space, the existence of an iterative fixed point often
implies its uniqueness. It is common to combine contraction conditions and iteration
methods to construct a Cauchy sequence, ensuring its convergence through the space’s
completeness, with the limit serving as the desired fixed point. Therefore, we cannot
get multiple fixed points in such a setting, as the uniqueness of the fixed point primarily
stems from the contraction conditions and implicitly from the uniqueness of the limit, a
single point in ordinary metric spaces. We address this issue by conducting our study
on a generalized metric space, where we consider the MP-metric space and exploit its
properties to show the existence of multiple iterative fixed points under certain conditions.
Compared to other results in the field, we observe that the finding presented in this work
offers a distinct advantage: it establishes the existence of multiple fixed points for a given
function while leveraging iterative methods and their notable practical benefits. The
concept of multiple iterative fixed points is a potentially valuable tool in both pure and
applied mathematics, as it facilitates the analysis of systems where uniqueness is either
undesirable or cannot be guaranteed.

2. Preliminaries

Numerous studies have highlighted the advantages of studying fixed points in general-
ized metric spaces, which help overcome some of the limitations of ordinary metrics and
provide a more suitable framework for achieving desired results. For example, one can
refer to [16],[17] ,[18] , [19], and [20] for results on fixed-point theory in generalized metric
spaces, such as G-metric and b-metric spaces.



Y. Alzubaidi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6256 3 of 11

In this section, we review the generalized MP-metric space, highlight its key properties,
and present essential theorems relevant to this study.

Definition 1 ([21]). Let P ∗(X) denote the set of all non-empty finite subsets of X. Then,

d : P ∗(X) −→ [0,∞)

is called an MP-metric (a multiple point-metric) if for all A,B ∈ P ∗(X), the following
hold:

• (A1) d(A) = 0 ⇐⇒ |A| = 1,

• (A2) A ⊆ B =⇒ d(A) ≤ d(B),

• (A3) A ∩B ̸= ϕ =⇒ d(A ∪B) ≤ d(A) + d(B),

• (A4) d(A ∪ {b}) = d(A) for all b ∈ B =⇒ d(A ∪B) = d(A),

where |A| represents the cardinality of the set A. Furthermore, (X, d) is called an MP-
metric space.

To simplify the presentation below, it is assumed that d(a1, ..., ak), d({a1, ..., ak}), and
d(A) for A = {a1, ..., ak} are identical. If we let d(2)(a, b) = d({a, b}), we can check that
d(2) is a standard metric that is referred to as the associated metric. The MP-metric
concept can be considered an extension to some other types of metrics, including the
standard metric (see [21]).

Example 1 ([21]). A natural MP-metric d : P ∗(R) −→ [0,∞) on R can be defined in
the following way: d(A) = Max(A) − Min(A). The restriction of this metric to two
points gives the usual distance metric on R, where d(2)(a, b) = d({a, b}) = Max({a, b})−
Min({a, b}) = |a− b|.

Next, we present the concept of convergence in MP-metric spaces. It allows us to
directly obtain a generalized form of a sequence’s limit, where the limit can be a set (a
compact set) rather than a single point. Such a way of recognizing the limit will be crucial
in achieving our main result.

Definition 2 ([21]). Let (X, d) be an MP-metric space and A ∈ P ∗(X). Then

D(A) = {x ∈ X : d(A ∪ {x}) = d(A)}.

is the set of all d-dependent points on A.

Example 2 ([21]). (i)- Let (X, d) be the MP-metric space introduced in Example 1 and
a ≤ b. Then, D({a, b}) = {x ∈ X : d({a, b, x}) = d({a, b})} = [a, b].

(ii)- In any MP-metric space and for any singleton set, it follows from (A1) that
D({a}) = {a}, since D({a}) = {x ∈ X : d({a, x}) = d({a}) = 0} = {a}.



Y. Alzubaidi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6256 4 of 11

Definition 3 ([21]). Let (X, d) be an MP -metric space and D ⊆ X. Then, we say that a

sequence xn is d-convergent to D and write
d
lim
n→∞

xn = D if there is A ∈ P ∗(X) such that

D = D(A) and:
(i)- lim

ni→∞
d({xn1 , xn2 ..., xnk

} ∪A) = d(A) for all k ∈ N,
(ii)- lim inf

n−→∞
d(xn, a) = 0 for all a ∈ A.

We will say that xn converges toD(A), and we will write limxn = D(A) to indicate that
A meets the convergence conditions in the preceding formulation. The above definition
preserves the characteristics of convergence in ordinary metric spaces and, at the same
time, provides a natural generalization to the concept of limit.

Proposition 1 ([21]). Let (X, d) be an MP-metric space, and let (X, d(2)) be the associated
metric space. A sequence xn converges to a in (X, d(2)) if and only if it is d-convergent to
D({a}) = {a} in (X, d).

The MP-metric, like the standard metric, is considered continuous with respect to all
of its components.

Proposition 2 ([21]). Let (X, d) be an MP-metric space, and let (X, d(2)) be the associated
metric space.

(i)- If xn converges to a, then lim
n−→∞

d({xn, a1..., ak}) = d({a, a1..., ak}).
(ii)- If xni converges to ai, ∀i = 1, ..., k, then lim

ni−→∞
d({xn1 , ..., xnk

}) = d({a1..., ak}).

The Cauchy convergence and completeness concepts are fundamental tools in any type
of metric space. The next definition provides a generalization of the notion of Cauchy
sequences that aligns with the concept of convergence in the generalized MP-metric spaces.

Definition 4 ([21]). Assume that (X, d) is an MP-metric space. Then, xn is called r-
Cauchy for some r ≥ 0 if there exists N ∈ N such that

lim
m−→∞

sup
ni>m

d(xn1 , xn2 , ..., xnk
) = r for all k ≥ N.

The convergence in MP-metric spaces implies some essential results, such as bounded-
ness, r-Cauchy, and other findings that are analogous to those in standard metric theory.

Theorem 1 ([21]). Let (X, d) be an MP-metric space. If limxn = D(A) for some A ∈
P ∗(X), then (xn) is r-Cauchy, and moreover r = d(A).

Proposition 3 ([21]). Let (X, d) be an MP-metric space. If xn is r-Cauchy for some
r ≥ 0, then xn is bounded.

In an obvious way, the completeness of the MP-metric spaces is defined, where we
require the equivalence between the convergence in MP-metric spaces and the satisfaction
of the r-Cauchy criterion.



Y. Alzubaidi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6256 5 of 11

Definition 5. An MP-metric space (X, d) is said to be complete MP-metric space if, for
all r-Cauchy sequence (xn) in (X, d) with r ≥ 0, there exists a finite nonempty subset
A ∈ P ∗(X) such that d(A) = r and the sequence (xn) is d-convergent to D(A); that is,

d
lim
n→∞

xn = D(A).

The following theorem provides an example of a complete MP-metric space. It shows
that the Euclidean space is complete with respect to the MP-metric introduced in Example
1. This result is valid for higher dimensions when appropriate MP-metrics are implemented
in Rn (see [21]).

Theorem 2 ([21]). Let (R, d) be the MP-metric space introduced in Example 1. If xn is
r-Cauchy for some r > 0 in (R, d), then there are a, b ∈ R such that xn is d-convergent to
D({a, b}).

Another notion that we will need in the next section is the notion of the boundedness
of an MP-metric space. Therefore, we introduce the following definition.

Definition 6. An MP-metric space (X, d) is called bounded MP-metric space if there is
a positive constant C such that d(A) < C for all A ∈ P ∗(X).

3. Construction and Main Results

We start this section by introducing some constructive definitions and propositions
that will help simplify the presentation of the main result. For a mapping T , we use Tn

to denote applying the mapping T n times. Also, we note that for any A ∈ P ∗(X), we
can always enumerate its elements and write A = {a1, a2, ..., ak}.

Definition 7. Let (X, d) be an MP-metric space and T is a self mapping on X, and
A = {a1, ..., ak} ∈ P ∗(X). Then for all j ∈ N ∪ {0}, we define the set

Sj
T (A) =

∞⋃
n=j

Tn(A), (1)

and for j > 0 and 1 ≤ l ≤ k, we define the set

Sj,l
T (A) = (

∞⋃
n=j

Tn(A)) ∪ T j−1({al, ..., ak}) (2)

The sets introduced above have some properties that are directly derived from their
definitions. The following proposition discusses some of these properties, including the
effect of the mapping T.

Proposition 4. Let (X, d) be an MP-metric space, and let T be a self mapping on X,
and A = {a1, ..., ak} ∈ P ∗(X). Then

(i) T (Sj
T (A)) = Sj+1

T (A) for all j ∈ N ∪ {0},



Y. Alzubaidi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6256 6 of 11

(ii) T (Sj,l
T (A)) = Sj+1,l

T (A) for all j ∈ N and 1 ≤ l ≤ k,

(iii) Sj
T (A) = Sj+1,1

T (A) for all j ∈ N,

(iv) Sj1
T (A) ⊆ Sj2

T (A) for all j1, j2 ∈ N ∪ {0} and j2 ≤ j1

(v) Sj,l1
T (A) ⊆ Sj,l2

T (A) for all j ∈ N, 1 ≤ l1, l2 ≤ k and j2 ≤ j1

(vi) Sj1,l1
T (A) ⊆ Sj2,l2

T (A) for all j1, j2 ∈ N, 1 ≤ l1, l2 ≤ k, j2 ≤ j1 and j2 ≤ j1

Proof. The results of this proposition follow directly from straightforward calculations
using Definition 7. However, for the convenience of the reader, we provide a proof for part
(iii); the remaining parts can be established in a similar manner.

Sj
T (A) =

∞⋃
n=j

Tn(A) = (

∞⋃
n=j+1

Tn(A)) ∪ T j(A)

= (
∞⋃

n=j+1

Tn(A)) ∪ T j+1−1({a1, ..., ak}) = Sj+1,1
T (A).

Note that the elements of the set S0
T (A) can be written as {a1, ..., ak, T (a1), ..., T (ak),

T 2(a1), ...}, suggesting that we may think of S0 as a sequence while Sj,l
T (A) as tail subse-

quences for all j ∈ N. In particular, we have the following definition.

Definition 8. Let (X, d) be an MP-metric space and T is a self mapping on X, and
A = {a1, ..., ak} ∈ P ∗(X). Then for all j ∈ N ∪ {0}, we define the sequence (αn) as

αn = αn(A, T ) = T j(an−jk) (3)

where j = Int(n−1
k ) (the function of the greatest integer less than or equal)

The following proposition describes the relationship between the sequence αn(A, T )

and the sets Ss,l
T (A).

Proposition 5. Let (X, d) be an MP-metric space, and let T be a self mapping on X,
and A = {a1, ..., ak} ∈ P ∗(X). Then

(i) Ss,l
T (A) = {αn : n ≥ m} where m = l + (s− 1)k, (s = Int(m−1

k ) + 1 )

(ii) Ss
T (A) = {αn : n > sk},

(iii) S0
T (A) = {αn : n ∈ N},

Proof. To show (i), let x ∈ Ss,l
T (A) = Ss

T (A) ∪ T s−1({al, al+1..., ak}). Therefore, either
x = T t(ar) for some t ≥ s and 1 ≤ r ≤ k, or x = T s−1(ar) for some r ∈ {l, ..., k}.
In the first case, we choose n = r + tk ≥ l + (s − 1)k = m, while in the second case,
we set n = r + (s − 1)k ≥ l + (s − 1)k = m. Hence, we get x = αn and n ≥ m.



Y. Alzubaidi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6256 7 of 11

For the other inclusion, let αn = T t(an−tk) where t = Int(n−1
k ) for some n ≥ m. We

set s = 1 + Int(m−1
k ) and l = m − (s − 1)k to get t ≥ s − 1. If t ≥ s , then αn =

T t(an−tk) ∈ Ss
T (A) ⊆ Ss,l

T (A). If t = s − 1, then n − tk ≥ m − (s − 1)k = l, and hence

αn = T t(an−tk) ∈ T s−1({al, al+1..., ak}) ⊆ Ss,l
T (A). (ii) and (iii) follow from (i) using

Proposition 4-(iii).

Proposition 6. Let (X, d) be an MP-metric space, and let T be a self mapping on X,
and A = {a1, ..., ak} ∈ P ∗(X). If (αnu) is a subsequence of (αn(A, T )). Then there are
a ∈ A and sv ∈ N such that T sv(a) is a subsequence of (αnu).

Proof. This result is a direct consequence of the finiteness of A, as assuming the
opposite leads to a contradiction with (αnu) being a subsequence. The conclusion of this
proposition is equivalent to the following statement:

∃ ai ∈ A s.t ∀ M ∈ N ∃ sv, nu > M, s.t. T sv(ai) = αnu.

Thus, its negation is:

∀ ai ∈ A ∃ Mi ∈ N s.t. T sv(ai) ̸= αnu. ∀ sv, nu > Mi.

Using the finiteness of A = {a1, ..., ak}, we define

M = max
i=1,...,k

Mi.

Therefore,
T sv(ai) ̸= αnu. ∀ sv, nu > M, and ∀ ai ∈ A,

which contradicts the assumption that (αnu) is a subsequence of (αn(A, T )) = (T j(an−jk))
where it directly conflicts with the construction of (αnu) as a subsequence formed by
applying the iteration of T over elements of A ( see Definition 8, Equation (3)).

We now present our main result, which provides sufficient conditions for a mapping to
attain several iterative fixed points.

Theorem 3. Let (X, d) be a bounded complete MP-metric space, and let T : X → X be
a continuous mapping. If there exist positive constants r, p, and q, and N ∈ N such that
p+ q ≤ 1 and the following inequalities hold

r ≤ sup
xi∈Ss,l

T (A)

d(T ({x1, ..., xu})) ≤ q sup
xi∈Ss,l

T (A)

d({x1, ..., xu}) + pr for all u ≥ N. (4)

for some A ∈ P ∗(X), then T has at least two distinct iterative fixed points.

Proof. Assume A = {a1, ..., ak} satisfying (4) and set αn = T s(an−ks) with s =
Int(n−1

k ) as defined in (3). Then

sup
ni≥m

d(αn1 , ..., αnu) = sup
ni≥m

d(T s1(an1−ks1), ..., T
su(anu−ksu))

= sup
xi∈Ss,l

T (A)

d({x1, ..., xu}) ( by Proposition 5-(i). )

= sup
xi∈Ss−1,l

T (A)

d(T ({x1, ..., xu})) (by Proposition 4-(ii). )

(5)



Y. Alzubaidi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6256 8 of 11

Using (4) and applying the standard iteration keeping in mind the properties of the sets

Ss,l
T , we get

sup
xi∈Ss−1,l

T (A)

d(T ({x1, ..., xu})) ≤ q sup
xi∈Ss−1,l

T (A)

d({x1, ..., xu}) + pr for all u ≥ N.

= q sup
xi∈Ss−2,l

T (A)

d(T ({x1, ..., xu})) + pr

≤ q[q sup
xi∈Ss−2,l

T (A)

d({x1, ..., xu}) + pr] + pr

= q2 sup
xi∈Ss−2,l

T (A)

d({x1, ..., xu}) + qpr + pr

= q2 sup
xi∈Ss−3,l

T (A)

d(T ({x1, ..., xu})) + (q0 + q1)pr.

(6)

Repeating the above process, we get

sup
xi∈Ss−1,l

T (A)

d(T ({x1, ..., xu})) ≤ qs sup
xi∈S0

T (A)

d({x1, ..., xu}) + (q0 + q1 + ...+ qs−1)pr

≤ qs sup
xi∈S0

T (A)

d({x1, ..., xu}) + r for all u ≥ N.
(7)

where in the last step we used

(q0 + q1 + ...+ qs−1)pr ≤ pr

1− q
≤ r, since p+ q ≤ 1 ⇐⇒ p

1− q
≤ 1.

Combining (4), (5) and (7), we get

r ≤ sup
ni≥m

d(αn1 , ..., αnu) ≤ qs sup
xi∈S0

T (A)

d({x1, ..., xu}) + r, for all u ≥ N. (8)

Using the boundedness of (X, d) and the fact m → ∞ ⇐⇒ s → ∞, we get

lim
m→∞

sup
ni≥m

d(αn1 , ..., αnu) = r, for all u ≥ N. (9)

Therefore, (αn) is r-Cauchy sequence and hence by the completeness of the MP-metric
space (X, d) there is B ∈ P ∗(X) such that

d
lim
n→∞

αn = D(B) (10)

in particular lim inf
n→∞

d(αn, b) = 0 for all b ∈ B (see Definition 3). Hence, there is a

subsequence (αnm) such that lim
nm→∞

d(αnm , b) = 0. By Proposition 6, and since A is finite,

there is a ∈ A such that (T sl(a)) is a subsequence of (αnm). Thus, using the continuity of
d and T , we obtain

0 = lim
sl→∞

d(T sl(a), b) = lim
sl→∞

d(T (T sl−1(a)), b) = d(T ( lim
sl→∞

T sl−1(a)), b) = d(T (b), b)



Y. Alzubaidi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6256 9 of 11

That is, b is a fixed point for all b ∈ B.
It remains to estimate the number of the fixed points. Using the properties of the

MP-metric, we have
d(αn1 , ..., αnu) ≤ d({αn1 , ..., αnu} ∪B) (11)

which in turn gives

r = lim
m→∞

sup
ni≥m

d(αn1 , ..., αnu) ≤ lim
m.→∞

sup
ni≥m

d({αn1 , ..., αnu} ∪B) (12)

Moreover, in view of Equation (10) and Definition 3, we observe that lim
ni→∞

d({αn1 , ..., αnu}∪
B) exists and is equal to d(B). Hence, lim

m→∞
sup
ni≥m

d({αn1 , ..., αnu}∪B) also exists, and we

have that

lim
m→∞

sup
ni≥m

d({αn1 , ..., αnu} ∪B) = lim
ni→∞

d({αn1 , ..., αnu} ∪B) = d(B) (13)

Combining (12) and (13), we get
r ≤ d(B)) (14)

As a result, d(B) > 0 (since r is positive), which according to axiom (A1) implies that B
contains at least two distinct elements, that are fixed points. This concludes the proof.

Example 3. Consider the MP-Metric d : P ∗([0, 1]) −→ [0,∞) defined as d(A) = Max(A)−
Min(A). The space ([0, 1], d) is a bounded complete MP-metric space. If xn is r-Cauchy
for some r > 0 in ([0, 1], d), then there are a, b ∈ [0, 1] such that xn is d-convergent to
D({a, b}). Let F : [0, 1] −→ [0, 1] be defined as F (x) = xt for some t > 1. Then, choosing
A = {1

2 , 1}, we have the sequence (αn) = {1
2 , 1, (

1
2)

t, 1, (12)
t2 , 1, ...}. Therefore, if we let

r = 1, N = 2 and p = 1− q where q can be any number in (0, 1), we get

1 ≤ sup
xi∈Ss,l

F (A)

d(F ({x1, ..., xu})) ≤ q sup
xi∈Ss,l

F (A)

d({x1, ..., xu}) + (1− q) for all u ≥ 2.

Thus, F and ([0, 1], d) satisfy the conditions of Theorem 3. We can check that

d
lim
n→∞

αn = D({0, 1}).

It is clear that 0 and 1 are fixed points for F . Moreover, it can be observed that they can
be obtained as iterative fixed points: lim

n→∞
Fn(12) = 0, and lim

n→∞
Fn(1) = 1.

4. Conclusion

In this research paper, we examine a class of functions that possess multiple fixed
points, which can be obtained through iterative methods. To achieve this objective, we
conducted our study within the framework of generalized MP-metric spaces, leveraging



Y. Alzubaidi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6256 10 of 11

their comprehensive convergence concepts. Furthermore, by implementing an iterative
process, we established the existence of multiple fixed points for this class of functions
that satisfy general contraction conditions.

Our results open several avenues for further research.For instance, potential applica-
tions can be explored in function spaces where an MP-metric d on a set X is extended
to an MP-metric δ on certain spaces of functions over X by employing integration and
setting

δ(f1, f2, ..., fk) =

∫
X
d(f1(x), f2(x), ..., fk(x)).

In this context, we consider two functions to be equal if they coincide almost everywhere.
Furthermore, we can verify that the space of continuous functions C[a, b], equipped with
the MP-metric δ, is not a complete MP-metric space. Thus, a prominent area of research
is the investigation of the MP-completion of the space C[a, b]. This exploration will lay a
foundation for studying MP-metric spaces of functions, while also facilitating the exami-
nation of potential applications for our results, such as studying the existence of multiple
solutions for certain systems.

Another potential direction for future work is the investigation of the geometric aspects
of multiple fixed points. In this regard, the concept of fixed discs, where individual points
are replaced by discs, becomes particularly relevant. This idea has been studied in both
standard metric spaces and some generalized metric spaces (see, for example, [22] and
[23]). In connection with our work, the theory of fixed discs appears to be compatible
with the framework of MP-metric spaces and aligns naturally with the set-valued nature
of limits within this setting.

Acknowledgements

The author extends his appreciation to Umm Al-Qura University, Saudi Arabia, for
funding this research work through grant number: 25UQU4230209GSSR01

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