EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6157
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Qualitative Properties of LΘ-Solutions for Coupled
Systems of Hadamard-Type Fractional Integral

Equations in Banach Spaces

Mohamed M.A. Metwali1,2,∗, Shami A. M. Alsallami3

1 Department of Mathematics, College of Science and Humanities in AlKharj, Prince Sattam
Bin Abdulaziz University, AlKharj 11942, Saudi Arabia
2 Department of Mathematics and Computer Science, Faculty of Science,
Damanhour University, Damanhour, 22511, Egypt
3 Mathematics Department, College of Sciences, Umm Al-Qura University, Makkah 24381,
Saudi Arabia

Abstract. In this manuscript, the measure of noncompactness (MNC), Darbo and Banach con-
traction fixed point theorems (FPT ), as well as fractional calculus, are used to carry out the anal-
ysis of the solvability of a general but abstract coupled system of quadratic Hadamard-fractional
integral equations in Orlicz spaces LΘ. Several qualitative properties of the solution to the studied
coupled system are established, such as the existence, monotonicity, and uniqueness, in addition
to continuous dependence on the data. We conclude with some examples that illustrate our hy-
pothesis.
2020 Mathematics Subject Classifications: 47H30, 45G10, 47N20

Key Words and Phrases: Fixed-point theorem (FPT ), Orlicz spaces LΘ, coupled system of
integral equations, (MNC) measure of noncompactness

1. Introduction

Coupled systems of differential and integral equations are often used to formulate
physical and biological models. The study of coupled systems of integral equations is of
significant interest to numerous fields of science, such as multimedia processing [1], nuclear
physics [2], diffusion equations [3], electromagnetics [4], and heat conduction [5].

The aim of the present paper is to analyze and demonstrate the solutions of the coupled
system:

x(t) = h1(t) + f1

(
t, Λ1(y)(t),

G1(y)(t)
Γ(β) ·

∫ t
1

(
log t

s

)β−1
R1(y)(s)

s ds

)
y(t) = h2(t) + f2

(
t, Λ2(x)(t),

G2(x)(t)
Γ(β) ·

∫ t
1

(
log t

s

)β−1
R2(x)(s)

s ds

)
, t ∈ [1, e],

(1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6157

Email addresses: m.metwali@psau.edu.sa (M. Metwali), sasallami@uqu.edu.sa (S. Alsallami)

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



M. Metwali, S. Alsallami / Eur. J. Pure Appl. Math, 18 (2) (2025), 6157 2 of 15

where 0 < β < 1, in Orlicz spaces LΘ, and the operators Gi, Λi, Ri, i = 1, 2, operate on
some arbitrary LΘ.

We establish and present assumptions that allow us to solve and study the coupled
system (1) under general growth conditions. As a result, we examine some qualitative
properties of the problem (1), such as existence, monotonicity, and uniqueness, in addition
to the continuous dependence on the data in the spaces LΘ (cf. [6]).

Several authors examined various types of coupled systems of integral equations in the
literature, including the space C(J) (cf. [7–10]) and the Banach algebras (cf. [11, 12], for
instance), where the outcomes have been made under conditions that are ”continuous,”
i.e., stronger than the ones provided in this article. Additionally, polynomial growth was
used on the studied functions to obtain Lp-solutions for the coupled systems in [13, 14].
As a result of eliminating these limitations, we extended these results to examine the
coupled system (1) using the technique presented in [15] that is not a Banach algebra,
using appropriately and various Orlicz spaces (LΘ1 , LΘ2 , LΘ3), which are not a Banach
algebra.

Using Orlicz spaces LΘ as the solution space, we can study operators with strong
nonlinear properties (such as exponential growth, for example). This allows us to examine
the solutions in LΘ rather than continuous results. Statistical physics and physics models
may inspire this (cf. [16, 17]). Recalling the thermodynamics model

y(s) +

∫
I
A(s, t)ey(t) dt = 0

contains exponential nonlinearity (cf. [18]). Furthermore, the quadratic integral equations
(QIE) were studied in the Banach-Orlicz algebra [19] and in various Orlicz spaces in [15, 20]
employing the approach of the fixed point theorems (FPT ) in conjunction with a suitable
(MNC) measure of noncompactness (MNC) concerning different assumptions, see also
[21, 22]. The measures of noncompactness (MNC) have been employed in the study of
numerous models of integral equations; (cf. [23–25]). These cases are unified and included
as special cases of the coupled system (1). Let us recall that, in [26], two existence theorems
of the coupled system

x(t) = g1(t) + f1

(
t, y(t), λ · V1y(t)

∫ b
a K

(
t, s

)
h1(s, y(s)) ds, λ ·G1y(t)

∫ b
a u1

(
t, s, y(s)

)
ds

)
y(t) = g2(t) + f2

(
t, x(t), λ · V2x(t)

∫ b
a K

(
t, s

)
h2(s, x(s)) ds, λ ·G2x(t)

∫ b
a u2

(
t, s, x(s)

)
ds

)
have been studied in arbitrary LΘ, in two separately cases ∆′ and ∆3-conditions using
Darbo’s(FPT ) with a (MNC). The authors in [27] studied the existence, in addition to
the uniqueness of monotonic solutions of the Hadamard fraction equations

x(t) =

n∏
i=1

(
hi(t)+G2i(x)(t)+

G1i(x)(t)

Γ(αi)
·
∫ t

1

(
log

t

s

)αi−1G3i(x)(s)

s
ds

)
, t ∈ [1, e], 0 < αi < 1



M. Metwali, S. Alsallami / Eur. J. Pure Appl. Math, 18 (2) (2025), 6157 3 of 15

in Orlicz spaces see also [28]. The current manuscript is motivated and induced by the
extension and generalization of the results introduced in the previous literature to prove
some qualitative properties of the solutions for an abstract but general coupled system of
quadratic Hadamard-fractional integral equations (1), including existence, monotonicity,
and uniqueness, in addition to continuous dependence on the data in LΘ-spaces. We
use the technique of (MNC) concerning (FPT ) and the theory of fractional calculus to
obtain the findings. We present a few constructed examples that support and illustrate
our findings.

2. Preliminaries

Let R+ = [0,∞) ⊂ R = (−∞,∞) and J = [1, e], e ≈ 2.718.

Definition 1. [17] The function Θ(u) =
∫ |u|
0 p(t) dt, defined on R+ is called a Young

function (Y.F.) if:

• The function p is nondecreasing, right-continuous, positive, and defined on R+;

• limt→∞Θ(t) = ∞ and Θ(0) = limt→0Θ(t) = 0.

The complementary (Y.F.) function Θ∗ of the function Θ is known as

Θ∗(t) = sup
s≥0

(
ts−Θ(s)

)
, ∀ t ≥ 0.

Furthermore, the function Θ is known as N -function if:

• limt→0
Θ(t)
t = 0 and limt→∞

Θ(t)
t = ∞;

• Θ(s) = 0 ⇔ s = 0 and Θ(s) > 0 if s > 0.

Definition 2. [26] The space LX = LΘ(J)× LΘ(J) is a Banach space under the norm

∥(x, y)∥X = ∥x∥Θ + ∥y∥Θ,

where x, y ∈ LΘ(J), and LΘ = LΘ(J) is called the Orlicz space of the functions f under
the norm

∥f∥Θ = inf
ϵ>0

{∫
J
Θ

(
f

ϵ

)
ds ≤ 1

}
.

Let EX = EΘ(J) × EΘ(J) be the closure in LX, where EΘ = EΘ(J) be the closure in
LΘ(J) such that

lim
δ→0

sup
measD<δ

sup
f∈EΘ

∥f · χD∥Θ = 0,

where χD and ”meas” are the characteristic function of a measurable subset D ⊂ J and
the Lebesgue measure, respectively. For multiplications of operators, we have:



M. Metwali, S. Alsallami / Eur. J. Pure Appl. Math, 18 (2) (2025), 6157 4 of 15

Lemma 1. ([29, Theorem 10.2] Let Θ1,Θ2 and Θ be arbitrary N -functions. The following
hypotheses are identical:

(i) For every u1 ∈ LΘ1 and u2 ∈ LΘ2, u1 · u2 ∈ LΘ.

(ii) ∃ k > 0 such that for all measurable functions u1, u2, we obtain ∥u1u2∥Θ ≤ k∥u1∥Θ1∥u2∥Θ2.

(iii) ∃ l > 0, u0 ≥ 0 s.t. ∀ t ≥ u0, Θ
(
st
l

)
≤ Θ1(s) + Θ2(t).

(iv) lim supt→∞
Θ−1

1 (t)Θ−1
2 (t)

Θ(t) < ∞.

Denote by W = W (J) the set of Lebesgue measurable functions on the interval J .
The functions are equal almost everywhere in the set W concerned with the metric

d(y, x) = inf
ρ>0

[ρ+meas{s : |y(s)− x(s)| ≥ ρ}],

becoming a complete metric space. It should be noted that the convergence in measure
on the interval J is the same as the convergence concerning the above metric d (cf. [30]).

Corollary 1. [26] Assume that U ⊂ LX is a bounded set and the functions x, y ∈ LΘ are
almost everywhere. nondecreasing (or almost everywhere nonincreasing) functions on the
interval J . Therefore, the pair (x, y) = u ∈ U becomes almost everywhere nondecreasing
(or almost everywhere nonincreasing) on the interval J , in addition to the set U being
compact in measure in LX.

Definition 3. [31] Assume that U ⊂ LX is a bounded set. The Hausdorff MNC βH(X)
(cf. [31]) is known as

βH(U) = inf{r > 0 : ∃ Y ⊂ LX s.t. U ⊂ Y + Br },

where Br = {x ∈ LX : ∥x∥X ≤ r}, r > 0.

Definition 4. [26] Assume that, ∅ ≠ U = (X1, X2) ⊂ LX, with X1, X2 ⊂ LΘ are bounded
sets and for ϵ > 0, then

c(U) = c(X1, X2) = c(X1) + c(X2)

= lim sup
ε→0

sup
mesD≤ε

sup
x1∈X1

∥x1 · χD∥Θ + lim sup
ε→0

sup
mesD≤ε

sup
x2∈X2

∥x2 · χD∥Θ

is known as the measure of equiintegrability in LX.

Corollary 2. [26] For a compact in measure and bounded set ∅ ≠ U ⊂ LX, we have

c(U) = βH(U).

Theorem 1. [26] Assume that ∅ ≠ C ⊂ LX is a closed, bounded, and convex in addition
to the continuous map T : C → C verifying

βH(T (U)) ≤ k βH(U), 0 ≤ k < 1, (Contraction condition)

for any ∅ ≠ U ⊂ C. Then T has at least one fixed point in C.



M. Metwali, S. Alsallami / Eur. J. Pure Appl. Math, 18 (2) (2025), 6157 5 of 15

Proposition 1. [32] Suppose that β ∈ (0, 1), t ∈ R+, and Θ is a Young function (YF),
then we get:

(a) For
∫ t
0 Θ(s−β) ds < ∞. If β2 < β, then the integral∫ t

0
Θ(s−β2) ds

is finite as well.

(b) The set

⋓(t) =

k > 0 :
1

k
1

1−β

∫ tk
1

1−β

0
Θ(sβ−1) ds ≤ 1


is increasing and continuous functions with ⋓(0) = 0.

Definition 5. [33] The Hadamard type fractional integral of order β > 0 for a given
integrable function y is known as

Kβy(t) =
1

Γ(β)

∫ t

1

(
log

t

s

)β−1 y(s)

s
ds, t > 1, β > 0,

where Γ(β) =
∫∞
0 e−ννβ−1 dν.

Proposition 2. [34] The operator Kβ maps the a.e. nonnegative-nondecreasing functions
into itself.

Lemma 2. [28] Suppose, that M∗ and M are complementary N -functions and Θ is N -
function with

∫ t
0 M(sβ−1) ds < ∞, 0 < β < 1. Moreover, put

k(t) =
1

ϵ
1

1−β

∫ tϵ
1

1−β

0
M(sβ−1) ds ∈ EΘ, s ∈ J, ϵ > 0,

then the Hadamard operator Kβ : LM∗ → LΘ is continuous and verifying

∥Kβx∥Θ ≤ 2

Γ(β)
∥k∥Θ∥x∥M∗ .

3. Main results.

Next, we discuss the solvability of the coupled system (1) in LΘ. Define the operator
T as follows

T (x, y)(t) =
(
T1y(t), T2x(t)

)
, t ∈ J,

where

T1y = h1 + Ff1

(
Λ1(y), U1(y)

)
, T2x = h2 + Ff2

(
Λ2(x), U2(x)

)
,



M. Metwali, S. Alsallami / Eur. J. Pure Appl. Math, 18 (2) (2025), 6157 6 of 15

Ffi

(
Λi(w), Ui(w)

)
= fi

(
t, Λ2(w), Ui(w)

)
, Ui(w) = Gi(w) ·Ai(w),

and
Ai(w)(t) = KβRi(w),

s.t. Kβ is Hadamard operator 5 and Gi, Ffi ,Λi, Ri, are different operators operate on
different Orlicz spaces i = 1, 2. First, we inspect the existence of monotonic-LΘ solutions
for the coupled system (1).

Definition 6. The ordered pair u = (x, y) ∈ LX s.t. x, y ∈ LΘ is called a solution of the
coupled system (1), if u verifies the coupled system (1).

3.1. The existence of solutions.

Let M, M∗ be complementary N -functions and Θ,Θ1,Θ2 be N -functions. Further-
more, put the assumptions for i = 1, 2:

(G1) ∃ k1 > 0 s.t. for every u1 ∈ LΘ1 and u2 ∈ LΘ2 we have ∥u1u2∥Θ ≤ k1∥u1∥Θ1∥u2∥Θ2 ,

(G2) hi ∈ EΘ(J) are a.e. nondecreasing functions on the interval J ,

(G3) fi(t, x, y) : J × R × R → R be continuous in x and y for almost all t and
measurable in t ∈ J . Furthermore, suppose that t → fi(t, x, y) are nondecreasing-
positive function and ∃ α1, α2 ≥ 0, and functions ci ∈ LΘ s.t.

|fi(t, x, y)| ≤ ci(t) + α1|x|+ α2|y|. (2)

(G4) The operators Λi : EΘ → EΘ, Gi : EΘ → EΘ1 , and Ri : EΘ → EM∗ , and they are
continuous. Moreover, let Λi, Gi, Ri take the set of all a.e. nondecreasing functions
into itself and assume that for any w ∈ EΘ we get Λi(w) ∈ EΘ, Gi(w) ∈ EΘ1 , and
Ri(w) ∈ EM∗ .

(G5) There exist positive functions ai ∈ LΘ, gi ∈ LΘ1 , bi ∈ LM∗ s.t. for t ∈ J ,

|Λi(w)(t)| ≤ ai(t)∥w∥Θ, |Gi(w)(t)| ≤ gi(t)∥w∥Θ, |Ri(w)(t)| ≤ bi(t)∥w∥Θ.

(G6) Assume that k(t) = 1

ϵ
1

1−β

∫ tϵ
1

1−β

0 M(sβ−1) ds ∈ EΘ2 for a.e. s ∈ J and ϵ > 0.

(G7) Let
(
α1∥a∗∥ − 1

)2
>

8α2k1∥k∥Θ2
Γ(β) ∥g∗∥Θ1∥b∗∥M∗

(
∥h1∥Θ + ∥h2∥Θ + ∥c1∥Θ + ∥c2∥Θ

)
,

and (
α1∥a∗∥Θ +

2 · r · α2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗

)
< 1,

where r is the positive solution of the equation

∥h1∥Θ+∥h2∥Θ+∥c1∥Θ+ ∥c2∥Θ−
(
1−α1∥a∗∥Θ

)
·r+2α2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗ ·r2 = 0

and ∥a∗∥Θ = max
{
∥ai∥Θ

}
, ∥b∗∥M∗ = max

{
∥bi∥M∗

}
and ∥g∗∥Θ1 = max

{
∥gi∥Θ1

}
.



M. Metwali, S. Alsallami / Eur. J. Pure Appl. Math, 18 (2) (2025), 6157 7 of 15

Theorem 2. Let the assumptions (G1)-(G7) hold, then there exists a.e. nondecreasing-
solution u = (x, y) ∈ EΘ of (1).

Proof. Step I. In what follows, put i = 1, 2. Lemma 2 and assumption (G6) imply
that the operator Kβ : LM∗ → LΘ2 is continuous and assumptions (G3) and (G4)
indicate that Ffi ,Λi : EΘ → EΘ, Gi : EΘ → EΘ1 and Ri : EΘ → EM∗ . Then the
operators Ai = KβRi : EΘ → EΘ2 is continuous. By assumptions (G1) and (G4) the
operators Ui = Gi·Ai : EΘ → EΘ is continuous. Assumption (G2) gives that, the operators
Ti : EΘ → EΘ are continuous. Therefore, T =

(
T1, T2

)
acts from EX into itself and is

continuous.

Step II. We shall prove that T : Br(EX) → EX is continuous, where

Br(EX) = {u = (x, y) ∈ LX : x, y ∈ EΘ, ∥u∥X ≤ r}.

For arbitrary u = (x, y) ∈ Br(EX), x, y ∈ EΘ, and recalling Remark 2, we have

∥T1y∥Θ ≤ ∥h1∥Θ + ∥f1(t, Λ1(y), U1(y))∥Θ
≤ ∥h1∥Θ + ∥c1∥Θ + α1∥Λ1(y)∥Θ + α2∥U1y∥Θ
≤ ∥h1∥Θ + ∥c1∥Θ + α1

∥∥∥a1 · ∥y∥Θ∥∥∥
Θ
+ α2∥G1(y) ·A1(y)∥Θ

≤ ∥h1∥Θ + ∥c1∥Θ + α1∥a1∥Θ · ∥y∥Θ + α2k1∥G1(y)∥Θ1 · ∥A1(y)∥Θ2

≤ ∥h1∥Θ + ∥c1∥Θ + α1∥a1∥Θ · ∥y∥Θ + α2k1

∥∥∥g1 · ∥y∥Θ∥∥∥
Θ1

∥∥∥KβR1(y)
∥∥∥
Θ2

≤ ∥h1∥Θ + ∥c1∥Θ + α1∥a1∥Θ · ∥y∥Θ + α2k1∥g1∥Θ1 · ∥y∥Θ
2∥k∥Θ2

Γ(β)
∥R1(y)∥M∗

≤ ∥h1∥Θ + ∥c1∥Θ + α1∥a1∥Θ · ∥y∥Θ +
2α2k1∥k∥Θ2

Γ(β)
∥g1∥Θ1 · ∥y∥Θ · ∥b1∥M∗∥y∥Θ

= ∥h1∥Θ + ∥c1∥Θ + α1∥a1∥Θ · ∥y∥Θ +
2α2k1∥k∥Θ2

Γ(β)
∥g1∥Θ1∥b1∥M∗ · ∥y∥2Θ.

Similarly, for x ∈ EΘ, we have

∥T2x∥Θ ≤ ∥h2∥Θ + ∥c2∥Θ + α1∥a2∥Θ · ∥x∥Θ +
2α2k1∥k∥Θ2

Γ(β)
∥g2∥Θ1∥b2∥M∗ · ∥x∥2Θ.

Then for u ∈ EX, we have

∥Tu∥X = ∥T1y∥Θ + ∥T2x∥Θ

≤ ∥h1∥Θ + ∥h2∥Θ + ∥c1∥Θ + ∥c2∥Θ + α1∥a1∥Θ · ∥y∥Θ +
2α2k1∥k∥Θ2

Γ(β)
∥g1∥Θ1∥b1∥M∗ · ∥y∥2Θ

+α1∥a2∥Θ · ∥x∥Θ +
2α2k1∥k∥Θ2

Γ(β)
∥g2∥Θ1∥b2∥M∗ · ∥x∥2Θ

≤ ∥h1∥Θ + ∥h2∥Θ + ∥c1∥Θ + ∥c2∥Θ + α1∥a∗∥Θ
(
∥x∥Θ + ∥y∥Θ

)



M. Metwali, S. Alsallami / Eur. J. Pure Appl. Math, 18 (2) (2025), 6157 8 of 15

+
2α2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗

(
∥x∥Θ + ∥y∥Θ

)2
≤ ∥h1∥Θ + ∥h2∥Θ + ∥c1∥Θ + ∥c2∥Θ + α1∥a∗∥Θ · ∥u∥X +

2α2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗ · ∥u∥2X

≤ ∥h1∥Θ + ∥h2∥Θ + ∥c1∥Θ + ∥c2∥Θ + α1∥a∗∥Θ · r + 2α2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗ · r2 ≤ r,

where ∥a∗∥Θ = max
{
∥ai∥Θ

}
, ∥b∗∥M∗ = max

{
∥bi∥M∗

}
and ∥g∗∥Θ1 = max

{
∥gi∥Θ1

}
, i =

1, 2. Recalling assumption (G7), we have T : Br(EX) → EX is continuous.

Step III. Let Qr ⊂ Br(EX) include all monotonic (a.e. nondecreasing) functions on
the interval J . Then ∅ ≠ Qr is closed, convex, and bounded, in EX in addition to be
compact in measure regarding Corollary 1.

Step IV. The operator T keeps the monotonicity property for the functions. For
i = 1, 2, let us choose u = (x, y) ∈ Qr, where x and y are nondecreasing on J . Proposition
2 implies that the operator Kβ takes the a.e. nonnegative-nondecreasing functions into
itself. Therefore, the operators Ai = KβRi and Λi, and Ui = Gi ·Ai are a.e. nondecreasing
on the interval J (by using (G4)). Assumptions (G2) and (G3) grant us that the operators
T1, T2 are a.e. nondecreasing on J . Those grant us that T = (T1, T2) : Qr → Qr is
continuous.

Step V. We need to show that βH(TX) ≤ kβH(X), k ∈ [0, 1). For any u = (x, y) ∈
U ⊂ Qr and a set D ⊂ J , with measD ≤ ε, ε > 0. By assumption (G5), we have

∥Λi(z) · χD∥Θ ≤ ∥Λi(x · χD)∥Θ ≤
∥∥a1 · ∥x · χD∥Θ

∥∥
Θ
≤ ∥ai∥Θ∥x · χD∥Θ

and similarly
∥Gi(z) · χD∥Θ ≤

∥∥gi∥∥Θ∥x · χD∥Θ.

Therefore, we have

∥T1(y) · χD∥Θ ≤ ∥h1 · χD∥Θ +
∥∥∥Ff1

(
Λ1(y), U1(y)

)
· χD

∥∥∥
Θ

≤ ∥h1 · χD∥Θ + ∥c1 · χD∥Θ + α1

∥∥∥Λ1(y) · χD

∥∥∥
Θ
+ α2∥G1(y) ·A1(y) · χD∥Θ

≤ ∥h1 · χD∥Θ + ∥c1 · χD∥Θ + α1∥a1∥Θ · ∥y · χD∥Θ + α2k1∥G1(y) · χD∥Θ1 · ∥A1(y)∥Θ2

≤ ∥h1 · χD∥Θ + ∥c1 · χD∥Θ + α1∥a1∥Θ · ∥y · χD∥Θ + α2k1∥g1∥Θ1 · ∥y · χD∥Θ
2∥k∥Θ2

Γ(β)
∥b1∥M∗∥y∥Θ

≤ ∥h1 · χD∥Θ + ∥c1 · χD∥Θ + α1∥a1∥Θ · ∥y · χD∥Θ +
2α2k1∥k∥Θ2

Γ(β)
∥g1∥Θ1 · ∥y · χD∥Θ∥b1∥M∗ · r.

Similarly, we have

∥T2x·χD∥Θ ≤ ∥h2·χD∥Θ+∥c2·χD∥Θ+ α1∥a2∥Θ·∥x·χD∥Θ+
2α2k1∥k∥Θ2

Γ(β)
∥g2∥Θ1 ·∥x·χD∥Θ∥b1∥M∗ ·r.



M. Metwali, S. Alsallami / Eur. J. Pure Appl. Math, 18 (2) (2025), 6157 9 of 15

Then

∥Tu · χD∥X = ∥T1y · χD∥Θ + ∥T2x · χD∥Θ
≤ ∥h1 · χD∥Θ + ∥h2 · χD∥Θ + ∥c1 · χD∥Θ + ∥c2 · χD∥Θ + α1∥a∗∥Θ ·

(
∥x · χD∥Θ + ∥y · χD∥Θ

)
+
2 · rα2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗ ·

(
∥x · χD∥Θ + ∥y · χD∥Θ

)
≤ ∥h1 · χD∥Θ + ∥h2 · χD∥Θ + ∥c1 · χD∥Θ + ∥c2 · χD∥Θ + α1∥a∗∥Θ · ∥u · χD∥X

+
2 · rα2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗ · ∥u · χD∥X.

Since hi, ci ∈ EΘ, i = 1, 2, we get

lim
ε→0

{
sup

mes D≤ε

[
sup
u∈X

{
∥hiχD∥Θ + ∥ciχD∥Θ = 0

}]}
.

Recalling Definition 4, we obtain

c(T (U)) ≤
(
α1∥a∗∥Θ +

2 · rα2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗

)
· c(U).

Since ∅ ̸= U ⊂ Qr is bounded in addition to compact in measure, then we shall apply
Corollary 2 to obtain

βH(T (U)) ≤
(
α1∥a∗∥Θ +

2 · rα2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗

)
· βH(U).

Since

(
α1∥a∗∥Θ +

2·rα2k1∥k∥Θ2
Γ(β) ∥g∗∥Θ1∥b∗∥M∗

)
< 1, we get our verification and Theorem

1 achieves our proof.

3.2. Uniqueness of the solution.

Next, we demonstrate that the coupled system (1) has exactly one solution.

Theorem 3. Assume that the assumptions of Theorem 2 hold with replacing the inequality
(2) with the following

|fi(t, 0, 0)| ≤ ci(t), |fi(t, x, y)−fi(t, x̄, ȳ)| ≤ α1|x−x̄|+α2|y−ȳ|, u = (x, y), ū = (x̄, ȳ) ∈ Qr,
(3)

for i = 1, 2, and in addition, assume that

C =

(
α1∥a∗∥Θ +

4r · α2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗

)
< 1, (4)

where r,Qr are defined in Theorem 2. Then the coupled system (1) has a unique solution
u ∈ LX in Qr.



M. Metwali, S. Alsallami / Eur. J. Pure Appl. Math, 18 (2) (2025), 6157 10 of 15

Proof. Using the inequalities (3) for i = 1, 2, we obtain∣∣∣∣|fi(t, x, y)| − |fi(t, 0, 0)|
∣∣∣∣ ≤ |fi(t, x, y)− fi(t, 0, 0)| ≤ α1|x|+ α2|y|

⇒ |fi(t, x, y)| ≤ |fi(t, 0, 0)|+ α1|x|+ α2|y| ≤ ci(t) + α1|x|+ α2|y|.

Thus, Theorem 2 indicates that, there is a.e. nondecreasing solution u ∈ EX of (1) in Qr.
Now, let u = (x, y), ū = (x̄, ȳ) ∈ Qr be any two distinct solutions of the coupled system
(1), then we have

∥x− x̄∥Θ ≤
∥∥∥f1(t,Λ1(y), U1(y)

)
− f1

(
t,Λ1(ȳ), U1(ȳ)

)∥∥∥
Θ

≤ α1∥Λ1(y)− Λ1(ȳ)∥Θ + α2∥U1(y)− U1(ȳ)∥Θ
≤ α1

∥∥∥a1∥y∥Θ − a1∥ȳ∥Θ
∥∥∥
Θ
+ α2∥G1(y)A1(y)−G1(ȳ)A1(ȳ)∥Θ

≤ α1∥a1∥Θ
∣∣∥y∥Θ − ∥ȳ∥Θ

∣∣+ α2

∥∥G1(y)A1(y)−G1(ȳ)A1(y)
∥∥
Θ
+ α2

∥∥G1(ȳ)A1(y)−G1(ȳ)A1(ȳ)
∥∥
Θ

≤ α1∥a1∥Θ
∥∥y − ȳ

∥∥
Θ
+ α2k1

∥∥G1(y)−G1(ȳ)
∥∥
Θ1

∥A1(y)∥Θ2 + α2k1
∥∥G1(ȳ)∥Θ1

∥∥A1(y)−A1(ȳ)
∥∥
Θ2

≤ α1∥a1∥Θ
∥∥y − ȳ

∥∥
Θ

+α2k1
∥∥g1∥∥Θ1

∥y − ȳ∥Θ
2∥k∥Θ2

Γ(β)
∥b1∥M∗∥y∥Θ + α2k1

∥∥g1∥∥Θ1
∥y∥Θ

2∥k∥Θ2

Γ(β)
∥R1(y)−R1(ȳ)

∥∥
M∗

≤ α1∥a1∥Θ
∥∥y − ȳ

∥∥
Θ

+
2α2k1∥k∥Θ2

Γ(β)

∥∥g1∥∥Θ1
∥b1∥M∗∥y∥Θ∥y − ȳ∥Θ +

2α2k1∥k∥Θ2

Γ(β)

∥∥g1∥∥Θ1
∥y∥Θ∥b1∥M∗

∥∥y − ȳ
∥∥
Θ

=

(
α1∥a1∥Θ +

4r · α2k1∥k∥Θ2

Γ(β)
∥g1∥Θ1∥b1∥M∗

)∥∥y − ȳ
∥∥
Θ
.

Similarly, ∥y − ȳ∥Θ ≤
(
α1∥a2∥Θ +

4r · α2k1∥k∥Θ2

Γ(β)

∥∥g2∥∥Θ1
∥b2∥M∗

)∥∥x− x̄
∥∥
Θ
.

Therefore,

∥u− ū∥X =
∥∥(x− x̄, y − ȳ)

∥∥
X = ∥x− x̄∥Θ + ∥y − ȳ∥Θ

≤
(
α1∥a1∥Θ +

4r · α2k1∥k∥Θ2

Γ(β)
∥g1∥Θ1∥b1∥M∗

)∥∥y − ȳ
∥∥
Θ

+

(
α1∥a2∥Θ +

4r · α2k1∥k∥Θ2

Γ(β)

∥∥g2∥∥Θ1
∥b2∥M∗

)∥∥x− x̄
∥∥
Θ

≤
(
α1∥a∗∥Θ +

4r · α2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗

)(∥∥y − ȳ
∥∥
Θ
+
∥∥x− x̄

∥∥
Θ

)
= C · ∥u− ū∥X.

Equation (4) grants us that u = ū (a.e.), and we get our verification.



M. Metwali, S. Alsallami / Eur. J. Pure Appl. Math, 18 (2) (2025), 6157 11 of 15

3.3. Continuous dependence on the functions h1, and h2.

Next, we may discuss the continuous dependence of the obtained solutions for the
coupled system (1) on the functions hi, i = 1, 2.

Definition 7. A solution u = (x, y) ∈ LX of (1) is continuously dependent on the function
h1, h2 if ∀ ϵ > 0, ∃δ > o such that ∥h1− h̄1∥Θ+∥h2− h̄2∥Θ ≤ δ implies that ∥u− ū∥Θ ≤ ϵ,
where

x̄(t) = h̄1(t) + f1

(
Λ1(ȳ)(t) +

G1(ȳ)(t)
Γ(β) ·

∫ t
1

(
log t

s

)β−1
R1(ȳ)(s)

s ds

)
ȳ(t) = h̄2(t) + f2

(
Λ2(x̄)(t) +

G2(x̄)(t)
Γ(β) ·

∫ t
1

(
log t

s

)β−1
R2(x̄)(s)

s ds

)
, t ∈ [1, e].

(5)

Theorem 4. Assume that the assumptions of Theorem 3 hold. Then the solutions u ∈ LX
of the system (1) depend continuously on the functions h1, h2.

Proof. Let u, ū be any two different solutions of (1), then similarly as done in Theorem
3, we have

∥u− ū∥X ≤ ∥h1 − h̄1∥Θ + ∥h2 − h̄2∥Θ

+

(
α1∥a∗∥Θ +

4r · α2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗

)(∥∥y − ȳ
∥∥
Θ
+
∥∥x− x̄

∥∥
Θ

)
= ∥h1 − h̄1∥Θ + ∥h2 − h̄2∥Θ +

(
α1∥a∗∥Θ +

4r · α2k1∥k∥Θ2

Γ(β)
∥g∗∥Θ1∥b∗∥M∗

)
∥u− ū∥X

≤ ∥h1 − h̄1∥Θ + ∥h2 − h̄2∥Θ + C∥u− ū∥X,

where C is given by (4). Then, we get

∥u− ū∥X ≤
(
1− C

)−1(
∥h1 − h̄1∥Θ + ∥h2 − h̄2∥Θ

)
.

Therefore, if ∥h1 − h̄1∥Θ + ∥h2 − h̄2∥Θ+ ≤ δ(ϵ), then ∥u− ū∥Θ ≤ ϵ, where

δ(ϵ) = ϵ · (1− C).

4. Remarks and Example

We would like to conclude with some significant remarks and examples that highlight
the applicability of the results we have found.

Remark 1. The neutron transport [35], the traffic theory [36], the kinetic theory of gases
[37], and astrophysics [38] are more efficient utilization of the quadratic integral equation
through Hadamard fractional operators.



M. Metwali, S. Alsallami / Eur. J. Pure Appl. Math, 18 (2) (2025), 6157 12 of 15

Remark 2. We can determine the acting and continuation assumptions for the operators
of the form Gi(w) = li(t) · w(t), li ∈ LΘ, over several Orlicz spaces in (cf. [17] and
assumption (G3)).

Example 1. Select the N -functions M(s) = M∗(s) = s2 and Θ2(s) = exp |s| − |s| − 1.
We need to show that, the operator Kβ : LM∗ → LΘ2 is continuous and the outcomes of
Lemma 2 is verified.
Indeed: Let t ∈ [1, e] and for any β ∈ (0, 1), we get

k(t) =

∫ t

0
M

(
uβ−1

)
du =

∫ t

0
u2β−2 du =

t2β−1

2β − 1
.

That gives us the verification of Proposition 1. Furthermore,∫ e

1
Θ2

(
k(t)

)
ds =

∫ e

1

(
e

t2β−1

2β−1 − t2β−1

2β − 1
− 1

)
dt,

which is finite. Then for x ∈ LM∗, we have Jβ : LM∗ → LΘ2 is continuous.

For additional details and many instances of the N -functions M,M∗, and Θ2 that
satisfy Lemma 2, refer to [17, Theorem 15.4].

Example 2. For i = 1, 2, let β = 1
2 , Λi(z) = ai(t) · z(t), Gi(z) = gi(t) · z(t), and

Ri(z) = ai(t) · z(t), where hi ∈ LΘ, ai ∈ LΘ, gi ∈ LΘ1, and bi ∈ LM∗ , then the coupled
system 

x(t) = h1(t) + f1

(
t, a1(t) · y(t), g1(t)·y(t)

Γ( 1
2
)

∫ t
0

√
log t

s
b1(t)·y(t)

s ds

)
y(t) = h2(t) + f2

(
t, a2(t) · x(t), g1(t)·x(t)

Γ( 1
2
)

∫ t
0

√
log t

s
b2(t)·x(t)

s ds

)
,

have a solution u = (x, y) ∈ LX, where t ∈ J .

5. Conclusion

There have been several qualitative properties developed in this paper, consisting of
existence, monotonicity, and uniqueness, in addition to the continuous dependence of the
data, which are all indications for an abstract and general coupled system of quadratic
Hadamard-fractional integral equations. To perform our analysis, we used the (MNC)
measure of noncompactness, as well as the (FPT ) fixed-point theorem and the fractional
calculus in the Orlicz spaces LΘ. Finally, we concluded with a few remarks as well as a
few examples that illustrate and support our hypothesis. Future research will concentrate
on the qualitative properties of the solutions for numerous fractional problems in distinct
function spaces, such as Lebesgue spaces or Orlicz spaces. Furthermore, we shall check
the numerical results for the issues considered.



M. Metwali, S. Alsallami / Eur. J. Pure Appl. Math, 18 (2) (2025), 6157 13 of 15

Acknowledgements

The authors extend their appreciation to Umm Al-Qura University, Saudi Arabia for
funding this research work through grant number: 25UQU4290491GSSR03.

Author contributions

All the authors contributed equally in preparing, writing, and obtaining the paper’s
results.

Funding

This research work was funded by Umm Al-Qura University, Saudi Arabia, under
grant number: 25UQU4290491GSSR03.

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