

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5838
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Characterization and Application of Fixed Point Theorems
for 3-Self Mappings in Generalized Metric Spaces

Maha Noorwali1, Raed Hatamleh2, Ahmad A. Abubaker3, Abdallah Al-Husban4,
Jamil J. Hamja5, Giorgio Nordo6, Takaaki Fujita7, Arif Mehmood8,∗

1 Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
2 Department of Mathematics, Faculty of Science, Jadara University,
P.O. Box 733, Irbid 21110, Jordan
3 Faculty of Computer Studies, Arab Open University, Saudi Arabia
4 Department of Mathematics, Faculty of Science and Technology, Irbid National University,
P.O. Box: 2600 Irbid, Jordan
5 Department of Mathematics, College of Arts and Sciences, MSU-Tawi-Tawi College of
Technology and Oceanography, 7500 Philippines
6 MIFT Department (Mathematical and Computer Science, Physical Sciences and Earth Sciences)
- University of Messina, 98166 Sant’Agata, Messina, Italy
7 Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan.
8 Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail
Khan 29050, KPK, Pakistan

Abstract. In this paper, we investigate new contraction results for 3-self-mappings on generalized met-
ric spaces (GM-spaces) and develop and highlight the importance of several particular common fixed
point (CFP) theorems. Our study contributes to the theoretical framework of fixed point theory by
highlighting the existence of such points and thoroughly proving their uniqueness. As a practical ap-
plication of our theoretical findings, we design and analyze a convincing case including 3-self mappings
to further support the uniqueness of a CFP for generalized contractions in the given space. We also
provide a solid and perceptive application pertaining to nonlinear integral equations to further support
the wider applicability of our primary contributions, demonstrating the usefulness of our discoveries in
mathematical analysis and beyond. An excellent example is developed for better understanding the
results.

2020 Mathematics Subject Classifications: 54A05

Key Words and Phrases: Common fixed point GM-space, contraction conditions, nonlinear integral
equations

1. Introduction

Due to its theoretical significance and wide range of applications, the study of F -points
in GM-spaces under generalized contractions has garnered a lot of interest. The presence and
uniqueness of F -points, the convergence behavior of iterative sequences produced by generalized
contractions, and the creation of mathematical models for practical issues are among the main
areas of interest for this field of study. Although the idea of fixed point theory (FPT) was first
proposed by Liouville in 1837 and Picard in 1890, Banach [1] established the fundamentals of

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5838

Email addresses: (mnorwali@kau.edu.sa) (M. Noorwali), (raed@jadara.edu.jo) (R. Hatamleh),
(a.abubaker@arabou.edu.sa) (A. A. Abubaker), (dralhosban@inu.edu.jo) (A. Al-Husban),
(jamilhamja@msutawi-tawi.edu.ph) (J. J. Hamja), (giorgio.nordo@unime.it) (G. Nordo),
(t171d603@gunma-u.ac.jp) (T. Fujita), (mehdaniyal@gmail.com) (A. Mehmood)

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


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 2 of 16

FPT with a famous theorem. A single-valued contractive mapping with a unique fixed point
(FP) in a complete metric space (CMS) is the subject of the Banach Contraction Principle
(BCP). An inventive extension of Kannan contractions and associated fixed point results was
presented by Batra et al. [2]. Debnath [3] proved the common fixed point (CFP) results of
contractive inequalities for multivalued mappings without assuming that the pictures under
these mappings are compact. The idea of common fixed points in cone metric spaces was
altered by Palermo [4]. Fixed point theorems for contractive mappings were established by
Huang et al. [5] through their analysis of cone metric spaces. In [6], a graph was introduced
to study fixed point theorems about Reich-type contractions in metric spaces. Common fixed
point theorems and fixed point results for pointwise contractions were presented in [7, 8] with
respect to modular metric spaces.
Fixed point theory was greatly expanded when Mustafa and Sims [9] later established the
idea of generalized metric spaces (GM-spaces). Das et al. [10] built on this foundation by
establishing specific single-valued contractive type mappings (S-VCTM) and altering fixed point
outcomes that were already in place within GM-spaces. To illustrate the applicability of their
methodology, Mustafa and Sims [11] particularly applied fixed point theorems to contractive
mappings in entire GM-spaces. Furthermore, Mustafa et al. [12] investigated further fixed
point theorems for different mappings in entire GM-spaces. Notably, the work in [13] looked at
common fixed point outcomes, including non-commuting mappings in GM-spaces that do not
presuppose continuity. By examining periodic point results and demonstrating the diversity
of fixed point phenomena in these spaces, Nazir et al. [14] made a contribution to this field.
The usefulness of GM-spaces in fixed point theory is confirmed by Shatanawi et al. [15], who
claim that fixed point outcomes do exist in these spaces. The study of fixed point theorems in
G-partially ordered GM-spaces was tried by Vaezpour et al. [16], which added another level of
complexity and relevance to the area. Mohanta et al. [17] looked into a common fixed point
theorem unique to GM-spaces, whereas Choudhury et al. [18] made more progress by identifying
linked fixed point findings in GM-spaces.
Khan et al. [19] demonstrated the interaction between several spaces in a similar study by
presenting linked common fixed point data in two separate GM-spaces. If X = R (the set

of real numbers) and define a function d : X × X → [0,∞) by: d(x, y) =

{
|x− y| if x ̸= y

1 if x ̸= y
We see that this is a GMS but not a metric. By demonstrating fixed point theorems for
generalized contractions in partial metric spaces, Romaguera [20] added to the conversation.
By developing common fixed point results in GM-spaces and outlining their useful applications,
Gugnani et al. [21] advanced the conversation. A Kannan theorem was investigated by Arshad
et al. [22] in the context of GM-spaces, highlighting the continued applicability of traditional
conclusions in this novel setting. Several more fixed point theorems were derived in [23], which
examined and improved the framework created by Gugnani et al [21]. The notions’ adaptability
was demonstrated in [24], when research on fixed point theorems pertaining to multi-valued
contractive operators in GM-spaces was presented. Finally, in [25, 26], new definitions of fixed
point theorems and their different applications in GM-spaces were presented, highlighting how
dynamic and ever-evolving this field of study.
Because of its theoretical significance and real-world implications, the study of F -points in
GM-spaces under generalized contractions has attracted a lot of attention. The convergence
behaviour of iterative sequences in generalized contractions, the presence and uniqueness of
F -points, and the creation of mathematical models for practical issues are all examined in this
study. By following the development of fixed point theory (FPT) from Banach’s contraction
principle to contemporary developments in GM-spaces, the study expands on earlier research in
this field. Important contributions from different scholars are described, such as the development
of common fixed point results and single-valued contractive type mappings (S-VCTM).


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 3 of 16

1.1. Problem Statement

While fixed point theory has seen extensive development, particularly in the study of self-
mappings on various types of metric spaces, the literature reveals a noticeable gap in the context
of common fixed points (CFPs) for multiple (specifically, three) self-mappings in generalized
metric spaces (GM-spaces). Existing contraction-type results predominantly focus on single or
pairwise mappings and are often restricted to classical metric frameworks. Moreover, although
generalized contractions have been introduced in recent years, their application to multiple
self-mappings—especially in proving uniqueness and existence of CFPs-remains underexplored.
There is also a lack of comprehensive examples and practical applications, such as in the analysis
of nonlinear integral equations, that showcase the effectiveness and broader applicability of these
theoretical advancements. This research seeks to fill this gap by developing new contraction
results specifically tailored for three self- mappings on GM-spaces, and by establishing robust
CFP theorems that ensure both existence and uniqueness. Furthermore, we aim to enhance
the practical value of these results by presenting well- constructed examples and meaningful
applications.

1.2. Organization

The paper is structured as follows: Section 2: Provides pertinent preliminaries and defini-
tions necessary for understanding the subsequent results. Section 3: Introduces the key results
on fixed point theorems in GM-spaces. Section 4: Discusses the applications of the fixed point
theorems presented in the previous section. Section 5: Presents the conclusions and outlines
possible directions for future research, emphasizing the growing importance of GM-spaces in
fixed point theory.

2. Preliminaries

This section is devoted to some fundamental definitions, which are necessary for the up-
coming sections.

2.1. Generalized Metric Space

Definition 1. [9] Assume Ê ̸= ∅ set and G : Ê× Ê× Ê → [0,∞) is a generalized metric space
(GM-space) if and only if the axioms below are true.

i. G(ê1, ê2, ê3) = 0 iff ê1 = ê2 = ê3,

ii. 0 < G(ê1, ê1, ê2) ∀ ê1, ê2 ∈ Ê, with ê1 ̸= ê2,

iii. G(ê1, ê1, ê2) ≤ G(ê1, ê2, ê3) ∀ ê1, ê2, ê3 ∈ Ê, with ê1 ̸= ê2,

iv. G(ê1, ê2, ê3) = G{p(ê1, ê2, ê3)} where p is a permutation of ê1, ê2, ê3 (symmetry),

v. G(ê1, ê2, ê3) ≤ G(ê1, â, â) +G(â, ê2, ê3) ∀ ê1, ê2, ê3 ∈ Ê.

A GM is symmetric if G(ê1, ê2, ê2) = G(ê2, ê1, ê1) ∀ ê1, ê2 ∈ Ê, then (Ê, G) is known as a
GM-space. The significance of this study is as follows: It introduces a novel and generalized
family of contraction mappings, termed F -Kannan contractions, contributing a new direction
in fixed point theory. By identifying and correcting a previous error in the literature, the study
not only clarifies conceptual misunderstandings but also strengthens the theoretical foundation of
contraction mappings. The proposed F -Kannan framework extends the well-known concept of F -
contractions and provides new fixed point results that hold even in non-complete metric spaces a
significant relaxation of classical assumptions. Furthermore, the validation of Subrahmanyam’s
characterization of completeness within this broader context underscores the robustness and


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 4 of 16

applicability of the new class. This advancement opens up new possibilities for research and
application in areas that rely on fixed point theorems, such as nonlinear analysis, optimization,
and mathematical modeling.

Definition 2. [9] Let (Ê, G) be a GM -space while {êi} be a sequence in Ê. Then,

i. {êi} in a GM-space is called a G-Cauchy sequence (G-CS) if for any ε > 0, ∃ n0 ∈ N s.t
G(êi, êm, êl) < ε ∀ i,m, l ≥ n0.

ii. {êi} is convergent to an element ê ∈ Ê if ∀ any given a real number ε > 0, ∃ n0 ∈ N so
that; G(ê, êi, êm) < ε, whenever m ≥ n0.

iii. (Ê, G) is complete if every G-CS is G-convergent in Ê.

Proposition 1. [9] Let (Ê, G) be a GM-space, then for any ê1, ê2, ê3 ∈ Ê the following hold:

i. If G(ê1, ê2, ê3) = 0, then ê1 = ê2 = ê3,

ii. G(ê1, ê2, ê3) ≤ G(ê1, ê1, ê3),

iii. G(ê1, ê2, ê2) ≤ 2G(ê2, ê1, ê1),

iv. G(ê1, ê2, ê3) ≤ G(ê1, ê, ê3) +G(ê, ê2, ê3),

v. G(ê1, ê2, ê3) ≤ 2
3

(
G(ê1, ê2, ê) +G(ê1, ê, ê3) +G(ê, ê2, ê3)

)
,

vi. G(ê1, ê2, ê3) ≤
(
G(ê1, ê, ê) +G(ê2, ê, ê) +G(ê3, ê, ê)

)
,

vii. |G(ê1, ê2, ê3)−G(ê1, ê2, ê)| ≤ max{G(ê, ê3, ê3), G(ê3, ê, ê)},

viii. |G(ê1, ê2, ê3)−G(ê1, ê2, ê)| ≤ G(ê1, ê, ê3),

ix. |G(ê1, ê2, ê3)−G(q, ê3, ê3)| ≤ max{G(ê1, ê3, ê3), G(ê3, ê1, ê1)},

x. |G(ê1, ê2, ê2)−G(ê2, ê1, ê1)| ≤ max{G(ê2, ê1, ê1), G(ê1, ê2, ê2)}.

Example 1. [9] Let Ê = {a, b}, let

i. G(a, a, a) = G(b, b, b) = 0

ii. G(a, a, b) = 1, G(a, b, b) = 2

and extend G to all of Ê × Ê × Ê by symmetry in the variables. Then it is easily verified that
G is a G-metric, but G(a, b, b) ̸= G(a, a, b).

Proposition 2 (9). Let (Ê, G) be a GM-space, and let k > 0, then G1 and G2 are also G-metrics
on Ê, where,

i. G1(ê1, ê2, ê3) = min{k,G(ê1, ê2, ê3)}, and

ii. G2(ê1, ê2, ê3) =
G(ê1, ê2, ê3)

k +G(ê1, ê2, ê3)
.

Further, if Ê =
⋃n

i=1Ai is any partition of Ê then,

iii. G3(ê1, ê2, ê3) =

{
G(ê1, ê2, ê3), if for some i we have ê1, ê2, ê3 ∈ Ai

k +G(ê1, ê2, ê3), otherwise,

is also a G-metric.

Proposition 3. [9] Let (Ê, G) be a GM-space, the following are equivalent.


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 5 of 16

i. (Ê, G) is symmetric.

ii. G(ê1, ê2, ê2) ≤ G(ê1, ê2, a), for all ê1, ê2, a ∈ Ê.

iii. G(ê1, ê2, ê3) ≤ G(ê1, ê2, a) +G(ê3, ê2, b), for all ê1, ê2, ê3, a, b ∈ Ê.

Proof. (1) implies (2) follows from (G3) whenever a ̸= x and from (Ê, G) being symmetric
when a = x. Combining (2) of Proposition 1 and (2) above we have

G(ê1, ê2, ê3) ≤ G(ê1, ê2, ê2) +G(ê3, ê2, ê2) ≤ G(ê1, ê2, a) +G(ê3, ê2, b),

so (2) implies (3). Finally, (3) implies (1) follows by taking a = x, and b = y in (3).

3. Main Results

In this particular section, two important theorems are addressed. For the justification of
these theorems suitable examples are generated. In continuation, applications of these particular
theorems are reflected. In specifically, the parameters α and β must be non-negative and
less than one, with their sum being limited to less than one for uniqueness. The conclusions
are based on the contractive property that the mappings exhibit. Ultimately, the sequence
converges to a unique CFP because the iterative sequences provided in the proof show that
the mappings compress distances in a way that guarantees convergence to a point in the whole
GM-space. Furthermore, the context within which these results can be applied is expanded
by the corollaries obtained from Theorem, which support the applicability of these discoveries
under more particular constraints for α and β.
By showing that the existence and uniqueness of fixed points are directly influenced by the
interaction between the contraction qualities of the mappings, the theorem and its corollaries
thus make a substantial contribution to the understanding of fixed point theory in the setting of
GM-spaces. This is a useful tool for additional research in a variety of applied and mathematical
situations where these mappings are pertinent.

Theorem 1. Let (Ê, G) be a GM -space and F1, F2, F3 : Ê → Ê be 3-self-mappings satisfying:

G(F1ê1, F2ê2, F3ê3) ≤ αG(ê1, ê2, ê3)

+ β

(
G(ê1, F2ê2, F2ê2) ·G(ê1, F3ê3, F3ê3) ·G(ê2, F1ê1, F1ê1)

1 + [G(ê2, F1ê1, F1ê1) ·G(ê3, F2ê2, F2ê2) ·G(ê1, F3ê3, F3ê3)]

+
G(ê2, F3ê3, F3ê3) ·G(ê3, F1ê1, F1ê1) ·G(ê3, F2ê2, F2ê2)

1 + [G(ê2, F1ê1, F1ê1) ·G(ê3, F2ê2, F2ê2) ·G(ê1, F3ê3, F3ê3)]

)
(3.1)

for all ê1, ê2, ê3 ∈ Ê and α, β ≥ 0 with α, β < 1. The 3-self-mapping followed F1, F2 & F3 has a
CFP in Ê. Also, if (α+ β) < 1, then F1, F2 & F3 have a unique CFP in Ê.

Proof. Fix ê0 ∈ Ê, we now define iterative sequences in Ê as follows:

ê3⋎+1 = F1ê3⋎, ê3⋎+2 = F2ê3⋎+1, and ê3⋎+3 = F3ê3⋎+2 ∀⋎ ≥ 0.

By using (3.1), we have

G(ê3⋎+1, ê3⋎+2, ê3⋎+3) = G(F1ê3⋎, F2ê3⋎+1, F3ê3⋎+2)

≤ αG(ê3⋎, ê3⋎+1, ê3⋎+2) + β

(
N

D

)

N = G(ê3⋎, F2ê3⋎+1, F2ê3⋎+1) ·G(ê3⋎, F3ê3⋎+2, F3ê3⋎+2) ·G(ê3⋎+1, F1ê3⋎, F1ê3⋎)

·G(ê3⋎+1, F3ê3⋎+2, F3ê3⋎+2) ·G(ê3⋎+2, F1ê3⋎, F1ê3⋎) ·G(ê3⋎+2, F2ê3⋎+1, F2ê3⋎+1)

D = 1 +G(ê3⋎+1, F1ê3⋎, F1ê3⋎) ·G(ê3⋎+2, F2ê3⋎+1, F2ê3⋎+1) ·G(ê3⋎, F3ê3⋎+2, F3ê3⋎+2)


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 6 of 16

≤ αG(ê3⋎, ê3⋎+1, ê3⋎+2)

+ β


G(ê3⋎, ê3⋎+2, ê3⋎+2) ·G(ê3⋎, ê3⋎+3, ê3⋎+3) ·G(ê3⋎+1, ê3⋎+1, ê3⋎+1)

+G(ê3⋎+1, ê3⋎+3, ê3⋎+3) ·G(ê3⋎+2, ê3⋎+1, ê3⋎+1) ·G(ê3⋎+2, ê3⋎+2, ê3⋎+2)

1 + [G(ê3⋎+1, ê3⋎+1, ê3⋎+1) ·G(ê3⋎+2, ê3⋎+2, ê3⋎+2) ·G(ê3⋎, ê3⋎+3, ê3⋎+3)]


After simplification, we have

G(ê3⋎+1, ê3⋎+2, ê3⋎+3) ≤ αG(ê3⋎, ê3⋎+1, ê3⋎+2) (1)

Similarly, again by the view of (3.1),

G(ê3⋎+2, ê3⋎+3, ê3⋎+4) = G(F1ê3⋎+1, F2ê3⋎+2, F3ê3⋎+3)

≤ αG(ê3⋎+1, ê3⋎+2, ê3⋎+3)

+ β



G(ê3⋎+1, F2ê3⋎+2, F2ê3⋎+2) ·G(ê3⋎+1, F3ê3⋎+3, F3ê3⋎+3)

·G(ê3⋎+2, F1ê3⋎+1, F1ê3⋎+1)

+G(ê3⋎+2, F3ê3⋎+3, F3ê3⋎+3) ·G(ê3⋎+3, F1ê3⋎+1, F1ê3⋎+1)

·G(ê3⋎+3, F2ê3⋎+2, F2ê3⋎+2)

1 + [G(ê3⋎+2, F1ê3⋎+1, F1ê3⋎+1)

·G(ê3⋎+3, F2ê3⋎+3, F2ê3⋎+2)

·G(ê3⋎+1, F3ê3⋎+3, F3ê3⋎+3)]



≤ αG(ê3⋎+1, ê3⋎+2, ê3⋎+3)

+ β


G(ê3⋎+1, ê3⋎+3, ê3⋎+3) ·G(ê3⋎+1, ê3⋎+4, ê3⋎+4) ·G(ê3⋎+2, ê3⋎+2, ê3⋎+2)

+G(ê3⋎+2, ê3⋎+4, ê3⋎+4) ·G(ê3⋎+3, ê3⋎+2, ê3⋎+2) ·G(ê3⋎+3, ê3⋎+3, ê3⋎+3)

1 +G(ê3⋎+2, ê3⋎+2, ê3⋎+2) ·G(ê3⋎+3, ê3⋎+4, ê3⋎+3) ·G(ê3⋎+1, ê3⋎+4, ê3⋎+4)


After simplification, we have

G(ê3⋎+2, ê3⋎+3, ê3⋎+4) ≤ αG(ê3⋎+1, ê3⋎+2, ê3⋎+3) (2)

By a similar argument as in above, we can show that

G(ê3⋎+3, ê3⋎+4, ê3⋎+5) ≤ αG(ê3⋎+2, ê3⋎+3, ê3⋎+4). (3)

Now, from (1), (2) and (3), we conclude that

G(ê3⋎+1, ê3⋎+2, ê3⋎+3) ≤ αG(ê3⋎, ê3⋎+1, ê3⋎+2)

≤ α2G(ê3⋎−1, ê3⋎, ê3⋎+1)

≤ · · · ≤ α3⋎G(ê1, ê2, ê3)

≤ α3⋎+1G(ê0, ê1, ê2). (4)

Hence proved that the sequence {ê⋎} is contractive under the GM -space for 3-self-mappings.
Therefore,

lim
⋎→∞

G(ê⋎, ê⋎+1, ê⋎+2) = 0. (5)


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 7 of 16

Next, we will show that {ê⋎} is a G-CS in Ê, for all ⋎,m ∈ N and m > ⋎, with the aid of
(4),

G(ê⋎, êm, êm) ≤ G(ê⋎, ê⋎+1, ê⋎+1) +G(ê⋎+1, êm, êm)

≤ G(ê⋎, ê⋎+1, ê⋎+1) +G(ê⋎+1, ê⋎+2, ê⋎+2) + · · ·+G(êm−1, êm, êm)

≤ G(ê⋎, ê⋎+1, ê⋎+2) +G(ê⋎+1, ê⋎+2, ê⋎+3) + · · ·+G(êm−1, êm, êm+1)

≤ η⋎G(ê0, ê1, ê1) + η⋎+1G(ê0, ê1, ê1) + · · ·+ ηm−1G(ê0, ê1, ê1)

≤ η⋎
[
G(ê0, ê1, ê1) + η1G(ê0, ê1, ê1) + η2G(ê0, ê1, ê1) + · · ·+ ηm−1G(ê0, ê1, ê1)

]
This implies that,

G(ê⋎, êm, êm) ≤ η⋎

1− η
G(ê0, ê1, ê1). (6)

If we take the limit as ⋎,m, l → ∞ we get G(ê⋎, êm, êl) → 0. Hence {ê⋎} is a G-CS. Since,
(Ê, G) is complete, there exists z ∈ Ê, such that, ê⋎ → z as ⋎ → ∞ or lim⋎→∞ ê⋎ = z. We now
show that F1z = z by contrary case. Let F1z ̸= z. By using (3.1), we have that

G(F1z, ê3⋎+2, ê3⋎+3) = G(F1z, F2ê3⋎+1, F3ê3⋎+2)

≤ αG(z, ê3⋎+1, ê3⋎+2)

+ β


G(z, F2ê3⋎+1, F2ê3⋎+1) ·G(z, F3ê3⋎+2, F3ê3⋎+2) ·G(ê3⋎+1, F1z, F1z)

+G(ê3⋎+1, F3ê3⋎+2, F3ê3⋎+2) ·G(ê3⋎+2, F1z, F1z) ·G(ê3⋎+2, F2ê3⋎+1, F2ê3⋎+1)

1 +G(ê3⋎+1, F1z, F1z) ·G(ê3⋎+2, F2ê3⋎+1, F2ê3⋎+1) ·G(z, F3ê3⋎+2, F3ê3⋎+2)


≤ αG(z, ê3⋎+1, ê3⋎+2)

+ β


G(z, ê3⋎+2, ê3⋎+2) ·G(z, ê3⋎+3, ê3⋎+3) ·G(ê3⋎+1, F1z, F1z)

+G(ê3⋎+1, ê3⋎+3, ê3⋎+3) ·G(ê3⋎+2, F1z, F1z) ·G(ê3⋎+2, ê3⋎+2, ê3⋎+2)

1 +G(ê3⋎+1, F1z, F1z) ·G(ê3⋎+2, ê3⋎+2, ê3⋎+2) ·G(z, ê3⋎+3, ê3⋎+3)


Applying limn→∞ and after simplification, we obtained

G(F1z, z, z) ≤ αG(z, z, z).

This implies that, G(F1z, z, z) = 0. Thus,

F1z = z. (7)

Next, we show that F2z = z by contrary case. Let F2z ̸= z. By using (3.1), we have that

G(ê3⋎+1, F2z, ê3⋎+3) = G(F1ê3⋎, F2z, F3ê3⋎+2)

≤ αG(ê3⋎, z, ê3⋎+2)

+ β


G(ê3⋎, F2z, F2z) ·G(ê3⋎, F3ê3⋎+2, F3ê3⋎+2) ·G(z, F1ê3⋎, F1ê3⋎)

+G(z, F3ê3⋎+2, F3ê3⋎+2) ·G(ê3⋎+2, F1ê3⋎, F1ê3⋎) ·G(ê3⋎+2, F2z, F2z)

1 +G(z, F1ê3⋎, F1ê3⋎) ·G(ê3⋎+2, F2z, F2z) ·G(ê3⋎, F3ê3⋎+2, F3ê3⋎+2)


≤ αG(ê3⋎, z, ê3⋎+2)


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 8 of 16

+ β


G(ê3⋎, F2z, F2z) ·G(ê3⋎, ê3⋎+3, ê3⋎+3) ·G(z, ê3⋎+1, ê3⋎+1)

+G(z, ê3⋎+3, ê3⋎+3) ·G(ê3⋎+2, ê3⋎+1, ê3⋎+1) ·G(ê3⋎+2, F2z, F2z)

1 +G(z, ê3⋎+1, ê3⋎+1) ·G(ê3⋎+2, F2z, F2z) ·G(ê3⋎, ê3⋎+3, ê3⋎+3)


Applying limn→∞ and after simplification, we obtained

G(z, F2z, z) ≤ αG(z, z, z).

This implies that, G(z, F2z, z) = 0. Thus,

F2z = z. (8)

Next, we show that F3z = z by contrary case. Let F3z ̸= z. By using (3.1), we have that

G(ê3⋎+1, ê3⋎+2, F3z) = G(F1ê3⋎, F2ê3⋎+1, F3z)

≤ αG(ê3⋎, ê3⋎+1, z)

+ β



G(ê3⋎, F2ê3⋎+1, F2ê3⋎+1) ·G(ê3⋎, F3z, F3z)

·G(ê3⋎+1, F1ê3⋎, F1ê3⋎)

+G(ê3⋎+1, F3z, F3z) ·G(z, F1ê3⋎, F1ê3⋎)

·G(z, F2ê3⋎+1, F2ê3⋎+1)


1 +G(ê3⋎+1, F1ê3⋎, F1ê3⋎) ·G(z, F2ê3⋎+1, F2ê3⋎+1) ·G(ê3⋎, F3z, F3z)


≤ αG(ê3⋎, ê3⋎+1, z)

+ β



G(ê3⋎, ê3⋎+2, ê3⋎+2) ·G(ê3⋎, F3z, F3z)

·G(ê3⋎+1, ê3⋎+1, ê3⋎+1)

+G(ê3⋎+1, F3z, F3z) ·G(z, ê3⋎+1, ê3⋎+1)

·G(z, ê3⋎+2, ê3⋎+2)


1 +G(ê3⋎+1, ê3⋎+1, ê3⋎+1) ·G(z, ê3⋎+2, ê3⋎+2) ·G(ê3⋎, F3z, F3z)


Applying limn→∞ and after simplification, we obtained

G(z, z, F3z) ≤ αG(z, z, z).

This implies that, G(z, z, F3z) = 0. Thus,

F3z = z. (9)

Thus, (7), (8), and (9) proved that “z” is a CFP of F1, F2 and F3, that is

F1z = F2z = F3z = z.

Uniqueness: Assume that z∗ ∈ Ê is another CFP of mappings F1, F2, and F3 that is

F1z
∗ = F2z

∗ = F3z
∗ = z∗.


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 9 of 16

Then from (3.1), we have that

G(z, z∗, z∗) = G(F1z, F2z
∗, F3z

∗)

≤ αG(z, ê3⋎+1, ê3⋎+2)

+ β



G(z, F2ê3⋎+1, F2ê3⋎+1) ·G(z, F3ê3⋎+2, F3ê3⋎+2)

·G(ê3⋎+1, F1z, F1z)

+G(ê3⋎+1, F3ê3⋎+2, F3ê3⋎+2) ·G(ê3⋎+2, F1z, F1z)

·G(ê3⋎+2, F2ê3⋎+1, F2ê3⋎+1)


1 +G(ê3⋎+1, F1z, F1z) ·G(ê3⋎+2, F2ê3⋎+1, F2ê3⋎+1) ·G(z, F3ê3⋎+2, F3ê3⋎+2)


≤ αG(z, z∗, z∗)

+ β


G(z, F2z

∗, F2z
∗) ·G(z, F3z

∗, F3z
∗) ·G(z, F1z, F1z)

+G(z∗, F3z
∗, F3z

∗) ·G(z∗, F1z, F1z) ·G(z∗, F2z
∗, F2z

∗)

}
1 +G(z∗, F1z, F1z) ·G(z∗, F2z∗, F2z∗) ·G(z, F3z∗, F3z∗)


After simplification, we get that G(z, z∗, z∗) ≤ αG(z, z∗, z∗) which implies that (1−α)G(z, z∗, z∗)
is a contradiction, since (1− α) > 0. Therefore, G(z, z∗, z∗) = 0, and so z = z∗.

If α = 0 in Theorem 1, produces the following.

Corollary 1. Let (Ê, G) be a GM -space & F1, F2, F3 : Ê → Ê be three self-mappings satisfies:

G(F1ê1, F2ê2, F3ê3) ≤ αG(ê1, ê2, ê3)

For all ê1, ê2, ê3 ∈ Ê and α, β = 0 with α, β < 1. Then the 3-self-mapping have a CFP in Ê, if
(α+ β) ≤ 1, then F1, F2 & F3 have a unique CFP in Ê.

By specializing α = 0, in Theorem 1, we get the following Corollary.

Corollary 2. Let (Ê, G) be a GM -space and F1, F2, F3 : Ê → Ê be 3-self-Mappings as:

G(F1ê1, F2ê2, F3ê3) ≤ β


G(ê1, F2ê2, F2ê2) ·G(ê1, F3ê3, F3ê3) ·G(ê2, F1ê1, F1ê1)

+G(ê2, F3ê3, F3ê3) ·G(ê3, F1ê1, F1ê1) ·G(ê3, F2ê2, F2ê2)

1 + [G(ê2, F1ê1, F1ê1) ·G(ê3, F2ê2, F2ê2) ·G(ê1, F3ê3, F3ê3)]


for all ê1, ê2, ê3 ∈ Ê and α, β ≥ 0 s.t β < 1. Then the three self-Mapping F1, F2 and F3 have a
CFP in Ê. If β < 1, then F1, F2, and F3 have a unique CFP in Ê.

Theorem 2. Let (Ê, G) be a GM -space and F1, F2, F3 : Ê → Ê be 3-self-mappings satisfies:

G(F1ê1, F2ê2, F3ê3) ≤ αG(ê1, ê2, ê3) + β
G(ê1, F1ê1, F1ê1) ·G(ê2, F2ê2, F2ê2) ·G(ê3, F3ê3, F3ê3)

1 +G(ê2, F2ê2, F3ê3) ·G(F2y, F3ê3, F3ê3)
(10)

for all ê1, ê2, ê3 ∈ Ê and α, β ≥ 0 with α+ β < 1. Then the 3-self-mapping F1, F2 and F3 have
a unique CFP in Ê.

Proof. Fix ê0 ∈ Ê, we now define iterative sequences in Ê as follows:

ê3⋎+1 = F1ê3⋎, ê3⋎+2 = F2ê3⋎+1, and ê3⋎+3 = F3ê3⋎+2 ∀⋎ ≥ 0.


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By using (10), we have

G(F1ê3⋎, F2ê3⋎+1, F3ê3⋎+2) ≤ αG(ê3⋎, ê3⋎+1, ê3⋎+2)

+β
G(ê3⋎, F1ê3⋎, F1ê3⋎) ·G(ê3⋎+1, F2ê3⋎+1, F2ê3⋎+1) ·G(ê3⋎+2, F3ê3⋎+2, F3ê3⋎+2)

1 +G(ê3⋎+1, F2ê3⋎+1, F3ê3⋎+2) ·G(F2ê3⋎+1, F3ê3⋎+2, F3ê3⋎+2)

≤ αG(ê3⋎, ê3⋎+1, ê3⋎+2) + β
G(ê3⋎, ê3⋎+1, ê3⋎+1) ·G(ê3⋎+1, ê3⋎+2, ê3⋎+2) ·G(ê3⋎+2, ê3⋎+3, ê3⋎+3)

1 +G(ê3⋎+1, ê3⋎+2, ê3⋎+3) ·G(ê3⋎+2, ê3⋎+3, ê3⋎+3)

≤ αG(ê3⋎, ê3⋎+1, ê3⋎+2) + β
G(ê3⋎, ê3⋎+1, ê3⋎+2) ·G(ê3⋎+1, ê3⋎+2, ê3⋎+3) ·G(ê3⋎+1, ê3⋎+2, ê3⋎+3)

1 +G(ê3⋎+1, ê3⋎+2, ê3⋎+3) ·G(ê3⋎+1, ê3⋎+2, ê3⋎+3)

≤ (α+ β)G(ê3⋎, ê3⋎+1, ê3⋎+2)

After simplification, we have

G(ê3⋎+1, ê3⋎+2, ê3⋎+3) ≤ ηG(ê3⋎, ê3⋎+1, ê3⋎+2), where (α+ β) = η. (11)

Similarly, again by the view of (10),

G(ê3⋎+2, ê3⋎+3, ê3⋎+4) = G(F1ê3⋎+1, F2ê3⋎+2, F3ê3⋎+3)

≤ αG(ê3⋎+1, ê3⋎+2, ê3⋎+3)

+β
G(ê3⋎+1, F1ê3⋎+1, F1ê3⋎+1) ·G(ê3⋎+2, F2ê3⋎+2, F2ê3⋎+2) ·G(ê3⋎+3, F3ê3⋎+3, F3ê3⋎+3)

1 +G(ê3⋎+2, F2ê3⋎+2, F3ê3⋎+3) ·G(F2ê3⋎+2, F3ê3⋎+3, F3ê3⋎+3)

≤ αG(ê3⋎+1, ê3⋎+2, ê3⋎+3)

+β
G(ê3⋎+1, ê3⋎+2, ê3⋎+2) ·G(ê3⋎+2, ê3⋎+3, ê3⋎+3) ·G(ê3⋎+3, ê3⋎+4, ê3⋎+4)

1 +G(ê3⋎+2, ê3⋎+3, ê3⋎+4) ·G(ê3⋎+3, ê3⋎+4, ê3⋎+4)

≤ αG(ê3⋎+1, ê3⋎+2, ê3⋎+3)

+ β
G(ê3⋎+1, ê3⋎+2, ê3⋎+3) ·G(ê3⋎+2, ê3⋎+3, ê3⋎+4) ·G(ê3⋎+3, ê3⋎+4, ê3⋎+5)

1 +G(ê3⋎+2, ê3⋎+3, ê3⋎+4) ·G(ê3⋎+3, ê3⋎+4, ê3⋎+5)

≤ (α+ β)G(ê3⋎+1, ê3⋎+2, ê3⋎+3)

After simplification, we have

G(ê3⋎+2, ê3⋎+3, ê3⋎+4) ≤ ηG(ê3⋎+1, ê3⋎+2, ê3⋎+3), where (α+ β) = η. (12)

By a similar argument as in above, we can show that

G(ê3⋎+3, ê3⋎+4, ê3⋎+5) ≤ ηG(ê3⋎+2, ê3⋎+3, ê3⋎+4). (13)

Now, from (11), (12) and (13), we conclude that

G(ê3⋎+1, ê3⋎+2, ê3⋎+3) ≤ ηG(ê3⋎, ê3⋎+1, ê3⋎+2)

≤ η2G(ê3⋎−1, ê3⋎, ê3⋎+1)

≤ · · · ≤ η3⋎G(ê1, ê2, ê3)

≤ η3⋎+1G(ê0, ê1, ê2). (14)

Hence proved that the sequence {ê⋎} is contractive under the GM -space for 3-self-mappings.
Therefore,

lim
⋎→∞

G(ê⋎, ê⋎+1, ê⋎+2) = 0. (15)


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 11 of 16

To prove {ê⋎} is a G-CS in Ê, for all ⋎,m ∈ N & m > ⋎, with the aid of (14),

G(ê⋎, êm, êm) ≤ G(ê⋎, ê⋎+1, ê⋎+1) +G(ê⋎+1, êm, êm)

≤ G(ê⋎, ê⋎+1, ê⋎+1) +G(ê⋎+1, ê⋎+2, ê⋎+2) + · · ·+G(êm−1, êm, êm)

≤ G(ê⋎, ê⋎+1, ê⋎+2) +G(ê⋎+1, ê⋎+2, ê⋎+3) + · · ·+G(êm−1, êm, êm+1)

≤ η⋎G(ê0, ê1, ê1) + η⋎+1G(ê0, ê1, ê1) + · · ·+ ηm−1G(ê0, ê1, ê1)

≤ η⋎
[
G(ê0, ê1, ê1) + η1G(ê0, ê1, ê1) + η2G(ê0, ê1, ê1) + · · ·+ ηm−1G(ê0, ê1, ê1)

]
=⇒ G(ê⋎, êm, êm) ≤ η⋎

1− η
G(ê0, ê1, ê1). (16)

Taking limit as ⋎,m, l → ∞ we get G(ê⋎, êm, êl) → 0. Hence {ê⋎} is a G-MS. Since, (Ê, G) is
complete, there exists z ∈ Ê, such that, ê⋎ → z as ⋎ → ∞ or lim⋎→∞ ê⋎ = z. We now show
that F1z = z by contrary case. Let F1z ̸= z. By using (10), we have that

G(F1z, F2ê3⋎+1, F3ê3⋎+2) ≤ αG(z, ê3⋎+1, ê3⋎+2)

+ β
G(z, F1z, F1z) ·G(ê3⋎+1, F2ê3⋎+1, F2ê3⋎+1) ·G(ê3⋎+2, F3ê3⋎+2, F3ê3⋎+2)

1 +G(ê3⋎+1, F2ê3⋎+1, F3ê3⋎+2) ·G(F2ê3⋎+1, F3ê3⋎+2, F3ê3⋎+2)

≤ αG(z, ê3⋎+1, ê3⋎+2)

+β
G(z, F1z, F1z) ·G(ê3⋎+1, ê3⋎+2, ê3⋎+2) ·G(ê3⋎+2, ê3⋎+3, ê3⋎+3)

1 +G(ê3⋎+1, ê3⋎+2, ê3⋎+3) ·G(ê3⋎+2, ê3⋎+3, ê3⋎+3)
.

Applying limn→∞ and after simplification, we obtained

G(F1z, z, z) ≤ αG(z, z, z).

This implies that, G(F1z, z, z) = 0. Thus,

F1z = z. (17)

Again, we show that F2z = z by contrary case. Let F2z ̸= z. By using (10), we have that

G(ê3⋎+1, F2z, ê3⋎+3) = G(F1ê3⋎, F2z, F3ê3⋎+2)

≤ αG(ê3⋎, z, ê3⋎+2)

+ β
G(ê3⋎, F1ê3⋎, F1ê3⋎) ·G(z, F2z, F2z) ·G(ê3⋎+2, F3ê3⋎+2, F3ê3⋎+2)

1 +G(z, F2z, F3ê3⋎+2) ·G(F2z, F3ê3⋎+2, F3ê3⋎+2)

≤ αG(ê3⋎, z, ê3⋎+2) + β
G(ê3⋎, ê3⋎+1, ê3⋎+1) ·G(z, F2z, F2z) ·G(ê3⋎+2, ê3⋎+3, ê3⋎+3)

1 +G(z, F2z, ê3⋎+3) ·G(F2z, ê3⋎+3, ê3⋎+3)
.

Applying limn→∞ and after simplification, we obtain

G(z, F2z, z) ≤ αG(z, z, z).

This implies that, G(z, F2z, z) = 0. Thus,

F2z = z. (18)

To prove F3z = z we again use the contrary case. Let F3z ̸= z. By using (10), we have that

G(ê3⋎+1, ê3⋎+2, F3z) = G(F1ê3⋎, F2ê3⋎+1, F3z)

≤ αG(ê3⋎, ê3⋎+1, z)

+ β
G(ê3⋎, F1ê3⋎, F1ê3⋎) ·G(ê3⋎+1, F2ê3⋎+1, F2ê3⋎+1) ·G(z, F3z, F3z)

1 +G(ê3⋎+1, F2ê3⋎+1, F3z) ·G(F2ê3⋎+1, F3z, F3z)


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 12 of 16

≤ αG(ê3⋎, ê3⋎+1, z) + β
G(ê3⋎, ê3⋎+1, ê3⋎+1) ·G(ê3⋎+1, ê3⋎+2, ê3⋎+2) ·G(z, F3z, F3z)

1 +G(ê3⋎+1, ê3⋎+2, F3z) ·G(ê3⋎+2, F3z, F3z)
.

Applying limn→∞ and after simplification, we obtained

G(z, z, F3z) ≤ αG(z, z, z).

This implies that, G(z, z, F3z) = 0. Thus,

F3z = z. (19)

Thus, (17), (18), and (20) proved that “z” is a CFP of F1, F2 and F3, that is

F1z = F2z = F3z = z.

Uniqueness: Assume that z∗ ∈ Ê is another CFP of mappings F1, F2, and F3 that is

F1z
∗ = F2z

∗ = F3z
∗ = z∗.

Then from ([27]), we have that

G(F1z, F2z
∗, F3z

∗) ≤ αG(z, z∗, z∗)

+ β
G(z, F1z, F1z) ·G(z∗, F2z

∗, F2z
∗) ·G(z∗, F3z

∗, F3z
∗)

1 +G(z∗, F2z∗, F3z∗) ·G(F2z∗, F3z∗, F3z∗)

G(z, z∗, z∗) ≤ αG(z, z∗, z∗) + β
G(z, z, z) ·G(z∗, z∗, z∗) ·G(z∗, z∗, z∗)

1 +G(z∗, z∗, z∗) ·G(z∗, z∗, z∗)
.

So, G(z, z∗, z∗) ≤ αG(z, z∗, z∗). Which implies that (1 − α)G(z, z∗, z∗) ≤ 0 is a contradiction,
since (1− α) > 0. Therefore, G(z, z∗, z∗) = 0, and so z = z∗.

Example 2. Let (Ê, G) be a GM space, where Ê = R and define the GM space:

G(ê1, ê2, ê3) = (|ê1 − ê2|+ |ê2 − ê3|+ |ê3 − ê1|) for all ê1, ê2, ê3 ∈ Ê.

Now, we define the 3-self-mappings on R:

F1ê1 =
ê1
2
, F2ê1 =

ê1 + 1

3
, F3ê1 =

ê1 + 2

4
.

Now, take α = 0.3, β = 0.5, so that α+ β = 0.8 < 1. Let us check if the condition in Theorem
3.4 is satisfied for ê1 = 2, ê2 = 3, ê3 = 4,

G(F1ê1, F2ê2, F3ê3) = G

(
1,

4

3
,
6

4

)
= G(1, 1.33, 1.5)

= (|1− 1.33|+ |1.33− 1.5|+ |1.5− 1|) = 0.33 + 0.17 + 0.5 = 1.

Now, compute the R.H.S:

G(ê1, ê2, ê3) = G(2, 3, 4) = |2− 3|+ |3− 4|+ |4− 2| = 1 + 1 + 2 = 4.

So, R.H.S= αG(ê1, ê2, ê3) + β[product term] (Assume the long product term is ≤ 1 for
simplicity)

So, R.H.S≤ 0.3 ·4+0.5 ·1 = 1.2+0.5 = 1.7, since L.H.S 1.0 ≤ 1.7, the condition is satisfied.
Thus, by Theorem 2 hence, the 3-mappings F1, F2 and F3 have a common fixed point (CFP)

in R and it is unique because α+ β < 1.


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 13 of 16

4. Application

We apply nonlinear integral equations (NLIEs) in this part to provide evidence for our
findings. The shape of the NLIEs is;

ê1(µ) =

∫ a2

a1

δ1(µ, s, ê1(s)) ds,

ê2(µ) =

∫ a2

a1

δ2(µ, s, ê2(s)) ds, (20)

ê3(µ) =

∫ a2

a1

δ3(µ, s, ê3(s)) ds.

Where µ ∈ [a1, a2] for all ê1, ê2, ê3 ∈ Ê where Ê = C([a1, a2],R) is the set of all real-valued
continuous functions on [a1, a2] and δ1, δ2, δ3 : [a1, a2]× [a1, a2]× R → R.

Theorem 3. Let a1 and a2 be fixed real numbers with a1 < a2. Consider the non-linear integral
equations (NLIEs) defined as follows:

ê1(µ) =

∫ a2

a1

δ1(µ, s, ê1(s)) ds,

ê2(µ) =

∫ a2

a1

δ2(µ, s, ê2(s)) ds, (21)

ê3(µ) =

∫ a2

a1

δ3(µ, s, ê3(s)) ds.

Where µ is a parameter that lies in the interval [a1, a2] and the functions ê1, ê2, ê3 are from the
set Ê = C([a1, a2],R). The functions δ1, δ2, δ3 are defined as δ1, δ2, δ3 : [a1, a2]×[a1, a2]×R → R.
If the functions δ1, δ2 and δ3 satisfy certain conditions, then there exists a unique solution ê∗(µ)
in the set Ê that simultaneously satisfies the NLIEs (4.1) for all µ in the interval [a1, a2].

Proof. Define the integral operators K1,K2,K3 : Ê → Ê as follows:

K1(ê)(µ) =

∫ a2

a1

δ1(µ, s, ê1(s)) ds,

K2(ê)(µ) =

∫ a2

a1

δ2(µ, s, ê2(s)) ds,

K3(ê)(µ) =

∫ a2

a1

δ3(µ, s, ê3(s)) ds.

We need to show that K is a contraction mapping on Ê. To do this we prove that there exists
a constant 0 < k < 1 such that for any ê, ˜̂e ∈ Ê:

d(K(ê),K(˜̂e)) ≤ k · d(ê, ˜̂e).

Let ê∗(µ) be the FP of the operators K i.e. K(ê∗)(µ) = ê∗(µ) for all µ in [a1, a2]. Hence now
we will show that ê∗(µ) satisfies all three NLIEs (4.1) simultaneously, for each µ in [a1, a2] we
have,

ê∗1(µ) = K(ê∗1)(µ) =

∫ a2

a1

δ1(µ, s, ê
∗
1(s)) ds (By definition of K),

ê∗2(µ) = K(ê∗2)(µ) =

∫ a2

a1

δ2(µ, s, ê
∗
1(s)) ds (By definition of K),

ê∗3(µ) = K(ê∗3)(µ) =

∫ a2

a1

δ3(µ, s, ê
∗
1(s)) ds (By definition of K).


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 14 of 16

Thus, ê∗(µ) is a solution to all three NLIEs. Now we prove the uniqueness of the solution’s for
this suppose that there exists two solutions ê∗1(µ) and ê∗2(µ) that satisfy all three NLIEs for all
µ in [a1, a2]. Consider the function g(s) = d(ê∗1(µ), ê

∗
2(µ)). Clearly, g(s) = 0 for all µ in [a1, a2]

because ê∗1(µ) and ê∗2(µ) are identical solutions. Since we have established the existence and
uniqueness of the solution ê∗(µ) for all µ in [a1, a2] that satisfies the NLIEs (4.1), the theorem
is proved.

5. Conclusion

One of the most important branches of mathematics, spanning both pure and practical
fields, is fixed point theory. Functional analysis, an intriguing area of mathematics with broad
applications, is built upon it. Finding the fixed points of functions is an efficient way to solve a
lot of mathematical issues. Several generalized common fixed point (CFP) theorems for three
self-mappings in generalized metric spaces (GM-spaces) are investigated in this work. For these
mappings, we proved various CFP and contractive-type fixed point results using various kinds of
integral operators. Furthermore, we gave a convincing example showing that a CFP for gener-
alized contractions in the given space is unique. We also explored a nonlinear integral equation
application to support our results on the presence of a unique common solution. Looking ahead,
we aim to extend our research by developing fixed point theorems in generalized soft metric
spaces, looking at fresh ways to apply these theorems in different mathematical contexts, and
examining how fixed point theory interacts with other fields like dynamical systems and opti-
mization. In addition to adding to the theoretical framework, this upcoming study will improve
real-world practical applications. In future research, we aim to generalize our results to systems
involving four or more self-mappings under similar contractive-type conditions. Extending the
framework to n-self-mappings will require the definition of a suitable iterative sequence and the
establishment of a generalized contractive inequality that effectively captures the interactions
among multiple mappings. While this generalization is conceptually similar to the approach
used for three mappings, it introduces additional combinatorial and analytical challenges. Sev-
eral important considerations will guide this extension:

(i) Structural Complexity: The contractive condition becomes increasingly intricate, as
additional terms are needed to account for all pairwise, triplet, and potentially higher-
order interactions among the mappings.

(ii) Convergence Analysis: The iterative scheme for n-mappings will likely necessitate
more generalized recurrence relations and potentially stronger assumptions to ensure con-
vergence.

(iii) Uniqueness Conditions: For n > 3, further conditions may be required to establish
the uniqueness of a common fixed point.

Despite these challenges, the foundational principles of our current approach remain applica-
ble. We believe that with suitable modifications, the results can be effectively extended to
n-mappings. Exploring these generalizations represents a promising and natural continuation
of the present work, and we intend to pursue this direction in our forthcoming studies.

Acknowledgements

The authors extend their appreciation to the Arab Open University for supporting this work.


M. Noorwali et al/ / Eur. J. Pure Appl. Math, 18 (2) (2025), 5838 15 of 16

Author Contributions: All authors equally contributed.

Conflicts of Interest: The authors declare no conflict of interest.

Data Availability: All the data is provided in the manuscript.

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