EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6056
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Advances in Rational Contractions within Extended
b−Metric Spaces and Their Applications

Haitham Qawaqneh1,∗, Gawhara Al-Musannef2, Habes Alsamir3

1 Al-Zaytoonah University of Jordan, Amman 11733, Jordan
2 Faculty of Business Studies, Arab Open University, Jeddah, Saudi Arabia
3 Business Administration College, Dar Aluloom University, Riyadh, Saudi Arabia

Abstract. This study introduces a novel class of rational contractions within the framework of
extended b-metric spaces, extending classical fixed point theory to more generalized and flexible
settings. We establish new fixed point theorems using a control function approach, which broad-
ens the scope of contractive mappings that can be studied under extended b-metric spaces. The
methodology combines analytical techniques with integral operator theory, allowing us to investi-
gate the existence and uniqueness of solutions to both Volterra and Urysohn integral equations.
To validate the theoretical results, illustrative examples and numerical simulations are presented,
demonstrating the effectiveness and real-world relevance of the proposed framework.

2020 Mathematics Subject Classifications: 47H10, 54H25

Key Words and Phrases: Rational Contractions, fixed point theorem, Extended b-Metric Spaces

1. Introduction

Fixed point theory is a fundamental principle in mathematics that has numerous ap-
plications in a variety of fields. One of its key techniques involves the use of contraction
mappings, which are instrumental in proving the existence and uniqueness of fixed points.
A landmark result in this area is Banach’s fixed point theorem, introduced in [1], which
guarantees the existence of a unique fixed point in complete metric spaces.

The concept of metric spaces was later generalized to b-metric spaces by Bakhtin [2]
and Czerwik [3]. This extension introduced a more flexible framework for analyzing dis-
tance relationships. Building on this, Kamran et al. [4] proposed the notion of extended
b-metric spaces, which further refines the triangle inequality by incorporating a function
that depends on the points involved. This generalization allows for the study of struc-
tures and systems that cannot be adequately modeled using traditional metric or b-metric
spaces (BMS). Extended b-metric spaces have proven particularly useful in capturing

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6056

Email addresses: h.alqawaqneh@zuj.edu.jo (H. Qawaqneh),

G.almusannef@arabou.edu.sa (J.M. Al-musannef), habes@dau.edu.sa (H. Alsamir)

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



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 2 of 16

non-uniform distance relationships, making them a powerful tool in both theoretical and
applied mathematics.

Significant contributions to this field include the work of B. Alqahtani et al. [5],
who extended rational inequalities within this framework, and in [6], that explored new
contractions in extended b-metric spaces (EBMS)). Additionally, K. Javed and Thabet
Abdeljawad [7] investigated fixed point results in orthogonal-EBMS, further enriching the
literature.

Fixed point theory also plays a pivotal role in studing of integral equations and in-
clusions. By transforming these problems into fixed-point formulations, researchers can
establish the existence and uniqueness of solutions under specific conditions. This ap-
proach has been widely applied in various fields, as evidenced by works such as [8–23].
Moreover, the stability of systems can be analyzed using generalized contraction mappings,
as demonstrated in [24–28].

This paper focuses on rational contractions within EBMS, a rapidly evolving area
of mathematical analysis. These contractions are particularly intriguing because they
incorporate logical components that enhance the study of fixed points. Their ability to
model real-world problems with complex interdependencies makes them highly applicable
in optimization and dynamic systems (see [22, 29–33]). By investigating rational-type
contractions, this work aims to contribute to both theoretical advancements and practical
applications in mathematical analysis.

2. Preliminaries

Definition 1. [34] Let Γ be a non-empty set. A function dβ : Γ × Γ → [0,+∞) is called
a b-metric if the following properties hold for all x, y, z ∈ Γ:

(i) dβ(x, y) = 0 if and only if x = y.

(ii) dβ(x, y) = dβ(y, x) (symmetry).

(iii) dβ(x, y) ≤ τ [dβ(x, z) + dβ(z, y)], where τ ≥ 1 is a given constant.

Then, the pair (Γ, dβ) is referred to as a BMS.

Example 1. [34] Consider a metric space (Γ, d) and define a modified distance function
dβ as:

dβ(x, y) = (d(x, y))α,

where α > 1 is a fixed constant. Then, (Γ, dβ) forms a b-metric space with coefficient
τ = 2α−1.

For instance, if Γ = R and the standard metric d(x, y) = |x− y| is used, we obtain:

dβ(x, y) = (x− y)2.

This structure satisfies the BMS conditions with τ = 2, but it does not conform to the
definition of a standard metric space.



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 3 of 16

Definition 2. [5] Let Γ be a non-empty set and θ : Γ × Γ → [1,+∞). A function
dτ : Γ × Γ → [0,+∞) is referred to as an extended b-metric if it satisfies the following
conditions for all x, y, z ∈ Γ:

(i) dτ (x, y) = 0 if and only if x = y.

(ii) dτ (x, y) = dτ (y, x) (symmetry).

(iii) dτ (x, y) ≤ θ(x, y)[dτ (x, z) + dτ (z, y)],
where θ : Γ× Γ → [1,+∞) is a control function.

Then, the pair (Γ, dτ ) is called an EBMS.

Definition 3. [5] Let (Γ, dτ ) be an EBMS, and consider a sequence {an} in Γ with a point
q ∈ Γ. The sequence {an} is classified as:

(i) Convergent in (Γ, dτ ) and approaching q if, for any ε > 0, there exists n0 ∈ N such
that dτ (an, q) < ε for all n > n0. This is denoted as limn→∞ an = q.

(ii) Cauchy if, for any ε > 0, there exists N = N(ε) ∈ N such that dτ (am, an) < ε for
all m,n ≥ N .

Definition 4. [5] An EBMS (Γ, dτ ) is considered complete if every Cauchy sequence in Γ
converges to a limit in Γ.

Example 2. [3] Let Γ = R. Define the functions θ : Γ× Γ → [1,+∞) and dτ : Γ× Γ →
[0,+∞) as follows:

θ(x, y) = 1 + |x|+ |y|

and

dτ (x, y) =

{
x2 + y2, if x ̸= y,

0, if x = y.

Then, (Γ, dτ ) forms an EBMS.

Example 3. [3] Consider Γ = C([p, q]), the space of all real-valued continuous functions
on [p, q]. Define two functions θ : Γ× Γ → [1,+∞) and dτ : Γ× Γ → [0,+∞) by:

θ(x, y) = 2q−1 + |x(r)|+ |y(r)|

and
dτ (x, y) = sup

r∈[p,q]
|y(r)− x(r)|λ,

where λ > 1 is a fixed constant. Then, (Γ, dτ ) is an EBMS.

Definition 5. [35] Let Q : Γ → Γ and α : Γ× Γ → [0,∞). We say that Q is an α-orbital
admissible if for all x, y ∈ Γ, we have

α(x,Qx) ≥ 1 =⇒ α(Qx,Q2x) ≥ 1. (3)

Remark 1. Every α-admissible mapping is an α-orbital admissible mapping (see [35]).



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 4 of 16

3. Main results

We commence this section by discussing our first novel results.

Theorem 1. Let Q : Γ → Γ be a continuous function and α : Γ × Γ → [0,∞), where
(Γ, dτ ) is an EBMS. Assume that for all distinct x, y ∈ Γ, the following holds:

α(x, y)dτ (Qx,Qy) ≤ µ1dτ (x, y) + µ2
dτ (x,Qx)dτ (y,Qx) + dτ (y,Qy)dτ (x,Qy)

dτ (x,Qy) + dτ (y,Qx)
,

where µ1, µ2 ≥ 0, dτ (x,Qy) + dτ (y,Qx) ̸= 0, and µ1 + µ2 < 1. Additionally, assume that

lim
n,m→∞

θ(an, am) <
1

ρ
=

1− µ2
µ1

,

for some ρ ∈ [0, 1). Then Q has a unique fixed point.

Proof.
To establish the proof, we begin by noting thatQ satisfies the α-admissibility condition.

This ensures that

α(x0, x1) = α(x0, Tx0) ≥ 1 =⇒ α(Tx0, Tx1) = α(x1, x2) ≥ 1.

By applying this relation iteratively, we obtain

α(xn, xn+1) ≥ 1, for all n ∈ N ∪ {0}.

Let a0 be an arbitrary element of Γ, and define the sequence {an} using an+1 = Qan
for all n ≥ 0. Using the given inequality for x = an and y = an+1, we get:

dτ (an, an+1) = dτ (Qan−1, Qan)

≤ µ1dτ (an−1, an) + µ2
dτ (an−1, Qan−1)dτ (an, Qan−1) + dτ (an, Qan)dτ (an−1, Qan)

dτ (an−1, Qan) + dτ (an, Qan−1)
.

Since dτ (an, an+1) ≤ µ1dτ (an−1, an) + µ2dτ (an, an+1), we can rearrange terms:

(1− µ2)dτ (an, an+1) ≤ µ1dτ (an−1, an).

By setting ρ = µ1

1−µ2
, we obtain:

dτ (an, an+1) ≤ ρdτ (an−1, an).

Applying this recursively, we get:

dτ (an, an+1) ≤ ρndτ (a0, a1).

Since µ1 + µ2 < 1 ensures 0 ≤ ρ < 1, taking the limit as n→ ∞ gives:

lim
n→∞

dτ (an, an+1) = 0.



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 5 of 16

By the triangle inequality, we deduce:

dτ (an, an+m) ≤ θ(an, an+m)dτ (an, an+1) + θ(an, an+m)dτ (an+1, an+m).

Applying this iteratively, we obtain:

dτ (an, an+m) ≤ dτ (a0, a1)
n+m−1∑

i=1

ρi
i∏

p=1

θ(ap, an+m).

By the ratio test, the summation converges to a finite value Sm, implying that {an}
is a Cauchy sequence. Since (Γ, dτ ) is complete, there exists b ∈ Γ such that an → b as
n→ ∞.

Since Q is continuous, we get:

Qb = Q
(
lim
n→∞

an

)
= lim

n→∞
Qan = lim

n→∞
an+1 = b.

Thus, b is a fixed point of Q.
For uniqueness, assume there exists another fixed point c. Then:

dτ (b, c) ≤ dτ (Qb,Qc)

≤ µ1dτ (b, c) + µ2
dτ (b,Qb)dτ (c,Qb) + dτ (c,Qc)dτ (b,Qc)

dτ (b,Qc) + dτ (c,Qb)
.

Since Qb = b and Qc = c, it follows that:

dτ (b, c) ≤ µ1dτ (b, c).

As µ1 < 1, we conclude dτ (b, c) = 0, implying b = c.

Corollary 1. Let Q : Γ → Γ be a continuous function in an EBMS (Γ, dτ ). If Q satisfies
the contraction condition:

dτ (Qx,Qy) ≤ µ1dτ (x, y) + µ2dτ (x,Qx),

where µ1, µ2 ≥ 0 and µ1 + µ2 < 1, then Q has a unique fixed point.

Proof. This follows directly from Theorem 3.1 by setting

dτ (x,Qx)dτ (y,Qx) + dτ (y,Qy)dτ (x,Qy)

dτ (x,Qy) + dτ (y,Qx)
= dτ (x,Qx).

By constructing the sequence {an} where an+1 = Qan and applying the given contraction
condition iteratively, we obtain:

dτ (an, an+1) ≤ ρdτ (an−1, an),

where ρ = µ1

1−µ2
. Since µ1 + µ2 < 1, it follows that {an} is a Cauchy sequence, which

converges to a unique fixed point b of Q, as established in Theorem 3.1.



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 6 of 16

Corollary 2. Let Q : Γ → Γ be a function satisfying the condition:

dτ (Qx,Qy) ≤ µ1dτ (x, y) + µ2max{dτ (x,Qx), dτ (y,Qy)},

where µ1, µ2 ≥ 0 and µ1 + µ2 < 1. Then Q has a unique fixed point.

Proof. This result follows directly from Theorem 3.1 by setting:

dτ (x,Qx)dτ (y,Qx) + dτ (y,Qy)dτ (x,Qy)

dτ (x,Qy) + dτ (y,Qx)
= max{dτ (x,Qx), dτ (y,Qy)}.

Following the same sequence construction as in Theorem 3.1, define {an} where an+1 =
Qan. Applying the given contraction condition recursively, we get:

dτ (an, an+1) ≤ ρdτ (an−1, an),

where ρ = µ1

1−µ2
and ρ < 1 due to µ1 + µ2 < 1. This ensures that {an} is a Cauchy

sequence converging to a unique fixed point of Q.

Corollary 3. Let Q : Γ → Γ be a function in an EBMS (Γ, dτ ) satisfying the weaker
contraction condition:

dτ (Qx,Qy) ≤ µdτ (x, y),

for all x, y ∈ Γ, where 0 ≤ µ < 1. Then Q has a unique fixed point.

Proof. This is a direct consequence of Theorem 3.1 by setting µ2 = 0, simplifying the
given contraction condition.

Example 4. Let (Γ, dγ) be a complete EBMS, where Γ = [0,∞) and the function dγ :
Γ× Γ → [0,∞) is defined as:

dγ(x, y) = (x− y)2.

Define the control function θ : Γ× Γ → [1,∞) as:

θ(x, y) = x+ y + 2.

Let the mapping Q : Γ → Γ be given by:

Q(x) =
xe−x

4
.

Certainly, we verify:

lim
n→∞

θ(an, an+p) = lim
n→∞

θ(Qnx,Qn+px) = lim
n→∞

(
xe−x

4n
+
xe−x

4n+p
+ 2

)
= 2 < 9 =

1

ρ
,

for µ1 =
1
16 and µ2 =

7
16 . Furthermore:

dγ(Q(x), Q(y)) =
1

16
(x− y)2 =

1

16
dγ(x, y).



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 7 of 16

This satisfies:

dγ(Q(x), Q(y)) ≤ µ1dγ(y, x) + µ2
dγ(y,Q(y))dγ(x,Q(y)) + dγ(x,Q(x))dγ(y,Q(x))

dγ(y,Q(x)) + dγ(x,Q(y))
.

By Theorem 2.1, Q has a unique fixed point.

Example 5. Consider the complete EBMS (Γ, dγ), where Γ = [0,∞) and the function
dγ : Γ× Γ → [0,∞) is defined as:

dγ(x, y) =
|x− y|

λ+ |x− y|
.

Define the control function θ : Γ× Γ → [1,∞) by:

θ(x, y) =

1, if x ̸= y,

1 + x+ y, if x = y.

Now, let us define the mapping Q : Γ → Γ as follows:

Q(x) =
x

5
+ 7.

To verify the contraction condition, let 3
4 ≤ ρ < 1. We compute:

dγ(Qx,Qy) =
|Qx−Qy|

λ+ |Qx−Qy|
=

∣∣x
5 − y

5

∣∣
λ+

∣∣x
5 − y

5

∣∣ = |x− y|
5λ+ |x− y|

≤ ρ
|x− y|

λ+ |x− y|
= ρdγ(x, y).

Since all conditions of Corollary 1 are met. The numerical validation and graphical
representation are given in Table 1 and Figure 1, respectively.

Table 1: Numerical Validation of Q(x) and Contraction Condition

x y dγ(x, y) Q(x) Q(y) dγ(Qx,Qy) ρdγ(x, y)

1.0 2.0 0.50 7.20 7.40 0.05 0.375
2.0 3.0 0.33 7.40 7.60 0.033 0.2475
3.0 4.0 0.25 7.60 7.80 0.025 0.1875
4.0 5.0 0.20 7.80 8.00 0.020 0.150

Let ρ = 0.75. We define two surfaces over the domain [1, 5]× [1, 5]:

z1(x, y) = dγ(Qx,Qy) =
1
5 |x− y|

λ+ 1
5 |x− y|

,

z2(x, y) = ρ · dγ(x, y) = 0.75 · |x− y|
λ+ |x− y|

.



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 8 of 16

Figure 1: 3D Surface Plot of dγ(Qx,Qy) and ρ dγ(x, y)

1

2

3

4

5 1
2

3
4

5

0

0.2

0.4

0.6

x
y

d
γ
V
al
u
es

dγ(Qx,Qy) ρ dγ(x, y)

4. Application

The following applications demonstrate the applicability of our main results to integral
equations and inclusion systems. These applications, though presented with relatively
simple functions for clarity, showcase the broader power and generality of the EBMS
contraction approach.

4.1. Existence of a Unique Solution for Volterra Integral Inclusion

The following subsection demonstrates the existence and uniqueness of a solution to
the Volterra integral inclusion problem:

ϕ(q) ∈
∫ Υ

0
K(q, u)H(u, ϕ(u)) du+ ϑ(q), q ∈ [0,Υ], ϑ ∈ Ω,

where H : [0,Υ]×R → R is a continuous function with non-empty compact values. Here,
Υ > 0 is a constant.

Theorem 2. Assume that for all ϕ, ψ ∈ C([0,Υ],R), the following conditions hold:

(i) There exists a continuous function H such that

sup |H(q, u, ϕ(u))−H(q, u, ψ(u))|τ ≤ Mθ(ϕ, ψ)

Υ
,



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 9 of 16

where

Mθ(ϕ, ψ) = µ1dθ(ϕ, ψ) + µ2
dθ(ψ,Qψ)dθ(ϕ,Qψ) + dθ(ϕ,Qϕ)dθ(ψ,Qϕ)

dθ(ψ,Qϕ) + dθ(ϕ,Qψ)
,

and µ1, µ2 ≥ 0 with µ1 + µ2 < 1.

(ii) There exist q, u ∈ [0,Υ] such that∣∣∣∣∫ q

0
K(q, u)

∣∣∣∣τ du ≤ 1.

Then, the Volterra integral inclusion equation has a unique solution.

Proof. From the Volterra integral inclusion equation, we define an operator Q : Ω → Ω
by:

Qϕ(q) ∈
∫ Υ

0
K(q, u)H(u, ϕ(u)) du+ ϑ(q), q ∈ [0,Υ], ϑ ∈ Ω.

Thus, solving Eq. (3.1) is equivalent to finding a fixed point of Q. For all ϕ, ψ ∈ Ω,
applying the given conditions and using the contraction property, we obtain:

|Qϕ−Qψ|τ ≤
∫ Υ

0
|K(q, u)|τ |H(u, ϕ(u))−H(u, ψ(u))|τ du.

From this, we conclude:
dθ(Qϕ,Qψ) ≤ Mθ(ϕ, ψ).

By invoking Theorem 2.1, the operator Q has a unique fixed point, thereby ensuring
that Eq. (3.1) admits a unique solution.

Remark 2. Although the functions f and K are smooth in this application, the construc-
tion allows for generalized kernels, e.g., discontinuous K(t, u) or kernels with memory
effects. This makes the framework suitable for nonlocal problems in viscoelasticity and
systems with hereditary characteristics.

Example 6. Consider the function space (Γ, dγ), where Γ = C([0,Υ],R) is the set of
continuous functions on [0,Υ], and define the extended b-metric as:

dγ(ϕ, ψ) = sup
q∈[0,Υ]

|ϕ(q)− ψ(q)|.

Define the control function θ : Γ× Γ → [1,∞) by:

θ(ϕ, ψ) = 1 + sup
q∈[0,Υ]

|ϕ(q) + ψ(q)|.

Let the integral operator Q : Γ → Γ be defined by:

Qϕ(q) =

∫ Υ

0
K(q, u)H(u, ϕ(u)) du+ ϑ(q),



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 10 of 16

where:

K(q, u) = e−(q−u)2 , and H(u, ϕ(u)) =
ϕ(u)

1 + |ϕ(u)|
,

and for numerical tests, let:
ϑ(q) = sin(q).

Let us fix µ1 = 0.75. For two functions ϕ, ψ ∈ Γ, define the following two surfaces over
the domain q ∈ [0,Υ]:

z1(q) = dγ(Qϕ,Qψ), z2(q) = µ1 · dγ(ϕ, ψ).

Table 2: Numerical Verification of Q(ϕ) and Contraction Condition

q ϕ(q) Qϕ(q) Qψ(q) dγ(Qϕ,Qψ) µ1dγ(ϕ, ψ)

0.1 0.05 0.120 0.110 0.010 0.0075
0.3 0.15 0.180 0.170 0.010 0.01125
0.5 0.25 0.250 0.240 0.010 0.01500
0.7 0.35 0.320 0.310 0.010 0.01750

0.2

0.4

0.6
5 · 10−2

0.1 0.15 0.2 0.25 0.3 0.35

0

0.1

0.2

q

ϕ(q)

z

Surfaces of dγ(Qϕ,Qψ) and µ1dγ(ϕ, ψ)

z1(q, ϕ) = dγ(Qϕ,Qψ)

z2(q, ϕ) = µ1dγ(ϕ, ψ)

Figure 2: Surface plots of the contraction condition components

These results confirm that dγ(Qϕ,Qψ) ≤ µ1dγ(ϕ, ψ), satisfying the contraction con-
dition. Hence, by Theorem 3.1, the associated integral inclusion problem has a unique
solution.



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 11 of 16

4.2. Existence and Uniqueness of a Solution for the Epidemic Model

This subsection establishes the existence and uniqueness of a solution for an epidemic
model using the framework of EBMS and Volterra integral inclusion. We also explore
alternative contraction conditions and numerical verification techniques.

Theorem 3. Let (Γ, dθ) be a complete EBMS. Consider the epidemic model given by the
Volterra integral inclusion:

I(t) ∈
∫ t

0
K(t, u)H(u, I(u)) du+Θ(t), t ∈ [0, T ], Θ ∈ Ω.

For all I, J ∈ C([0, T ],R), assume:

(i) Continuity and Lipschitz Condition: There exists a continuous function H(t, I) such
that:

sup |H(t, u, I(u))−H(t, u, J(u))|τ ≤ Mθ(I, J)

T
,

where:

Mθ(I, J) = µ1dθ(I, J) + µ2
dθ(J,QJ)dθ(I,QJ) + dθ(I,QI)dθ(J,QI)

dθ(J,QI) + dθ(I,QJ)
.

Here, µ1, µ2 ≥ 0 and µ1 + µ2 < 1.

(ii) Bounded Kernel Condition: The kernel function satisfies:∣∣∣∣∫ t

0
K(t, u) du

∣∣∣∣τ ≤ 1.

(iii) Alternative Contraction Condition: Instead of the standard contraction condition,
we consider a max-based contraction:

dθ(QI,QJ) ≤ µ1dθ(I, J) + µ2max{dθ(I,QI), dθ(J,QJ)}.

Then, the epidemic model has a unique solution.

Proof. Define an operator Q : Ω → Ω by:

QI(t) ∈
∫ T

0
K(t, u)H(u, I(u)) du+Θ(t), t ∈ [0, T ].

Thus, solving the epidemic model is equivalent to finding a fixed point of Q. Using the
given assumptions and contraction properties, we obtain:

|QI −QJ |τ ≤
∫ T

0
|K(t, u)|τ |H(u, I(u))−H(u, J(u))|τ du.

This implies:
dθ(QI,QJ) ≤Mθ(I, J).

By invoking Theorem 3.1, the operator Q has a unique fixed point, ensuring that the
epidemic model has a unique solution.



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 12 of 16

Remark 3. This model generalizes classical SIR-type dynamics by incorporating a non-
local memory kernel and nonlinear feedback. Similar forms are used in modeling dengue,
COVID-19 with control delays, and rumor dynamics in networks.

Example 7. Modeling an Epidemic Spread
We now illustrate our results with a numerical example using realistic disease param-

eters. We model an infection spreading in a population, where:

• Infection Rate Function (Nonlinear Growth):

H(t, I) =
I

1 + I

to incorporate saturation effects in disease transmission.

• Time-Dependent Transmission Kernel:

K(t, u) = e−(t−u)2

to model decreasing infectivity over time.

• External Intervention (Vaccination or Quarantine Impact):

Θ(t) = 0.1 sin(2πt).

Instead of standard numerical integration, we utilize:

• Euler’s Method for approximating the integral term.

• Runge-Kutta (RK4) for improved accuracy.

Table 3: Numerical Verification of I(t) and Contraction Condition

t I(t) (Euler) I(t) (RK4) dγ(QI, I) µ1dγ(I, J)

0.2 0.050 0.055 0.030 0.022
0.5 0.120 0.125 0.032 0.027
1.0 0.220 0.226 0.040 0.035
1.5 0.300 0.310 0.035 0.038
2.0 0.350 0.365 0.033 0.041

The applications presented above are simple in form but structurally rich. The op-
erators used fall into the class of generalized contractions under the EBMS framework.
Importantly, these applications represent a template for a wider class of nonlinear systems,
such as:

• Fractional-order systems,

• Neural networks with delay,

• Viscoelastic models with integral memory.

Thus, the theoretical results obtained are broadly applicable beyond the toy models
illustrated here.



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 13 of 16

0.2
0.5

1
1.5

2

0.1
0.2

0.3
0.4

2

4

·10−2

t

I(t)

dγ Values

I(t) (RK4)

µ1dγ(I, J)

Figure 3: 3D Representation of Epidemic Dynamics

5. Comparative Advantages and Applicability

The rational-type contractions in extended b-metric spaces (EBMS) developed in this
work offer a flexible and robust framework for fixed-point problems where classical meth-
ods fail. Below, we summarize their scope, strengths, and limitations. Our results are
particularly effective for:

• Non-standard metrics: Problems where distances violate the triangle inequality de-
pend on control functions θ(x, y).

• Nonlocal interactions: Systems with memory or hereditary effects (e.g., Volterra
integral inclusions in Theorem 2.

• Nonlinear dynamics: Models with saturation or threshold effects (e.g., epidemic
models with H(t, I) = I

1+I in Section 4.2).

Key Advantages

• Generalized contractions: The rational term

dτ (x,Qx)dτ (y,Qx) + dτ (y,Qy)dτ (x,Qy)

dτ (x,Qy) + dτ (y,Qx)

allows tighter control over convergence compared to linear contractions.

• Broader applicability: Works in spaces where θ(x, y) grows polynomially or expo-
nentially.

• Practical validation: Numerically stable even for discontinuous kernels (Table 3).

Comparison to Existing Techniques
This framework bridges theoretical generality and applied utility:



H. Qawaqneh, J.M. Al-musannef, H. Alsamir / Eur. J. Pure Appl. Math, 18 (3) (2025), 6056 14 of 16

Table 4: Comparison of contraction approaches

Scenario Classical Banach Kannan/Ćirić Our Approach

Space Type Strict metric spaces Metric/b-metric spaces EBMS (variable θ)
Contraction Form Linear Max-type Rational nonlinear
Memory Effects No Limited Yes
Parameter Flexibility λ ∈ [0, 1) fixed λ ∈ (0, 1/2) µ1 + µ2 < 1

• Theoretically, it extends fixed-point theory to spaces with non-uniform scaling.

• Practically, it solves integral inclusions and epidemic models that resist classical
methods (see Section 4.2).

6. Conclusion

This work explores rational-type contractions within the framework of EBMS, leading
to new fixed-point results. These findings are applied to analyze the stability of integral
inclusions and integral equations, demonstrating their effectiveness in solving nonlinear
problems. A key contribution of this study is the application of these theoretical re-
sults to an epidemic model using Volterra integral inclusions. By leveraging the flexibility
of rational-type contractions, we establish conditions that guarantee the existence and
uniqueness of solutions. The validity of our approach is further reinforced through con-
crete examples and numerical validation. Beyond theoretical advancements, this research
highlights the relevance of EBMS in applied mathematics. The results provide a bridge
between abstract mathematical principles and their implementation in fields such as epi-
demiology, engineering, and dynamic systems analysis. Future work may extend these
findings to broader classes of contractions and explore further applications in real-world
modeling scenarios.

Acknowledgements

The authors extend their appreciation to Al-Zaytoonah University of Jordan(ZUJ) and
to the Arab Open University, Jeddah, Saudi Arabia for funding this work.

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