EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6106
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fixed Points of Mappings Contracting Perimeters of
Polygons: A Geometric Generalization of Banach’s

Principle

Muhammad Nazam1,∗, Umme Habiba1, Manuel De la Sen2,∗

1 Department of Mathematics, Allama Iqbal Open University, H-8, Islamabad, Pakistan
2 Department of Electricity and Electronics, Faculty of Science and Technology, University
of the Basque Country, Campus of Leioa, Leioa (Bizkaia), 48940, Spain

Abstract. In this paper, we investigate fixed points of mappings that contract the perimeters of
polygons. Our study is motivated by the idea that the perimeter, as a global geometric measure,
provides a more natural and flexible framework than the individual edge lengths when analyzing
contraction properties in metric spaces. We extend Petrov’s fixed point theorem from triangles to
polygons with an arbitrary number of vertices and establish conditions under which such mappings
admit unique fixed points. The methodology relies on generalizations of contraction mappings and
properties of metric spaces. Our main contributions include a new perimeter-based contraction
principle, a demonstration of its application in proving Banach’s contraction theorem, and examples
that validate the theoretical results. This generalization enriches the existing literature on fixed
point theory and opens avenues for further applications in geometric analysis and related areas.

2020 Mathematics Subject Classifications: 47H09, 47H10, 30L15

Key Words and Phrases: Fixed point, perimeter of polygons, contractions, complete metric
space

1. Introduction

The choice of considering the perimeter instead of the side lengths individually is mo-
tivated by both mathematical and geometric reasons. The perimeter provides a single
aggregate measure that captures the overall contraction of a polygon, avoiding the need
to track each side separately. This global perspective not only simplifies the analysis
but also aligns naturally with contraction principles, which are often defined in terms of
distances between objects rather than individual components. Furthermore, by contract-
ing perimeters, we establish a unified framework that generalizes earlier results (such as

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6106

Email addresses: muhammad.nazam@aiou.edu.pk (M. Nazam),
umme6427@gmail.com (U. Habiba), manuel.delasen@ehu.eus (M. De la Sen)

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



M. Nazam, U. Habiba, M. De la Sen / Eur. J. Pure Appl. Math, 18 (4) (2025), 6106 2 of 10

Petrov’s theorem for triangles) while offering broader applicability to polygons with ar-
bitrary numbers of vertices. This approach highlights the novelty and significance of our
contribution in comparison to other extensions of Banach’s fixed point theorem.
We consider a mapping contracting perimeters of polygons and demonstrate the conti-
nuity of such mapping. We prove a fixed point theorem for this mapping. Through an
example we illustrate that maximum number of the fixed points can be n− 1. We deduce
the well-known Banach fixed point theorem. Our work is generalization of Petrov fixed
point theorem [1]. By defining the concept of mappings contracting perimeters of triangles
and proving the fixed point theorem for such mappings, Petrov[1] (2023) provided a new
generalization of the Banach fixed point theorem (BFPT). The key distinction is that,
though the central hypothesis of his research is predicated on the concepts of Banach’s
classical theorem, the definition of this mapping is based on the mapping of three points
of space rather than two. Further a condition is imposed on mapping L, L(L(y)) ̸= y for
all y ∈ Y such that Ly ̸= y. One significant subclass of these mappings is the ordinary
contraction mapping, from which we are able to derive the classical Banach theorem as a
straightforward corollary right away. Banach developed the contraction principle in 1922
[2]. Though the concept of successive approximations had previously been mentioned in
the writings of Chebysher, Picard, and others, it was applied in approximation theory, so-
lutions to differential and integral equations, etc. The first person to correctly formulate
this outcome was Banach. Over time, there have been numerous ways to generalize the
Banach contraction principle. Additionally, metric extensions are typically verified against
three classical fixed point theorems[3]. These are the extension of Banach’s theorem to
nonexpansive mappings[4], Caristi’s theorem[5], and Nadler’s well-known set-valued exten-
sion of Banach’s theorem[6]. In present paper, we extend petrov’s idea and prove Banach’s
theorem as a sub-result. The fundamental distinction is that mentioned contraction map-
ping is predicated on n points in the space. A mapping contracting perimeter of polygon
that is not a contraction mapping is defined with |Y | = N0, where N0 is the cardinality
of set Y . Recent developments in fixed point theory and Ulam-type stabilities include re-
sults on dislocated quasi-metric and b-metric settings [7, 8], and on Ulam-type stabilities
for integral and integro-differential equations with delays [9–13]. Recent works have also
emphasized the importance of stability and qualitative analyses in related contexts. For
instance, several studies on Ulam stability have investigated nonlinear integral equations
and differential equations with delays [9–13]. These contributions highlight how stability
conditions can broaden the applicability of fixed point techniques. Motivated by these
directions, our paper proposes a geometric generalization where contraction acts on the
perimeters of polygons rather than point-to-point distances.

2. Comparison with other generalizations

The Banach contraction principle has been generalized in many directions, including
in b-metric spaces, dislocated quasi-metric spaces, ordered metric spaces, and multivalued
mapping settings. For example, Shoaib and Mehmood [7] established fixed point results
for mappings satisfying (δ, φ)-domination and continuity in dislocated quasi-metric spaces,



M. Nazam, U. Habiba, M. De la Sen / Eur. J. Pure Appl. Math, 18 (4) (2025), 6106 3 of 10

while Shoaib and Mir [8] studied interpolative multivalued α∗-dominated contractive map-
pings in dislocated b-metric spaces. These works extend Banach’s theorem by relaxing the
underlying distance structure or by considering generalized contraction conditions.

Our approach differs in a fundamental aspect: instead of directly contracting distances
between two points, we require contraction of the perimeter of polygons with n vertices.
This perimeter-based contraction condition naturally generalizes Petrov’s theorem (which
focused on triangles, n = 3) and provides a geometric perspective distinct from the analytic
contractive inequalities found in earlier literature. In particular, our framework allows us
to re-derive Banach’s contraction theorem as a corollary while also offering new geometric
insights that are not captured by distance-based or order-based generalizations.

Therefore, while earlier results broadened the applicability of Banach’s principle by
altering the ambient space or the contractive condition, the present work contributes
a complementary direction by showing that geometric contraction of polygons leads to
robust fixed point results. This highlights the uniqueness of our contribution relative to
other existing generalizations.

3. Mappings contracting perimeters of polygons

In this section, we recall some fundamental notions in metric fixed point theory and
give some properties of mappings contracting perimeters of polygons.

Definition 1. Let Z be any non-empty set. A function s : Z × Z → R is a metric on Z
if it satisfies the following axioms:

(1) s(l1, l2) ≥ 0;

(2) s(l1, l2) = 0 ⇐⇒ l1 = l2;

(3) s(l1, l2) = s(l2, l1) for all l1, l2 ∈ Z;

(4) s(l1, l2) ≤ s(l1, l) + s(l, l2) for all l1, l2, l ∈ Z,

the pair (Z, s) is known as metric space.

Definition 2. Let |Z| ≥ n and (Z, s) be a metric space. We refer to L : Z → Z as a
mapping contracting perimeters of polygons on Z, if there exists β ∈ [0, 1) such that the
inequality

s (Ll1, Ll2) + s (Ll2, Ll3) + · · ·+ s (Lln−1, Lln) ≤ β (s (l1, l2) + s (l2, l3) + · · ·+ s (ln−1, ln))
(1)

holds for every n points l1, l2, l3, · · · , ln ∈ Z.

Proposition 3. Mappings contracting perimeters of polygons are continuous.

Proof. Given a metric space (Z, s), and |Z| ≥ n, and L : Z → Z a mapping contracting
perimeters of polygons on Z and consider l0 as an isolated point in Z. The mapping L is



M. Nazam, U. Habiba, M. De la Sen / Eur. J. Pure Appl. Math, 18 (4) (2025), 6106 4 of 10

thus obviously continuous at l0. It can be shown that for each ϵ ≥ 0, there exists a δ ≥ 0
such that

s (Ll0, Ll) ≤ ϵ,

whenever
s (l0, l) ≤ δ.

Given l0 as an accumulation point. Therefore, for every δ ≥ 0, ∃ l ∈ Z such that s(li, l) ≤ δ
for all i ≥ 0

s(Ll0, Ll) ≤ s(Ll0, Ll) + s(Ll, Ll1) + s(Ll1, Ll2) + · · ·+ s(Lln−1, Ll0)

≤ β(s(lo, l) + s(l, l1) + s(l1, l2) + · · ·+ s(ln−1, l0))

≤ β(s(l0, l) + s(l, l0) + s(l0, l1) + s(l1, l0) + s(l0, l2) + · · ·+ s(ln−2, l0))

≤ 2β(n− 1)δ.

Set δ = ϵ/2(n− 1)β, the desired inequality is obtained.

Theorem 4. Given a complete metric space (Z, s), |Z| ≥ n, if the mapping L : Z → Z
satisfy the following conditions.

(i) L(L(l)) ̸= l ∀ l ∈ Z such that L(l) ̸= l.

(ii) L is a mapping contracting perimeters of polygons.

Then it admits a fixed point, moreover, maximum number of fixed points is (n− 1).

Proof. Let l0 ∈ Z,Ll0 = l1, Ll1 = l2, · · · , Lln = ln+1. Let li ̸= L(li) for each i =
0, 1, 2, .... So that, we have li ̸= li+1 = Lli. As li+2 = L(Lli) ̸= li, by condition (i) and
assuming that li+1 is not fixed, we obtain li+1 ̸= li+2 = Lli+1. Continuing the process, we
have li, li+1, · · · , li+n pairwise distinct. Moreover, set

p0 = s(l0, l1) + s(l1, l2) + s(l2, l3) + · · ·+ s(ln−1, l0)

p1 = s(l1, l2) + s(l2, l3) + s(l3, l4) + · · ·+ s(ln, l1)

· · ·
pk = s(lk, lk+1) + s(lk+1, lk+2) + · · ·+ s(ln+(k−1), lk),

by (1) we have, p1 ≤ βp0, p2 ≤ βp1, · · · pk ≤ βpk−1 and

p0 > p1 > p2 · · · > pk > · · · . (2)

Assume that j ≥ n is such that lj = li for 0 ≤ i ≤ j − (n − 1) then lj+1 = li+1, lj+2 =
li+2, lj+3 = li+3, · · · . Thus, pi = pj , which defies (2). Let’s verify that the {li} is a Cauchy
sequence. Clearly,

s(l1, l2) ≤ p0,

s(l2, l3) ≤ p1 ≤ βp0,



M. Nazam, U. Habiba, M. De la Sen / Eur. J. Pure Appl. Math, 18 (4) (2025), 6106 5 of 10

s(l3, l4) ≤ p2 ≤ β2p0,

· · ·
s(lk, lk+1) ≤ pk−1 ≤ βk−1p0,

by triangular inequality,

s(lk, lk+p) ≤ s(lk, lk+1) + s(lk+1, lk+2) + · · ·+ s(lk+p−1, lk+p),

≤ βk−1p0 + βkp0 + · · ·+ βk+p−2p0,

= βk−1(1 + β + · · ·+ βp−1)p0,

= βk−1(
1− βp

1− β
)p0.

Since, 0 ≤ β < 1, we have s(lk, lk+p) < βk−1 1
1−βp0. Hence, s(lk, lk+p) → 0 as k → ∞ where

p is positive. The sequence lk is therefore Cauchy. According to (Z, s) completeness, this
sequence’s limit is l∗ ∈ Z. Let us establish Ll∗ = l∗. Using inequality (1) and triangular
inequality, we have

s(l∗, Ll∗) ≤ s(l∗, lk) + s(lk, l
∗)

= s(l∗, lk) + s(Llk−1, l
∗)

≤ s(l∗, lk) + s(Llk−1, Ll
∗) + s(Llk−1, Llk) + · · ·+ s(Llk+n−3, Ll

∗)

≤ s(l∗, lk) + β(s(lk−1, l
∗) + s(lk−1, lk) + · · ·+ s(lk+n−3, l

∗)).

We obtain s(l∗, Ll∗) = 0, as every term in the preceding sum tends to zero as k → ∞.
Let us assume that there are a minimum of n distinct pairwise fixed points, denoted as
l1, l2, · · · , ln. Thus, Ll1 = l1, Ll2 = l2, · · · , Lln = ln, which is in opposition to (1).

The main result shows that if a mapping always reduces the perimeter of any polygon,
then applying the mapping repeatedly forces the polygon to shrink down to a single
point. That point is the fixed point of the mapping. This provides a geometric way of
understanding Banach’s contraction principle and extends Petrov’s triangle-based theorem
to polygons with any number of vertices.

Remark 5. Let us assume the following: the mapping L has a fixed point point l∗, a
limit of a sequence {ln}, consider l1 = Ll0,l2 = Ll1... such that ln ̸= l∗ for all n ∈ N .
Then there is only one fixed point, l∗. Suppose L has an additional fixed point l2∗ ̸= l∗.
It is evident that ln ̸= l2∗ ∀ n ≥ 1. Likewise, assume that l3∗, l4∗, · · · , ln∗ are fixed points.
Then ln ̸= l3∗ ̸= l4∗, · · · , ̸= ln∗ ∀ n ≥ 1. Thus, ∀ n ≥ 1, we have that the points
ln, l

∗, l2∗, · · · , l(n−1)∗ are pairwise distinct. Consider the ratio

Rn =
s(Lln, Ll

∗) + s(Ll∗, Ll2∗) + · · ·+ s(Ll(n−1)∗, Lln)

s(ln, l∗) + s(l∗, l2∗) + · · ·+ s(l(n−1)∗, ln)

=
s(ln+1, l

∗) + s(l∗, l2∗) + · · ·+ s(l(n−1)∗, ln+1)

s(ln, l∗) + s(l∗, l2∗) + · · ·+ s(l(n−1)∗, ln)
.

We obtain Rn → 1 as n → ∞, which does not satisfy 1.



M. Nazam, U. Habiba, M. De la Sen / Eur. J. Pure Appl. Math, 18 (4) (2025), 6106 6 of 10

Example 6. Let’s take an example of mapping L having precisely n− 1 fixed points. Let
Z = {l1, l2, · · · , ln}, s(l1, l2) = s(l2, l3) = s(l3, l4) = · · · = s(ln, l1) = 1 and let L : Z → Z
be such that L(l1) = l1, L(l2) = l2, · · · , L(ln−1) = ln−1, Lln = l1. It is evident that both of
the requirements of Theorem (4) are met.

Example 7. We will demonstrate that assumption (i) of Theorem (4) is necessary. Let
Y = {l1, l2, l3, · · · , ln}, s(l1, l2) = s(l2, l3) = · · · = s(ln, l1) and let L : Z → Z be such that
Ll1 = l2, Ll2 = l1, Ll3 = l4, Ll4 = l3, · · · , Lln = l1. It is evident that assumption (ii) of
Theorem (4) is satisfied, but the mapping L lacks a fixed point.

Given a metric space (Z, s), a mapping L : Z → Z is known as a contraction mapping
on Z if and only if there is a value for β ∈ [0, 1) such that

s(Lx,Ll) ≤ βs(x, l). (3)

where x, l ∈ Z.

Corollary 1. (Banach fixed point theorem) Let (Z, s) be a complete metric space, and let
L : Z → Z be a contraction mapping, then it admits a fixed point.

Proof. Let |Z| ≥ n. Assume that ∃ l ∈ Z which satisfy L(L(l)) = l. As a result,
s(x, Lx) = s(Lx, x) = s(Lx,L(Lx)), which contradicts to (3). Condition (i) of Theorem(4)
is thus met. Assume that l1, l2, l3 · · · ln ∈ Z are distinctive pairs. By using the contraction
definition, we have s(Ll1, Ll2) ≤ αs(l1, l2), s(Ll2, Ll3) ≤ αs(l2, l3), · · · , s(Lln−1, Lln)) ≤
αs(ln−1, ln) and which implies condition (ii) of Theorem(4). This concludes the evidence
for the existence of a fixed point. Assume there are x, y ∈ Z such that Lx = x and Ly = y.
Hence, s(Lx,Ly) = s(x, y) which contradicts to (3).

Example 8. Let us define a mapping L : Z → Z contracting perimeters of polygons for
a metric space Z, that is not a contraction mapping with |Z| = N0. Let Z = {l∗, l1, l2...}
and let b ∈ R+. Consider a metric s on Z such that

s(x, l) =



b
2⌊i/(n−1)⌋ if x = li, l = li+1, i ∈ N ;

s(li, li+1) + · · ·+ s(lj−1, lj) if x = li, l = lj , i+ 1 < j;

2(n− 1)b− s(l0, li) if x = li, l = l∗

0 if x = l,

where ⌊.⌋ is a floor function. Define a mapping L : Z → Z as Lli = li+1 for all i = 0, 1, ...
and Ll∗ = l∗. Since s(l2n, l2n+1) = s(Ll2n, Ll2n+1), n = 0, 1, .... We note that L does
not satisfy (1). Let us demonstrate that inequality(1) is true for each of the n distinct
pairwise points in the space Z. Consider the points l1, l2, · · · , ln−1, l

∗ ∈ Z. Using structure
of metric s we have

s(li, li+1) + s(li+1, li+2) + · · ·+ s(l∗, li+(n−1)) + s(li+(n−1), li+n)



M. Nazam, U. Habiba, M. De la Sen / Eur. J. Pure Appl. Math, 18 (4) (2025), 6106 7 of 10

= (n− 1)s(li, l
∗)

= (n− 1)(2(n− 1)a− s(li, l
∗)),

and

s(Lli, Lli+1)+s(Lli+1, Lli+2)+· · ·+s(Ll∗, Lli+(n−1))+s(Lli+(n−1), Lli+n) = (n−1)s(Lli+1, Ll
∗)

= (n− 1)(2(n− 1)b− s(li+1, l
∗)).

Moreover,

s(l0, li) =


2(n− 1)b(1− 1

2n ) if i = n(n− 1)

2(n− 1)b(1− 1
2n )−

b
2n−1 if i = n(n− 1)− 1,

n=1,2,..., observe also that s(t0, ti+1) = s(t0, ti) +
b

2⌊i/n−1⌋ . Consider the ratio

s(Lli, Lli+1) + s(Lli+1, Lli+2) + · · ·+ s(Ll∗, Lli+n−1) + s(Lli+n−1, Lli+n)

s(li, li+1) + s(li+1, li+2) + · · ·+ s(l∗, li+n−1) + s(li+n−1, li+n

=
(n− 1)(2(n− 1)b− s(l0, li+1))

(n− 1)(2(n− 1)b− s(l0, li))

=
2(n− 1)b− s(l0, li+1)

2(n− 1)b− s(l0, li)

=
2(n− 1)b− s(l0, li)− b/2⌊i/n−1⌋

2(n− 1)b− s(l0, li)

=


2(n−1)b−2(n−1)b(1−1/2n)−b/2⌊i/n−1⌋

2(n−1)b−2b(n−1)(1−1/2n) if i = n(n− 1);

2(n−1)b−2(n−1)b(1−1/2n)+a/2n−1−b/2⌊i/n−1⌋

2(n−1)b−2(n−1)b(1−1/2n)−b/2n−1 if i = n(n− 1)− 1

=


2n−3
2n−2 if i = n(n− 1)

n−1
n−2 if i = n(n− 1)− 1

.

Let li+1, li+2, · · · , li+n ∈ Z. We see that

s(li+1, li+2) + s(li+2, li+3) + · · ·+ s(li+n−1, li+n)

− (s(Lli+1, Lli+2) + s(Lli+2, Lli+3) + · · ·+ s(Lli+n−1, Lli+n))

= (n− 1)(b/2⌊i/(n−1)⌋ − b/2⌊m/(n−1)⌋).

Consider the ratio

Ri,k =
s(Lli+1, Lli+2) + s(Lli+2, Lli+3) + · · ·+ s(Lli+n−1, Lli+n)

s(li+1, li+2) + s(li+2, li+3) + · · ·+ s(li+n−1, li+n)

=
s(li+1, li+2) + s(li+2, li+3) + · · ·+ s(li+n−1, li+n)− (n− 1)(b/2⌊i/(n−1)⌋ − b/2⌊m/(n−1)⌋)

s(li+1, li+2 + s(li+2, li+3) + · · ·+ s(li+n−1, li+n)



M. Nazam, U. Habiba, M. De la Sen / Eur. J. Pure Appl. Math, 18 (4) (2025), 6106 8 of 10

=
1− (n− 1)(b/2⌊i/(n−1)⌋ − b/2⌊m/(n−1)⌋

s(li+1, li+2) + s(li+2, li+3) + · · ·+ s(li+n−1, li+n)
.

Note that i+ n− 2 < k, hence,

b/2⌊k/(n−1)⌋ ≤ b/2.b/2⌊i/(n−1)⌋. (4)

It can be demonstrated that by using the structure of the space (L, s),

s(li+k, l
∗) ≤ 2(n− 1)s(li+1, li+2).

Clearly s(li+k, li+m) ≤ s(li+k, l
∗). Hence s(li+m, li+k) ≤ 2(n− 1)s(li+m, li+m+1). It can be

concluded by using the last inequality

s(li+1, li+2) + s(li+2, li+3) + · · ·+ s(li+n−1, li+n) = (n− 1)s(li+m, li+k)

≤ 2(n− 1)2s(li+1, li+2)

=
2(n− 1)2b

2⌊i/(n−1)⌋ .

By applying the inequality (4), we get

Ri,k ≤ 1− (n− 1)(
b/2⌊i/(n−1)⌋ − b/2.2⌊i/(n−1)⌋

2(n− 1)2b/2⌊i/(n−1)⌋ )

=
4n− 5

4n− 4
.

Hence the inequality (1) holds true.

The following diagrams effectively demonstrate:

1. How a sequence of polygons shrinks under a contracting transformation.

2. The idea that a sequence of transformations converges to a unique fixed point.

3. The concept of the Fixed Point Theorem in action.

Polygon P0

Polygon P1

Fixed Point P ∗



M. Nazam, U. Habiba, M. De la Sen / Eur. J. Pure Appl. Math, 18 (4) (2025), 6106 9 of 10

Hexagon H0

Hexagon H1 Fixed Point H∗

Heptagon G0

Heptagon G1

Fixed Point G∗

4. Conclusion

In this paper, we studied fixed point results for mappings that contract the perimeters
of polygons. By extending Petrov’s theorem from the case of triangles (n = 3) to gen-
eral polygons with n vertices, we developed a new geometric perspective on contraction
mappings. Our results show that requiring the contraction of polygonal perimeters is suf-
ficient to ensure the existence of fixed points, thereby providing a natural generalization
of both Petrov’s theorem and the Banach contraction principle. We also demonstrated
that Banach’s contraction theorem can be re-derived as a corollary of our framework,
which highlights the strength and versatility of this perimeter-based approach. The com-
parison with other generalizations of Banach’s principle shows that, unlike distance- or
order-based extensions, our method provides geometric insights not captured in previous
studies. In addition to the theoretical contributions, several illustrative examples were
given to confirm the applicability of the main results. The presented framework opens
avenues for further work on fixed point theory in geometric and applied settings, such
as in dynamical systems, optimization, and problems involving polygonal or polyhedral
structures. In summary, the present study contributes a complementary and geometrically
motivated extension of fixed point theory. Future work may explore practical applications
of perimeter-contraction mappings, connections with Ulam stability problems, and the
extension of these ideas to higher-dimensional polyhedra.

Acknowledgements

The authors extend their sincere gratitude to the Honorable Reviewers of the European
Journal of Pure and Applied Mathematics for their valuable insights and constructive
feedback, which greatly enhanced the quality of our submission.



M. Nazam, U. Habiba, M. De la Sen / Eur. J. Pure Appl. Math, 18 (4) (2025), 6106 10 of 10

5. Authors’ contributions

M.N. tabled the main idea of this paper; U.H. wrote the first draft of this paper; M.N.
and M. D. S. reviewed and prepared the second draft; M. D. S. supervised the project.
All authors have read and agreed to the published version of the manuscript.

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