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Article

Some Results for a Class of Pantograph Integro-Fractional Stochastic Differential Equations

by
Sahar Mohammad Abusalim
1,
Raouf Fakhfakh
1,*,
Fatimah Alshahrani
2 and
Abdellatif Ben Makhlouf
3
1
Department of Mathematics, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia
2
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3
Department of Mathematics, Faculty of Sciences, Sfax University, BP 1171, Sfax 3000, Tunisia
*
Author to whom correspondence should be addressed.
Symmetry 2024, 16(10), 1362; https://doi.org/10.3390/sym16101362
Submission received: 4 September 2024 / Revised: 8 October 2024 / Accepted: 10 October 2024 / Published: 14 October 2024
(This article belongs to the Section Mathematics)

Abstract

:
Symmetrical fractional differential equations have been explored through a variety of methods in recent years. In this paper, we analyze the existence and uniqueness of a class of pantograph integro-fractional stochastic differential equations (PIFSDEs) using the Banach fixed-point theorem (BFPT). Also, Gronwall inequality is used to demonstrate the Ulam–Hyers stability (UHS) of PIFSDEs. The results are illustrated by two examples.

1. Introduction

Fractional derivatives were first introduced by Leibnitz and L’Hôpital in 1695. Numerous scientists have investigated this theory; see [1]. This idea will be demonstrated by an overview of the work carried out in this field. Abel’s work in fractional calculation fields, Euler’s studies from 1730, Lagrange’s applications from 1772, and Laplace’s idea of the fractal derivative from 1812 are all important. For additional information about the extensions made to fractional calculus by Atangana, Baleanu, and other scientists, see [2,3,4].
In recent decades, fractional calculus has grown in popularity. Its significance is shown in numerous domains, including biology, physics, economics, and automatic control. Fictitious calculus can provide more flexibility and precision when studying a variety of phenomena because it can differentiate and integrate non-integer orders; see [5,6,7,8]. For instance, Laskin in [5] introduced a new fractional Langevin-type stochastic differential equation. Koeller in [6] presented the applications of fractional calculus to the theory of viscoelasticity. Petras et al. in [7] studied the numerical solution of a two-compartmental pharmacokinetic model for oral drug administration. Yang et al. in [8] presented a fractional damped vibration problem. The methods and ideas for using fractional derivatives to solve symmetrical differential equations are covered in this area.
Fractional stochastic differential equations (FSDEs) have been the subject of an increasing amount of research in recent years [9,10,11]. These equations combine fractional calculus with stochastic processes to yield an impotent framework for modeling and evaluating complicated systems determined by memory and randomness. The study of FSDEs is very beneficial to many domains, including engineering and physics [6,12], providing valuable insight into the dynamics of complex systems.
The existence, uniqueness, and stability of pantograph stochastic differential equations are important for a variety of reasons. First, they provide a robust theoretical framework, assuring that solutions are well defined and predictable under specified initial conditions. This is critical for accurately simulating real-world processes in domains such as biology, economics, and engineering, where these equations are frequently used. Furthermore, stability analysis allows us to understand how solutions react to disturbances, which is critical for applications in control systems and other fields. Finally, these studies provide useful insights that might improve both theoretical knowledge and practical applications across a variety of disciplines.
UHS, or Ulam stability, is vital for various equations [13,14,15] as it offers analytical approximate solutions for many problems where exact solutions are inaccessible. It is important to note that stability is crucial; if a system is stable in the UHS sense, essential properties will hold near the exact solution. This is evident in fields such as optimization, economics, and biology. UHS emerged after Ulam’s notable talk at a conference in 1940 [16]. The UHS idea has been extensively studied [17,18,19]. For instance, Ben Makhlouf et al. in [17] proved the UHS of pantograph FSDEs by using generalized Gronwall inequality. Rhaima et al. in [18] investigated UHS for a class of integro-FSDEs. The authors in [19] proved UHS for a class of neutral FSDEs.
Motivated by the works [17,18], this paper presents some results for a class of PIFSDEs. The main findings of the article are as follows: (1) it proves the existence and uniqueness of the solution of PIFSDEs; (2) it studies the Ulam stability of PIFSDEs.

2. Notational Preliminaries

Let { F , E , F T , P } be a complete probability space, with F T = { F s } s [ 1 , T ] , T > 1 and W ( s ) represents a standard Brownian motion. Let X s = L 2 ( Ω , F s , P ) be the space of all mean square integrable and F s -measurable functions h : F R . Let
h m s = E h 2 .
Definition 1
([1]). The Riemann–Liouville fractional integral of order a > 0 for X L 1 ( [ 1 , b ] , R ) is given by
I υ R L X ( u ) = 1 Γ ( a ) 1 u u r a 1 X ( r ) d r .
Definition 2
([1]). The Hadamard fractional integral of order a > 0 for X L 1 ( [ 1 , b ] , R ) is given by
I a H X ( u ) = 1 Γ ( a ) 1 u log u r a 1 X ( r ) r d r .
Definition 3
([1]). Let X : [ 1 , b ] R . The Caputo–Hadamard fractional derivative of order a ( 0 , 1 ) of X is given by
D 1 a H C G ( u ) = 1 Γ ( 1 a ) u d d u 1 u log u r a G ( r ) G ( 1 ) r d r .
Definition 4
([1]). The Mittag-Leffler function with one-parameter is given by
E a ( x ) = m = 0 x m Γ ( m a + 1 ) ,
where a > 0 and x C .
Consider the following PIFSDE:
D 1 a H C β ( s ) i = 1 m R L I b i G i ( s , β ( s ) , β ( θ s ) ) = ζ 1 ( s , β ( s ) , β ( θ s ) ) + ζ 2 ( s , β ( s ) , β ( θ s ) ) d W ( s ) d s ,
s [ 1 , T ] , where θ ( 0 , 1 ) , 1 2 < a < 1 , 1 2 < b κ < 1 , κ = 1 , , m , β ( s ) = χ ( s ) for s [ θ , 1 ] and G κ , ζ j : [ 1 , T ] × R R , κ = 1 , , m , j = 1 , 2 are measurable.
Consider the hypothesis:
( H 1 )
There is L > 0 such that
κ = 1 m | G κ ( s , β 1 , β ˜ 1 ) G κ ( s , β 2 , β ˜ 2 ) | + κ = 1 2 | ζ κ ( s , β 1 , β ˜ 1 ) ζ κ ( s , β 2 , β ˜ 2 ) | L | β 1 β 2 | + | β ˜ 1 β ˜ 2 | ,
for every ( s , β 1 , β 2 ) [ 1 , T ] × R × R .
( H 2 )
The functions ζ 2 ( · , 0 , 0 ) , ζ 1 ( · , 0 , 0 ) and G i , i = 1 , , m satisfy the following conditions:
| | ζ 2 ( · , 0 , 0 ) | | = ess sup s [ 1 , T ] | ζ 2 ( s , 0 , 0 ) | < , 1 T | G i ( s , 0 , 0 ) | 2 d s < , 1 T | ζ 1 ( s , 0 , 0 ) | 2 d s < .

3. Existence and Uniqueness Results

Consider the Banach space D 2 ( [ θ , T ] ) of all processes β that are F T -adapted and measurable with
β D 2 ( [ θ , T ] ) = sup θ l T β ( l ) m s < ,
Consider the operator P χ : D 2 ( [ θ , T ] ) D 2 ( [ θ , T ] ) defined as follows:
P χ β ( s ) = χ ( 1 ) + j = 1 m 1 Γ ( b j ) 1 s ( s l ) b j 1 G j ( l , β ( l ) , β ( θ l ) ) d l + 1 Γ ( a ) 1 s log s l a 1 l ζ 1 ( l , β ( l ) , β ( θ l ) ) d l + 1 Γ ( a ) 1 s log s l a 1 l ζ 2 ( l , β ( l ) , β ( θ l ) ) d W ( l ) , s [ 1 , T ]
and
P χ β ( s ) = χ ( s ) , s [ θ , 1 ] .
Lemma 1.
χ D 2 ( [ θ , 1 ] ) , the operator P χ is well defined.
Proof. 
Given β D 2 ( [ θ , T ] ) , then
P χ β ( s ) m s 2 = E P χ β ( s ) 2 ( m + 3 ) χ ( 1 ) m s 2 + j = 1 m 1 Γ ( b j ) 2 E 1 s ( s l ) b j 1 G i ( l , β ( l ) , β ( θ l ) ) d l 2 + 1 Γ ( a ) 2 E 1 s log s l a 1 l ζ 1 ( l , β ( l ) , β ( θ l ) ) d l 2 + 1 Γ ( a ) 2 E 1 s log s l a 1 ζ 2 ( l , β ( l ) , β ( θ l ) ) l d W ( l ) 2
It follows from the Cauchy–Schwarz inequality that
E 1 s ( s l ) b i 1 G j ( l , β ( l ) , β ( θ l ) ) d l 2 1 s ( s l ) 2 b j 2 d l E 1 s G j ( l , β ( l ) , β ( θ l ) ) 2 d l 1 2 b j 1 ( s 1 ) 2 b j 1 E 1 s G j ( l , β ( l ) , β ( θ l ) ) 2 d l .
It follows from ( H 1 ) that
G j ( l , β ( l ) , β ( θ l ) ) 2 2 L 2 β ( l ) + β ( θ l ) 2 + 2 G j ( l , 0 , 0 ) 2 .
Therefore,
E 1 s G j ( l , β ( l ) , β ( θ l ) ) 2 d l 8 ( T 1 ) L 2 sup s [ θ , T ] E β ( s ) 2 + 2 1 s G j ( l , 0 , 0 ) 2 d l .
Thus,
E 1 s ( s l ) b j 1 G j ( l , β ( l ) , β ( θ l ) ) d l 2
( T 1 ) 2 b j 1 2 b j 1 8 ( T 1 ) L 2 sup s [ θ , T ] E β ( s ) 2 + 2 1 s G j ( l , 0 , 0 ) 2 d l .
It follows from the Cauchy–Schwarz inequality that
E 1 s ( log s l ) a 1 l ζ 1 ( l , β ( l ) , β ( θ l ) ) d l 2 1 s ( log s l ) 2 a 2 l 2 d l E 1 s ζ 1 ( l , β ( l ) , β ( θ l ) ) 2 d l 1 2 a 1 ( log T ) 2 a 1 E 1 s ζ 1 ( l , β ( l ) , β ( θ l ) ) 2 d l .
Using ( H 1 ) , we obtain:
ζ 1 ( l , β ( l ) , β ( θ l ) ) 2 2 L 2 β ( l ) + β ( θ l ) 2 + 2 ζ 1 ( l , 0 , 0 ) 2 .
Therefore,
E 1 s ( log s l ) a 1 l ζ 1 ( l , β ( l ) , β ( θ l ) ) d l 2
( log T ) 2 a 1 2 a 1 8 L 2 ( T 1 ) sup u [ θ , T ] E β ( u ) 2 + 2 1 s ζ 1 ( l , 0 , 0 ) 2 d l .
By applying the Itô isometry, we obtain
E 1 s log s l a 1 ζ 2 ( l , β ( l ) , β ( θ l ) ) l d W ( l ) 2 = E 1 s log s l 2 a 2 l 2 ζ 2 ( l , β ( l ) , β ( θ l ) ) 2 d l .
It follows from Assumption ( H 1 ) that
ζ 2 ( l , β ( l ) , β ( θ l ) ) 2 2 L 2 β ( l ) + β ( θ l ) 2 + 2 ζ 2 ( l , 0 , 0 ) 2 .
Then,
E 1 s log s l a 1 ζ 2 ( l , β ( l ) , β ( θ l ) ) l d W ( l ) 2 2 L 2 E 1 s log s l 2 a 2 l 2 β ( l ) + β ( θ l ) 2 d l + 2 ζ 2 ( l , 0 , 0 ) 2 1 s log s l 2 a 2 l 2 d l 8 L 2 2 a 1 log T 2 a 1 β D 2 ( [ θ , T ] ) + 2 2 a 1 log T 2 a 1 ζ 2 ( l , 0 , 0 ) 2 .
Therefore P χ is well defined. □
Theorem 1.
If ( H 1 ) and ( H 2 ) are satisfied, then Equation (1) possesses a unique solution.
Proof. 
Let us consider the norm · on D 2 ( [ θ , T ] ) by:
β { λ , λ 1 , , λ m } = sup t [ θ , T ] E β ( t ) 2 h ( t ) , β D 2 ( [ θ , T ] ) ,
where
h ( t ) = t λ i = 1 m E 2 b i 1 λ i ( t 1 ) 2 b i 1 for t [ 1 , T ] ,
with λ , λ 1 , , λ m > 0 and
( 4 m + 8 ) L 2 i = 1 m ( T 1 ) Γ ( b i ) 2 Γ ( 2 b i 1 ) λ i + T Γ ( a ) 2 Γ ( 2 a 1 ) λ 2 a 1 < 1
and
h ( t ) = 1 for t [ θ , 1 ] .
We know that D 2 ( [ θ , T ] ) , · { λ , λ 1 , , λ m } is a Banach space because · D 2 ( [ θ , T ] ) and · { λ , λ 1 , , λ m } are equivalent.
Let β 1 , β 2 D 2 ( [ θ , T ] ) .
For t [ θ , 1 ] , we have P χ β 1 ( t ) P χ β 2 ( t ) = 0 .
For t [ 1 , T ] , we obtain
E P χ β 1 ( t ) P χ β 2 ( t ) 2 ( m + 2 ) i = 1 m 1 Γ ( b i ) 2 E 1 t ( t s ) b i 1 [ G i ( s , β 1 ( s ) , β 1 ( θ s ) ) G i ( s , β 2 ( s ) , β 2 ( θ s ) ) ] d s 2 + 1 Γ ( a ) 2 E 1 t log t s a 1 s ζ 1 ( s , β 1 ( s ) , β 1 ( θ s ) ) ζ 1 ( s , β 2 ( s ) , β 2 ( θ s ) ) d s 2 + 1 Γ ( a ) 2 E 1 t log t s a 1 s ζ 2 ( s , β 1 ( s ) , β 1 ( θ s ) ) ζ 2 ( s , β 2 ( s ) , β 2 ( θ s ) ) d W ( s ) 2 .
Using the Cauchy–Schwartz inequality, we obtain
E | 1 t ( t s ) b i 1 [ G i ( s , β 1 ( s ) , β 1 ( θ s ) ) G i ( s , β 2 ( s ) , β 2 ( θ s ) ) ] d s | 2 2 L 2 ( T 1 ) 1 t ( t s ) 2 b i 2 E ( | β 1 ( s ) β 2 ( s ) | 2 + | β 1 ( θ s ) β 2 ( θ s ) | 2 ) d s
and
E 1 t log t s a 1 s ζ 1 ( s , β 1 ( s ) , β 1 ( θ s ) ) ζ 1 ( s , β 2 ( s ) , β 2 ( θ s ) ) d s 2 2 L 2 ( T 1 ) 1 t log t s 2 a 2 s 2 E ( | β 1 ( s ) β 2 ( s ) | 2 + | β 1 ( θ s ) β 2 ( θ s ) | 2 ) d s .
It follows from the isometry of Itô that
E 1 t log t s a 1 s ζ 2 ( s , β 1 ( s ) , β 1 ( θ s ) ) ζ 2 ( s , β 2 ( s ) , β 2 ( θ s ) ) d W ( s ) 2 = E 1 t log t s 2 a 2 s 2 ζ 2 ( s , β 1 ( s ) , β 1 ( θ s ) ) ζ 2 ( s , β 2 ( s ) , β 2 ( θ s ) ) 2 d s 2 L 2 1 t log t s 2 a 2 s 2 E ( | β 1 ( s ) β 2 ( s ) | 2 + | β 1 ( θ s ) β 2 ( θ s ) | 2 ) d s .
Therefore,
E P χ β 1 ( t ) P χ β 2 ( t ) 2 ( m + 2 ) i = 1 m 2 L 2 ( T 1 ) Γ ( b i ) 2 1 t ( t s ) 2 b i 2 E ( | β 1 ( s ) β 2 ( s ) | 2 + | β 1 ( θ s ) β 2 ( θ s ) | 2 ) d s + 2 L 2 ( T 1 ) Γ ( a ) 2 1 t log t s 2 a 2 s 2 E ( | β 1 ( s ) β 2 ( s ) | 2 + | β 1 ( θ s ) β 2 ( θ s ) | 2 ) d s + 2 L 2 Γ ( a ) 2 1 t log t s 2 a 2 s 2 E ( | β 1 ( s ) β 2 ( s ) | 2 + | β 1 ( θ s ) β 2 ( θ s ) | 2 ) d s ( 2 m + 4 ) L 2 i = 1 m ( T 1 ) Γ ( b i ) 2 1 t ( t s ) 2 b i 2 E ( | β 1 ( s ) β 2 ( s ) | 2 + | β 1 ( θ s ) β 2 ( θ s ) | 2 ) d s + T Γ ( a ) 2 1 t log t s 2 a 2 s 2 E ( | β 1 ( s ) β 2 ( s ) | 2 + | β 1 ( θ s ) β 2 ( θ s ) | 2 ) d s ( 2 m + 4 ) L 2 i = 1 m ( T 1 ) Γ ( b i ) 2 1 t ( t s ) 2 b i 2 [ h ( s ) E ( | β 1 ( s ) β 2 ( s ) | 2 ) h ( s ) + h ( θ s ) E ( | β 1 ( θ s ) β 2 ( θ s ) | 2 ) h ( θ s ) ] d s + T Γ ( a ) 2 1 t log t s 2 a 2 s 2 h ( s ) E ( | β 1 ( s ) β 2 ( s ) | 2 ) h ( s ) + h ( θ s ) E ( | β 1 ( θ s ) β 2 ( θ s ) | 2 ) h ( θ s ) d s ( 4 m + 8 ) L 2 i = 1 m ( T 1 ) Γ ( b i ) 2 β 1 β 2 λ , λ 1 , , λ m 2 1 t ( t s ) 2 b i 2 h ( s ) d s + T Γ ( a ) 2 β 1 β 2 λ , λ 1 , , λ m 2 1 t log t s 2 a 2 s 2 h ( s ) d s .
Similar to the proof of Theorem 3.2 in [18], we obtain
E P χ β 1 ( t ) P χ β 2 ( t ) 2 ( 4 m + 8 ) L 2 i = 1 m ( T 1 ) Γ ( b i ) 2 Γ ( 2 b i 1 ) λ i + T Γ ( a ) 2 Γ ( 2 a 1 ) λ 2 a 1 h ( t ) β 1 β 2 λ , λ 1 , , λ m 2 .
Consequently, we have
P χ β 1 P χ β 2 λ , λ 1 , , λ m K β 1 β 2 λ , λ 1 , , λ m
where
K 2 = ( 4 m + 8 ) L 2 i = 1 m ( T 1 ) Γ ( b i ) 2 Γ ( 2 b i 1 ) λ i + T Γ ( a ) 2 Γ ( 2 a 1 ) λ 2 a 1 .
Consequently, (1) has a unique solution β with β ( t ) = χ ( t ) for t [ θ , 1 ] . □

4. Stability Results

Firstly, we present a relevant definition before discussing our results.
Definition 5
([18]). Equation (1) is UHS with respect to ϖ if K ˜ > 0 exists so that ϖ > 0 and y D 2 ( [ θ , T ] ) , with y ( s ) = ξ ( s ) for s [ θ , 1 ] which satisfies
E y ( s ) P ξ y ( s ) 2 ϖ ; s [ 1 , T ] ,
there exists a solution β D 2 ( [ θ , T ] ) of (1), with β ( s ) = ξ ( s ) for s [ θ , 1 ] which satisfies
E y ( s ) β ( s ) 2 K ˜ ϖ , s [ 1 , T ] .
Theorem 2.
If Assumptions ( H 1 ) and ( H 2 ) are satisfied, then Equation (1) is UHS with respect to ϖ.
Proof. 
Consider ϖ > 0 . Suppose y ( s ) is a solution of (2). Let β ( s ) be the solution of (1) so that β ( s ) = ξ ( s ) when s [ θ , 1 ] . Therefore,
β ( s ) = ξ ( 1 ) + j = 1 m 1 Γ ( b j ) 1 s ( s l ) b j 1 G j ( l , β ( l ) , β ( θ l ) ) d l + 1 Γ ( a ) 1 s log s l a 1 l ζ 1 ( l , β ( l ) , β ( θ l ) ) d l + 1 Γ ( a ) 1 s log s l a 1 l ζ 2 ( l , β ( l ) , β ( θ l ) ) d W ( l ) .
Hence, we have
y ( s ) β ( s ) = ( y ( s ) ξ ( 1 ) j = 1 m 1 Γ ( b j ) 1 s ( s l ) b j 1 G j ( l , y ( l ) , y ( θ l ) ) d l 1 Γ ( a ) 1 s log s l a 1 l ζ 1 ( l , y ( l ) , y ( θ l ) ) d l 1 Γ ( a ) 1 s log s l a 1 l ζ 2 ( l , y ( l ) , y ( θ l ) ) d W ( l ) ) + j = 1 m 1 Γ ( b j ) 1 s ( s l ) b j 1 G j ( l , y ( l ) , y ( θ l ) ) G j ( l , β ( l ) , β ( θ l ) ) d l + 1 Γ ( a ) 1 s log s l a 1 l ζ 1 ( l , y ( l ) , y ( θ l ) ) ζ 1 ( l , β ( l ) , β ( θ l ) ) d l + 1 Γ ( a ) 1 s log s l a 1 l ζ 2 ( l , y ( l ) , y ( θ l ) ) ζ 2 ( l , β ( l ) , β ( θ l ) ) d W ( l ) .
Thus,
E y ( s ) β ( s ) 2 ( m + 3 ) ϖ + j = 1 m 1 Γ ( b j ) 2 E 1 s ( s l ) b j 1 [ G j ( l , y ( l ) , y ( θ l ) ) G j ( l , β ( l ) , β ( θ l ) ) ] d l 2 + 1 Γ ( a ) 2 E 1 s log s l a 1 l [ ζ 1 ( l , y ( l ) , y ( θ l ) ) ζ 1 ( l , β ( l ) , β ( θ l ) ) ] d l 2 + 1 Γ ( a ) 2 E 1 s log s l a 1 l [ ζ 2 ( l , y ( l ) , y ( θ l ) ) ζ 2 ( l , β ( l ) , β ( θ l ) ) ] d W ( l ) 2 .
It follows from the Cauchy–Schwarz inequality and the isometry of Itô that
E y ( s ) β ( s ) 2 ( m + 3 ) ϖ + j = 1 m 2 ( T 1 ) 2 b j 1 L 2 ( 2 b j 1 ) Γ ( b j ) 2 1 s E y ( l ) β ( l ) 2 + E y ( θ l ) β ( θ l ) 2 d l + 2 log T 2 a 1 L 2 ( 2 a 1 ) Γ ( a ) 2 1 s E y ( l ) β ( l ) 2 + E y ( θ l ) β ( θ l ) 2 d l + 2 L 2 Γ ( a ) 2 1 s log s l 2 a 2 l 2 E y ( l ) β ( l ) 2 + E y ( θ l ) β ( θ l ) 2 d l .
Thus,
E y ( s ) β ( s ) 2 q 1 ϖ + q 2 1 s E y ( l ) β ( l ) 2 + E y ( θ l ) β ( θ l ) 2 d l + q 3 1 s log s l 2 a 2 l E y ( l ) β ( l ) 2 + E y ( θ l ) β ( θ l ) 2 d l ,
where
q 1 = m + 3 , q 2 = 2 L 2 ( m + 3 ) j = 1 m ( T 1 ) 2 b j 1 ( 2 b j 1 ) Γ ( b j ) 2 + log T 2 a 1 ( 2 a 1 ) Γ ( a ) 2 and q 3 = 2 ( m + 3 ) L 2 Γ ( a ) 2 .
Let f ( s ) = sup l [ θ , s ] E | y ( l ) β ( l ) | 2 for s [ 1 , T ] .
We obtain E | y ( s ) β ( s ) | 2 f ( s ) and E | y ( θ s ) β ( θ s ) | 2 f ( s ) , for all s [ 1 , T ] .
Therefore, for s [ 1 , T ] , we obtain
E y ( s ) β ( s ) 2 q 1 ϖ + 2 q 2 1 s f ( l ) d l + 2 q 3 1 s log s l 2 a 2 l f ( l ) d l .
Then, for every l [ 1 , s ] ,
E y ( l ) β ( l ) 2 q 1 ϖ + 2 q 2 1 s f ( u ) d u + 2 q 3 1 s log s u 2 a 2 u f ( u ) d u .
Therefore, for each s [ 1 , T ] ,
f ( s ) q 1 ϖ + 2 q 2 1 s f ( l ) d l + 2 q 3 1 s log s l 2 a 2 l f ( l ) d l .
We obtain from Corollary 13.2 in [20]
f ( s ) q 1 ϖ + 2 q 2 1 s f ( l ) d l E 2 a 1 2 q 3 Γ ( 2 a 1 ) log s 2 a 1 ϖ q 4 + q 5 1 s f ( l ) d l ,
with
q 4 = q 1 E 2 a 1 2 q 3 Γ ( 2 a 1 ) log T 2 a 1
and
q 5 = 2 q 2 E 2 a 1 2 q 3 Γ ( 2 a 1 ) log T 2 a 1 .
It follows from the Gronwall inequality that
f ( s ) ϖ q 4 e q 5 ( T 1 ) .
Thus,
E y ( s ) β ( s ) 2 ϖ M , s [ 1 , T ] ,
with M = q 4 e q 5 ( T 1 ) . Thus, Equation (1) is UHS with respect to ϖ . □

5. Examples

Example 1.
Consider the PIFSDE:
D 1 a H C β ( s ) i = 1 2 R L I 4 + i 9 G i ( s , β ( s ) , β ( θ s ) ) = ζ 1 ( s , β ( s ) , β ( θ s ) ) + ζ 2 ( s , β ( s ) , β ( θ s ) ) d W ( s ) d s ,
1 < s 5 , where
G i ( s , β ( s ) , β ( θ s ) ) = 1 18 sin ( β ( θ s ) ) , i = 1 , 2 ζ 1 ( s , β ( s ) , β ( θ s ) ) = 1 15 sin ( β ( s ) ) s 2 + 1 , ζ 2 ( s , β ( s ) , β ( θ s ) ) = cos ( β ( θ s ) ) s 2 + 17 .
The assumptions ( H 1 ) and ( H 2 ) are satisfied with L = 1 2 . Using Theorem 1, we obtain the existence and uniqueness of the solutions of Equation (4). In addition, it follows from Theorem 2 that (4) is UHS with respect to ϖ.
Example 2.
Consider the PIFSDE:
D 1 a H C β ( s ) i = 1 2 R L I 3 + i 7 G i ( s , β ( s ) , β ( θ s ) ) = ζ 1 ( s , β ( s ) , β ( θ s ) ) + ζ 2 ( s , β ( s ) , β ( θ s ) ) d W ( s ) d s ,
1 < s 3 , where
G i ( s , β ( s ) , β ( θ s ) ) = 1 28 cos ( β ( θ s ) ) , i = 1 , 2 ζ 1 ( s , β ( s ) , β ( θ s ) ) = 1 25 cos ( β ( s ) ) s 6 + 7 , ζ 2 ( s , β ( s ) , β ( θ s ) ) = sin ( β ( θ s ) ) s 2 + 27 .
The assumptions ( H 1 ) and ( H 2 ) are satisfied with L = 1 5 . Using Theorem 1, we obtain the existence and uniqueness of the solutions of Equation (4). In addition, it follows from Theorem 2 that (4) is UHS with respect to ϖ.

6. Conclusions

In this work, we have explored the existence, uniqueness and UHS of PIFSDEs using a combination of valuable mathematical techniques, stochastic analysis and Banach fixed-point methods. We have demonstrated how our results are applicable with two examples. In a future paper, it would be fascinating to expand this research to include the effects of infinite time delays.

Author Contributions

Conceptualization, F.A.; Software, R.F.; Validation, S.M.A.; Resources, S.M.A.; Writing—original draft, A.B.M.; Writing—review and editing, R.F. and A.B.M.; Supervision, A.B.M.; Project administration, F.A.; Funding acquisition, F.A. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R358), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

References

  1. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
  2. Ahmad, B.; Alsaedi, A.; Ntouyas, S.K.; Tariboon, J. Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities; Springer: Cham, Switzerland, 2017. [Google Scholar]
  3. Atangana, A.; Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Therm. Sci. 2016, 20, 763–769. [Google Scholar] [CrossRef]
  4. Caputo, M.; Fabrizio, M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1, 73–85. [Google Scholar]
  5. Laskin, N. Fractional market dynamics. Physica A 2000, 287, 482–492. [Google Scholar] [CrossRef]
  6. Koeller, R. Applications of fractional calculus to the theory of viscoelasticity. ASME J. Appl. Mech. 1984, 51, 299–307. [Google Scholar] [CrossRef]
  7. Petras, I.; Magin, R.L. Simulation of drug uptake in a two compartmental fractional model for a biological system. Commun. Nonlinear Sci. Numer. Simul. 2011, 16, 4588–4595. [Google Scholar] [CrossRef] [PubMed]
  8. Yang, X.-J.; Srivastava, H.M. An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 2015, 29, 499–504. [Google Scholar] [CrossRef]
  9. Ahmad, M.; Zada, A.; Ahmad, J.; Salam, M.A.A.E. Analysis of Stochastic Weighted Impulsive Neutral ψ-Hilfer Integro-Fractional Differential System with Delay. Math. Probl. Eng. 2022, 2022, 1490583. [Google Scholar] [CrossRef]
  10. Shahid, S.; Saifullah, S.; Riaz, U.; Zada, A.; Moussa, S.B. Existence and Stability Results for Nonlinear Implicit Random Fractional Integro-Differential Equations. Qual. Theory Dyn. Syst. 2023, 22, 81. [Google Scholar] [CrossRef]
  11. Saifullah, S.; Shahid, S.; Zada, A. Analysis of Neutral Stochastic Fractional Differential Equations Involving Riemann–Liouville Fractional Derivative with Retarded and Advanced Arguments. Qual. Theory Dyn. Syst. 2024, 23, 39. [Google Scholar] [CrossRef]
  12. Miller, K.S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; Wiley: New York, NY, USA, 1993. [Google Scholar]
  13. Gajda, Z. On stability of additive mappings. Int. J. Math. Math. Sci. 1991, 14, 431–434. [Google Scholar] [CrossRef]
  14. Rassias, T.M. On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72, 297–300. [Google Scholar] [CrossRef]
  15. Rassias, T.M. On a modified Hyers–Ulam sequence. J. Math. Anal. Appl. 1991, 158, 106–113. [Google Scholar] [CrossRef]
  16. Hyers, D.H. On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27, 222–224. [Google Scholar] [CrossRef] [PubMed]
  17. Makhlouf, A.B.; Mchiri, L.; Rguigui, H. Ulam-Hyers stability of pantograph fractional stochastic differential equations. Math. Methods Appl. Sci. 2023, 46, 4134–4144. [Google Scholar]
  18. Rhaima, M.; Mchiri, L.; Makhlouf, A.B.; Ahmed, H. Ulam type stability for mixed Hadamard and Riemann–Liouville Fractional Stochastic Differential Equations. Chaos Solitons Fractals 2024, 78, 114356. [Google Scholar] [CrossRef]
  19. Makhlouf, A.B.; Sallay, J. Some results on proportional Caputo neutral fractional stochastic differential equations. Discret. Contin. Dyn. Syst.-S 2024, 17, 3102–3115. [Google Scholar] [CrossRef]
  20. Baleanu, D.; Machado, J.A.; Luo, A.C. Fractional Dynamics and Control; Springer Science and Business Media: New York, NY, USA, 2011. [Google Scholar]
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MDPI and ACS Style

Abusalim, S.M.; Fakhfakh, R.; Alshahrani, F.; Ben Makhlouf, A. Some Results for a Class of Pantograph Integro-Fractional Stochastic Differential Equations. Symmetry 2024, 16, 1362. https://doi.org/10.3390/sym16101362

AMA Style

Abusalim SM, Fakhfakh R, Alshahrani F, Ben Makhlouf A. Some Results for a Class of Pantograph Integro-Fractional Stochastic Differential Equations. Symmetry. 2024; 16(10):1362. https://doi.org/10.3390/sym16101362

Chicago/Turabian Style

Abusalim, Sahar Mohammad, Raouf Fakhfakh, Fatimah Alshahrani, and Abdellatif Ben Makhlouf. 2024. "Some Results for a Class of Pantograph Integro-Fractional Stochastic Differential Equations" Symmetry 16, no. 10: 1362. https://doi.org/10.3390/sym16101362

APA Style

Abusalim, S. M., Fakhfakh, R., Alshahrani, F., & Ben Makhlouf, A. (2024). Some Results for a Class of Pantograph Integro-Fractional Stochastic Differential Equations. Symmetry, 16(10), 1362. https://doi.org/10.3390/sym16101362

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