EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5605
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Langevin Fractional System Driven by Two ψ-Caputo
Derivatives With Random Effects

Mohamed Ziane1, Hussein Al-Taani2, Mohammad Abudayah2,∗, Oualid Zentar3,
Ma’mon Abu Hammad4

1 Department of Mathematics, University of Tiaret, Tiaret, Algeria and Laboratory of
Research in Artificial Intelligence and Systems (LRAIS), University of Tiaret, Algeria
2 School of Electrical Engineering and Information Technology, German Jordanian
University, Amman 11180, Jordan
3 Department of Computer Science, University of Tiaret, Tiaret, Algeria and Laboratory of
Research in Artificial Intelligence and Systems (LRAIS), University of Tiaret, Algeria.
4 Department of Mathematics, Al-Zaytoonah University of Jordan, Amman 11733, Jordan

Abstract. A nonlinear Langevin fractional system involving two ψ-Caputo derivatives with ran-
dom effects is investigated. First, a random version of Perov’s fixed-point theorem in generalized
Banach space endowed with the Bielecki-type vector-valued norm is employed to achieve a unique-
ness result. Second, the existence result is established using Sadovskii’s fixed point principle under
fairly general conditions on the nonlinear forcing terms. Finally, our findings are justified through
illustrative examples.

2020 Mathematics Subject Classifications: AMS 34A08, 47H08, 60H25

Key Words and Phrases: Langevin equation, ψ-Caputo derivative, random variable, vector-
valued norm, measure of noncompactness

1. Introduction

Fractional calculus and its applications have garnered significant attention from sci-
entists and researchers in recent years, not only in mathematics but also across various
scientific disciplines, including physics [20], chemical kinetics [26], fluid dynamics [21], vis-
coelastic [11], electrochemistry [19], elasticity [4], engineering [29],economics [28], financial
systems [22], biology [14], medicine [24], statistics [2], computing image [31], nonlinear
heat conduction [8], optimal control [9], etc. Moreover, many cosmic events that classical
differential equations cannot describe can be described by fractional differential equations.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5605

Email addresses: mohamed.ziane@univ-tiaret.dz (M. Ziane),
hussein.taani@gju.edu.jo (H. Al-Taani), mohammad.abudayah@gju.edu.jo (M. Abudayah),
oualid.zentar@univ-tiaret.dz (O. Zentar),m.abuhammad@zuj.edu.jo (M. Abu Hammad)

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



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 2 of 21

On the other side, Almeida [3] proposed a general definition of Caputo FD with respect
to functions which is more flexible, beneficial and play an important role in modeling
practical applications, see for instance [10]. Multitude scholars investigate several aspect
of the theory [6, 32, 33].

The classical Langevin equation, as proposed in [18], is crucial for demonstrating how
particles interact with their surrounding medium and the random forces or fluctuations
that lead to their unpredictable motion. Nevertheless, the reliance on the specific rela-
tionship between a particle’s position and velocity has prompted the development of the
fractional Langevin model, aimed at describing anomalous diffusion phenomena [17]. Also,
it’s important to highlight that certain phenomena are more accurately described by cou-
pled random systems. For example, in epidemiology, the migration of birds from various
regions worldwide can introduce infectious diseases. Therefore, the transmission rate of
these diseases increases as migratory birds flock together. Moreover, this scenario war-
rants consideration of the presence of random disturbances. While the above-mentioned
motivational models have a great advantage, the difficulty of the corresponding mathemat-
ical model may significantly increase, complicating the study of the existence of solutions.
Accordingly, exploring the qualitative aspects of ψ-Caputo nonlinear Langevin coupled
systems with random effects has become increasingly important.

Recently, the authors in [13, 33] studied theoretically some quantitative aspects for the
following problem:

(
cD

ϑ;ψ
a+

+ϖcD
ϑ−1;ψ
a+

)
z(ξ) = f(ξ, z(ξ)), ξ ∈ [a, b],

z(a) = z′(a) = 0,

where 1 < ϑ < 2, ϖ ∈ R, cDθ;ψ
a+

represents the Caputo fractional derivative FD with
respect to ψ of order θ ∈ {ϑ, ϑ − 1}, f : [a, b] × G → G is a given function and X is a
Banach space.

O. Zentar et al. in [34] investigated the existence of solutions for the following system:

Dϑ1
0+
z1(ξ, ω) = f1(ξ, z1(ξ, ω), z2(ξ, ω), ω), ξ ∈ (0, b],

Dϑ2
0+
z2(ξ, ω) = f2(ξ, z1(ξ, ω), z2(ξ, ω), ω), ξ ∈ (0, b],

lim
ξ→0+

ξ1−ϑ1z1(ξ, ω) = Z3(ω), ω ∈ Ω,

lim
ξ→0+

ξ1−ϑ2z2(ξ, ω) = Z4(ω), ω ∈ Ω,

where Z3,Z4 : Ω → G are random variables, Dϑi
0+

represents the standard Riemann-
Liouville FD of order ϑi ∈ (0, 1] for each i = 1, 2 and fi : [0, b] × G × G × Ω → G are
funcions and (G, ∥ · ∥) is a real separable Banach space.

Motivated by the preceding discussions, this paper presents new qualitative results for



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 3 of 21

the following random coupled Langevin system involving ψ-Caputo FD:

(
cD

ϑ1;ψ
a+

+ϖ1
cD

ϑ1−1;ψ
a+

)
z1(ξ, ω) = f1(ξ, z1(ξ, ω), z2(ξ, ω), ω), ξ ∈ I := [a, b],(

cD
ϑ2;ψ
a+

+ϖ2
cD

ϑ2−1;ψ
a+

)
z2(ξ, ω) = f2(ξ, z1(ξ, ω), z2(ξ, ω), ω), ξ ∈ I := [a, b],

z1(a, ω) = z′1(a, ω) = 0,

z2(a, ω) = z′2(a, ω) = 0,

(1)

where 1 < ϑi < 2, ϖi > 0. cD
θi;ψ
a+

(for i = 1, 2) is the FD with respect to ψ of order
θi ∈ {ϑi, ϑi − 1}, fi : I × G × G → G, (i = 1, 2) verifying some conditions that will be
precised later.

A notable feature of our research is the following:

• We utilize Perov’s fixed-point theorem with the Bielecki-type vector-valued norm to
establish a new uniqueness criterion.

• We established existence results by applying Sadovskii’s fixed-point principle in a
random setting, utilizing the measure of noncompactness (MNC) procedure and the
a priori estimate technique.

• The obtained findings generalize the results appearing in the existing research, such
as in [6, 13, 33].

This research is structured as follows. Section 2 presents some preliminary facts that
will be utilized in subsequent sections. The main results are provided in Section 3. Finally,
illustrative examples are presented in Section 4.

2. Preliminary Results

Throughout the paper, let (G, ∥ · ∥) be a separable Banach space, we endow the space
C(I,G) of G-valued continuous functions on I with the supnorm

∥u∥∞ = sup
ξ∈I

∥u(ξ)∥. (2)

L1(I,G) denotes the space of Bochner integrable functions u : I → G normed by

∥u∥L1 =

∫ b

a
∥u(s)∥ds, for all u ∈ L1(I,G).

L∞(I,R+) stands for the space all essentially bounded functions normed by

∥u∥L∞ = ess sup
ξ∈I

∥u(ξ)∥ = inf{M > 0; ∥u(ξ)∥ ≤M for almost every ξ ∈ I}.



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 4 of 21

Set
S1+(I,R) = {ψ : ψ ∈ C1(I,R) and ψ′(ξ) > 0 for all ξ ∈ I}.

Let ψ ∈ S1+(I,R) for ξ, s ∈ I, (s < ξ), we define

ϕ(ξ, s) = ψ(ξ)− ψ(s) and ϕ(ξ, s)ϑ = (ψ(ξ)− ψ(s))ϑ .

Definition 1. The Mittag-Leffler function is defined as follows:

Eϑ(u) =
∞∑
j=0

uj

Γ(jϑ+ 1)
, ϑ > 0.

where Γ(·) is the gamma function .

Definition 2. [3, 16] Let ψ ∈ S1+(I,R) and ϑ > 0. The ψ-fractional integral (FI) of a
function f of order ϑ is defined by

I
ϑ,ψ
a+
f(ξ) =

1

Γ(ϑ)

∫ ξ

a
ϕ(t, s)ϑ−1ψ′(s)f(s)ds, t > a,

Lemma 1. [3, 16] Let ϑ, γ > 0, then

I
ϑ;ψ
a+

ϕ(ξ, a)γ−1 =
Γ(γ)

Γ(ϑ+ γ)
ϕ(ξ, a)ϑ+γ−1.

Definition 3. [3] Let n− 1 < ϑ ≤ n with n ∈ N, ψ ∈ S1+(I,R). The ψ-Caputo FDs of a
function f of order ϑ is defined as(

cD
ϑ;ψ
a+
f
)
(ξ) = I

n−ϑ;ψ
a+

(
1

ψ′(ξ)

d

dξ

)n
f(ξ).

Now, for ζ > 0, we endow the space C(I,G) by the Bielecky norm

∥f∥B = sup
ξ∈I

e−ζϕ(ξ,a)∥f(ξ)∥. (3)

Lemma 2. [27, 30] The norms ∥ · ∥B defined by (3) and ∥ · ∥∞ are equivalent, i.e; there
exist c0 ∈ (0,∞) such that

∥ · ∥B ≤ ∥ · ∥∞ ≤ c0∥ · ∥B.

Lemma 3. [6] Let ϑ > 1 and ζ > 0. Then for all ξ ∈ I, one has

I
ϑ−1;ψ
a+

eζϕ(ξ,a) ≤ eζϕ(ξ,a)

ζϑ−1
.

If, y, v ∈ Rn, y = (y1, . . . , yn), v = (v1, . . . , vn), by y ≤ v we mean yi ≤ vi for all
i = 1, . . . , n. Also |y| = (|y1|, . . . , |yn|), max(y, v) = (max(y1, v1), . . . ,max(yn, vn)) and
Rn+ = {y ∈ Rn : yi > 0}. If c ∈ R, then y ≤ c means yi ≤ c for each i = 1, . . . , n.



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 5 of 21

Definition 4. Let F be a nonempty set. By a vector-valued metric on F we mean a map
d : F× F → Rn+ with the following properties:

(i) d(y, v) ≥ 0 for all y, v ∈ F; if d(y, v) = 0 then y = v;

(ii) d(y, v) = d(v, u) for all y, v ∈ F;

(iii) d(y, v) ≤ d(y, u) + d(u, v) for all u, v, y ∈ F.

For di, i = 1, . . . , n are metrics on F, the pair (F, d) is called a generalized metric space

(shortly, GMS) (or a vector-valued metric space) with d(y, v) :=

d1(y, v)...
dn(y, v)

.

Definition 5. We call a matrix M ∈ Mn×n(R) of real numbers convergent to zero if its
spectral radius ρ(M) < 1. In other words, this means that all the eigenvalues of M are in
the open unit disc i.e. |λ| < 1, for every λ ∈ C with det(M− λI) = 0, where I denote the
unit matrix.

Proposition 1. [23] Let M ∈ Mn×n(R+). The following statements are equivalent:

(i) Mr → 0 when r → ∞.

(ii) M is convergent to zero.

(iii) The matrix (I −M) is nonsingular and

(I −M)−1 = I +M+M2 + . . .+Mk + . . . .

(iv) (I −M) is nonsingular matrix and (I −M)−1 has positive elements.

Let G be a separable GMS and (Ω,F) be a measurable space. We denote B(G) the
Borel σ-algebra on Ω×G. Therefore, F×B(G) is the smallest σ-algebra on Ω×G which
contains all the sets F × S, where F ∈ F and S ∈ B(G).

Definition 6. Given two separable GMSs G and X, a mapping Q : Ω × G → X is called
a random operator if ω 7−→ Q(ω, u) is measurable for all u ∈ G. The random operator L

on G will be denoted by

Q(u)(ω) = Q(ω, u), ω ∈ Ω, u ∈ G.

Definition 7. The fixed point of a random operator Q is a measurable function u : Ω → G
such that

u(ω) = Q(ω, u(ω)) for all ω ∈ Ω.

Definition 8. Let f : I × G × Ω → X is called random Carathéodory if the following
statements are verified:



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 6 of 21

(i) The map u 7−→ f(ξ, u, ω) is continuous for all ξ ∈ I and ω ∈ Ω.

(ii) The map (ξ, ω) 7−→ f(ξ, u, ω) is jointly measurable for all u ∈ G.

Lemma 4. [25] Let G be a separable metric space and Q : Ω×G → G be a mapping such
that Q(ω, ·) is continuous for all ω ∈ Ω and Q(·, u) is measurable for all u ∈ G . Then the
map (ω, u) → Q(ω, u) is jointly measurable.

Definition 9. [12] Let G be a generalized Banach space and (O,≤) be a partially ordered
set. A map Λ : P(G) → O× O× . . .× O is called a generalized MNC on G, if

Λ(co O) = Λ(O) for every O ∈ P(G),

where Λ(O) :=

 Λ1(O)
...

Λn(O)

, P(G) denotes the family of all bounded subsets of G and coO

is the closed convex hull of O.

Definition 10. The application Λ is called:

(i) Monotone if O0,O1 ∈ P(G),O0 ⊂ O1 implies Λ (O0) ≤ Λ (O1).

(ii) Nonsingular if Λ({a} ∪ O) = Λ(O) for every a ∈ G and O ∈ P(G).

If O is a cone in a normed space, we say that the MNC is

(iii) Regular if the condition Λ(O) = 0 is equivalent to the compactness of O.

The most well-known example of a MNC possessing all previous properties is the
Hausdorff MNC defined by:

η(O) = inf {ϵ > 0 : O has a finite ϵ− net} .

Definition 11. [12] Let X,Y be two generalized normed spaces. A continuous map G :
X → Y is called a M-contraction (with respect to the generalized MNC Λ) if there exists a
matrix M ∈ Mn×n(R) converges to zero such that for every D ∈ P(X), one has

Λ(G(D)) ≤ MΛ(D).

Lemma 5. [15] If {xn}+∞
n=1 ⊂ L1(I,G) satisfies ∥xn(ξ)∥ ≤ ι(ξ) a.e. on I for all n ≥ 1

with some ι ∈ L1(I,R+). Then, the function η({xn(ξ)}+∞
n=1) is integrable and

η

({∫ ξ

0
xn(s)ds : n ≥ 1

})
≤
∫ ξ

0
η(xn(s) : n ≥ 1)ds. (4)

Theorem 1. [7, 25] Let X be a real separable generalized Banach space and (Ω,G) be a
measurable space and Q : Ω × X → X a continuous random operator, and let M(ω) ∈
Mn×n(R+) be a random variable matrix such that for every ω ∈ Ω, the matrix M(ω)
converges to zero and:

d(Q(ω, z1),Q(ω, z2)) ≤ M(ω)d(z1, z2), for each z1, z2 ∈ X and ω ∈ Ω.

Then, there exists a unique random fixed point of Q.



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 7 of 21

Theorem 2. [7, 25] Let G be a separable generalized Banach space, and let Q : Ω×G → G
be a condensing continuous random operator. Then either of the following holds:

(i) The random equation Q(ω, z) = z has a random solution, i.e., there is a measurable
function z : Ω → G such that Q(ω, z(ω)) = z(ω) for all ω ∈ Ω,
or

(ii) The set
W = {z : Ω → G is measurable κ(ω)Q(ω, z) = z}

is unbounded for some measurable function κ : Ω → G with µ(ω) ∈ (0, 1) on Ω.

Lemma 6. [5, Corollary 2.1.] Let αl > 0, l = 1, n, n ∈ N and ψ ∈ S1+(I,R). Assume that

(i) The functions gl are the bounded and monotonic increasing functions on [a, b),

(ii) s and u are nonnegative functions locally integrable on [a, b).

(iii) u(ξ) is a nondecreasing function for ξ ∈ [a, b),

If

s(ξ) ≤ u(ξ) +
n∑
l=1

gl(ξ)

∫ ξ

a
ϕ(ξ, s)αl−1s(s)ψ′(s)ds, (5)

then

s(ξ) ≤ u(ξ)

n∑
l=0

Eαl
(gl(ξ)Γ(αl)ϕ(ξ, a)

αl) .

Lemma 7. Let ψ ∈ S1+(I,R), γ > 0, 1 < ϑi < 2, ϖi > 0 and a constant random variable
ϱi,j : Ω → [0,∞), i, j = 1, 2. Then

ℵi,j(γ, ω) := sup
ξ∈I

4eϖiϕ(b,a)ϱi,j(ω)

Γ(ϑi − 1)

∫ ξ

a
ψ′(s)ϕ(s, a)ϑi−1e−γ(ξ−s)ds −−−−→ 0

γ→+∞
, i, j = 1, 2, (6)

Proof. From
ϕ(·, a)ϑi−1ψ′(·) ∈ L1(I,R), i = 1, 2.

So, there exists ℏ ∈ C(I,R) such that∫ ξ

a

∣∣∣ϕ(s, a)ϑi−1ψ′(s)− ℏ(s)
∣∣∣ ds < 1

2
ϵ.

Hence ∣∣∣∣∫ ξ

a
ϕ(s, a)ϑi−1ψ′(s)e−γ(ξ−s)ds

∣∣∣∣
≤
∫ ξ

a

∣∣∣ϕ(s, a)ϑi−1ψ′(s)− ℏ(s)
∣∣∣ e−γ(ξ−s)ds+ ∫ ξ

a
|ℏ(s)|e−γ(ξ−s)ds

≤ ϵ

2
+

1− e−γ(b−a)

γ
∥ℏ∥∞, i = 1, 2,



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 8 of 21

Consequently, ∫ ξ

a
ψ′(s)ϕ(s, a)ϑi−1e−γ(ξ−s)ds −→ 0 as γ −→ +∞, i = 1, 2.

This completes the proof of the Lemma.

3. Main results

Our first result establishes the existence and uniqueness result for the system (1),
where Perov’s fixed-point principle is applied.

Theorem 3. Suppose that

(A1) The functions fi are random Carathéodory on I×G×G× Ω.

(A2) There exists random variables Φi,j : Ω → (0,∞); i, j = 1, 2 such that:

∥fi(ξ, u1, u2, ω)− fi(ξ, v1, v2, ω)∥ ≤ Φi,1(ω)∥u1 − v1∥+Φi,2(ω)∥u2 − v2∥, i = 1, 2,

for u1, u2, v1, v2 ∈ G, (ξ, ω) ∈ I× Ω.

Then, system (1) admits a unique random solution.

Proof. Firstly, endowing the product Banach space J = C(I,G) × C(I,G) by the
vector-norm

∥(z1, z2)∥J =

(
∥z1∥∞
∥z2∥∞

)
. (7)

Next, according to [13, Theorem 3.1], system (1) is equivalent to the operator equation
H(z1, z2, ω) = (z1, z2) where H : J× Ω → J be the operator given by:

H(z1(ξ, ω), z2(ξ, ω), ω) = (H1(z1(ξ, ω), z2(ξ, ω), ω),H2(z1(ξ, ω), z2(ξ, ω), ω)) . (8)

where,

Hi(z1(ξ, ω), z2(ξ, ω), ω)

= (ϑi − 1)

∫ ξ

a
e−ϖiϕ(ξ,s)

(∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
fi(τ, z1(τ, ω), z2(τ, ω), ω)dτ

)
ψ′(s)ds, i = 1, 2.

(9)
Since the function fi, i = 1, 2 are absolutely continuous for all ω ∈ Ω and ξ ∈ I, then
(z1, z2) is a random solution for the problem (1) if and only if (z1, z2) = (H(z1, z2))(ξ, ω).

We need to demonstrate that the operator H is a contraction mapping on J using
Bielecki’s vector-norm.



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 9 of 21

Step 1. H is a random operator on J.

Using (A1), the functions ω → fi(ξ, z1, z2, ω) are measurable for i = 1, 2. In view of
Lemma 4, the products

ϕ(s, τ)ϑi−2fi(τ, z1(τ, ω), z2(τ, ω), ω), i = 1, 2,

are again measurable. Further, the integral is a limit of a finite sum of measurable func-
tions, therefore, the maps

ω → Hi(z1(ξ, ω), z2(ξ, ω), ω), i = 1, 2,

are measurable. Accordingly, H is a random operator on J× Ω into J.

Step 2. H is a contraction mapping on J.

For any ω ∈ Ω and each (z1, z2), (r1, r2) ∈ J, using (A2), we can get

∥Hi(z1(ξ, ω), z2(ξ, ω), ω)−Hi(r1(ξ, ω), r2(ξ, ω), ω)∥

≤ (ϑi − 1)

∫ ξ

a
ψ′(s)e−ϖiϕ(ξ,s)

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
∥fi(τ, z1(τ, ω), z2(τ, ω), ω)

−fi(τ, r1(τ, ω), r2(τ, ω), ω)∥dτds

≤
2∑
j=1

(ϑi − 1)

∫ ξ

a
ψ′(s)e−ϖiϕ(ξ,s)

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
Φi,j(ω)×

∥zj(τ, ω)− rj(τ, ω)∥dτds, i = 1, 2.

which, by (3), can be written as

∥Hi(z1(ξ, ω), z2(ξ, ω), ω)−Hi(r1(ξ, ω), r2(ξ, ω), ω)∥

≤
2∑
j=1

(ϑi − 1)Φi,j(ω)∥zj(·, ω)− rj(·, ω)∥B×∫ ξ

a
ψ′(s)e−ϖiϕ(ξ,s)

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
eζϕ(τ,a)dτds, i = 1, 2.

By Lemma 3, one obtains

∥Hi(z1(ξ, ω), z2(ξ, ω), ω)−Hi(r1(ξ, ω), r2(ξ, ω), ω)∥

≤
2∑
j=1

(ϑi − 1)Φi,j(ω)∥zj(·, ω)− rj(·, ω)∥B
∫ ξ

a
ψ′(s)

e−ϖiϕ(ξ,s)eζϕ(s,a)

ζϑi−1
ds,

≤
2∑
j=1

(ϑi − 1)Φi,j(ω)∥zj(·, ω)− rj(·, ω)∥B
e−ϖiψ(ξ)−ζψ(a)

(ζ +ϖi)ζϑi−1

∫ ξ

a
ψ′(s)(ζ +ϖi)e

(ζ+ϖi)ψ(s)ds

=

2∑
j=1

(ϑi − 1)e−ϖiψ(ξ)−ζψ(a)

(ζ +ϖi)ζϑi−1

[
e(ζ+ϖi)ψ(ξ) − e(ζ+ϖi)ψ(a)

]
Φi,j(ω)∥zj(·, ω)− rj(·, ω)∥B, i = 1, 2.



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 10 of 21

By e(ϖi+ζ)ψ(ξ) − e(ϖi+ζ)ψ(a) ≤ e(ϖi+ζ)ψ(ξ) and e−ϖiψ(ξ)−ζψ(a) ≤ e−(ϖi+ζ)ψ(a), we get

∥Hi(z1(ξ, ω), z2(ξ, ω), ω)−Hi(r1(ξ, ω), r2(ξ, ω), ω)∥

≤
2∑
j=1

(ϑi − 1)e(ϖi+ζ)ϕ(ξ,a)

(ζ +ϖi)ζϑi−1
Φi,j(ω)∥zj(·, ω)− rj(·, ω)∥B, i = 1, 2.

Hence
∥Hi(z1(·, ω), z2(·, ω), ω)−Hi(r1(·, ω), r2(·, ω), ω)∥B

≤
2∑
j=1

(ϑi − 1)eϖiϕ(ξ,a)

(ζ +ϖi)ζϑi−1
Φi,j(ω)∥zj(·, ω)− rj(·, ω)∥B, i = 1, 2.

Therefore, we have

d((H(z1, z2))(·, ω), (H(r1, r2))(·, ω)) ≤ Nζ(ω)d ((z1(·, ω), z2(·, ω)), (r1(·, ω), r2(·, ω))) ,

where:

Nζ(ω) =


(ϑ1 − 1)eϖ1ϕ(b,a)

(ζ +ϖ1)ζϑ1−1
Φ1,1(ω)

(ϑ1 − 1)eϖ1ϕ(b,a)

(ζ +ϖ1)ζϑ1−1
Φ1,2(ω)

(ϑ2 − 1)eϖ2ϕ(b,a)

(ζ +ϖ2)ζϑ2−1
Φ2,1(ω)

(ϑ2 − 1)eϖ2ϕ(b,a)

(ζ +ϖ2)ζϑ2−1
Φ2,2(ω)

 ,

and

d ((z1(·, ω), z2(·, ω)), (r1(·, ω), r2(·, ω))) =

(
∥z1(·, ω)− r1(·, ω)∥B

∥z2(·, ω)− r2(·, ω)∥B

)
.

Choosing ζ > 0 large enough, the matrix Nζ(ω) converges to zero. Then, according
to Theorem 1, H possesses a unique random fixed-point, serving as the unique random
solution to system (1).

Our second result investigates the existence result for the system (1), employing The-
orem 2 as a tool.

Theorem 4. Suppose that

(A1) The functions fi are random Carathéodory on I×G×G× Ω.

(A3) There exist Ψi : I× Ω → L∞(I,R+), i = 1, 2 such that

∥fi(ξ, u1, u2, ω)∥ ≤ Ψi(ξ, ω)(1 + ∥u1∥+ ∥u2∥), i = 1, 2,

for all (ξ, u1, u2, ω) ∈ I×G2 × Ω.

(A4) There exists a constant random variable ϱi,j : Ω → [0,∞), i, j = 1, 2 such that for
each U j ⊂ P(C(I,O)),

Λ(fi(ξ, U
1, U2, ω)) ≤

2∑
j=1

ϱi,j(ω)Λ(U
j(ξ)), for all (ξ, ω) ∈ I× Ω.



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 11 of 21

Then, the system (1) possesses at least one random solution.

For easy computations, let Ψ∗
i = sup

ω∈Ω
∥Ψi(·, ω)∥L∞ , i = 1, 2.

Proof. For R > 0, consider a closed ball

BR = {(z1, z2) ∈ J : ∥zi(·, ω)∥∞ < R, i = 1, 2}. (10)

The proof of Theorem 4 will proceed through several steps.

Step 1. H transforms bounded sets into bounded sets in J.

Let (z1, z2) ∈ BR and ξ ∈ I, then for i = 1, 2 we have :

∥Hi(z1(ξ, ω), z2(ξ, ω), ω)∥

≤ (ϑi − 1)e−ϖiϕ(ξ,a)

∫ ξ

a
eϖiϕ(s,a)

(∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
∥fi(τ, z1(τ, ω), z2τ, ω))∥dτ

)
ψ′(s)ds

By using hypothesis (A3), for each ξ ∈ I, we have

∥fi(τ, z1(τ, ω), z2τ, ω))∥ ≤ Ψi(τ, ω)(1 + ∥z1(τ, ω)∥+ ∥z2(τ, ω)∥)

≤ ∥Ψi(·, ω)∥L∞(1 + ∥z1(·, ω)∥∞ + ∥z1(·, ω)∥∞), i = 1, 2.
(11)

So, by the fact e−ϖiϕ(ξ,a) ≤ 1 for ξ ∈ I and using (11) to gathere Lemma 1 , we get

∥Hi(z1(ξ, ω), z2(ξ, ω), ω)∥

≤ (ϑi − 1)∥Ψi(·, ω)∥L∞(1 + ∥z1(·, ω)∥∞ + ∥z1(·, ω)∥∞)

∫ ξ

a
eϖiϕ(s,a)

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
dτψ′(s)ds

≤ (1 + 2R)∥Ψi(·, ω)∥L∞

∫ ξ

a
eϖiϕ(s,a)

ϕ(s, a)ϑi−1

Γ(ϑi − 1)
ψ′(s)ds

≤ (1 + 2R)∥Ψi(·, ω)∥L∞eϖiϕ(b,a)
ϕ(ξ, a)ϑi

ϑiΓ(ϑi − 1)
, i = 1, 2.

Hence

∥Hi(z1(·, ω), z2(·, ω), ω)∥∞ ≤ (1 + 2R)∥Ψi(·, ω)∥L∞eϖiϕ(b,a)
ϕ(b, a)ϑi

ϑiΓ(ϑi − 1)
, i = 1, 2.

This implies that:

∥H(z1(·, ω), z2(·, ω), ω)∥J = ∥H1(z1(·, ω), z2(·, ω), ω)∥∞ + ∥H2(z1(·, ω), z2(·, ω), ω)∥∞

≤
2∑
i=1

(1 + 2R)∥Ψi(·, ω)∥L∞eϖiϕ(b,a)
ϕ(b, a)ϑi

ϑiΓ(ϑi − 1)
.



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 12 of 21

This shows that H transforms bounded sets into bounded sets in J.

Step 2. H is continuous.

Let {z1,n, z2,n} be a sequence satisfying {z1,n, sz,n} → (z1, z2) in BR as n → ∞. For
each (ξ, ω) ∈ I× Ω, making use of (A1), we easily have

∥fi(τ, z1,n(τ, ω), z2,n(τ, ω), ω)− fi(τ, z1(τ, ω), z2(τ, ω), ω)∥ → 0, as n −→ ∞, i = 1, 2.

Next, in view of (A3), one gets

∥fi(τ, z1,n(τ, ω), z2,n(τ, ω), ω)− fi(τ, z1(τ, ω), z2(τ, ω), ω)∥

≤ ∥fi(τ, z1,n(τ, ω), z2,n(τ, ω), ω)∥+ ∥fi(τ, z1,n(τ, ω), z2,n(τ, ω), ω)∥

≤ 2Ψi(τ, ω) (1 + ∥z1(τ, ω)∥+ ∥z2(τ, ω)∥)

≤ 2(1 + 2R)Ψi(τ, ω), i = 1, 2.

Since, the functions τ 7→ ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
Ψi(τ, ω) and s 7→ ψ′(s)ϕ(s, a)ϑi−1

Γ(ϑi)
Ψi(s, ω), i =

1, 2 are Lebesgue integrable over [a, s] (resp. [a, ξ]). Then it follows from the Lebesgue
dominated convergence theorem that

∥Hi(z1,n(ξ, ω), z2,n(ξ, ω), ω)−Hi(z1(ξ, ω), z2(ξ, ω), ω)∥

≤ (ϑi − 1)eϖiϕ(b,a)

∫ ξ

a

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
×

∥fi(τ, z1,n(τ, ω), z2,n(τ, ω), ω)− fi(τ, z1(τ, ω), z2(τ, ω), ω)∥dτψ′(s)ds

−−−→
n→∞

0, for all ξ ∈ I, i = 1, 2.

Therefore,

∥Hi(·, z1,n(·, ω), z2,n(·, ω), ω)−Hi(·, z1(·, ω), z2(·, ω), ω)∥∞ −−−→
n→∞

0, i = 1, 2.

Accordingly, the operator H(·, ·) is continuous.

Step 3. H(BR) is equicontinuous.

For any ξ1, ξ2 ∈ I with ξ1 < ξ2 and (z1, z2) ∈ BR, we obtain

∥Hi(z1(ξ2, ω), z2(ξ2, ω), ω)−Hi(z1(ξ1, ω), z2(ξ1, ω), ω)∥ ≤ Ji,1 + Ji,2, i = 1, 2,

where

Ji,1 =
(ϑi − 1)

eϖiϕ(ξ2,a)

∫ ξ2

ξ1

ψ′(s)eϖiϕ(s,a)

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
∥fi(τ, z1(τ, ω), z2(τ, ω), ω)∥dτds,



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 13 of 21

and

Ji,2 = (ϑi− 1)

∫ ξ1

a
ψ′(s)

∣∣∣e−ϖiϕ(ξ2,s)− e−ϖiϕ(ξ1,s)
∣∣∣∥∥∥(Iϑi−1;ψ

a+
fi(τ, z1(τ, ω), z2(τ, ω), ω

)
(s)
∥∥∥ds,

From (A3) and using (11) and the fact e−ϖiϕ(ξ2,a) ≤ 1 and Lemma 1, we get

Ji,1 ≤ (1 + 2R)(ϑi − 1)∥Ψi(·, ω)∥L∞

∫ ξ2

ξ1

eϖiϕ(s,a)

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
dτψ′(s)ds

≤ (1 + 2R)∥Ψi(·, ω)∥L∞

∫ ξ2

ξ1

eϖiϕ(s,a)
ψ′(s)ϕ(s, a)ϑi−1

Γ(ϑi − 1)
ds

≤ (1 + 2R)∥Ψi(·, ω)∥L∞eϖiϕ(b,a)

[
ϕ(ξ2, a)

ϑi − ϕ(ξ1, a)
ϑi

ϑiΓ(ϑi − 1)

]
, i = 1, 2.

Thus,
Ji,1 −→ 0 when ξ2 −→ ξ1, i = 1, 2. (12)

On the other side,

Ji,2 = (ϑi − 1)
(
e−ϖiϕ(ξ1) − e−ϖiϕ(ξ2)

)
×∫ ξ1

a
eϖiϕ(s)

∥∥∥(Iϑi−1;ψ
a+

fi(τ, z1(τ, ω), z2(τ, ω), ω
)
(s)
∥∥∥ψ′(s)ds, i = 1, 2.

Thus,
Ji,2 −→ 0 when ξ2 −→ ξ1, i = 1, 2. (13)

From (12) and (13), we get

∥Hi(z1(ξ2, ω), z2(ξ2, ω), ω)−Hi(z1(ξ1, ω), z2(ξ1, ω), ω)∥ −−−−→
ξ2→ξ1

0, i = 1, 2.

This proves that, H(BR) is equicontinuous.

Step 4. H is ΘJ-condensing.

First, for every U1 × U2 ⊂ P(J), we define the MNC as

ΘJ(U
1 × U2) =

(
Θ(U1)

Θ(U2)

)
, (14)

where
Θ(U i) = sup

ξ∈I
e−γξΛ(U i(ξ)); γ > 0, i = 1, 2, (15)

The MNC ΘJ is well defined and gives a semiadditive, monotone, nonsingular and regular
MNC in J.



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 14 of 21

Secondly, let U1 × U2 ⊂ P(J) be such that

ΘJ
(
Hi(U

1 × U2)
)
≥ ΘJ(U

1 × U2), i = 1, 2. (16)

We will show that (16) implies the relative compactness of U1 × U2. There exists a
countable set {(Z1,n,Z2,n)}∞n=1 such that

Zi,n(ξ, ω) = Hi ({z1,n(ξ, ω), z2,n(ξ, ω), ω}) , i = 1, 2,

where {(z1,n, z2,n)}∞n=1 ⊂ J. From the properties of the MNC, one gets (for i = 1, 2)

Θ({Zi,n}∞n=1)

≤ Θ

({
(ϑi − 1)eϖiϕ(b,a)

∫ ξ

a

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
fi(τ, z1,n(τ, ω), z2,n(τ, ω))dτψ

′(s)ds

}+∞

n=1

)
.

(17)
Now, we will find an estimate for Θ({Zi,n}∞n=1), i = 1, 2. By using (A4), for all ξ ∈ I and
τ ≤ s ≤ ξ, one has

Λ

({
ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
fi(τ, z1,n(τ, ω), z2,n(τ, ω))

}+∞

n=1

)

≤ ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)

2∑
j=1

ϱi,j(ω)Λ({zj,n(τ, ω)}+∞
n=1)

≤
2∑
j=1

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
ϱi,j(ω)e

γτ sup
a≤τ≤s

e−γτΛ({zj,n(τ, ω)}+∞
n=1)

≤
2∑
j=1

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
ϱi,j(ω)e

γτΘ({zj,n(·, ω)}+∞
n=1).

Then, applying Lemma 5, we get for all ξ ∈ I, s ∈ [a, ξ] and τ ≤ s,

Λ

({
(ϑi − 1)eϖiϕ(b,a)

∫ ξ

a

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
fi(τ, z1,n(τ, ω), z2,n(τ, ω))dτψ

′(s)ds

}+∞

n=1

)

≤
2∑
j=1

Θ({zj,n(·, ω)}+∞
n=1)ϱi,j(ω)

4(ϑi − 1)eϖiϕ(b,a)

Γ(ϑi − 1)

∫ ξ

a

∫ s

a
ψ′(τ)ϕ(s, τ)ϑi−2eγτdτψ′(s)ds

≤
2∑
j=1

Θ({zj,n(·, ω)}+∞
n=1)ϱi,j(ω)

4(ϑi − 1)eϖiϕ(b,a)

Γ(ϑi − 1)

∫ ξ

a
ψ′(s)eγs

∫ s

a
ψ′(τ)ϕ(s, τ)ϑi−2dτds

≤
2∑
j=1

Θ({zj,n(·, ω)}+∞
n=1)ϱi,j(ω)

4eϖiϕ(b,a)

Γ(ϑi − 1)

∫ ξ

a
ψ′(s)eγsϕ(s, a)ϑi−1ds.



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 15 of 21

Multiplying both sides by e−γξ and taking sup
ξ∈I

, one obtains

sup
ξ∈I

e−γξΛ

({
(ϑi − 1)eϖiϕ(b,a)

∫ ξ

a

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
fi(τ, z1,n(τ, ω), z2,n(τ, ω))dτψ

′(s)ds

}+∞

n=1

)

≤
2∑
j=1

Θ({zj,n(·, ω)}+∞
n=1)ℵi,j(γ, ω).

where ℵi,j(γ, ω), i, j = 1, 2 are defined in (6).
Hence,

Θ

({
(ϑi − 1)eϖiϕ(b,a)

∫ ξ

a

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
fi(τ, z1,n(τ, ω), z2,n(τ, ω))dτψ

′(s)ds

}+∞

n=1

)

≤
2∑
r=1

Θ({zj,n(·, ω)}∞n=1)ℵi,j(γ, ω), i = 1, 2.

(18)
Next, by (17) and (18), we derive

Θ({Zi,n}∞n=1) ≤
2∑
r=1

Θ({zj,n(·, ω)}∞n=1)ℵi,j(γ, ω), i = 1, 2,

which implies

ΘJ(H({z1,n(·, ω), z2,n(·, ω), ω}+∞
n=1) =

(
Θ(H1({z1,n(·, ω), z2,n(·, ω), ω}+∞

n=1)

Θ(H2({z1,n(·, ω), z2,n(·, ω), ω}+∞
n=1)

)

≤ Ξγ(ω)

(
Θ({z1,n(·, ω)}∞n=1)

Θ ({z2,n(·, ω)}∞n=1)

)
,

where

Ξγ(ω) =

ℵ1,1(γ, ω) ℵ1,2(γ, ω)

ℵ2,1(γ, ω) ℵ2,2(γ, ω)

 .

By Lemma 7, one can choose γ such that the spectral radius ρ(Ξγ(ω)) < 1, therefore

Θ(Hi({z1,n(·, ω), z2,n(·, ω), ω}+∞
n=1) = 0, i = 1, 2.

This implies that

Θ(Hi({z1,n(ξ, ω), z2,n(ξ, ω), ω}+∞
n=1) = 0, for ξ ∈ I, i = 1, 2.



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 16 of 21

Finally,
ΘJ(U

1 × U2) = (0, 0),

which proves the compactness of the set U1 × U2.

Step 5. The set W (see Theorem 2 (2)) is bounded.

Let (z1, z2) ∈ J and (z1, z2) = κ(ω)H(z1, z2) for some κ(ω) ∈ (0, 1). Then, by the fact
e−ϱχ(ξ,a) ≤ 1 for all ξ ∈ I, we obtain

zi(ξ, ω) = κ(ω)

[
(ϑi − 1)e−ϖiϕ(ξ,a)

∫ ξ

a
eϖiϕ(s,a)

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
×

fi(τ, z1(τ, ω), z2(τ, ω), ω)dτψ
′(s)ds

]

≤ (ϑi − 1)

e−ϖiϕ(b,a)

∫ ξ

a

∫ s

a

ψ′(τ)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
fi(τ, z1(τ, ω), z2(τ, ω), ω)dτψ

′(s)ds, i = 1, 2.

Using Fubini’s Theorem, we have

∥zi(ξ, ω)∥ ≤ (ϑi − 1)

e−ϖiϕ(b,a)

∫ ξ

a
∥fi(τ, z1(τ, ω), z2(τ, ω), ω)∥

∫ ξ

τ

ψ′(s)ϕ(s, τ)ϑi−2

Γ(ϑi − 1)
dsψ′(τ)dτ

≤ (ϑi − 1)

e−ϖiϕ(b,a)Γ(ϑi)

∫ ξ

a
ψ′(τ)ϕ(ξ, τ)ϑi−1∥fi(τ, z1(τ, ω), z2(τ, ω), ω)∥dτ, i = 1, 2.

Using (A3), we get

∥zi(ξ, ω)∥ ≤ (ϑi − 1)

e−ϖiϕ(b,a)Γ(ϑi)

∫ ξ

a
ψ′(τ)ϕ(ξ, τ)ϑi−1Ψi(τ, ω)(1 + ∥z1(τ, ω)∥+ ∥z2(τ, ω)∥)dτ

≤ (ϑi − 1)

e−ϖiϕ(b,a)Γ(ϑi)

∫ ξ

a
ψ′(τ)ϕ(ξ, τ)ϑi−1Ψi(τ, ω)(∥z1(τ, ω)∥+ ∥z2(τ, ω)∥)dτ

+
(ϑi − 1)Ψi(b, ω)

e−ϖiϕ(b,a)ϑiΓ(ϑi)
ϕ(ξ, a)ϑi , i = 1, 2.

Therefore

∥z1(ξ, ω)∥+ ∥z2(ξ, ω)∥

≤ ς(ξ) +
(ϑ1 − 1)

e−ϖ1ϕ(b,a)Γ(ϑ1)

∫ ξ

a
ψ′(τ)ϕ(ξ, τ)ϑ1−1Ψ1(τ, ω)(∥z1(τ, ω)∥+ ∥z2(τ, ω)∥)dτ

+
(ϑ2 − 1)

e−ϖ2ϕ(b,a)Γ(ϑ2)

∫ ξ

a
ψ′(τ)ϕ(ξ, τ)ϑ2−1Ψ2(τ, ω)(∥z1(τ, ω)∥+ ∥z2(τ, ω)∥)dτ



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 17 of 21

where

ς(ξ) :=
(ϑ1 − 1)Ψ1(b, ω)

e−ϖ1ϕ(b,a)ϑ1Γ(ϑ1)
ϕ(ξ, a)ϑ1 +

(ϑ2 − 1)Ψ2(b, ω)

e−ϖ2ϕ(b,a)ϑ2Γ(ϑ2)
ϕ(ξ, a)ϑ2

Applying Lemma 6, we obtain

∥z1(ξ, ω)∥+ ∥z2(ξ, ω)∥ ≤ ς(b)
2∑
j=0

Eϑj
(
(ϑj − 1)eϖjϕ(b,a)ϕ(b, a)ϑj

)
:= D.

Hence

∥(z1(·, ω), z2(·, ω))∥J ≤ D̂ :=

(
D
D

)
Which achieves the desired estimate. Therefore, Theorem 2 ensures the existence of a
random solution for the system (1).

4. Examples

Let Ω = (−∞, 0) be endowed with the usual σ-algebra consisting of Lebesgue measur-
able subsets of (−∞, 0). Consider the separable Banach space

G = c0 = {s = (s1, s2, · · · , sn, · · · ) : sn → 0 as n→ ∞}

endowed with
∥s∥G = sup

n≥1
|sn|.

Example 1: Illustration of Theorem 3.

Let us take ϖi = ...., i = 1, 2.
For (ξ, ω) ∈ I× Ω, consider the nonlinear functions fi, i = 1, 2 be defined by

f1(ξ, z1(ξ, ω), z2(ξ, ω), ω) =

{
z1,n(ξ, ω)

|ω|(1 + |ω|)
+

sin(z2,n(ξ, ω))

1 + |ω|2

}
n≥1

,

f2(ξ, z1(ξ, ω), z2(ξ, ω), ω) =

{
arctan |s1,n(ξ, ω)|

1 + |ω|
+
e−|ω|z2,n(ξ, ω)

1 + |z2,n(ξ, ω)|

}
n≥1

(19)

Firstly, we easily see that, the functions fi, i = 1, 2, satisfy (A1). Secondly, we can check
that

∥f1(ξ, z1(ξ, ω), z2(ξ, ω), ω)− f1(ξ, r1(ξ, ω), r2(ξ, ω), ω)∥

≤ 1

|ω|(1 + |ω|)
∥z1,n(ξ, ω)− r1,n(ξ, ω)∥+ 1

1 + |ω|2
∥z2,n(ξ, ω)− r2,n(ξ, ω)∥,

and
∥f2(ξ, z1(ξ, ω), z2(ξ, ω), ω)− f2(ξ, r1(ξ, ω), r2(ξ, ω), ω)∥

≤ 1

1 + |ω|
∥z1,n(ξ, ω)− r1,n(ξ, ω)∥+ 1

e|ω|
∥z2,n(ξ, ω)− r2,n(ξ, ω)∥.



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 18 of 21

So, the hypotheses (A2) holds with

Φ1,1(ω) =
1

|ω|(1 + |ω|)
, Φ1,2(ω) =

1

1 + |ω|2
, for all ω ∈ Ω.

Φ2,1(ω) =
1

1 + |ω|
, Φ2,2(ω) =

1

e|ω|
, for all ω ∈ Ω.

An application of Theorem 3, we deduce that system (1) with (19) has a unique random
solution (z1, z2).

Example 2: Illustration of Theorem 4.

For (ς, ω) ∈ I× Ω and si = {si,n}n ∈ c0, consider the nonlinear forcing terms,



f1(ξ, z1(ξ, ω), z2(ξ, ω), ω)

=
2 sin(ω/5)

π
arctan(ξ)

{
sin |z1,n(ξ, ω)|+ loge(|z2,n(ξ, ω)|+ 1) + 5−n

}
n≥1

f2(ξ, z1(ξ, ω), z2(ξ, ω), ω) =
|ω|(e2ξ − 1)

(1 + |ω|)(eξ + 1)

{
arctan(|z1,n(ξ, ω)|) + |z2,n(ξ, ω)|+ π−n

}
n≥1

(20)
Obviously, fi, (i = 1, 2) satisfy hypothesis (A1).

To illustrate (A3), let ξ ∈ I and zi = {zi,n}n ∈ U ⊂ c0, i = 1, 2. Then

∥f1(ξ, z1(ξ, ω), z2(ξ, ω), ω)∥ ≤ 2 sin(ω/5)

π
arctan(ξ)

(
∥z1,n(ξ, ω)∥+ ∥z2,n(ξ, ω)∥+ 1

)
,

(21)
and

∥f2(ξ, z1(ξ, ω), z2(ξ, ω), ω)∥ ≤ |ω|(e2ξ − 1)

(1 + |ω|)(eξ + 1)

(
∥z1,n(ξ, ω)∥+ ∥z2,n(ξ, ω)∥+ 1

)
, (22)

Therefore, (H3) is verified with

Ψ1(ξ, ω) =
2 sin(ω/5)

π
arctan(ξ) and Ψ2(ξ, ω) =

|ω|(e2ξ − 1)

(1 + |ω|)(eξ + 1)
for all (ξ, ω) ∈ I×Ω.

Next, hypothesis (A4) is satisfied. Indeed, we recall that the Hausdorff MNC Θ in
(c0, ∥ · ∥c0) can be computed by means of the formula

Θ(U) = lim
n→∞

sup
z∈U

∥(I− Pn) z∥∞ ,

where U ∈ P(c0), Pn represents the projection onto the linear span of the first n vectors
in the standard basis (see [1]).



M. Ziane et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5605 19 of 21

Using (21) and (22) (see also Example in [32]), we get

Θ
(
fi(ξ, U

1, U2)
)
≤ ϱi,1(ω)Θ(U1) + ϱi,2(ω)Θ(U2), for all (ξ, ω) ∈ I× Ω.

where

ϱ1,1(ω) = ϱ1,2(ω) = sin(ω/5), ϱ2,1(ω) = ϱ2,2(ω) =
|ω|

1 + |ω|
, for all ω ∈ Ω.

The conclusion of Teorem 4 implies that problem (1) with (20) has at least one solution
(z1, z2).

5. Conclusion

The fractional Langevin system is a crucial mathematical model for describing the
random motion of particles. Consequently, we investigated a class of ψ-Caputo Langevin
systems with random effects in a generalized separable Banach space. By employing the
Bielecki-type vector-valued norm, we established a new uniqueness criterion. Additionally,
we imposed rather mild assumptions to obtain a new existence result by utilizing a recent
random version of Sadovski’s fixed-point theorem. As a result, numerous findings in the
literature can be recovered through our results.

Acknowledgements

The authors would like to express their gratitude to the referees for their insightful
comments, which have improved the paper.

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