Abstract
In this manuscript, we study an averaging principle for fractional stochastic pantograph differential equations (FSDPEs) in the \(\psi \)-sense accompanied by Brownian movement. Under certain assumptions, we are able to approximate solutions for FSPEs by solutions to averaged stochastic systems in the sense of mean square. Analysis of system solutions before and after the average allows extending the classical Khasminskii approach to random fractional differential equations in the sense of \(\psi \)-Caputo. For clarity, we present at the end an applied example to facilitate the clarification of the theoretical results obtained
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1 Introduction
The Khasminskii approach, developed by Russian mathematician Mark Khasminskii in the 1960s, is a mathematical technique utilized to investigate the characteristics of stochastic processes. This method focuses on analyzing the behavior of conditional expectations to gain insights into the stochastic process under consideration. By doing so, it enables the establishment of long-term bounds on the process’s behavior. The versatility of the Khasminskii approach is evident in its applications to diverse stochastic analysis problems, encompassing areas such as stochastic differential equations, optimal control, and filtering. Additionally, it has proven valuable in comprehending the behavior of random walks, queuing systems, and other stochastic processes. Notably, the Khasminskii approach excels in managing systems involving a large number of interacting particles, making it particularly advantageous for studying complex systems in physics, chemistry, and biology. Consequently, the Khasminskii approach has made significant contributions to the field of stochastic analysis and remains a crucial tool for understanding the behavior of stochastic processes [1]. Khasminiskii was interested in studying the convergence of idle systems on the drag time scale \(\varepsilon \rightarrow 0,\) in resolving intermediate arguments. He concluded that averaging principle lay in the study of equations lost in terms of the relevant average. Therefore, we get an easy way to solve these equations.
Brownian motion, named after the botanist Robert Brown who first observed it in 1827, is the random movement of particles suspended in a fluid or gas, resulting from collisions with the molecules of the surrounding medium. Brownian motion is a fundamental concept in the study of stochastic processes and has applications in physics, chemistry, finance, and many other fields see [2, 3].
Stochastic fractional differential equations (SFDEs) are a type of differential equation that combine both stochastic (random) and fractional calculus concepts. On the other hand, stochastic calculus deals with random processes and their properties, and it has found applications in many areas including finance, economics, physics, and engineering. SFDEs combine both of these concepts by considering a stochastic process whose evolution is described by a fractional differential equation. These equations have been used to model many complex phenomena, including turbulence, anomalous diffusion, and financial markets [4,5,6,7,8].
In recent years, several authors started to work on fractional operators with more general kernels. The connecyed fractional derivatives are either in Riemann–Liouville or Caputo sense. For example, some authors called those in the Caputo sense by the \(\psi \)-Caputo derivatives [9,10,11]. In this work, we investigate Pantograph equations in the frame of such generalized fractional derivatives. For the sake of completeness and the benefit of readers we refer to [12,13,14] and the cited works therein.
In fact, our work is an important generalization of what is obtained in the reference see [15]
The nature of solutions for fractional stochastic differential pantograph equations (FSDPEs) with the \(\psi \)-Caputo sense in Euclidean n-dimentional spaces, \( {\mathbb {R}} ^{n}\) [14, 16, 17], is of hight interest in many applications. In general, such systems can take the form
where \(\lambda \in \left( 0,\frac{{\mathcal {T}}-a}{{\mathcal {T}}}\right) \), \({\mathcal {D}}_{\varsigma }^{\mathfrak {\alpha };\psi }\) is the \(\psi \)-Caputo fractional derivative, \(\alpha \in (\frac{1}{2},1),\) for each \(\varsigma \ge a\), \(\vartheta _{1}:\left[ a,{\mathcal {T}}\right] \times {\mathbb {R}} ^{n}\rightarrow {\mathbb {R}} ^{n}\) and \(\vartheta _{2}:\left[ a,{\mathcal {T}}\right] \times {\mathbb {R}} ^{n}\rightarrow {\mathbb {R}} ^{n\times m}\) are mesurable continuous functions (CFs), \(B(\varsigma )\) is a m-dimensional standard Brownian motion on the complete probability space \(\left\{ \Omega ,{\mathbb {F}},P\right\} \). The initial value \(X_{0}\) is "second order random variable", it also an \({\mathbb {F}}_{0} \)-mesurable \( {\mathbb {R}} ^{n}\)-value random variable, satisfying \(E \left| X_{0}\right| ^{2}<\infty .\) Solutions of non-linear FSDPEs are almost impossible to solve and very difficult. For this reason, we use symmetrical methods and techniques in the widest field. That plays an important role in the evolution of partial calculus [16, 18, 19] of which the averaging principle is its theoretical support.
Pantograph equations are a class of differential equations that were first introduced by the German mathematician Heinrich Weber in 1845. The name "pantograph" comes from the fact that the equations resemble the mechanism of a pantograph, which is a device used for copying drawings and diagrams at a different scale.
The pantograph equation has the form: \(y^{\prime }(t)=a(t)y(t)+b(t)y(qt),\) Where y(t) is the unknown function, a(t) and b(t) are given functions, and q is a positive constant. Pantograph equations have applications in various areas of science and engineering, such as electrical engineering, signal processing, and control theory. In particular, they are used to model systems with time delays, which arise in many real-world applications [20,21,22,23,24,25].
In our work, we expanded the classical Khasminskii appraoch to fractional differential equations using the \(\psi \)-Caputo fractional derivative. Our manifestation is that mild solutions of tow systems before and after averaging are equivalent in the sense of mean square. Which clearly and rigorously proves the principle of fractional average. In other words, a simple and effective method is given to solve random differential pantograph Eq. (1.1) inaccurately.
We have organized the article into four primary sections. The first section covers general definitions of fractional calculus and its properties, along with a description of the specific equation type under consideration. Moving on to the second section, we introduce approximation methods and provide definitions for the generalized Caputo derivative, as well as conditionally including the classic Khasminskii theorem. In the third section, we present the key results that have been achieved. Finally, we provide a practical example that illustrates the outcomes of our research.
2 Preliminaries
This section includes some basic techniques, definitions, lemmas and theories that we need to proceed. For more details see [9,10,11, 17, 26,27,28,29].
Let \(\psi :[a_{1},a_{2}]\rightarrow {\mathbb {R}}\) be increasing via \(\psi ^{\prime }(\varsigma )\ne 0,\) \(\forall \varsigma .\) The symbol \(C(J,{\mathbb {R}})\) represents the Banach space of CFs \(\varkappa :J\rightarrow {\mathbb {R}}\) with the norm \(\Vert \varkappa \Vert =\sup \{|\varkappa (\varsigma )|:\varsigma \in J\}.\)
Definition 2.1
[17, 29] The Riemann–Liouville fractional integral of order \(\alpha >0\) for a function \(x:\left[ 0,+\infty \right) \rightarrow {\mathbb {R}} \) is defined as
where \(\Gamma \) is the Euler gamma function given by
Definition 2.2
[17, 29] The Riemann–Liouville fractional derivative of order \(\alpha >0\) for a function \(x:\left[ 0,+\infty \right) \rightarrow {\mathbb {R}} \) is defined as
Definition 2.3
[17, 18, 29] The \(\psi \)-Riemann–Liouville fractional integral (\(\psi \)-RLFI) of order \(\alpha >0\) for a CF \(\varkappa :\left[ a,{\mathcal {T}}\right] \rightarrow {\mathbb {R}} \) is referred to as
Definition 2.4
[17, 18, 29] The Caputo fractional derivative (CFD) of order \(\alpha >0\) for a \(\varkappa :\left[ 0,+\infty \right) \rightarrow {\mathbb {R}} \) is intended by
Definition 2.5
[9, 17, 18] The \(\psi \)-Caputo fractional derivative (\(\psi \)-CFD) of order \(\alpha >0\) for a CF \(\varkappa :\left[ a,{\mathcal {T}}\right] \rightarrow {\mathbb {R}} \) is the aim of
where \(\partial _{\psi }^{n}=\left( \frac{1}{\psi ^{\prime }(\varsigma )}\frac{d}{d\varsigma }\right) ^{n},n\in {\mathbb {N}}\).
Lemma 2.6
[9, 17] Let \(q,\ell >0\), and \(\varkappa \in C([a,b],{\mathbb {R}})\). Then \(\forall \varsigma \in [a,b]\) and by assuming \(F_{a}(\varsigma )=\psi (\varsigma )-\psi (a)\), we have
1. \({\mathcal {I}}_{a}^{q;\psi }{\mathcal {I}}_{a}^{\ell ;\psi }\varkappa (\varsigma )={\mathcal {I}}_{a}^{q+\ell ;\psi }\varkappa (\varsigma ),\)
2. \({\mathcal {D}}_{a}^{q;\psi }{\mathcal {I}}_{a}^{q;\psi }\varkappa (\varsigma )=\varkappa (\varsigma ),\)
3. \(\displaystyle {\mathcal {I}}_{a}^{q;\psi }(F_{a}(\varsigma ))^{\ell -1} =\frac{\Gamma (\ell )}{\Gamma (\ell +q)}(F_{a}(\varsigma ))^{\ell +q-1},\)
4. \(\displaystyle {\mathcal {D}}_{a}^{q;\psi }(F_{a}(\varsigma ))^{\ell -1} =\frac{\Gamma (\ell )}{\Gamma (\ell -q)}(F_{a}(\varsigma ))^{\ell -q-1},\)
5. \({\mathcal {D}}_{a}^{q;\psi }(F_{a}(\varsigma ))^{k}=0,\ k\in \{0,\ldots ,n-1\},\ n\in {\mathbb {N}},\ q\in (n-1,n]\).
Lemma 2.7
[9, 14, 17] Let \(n-1<\alpha _{1}\le n,\alpha _{2}>0,\ a>0,\ \varkappa \in \mathcal { {\mathcal {L}} }(a,{\mathcal {T}}),\ {\mathcal {D}}_{a_{1}^{+}}^{\alpha _{1};\psi }\varkappa \in \mathcal { {\mathcal {L}} }(a,{\mathcal {T}})\). Then the differential equation
has the unique solution
and
with \(w_{\ell }\in {\mathbb {R}},\ \ell =0,1,\ldots ,n-1\).
Furthermore,
and
To process the qualitative properties of solving equation (1.1), we propose some conditions on the functions of the coefficient, which will prevent us from solving them.
\(\left( \Lambda 1\right) \) For any \(x,y,z,w\in {\mathbb {R}} ^{n}\) and \(\varsigma \in \left[ a,{\mathcal {T}}\right] \), there exist three positive constants \(C_{1},\) \(C_{2}\) and \(C_{3}\) such that
where \(\left| .\right| \) is the norm of \( {\mathbb {R}} ^{n}\), \(x_{1}\vee x_{2}=\max \left\{ x_{1,}x_{2}\right\} .\)
By adopting the interest research of Zone [30], Zhang and Agarwal [31], note that by condition \(\left( \Lambda 1\right) \), FSDEs (1.1) have a unique solution
where \(X\left( \varsigma \right) \) is \({\mathbb {F}}\left( \varsigma \right) \)-adapted and E\(\left( \int _{a}^{{\mathcal {T}}}\left| X\left( \varsigma \right) \right| ^{2}d\varsigma \right) <\infty .\)
3 An Averaging Principle
In this part, the combination of existence and uniqueness in the next part gives us a demonstration and proof in the principle of the average of \(\psi \)-Caputo FSDPEs. Let’s consider the usual form of Eq. (1.1):
where the initial value \(X_{0}\), coefficients \(\vartheta _{1}\) and \(\vartheta _{2}\) similar conditions as in Eq. (1.1), and noted by \(\epsilon _{0}\) a fixed number, \(\epsilon \in \left[ 0,\epsilon _{0}\right] \) is a positive small parameter.
Before we continue with the average principle, we offer some measurable coefficients, \(\overline{\vartheta _{1}}: {\mathbb {R}} ^{n}\rightarrow {\mathbb {R}} ^{n},\overline{\vartheta _{2}}: {\mathbb {R}} ^{n}\rightarrow {\mathbb {R}} ^{n},\) satisfing \((\Lambda 1)\) and additive inequalities:
\((\Lambda 2)\) For any \({\mathcal {T}}_{1}\in \left[ a,{\mathcal {T}}\right] ,\) \(x,y\in {\mathbb {R}} ^{n}\), there existe two positive bounded functions \(\alpha _{i}({\mathcal {T}} _{1}),i=1,2\) such that
where \({\lim }_{{\mathcal {T}}_{1}\rightarrow \infty }\alpha _{i} ({\mathcal {T}}_{1})=0.\)
With sufficient help above, we explain that the original solution \(X_{\epsilon }(\varsigma )\) converges to \(Z_{\epsilon }(\varsigma )\), as \(\epsilon \rightarrow 0.\)
Now we touch on the main finding in this research.
Theorem 3.1
Assume that condition \((\Lambda 1)-(\Lambda 2)\) are satisfied. For a given arbitrarily small number \(\delta _{1}>0\) there exists \(L>a,\) \(\epsilon _{1}\in \left( 0,\epsilon _{0}\right] \) and \(\beta \in \left( 0,1\right) \) such that for all \(\epsilon \in \left( 0,\epsilon _{1}\right] ,\)
Proof
For any \(\varsigma \in \left[ a,u\right] \subset \left[ a,{\mathcal {T}}\right] ,\)
Using the elementary inequality
we have
Recalling inequality (3.5), we get
Assuggest \((\Lambda 1)\) and inequality of Cauchy–Schwarz applied, we get
where \(K_{11}=\dfrac{8\left( C_{2}^{2}+C_{3}^{2}\right) }{\Gamma \left( \alpha \right) ^{2}}.\) By the definition of upper limit integration,
integration by part is used,
then together with the hypothesis \(\left( \Lambda 2\right) \) and Cauchy-Schwarz inequality, we get
in which
For the second term, similar way,
Exploiting inequality of Doob’s martingale, Itô’s formula and condition \(\left( \Lambda 1\right) \),
where \(K_{21}=\dfrac{8\left( C_{2}^{2}+C_{3}^{2}\right) }{\Gamma \left( \alpha \right) ^{2}}.\) Reuse again,
Integrating by parts, produces
thanks to the hypothesis \(\left( \Lambda 2\right) ,\) we can conclude
where
Now, plug Eqs. (3.8)–(3.17) in to (3.6), for each \(u\in \left[ a,{\mathcal {T}}\right] ,\) we reach
with Gronwall–Bellman inequality [32], we get
This implies that we can select \(\beta \in \left( 0,1\right) \) and \(L>a,\) for every \(\varsigma \in \left[ a,L^{\epsilon ^{-\beta }}\right] \subseteq \left[ a,{\mathcal {T}}\right] \) having
where
is a constant. Hence, for any given number \(\delta _{1,}\)there exists \(\epsilon _{1}\in \left( 0,\epsilon _{0}\right] \) such that for each \(\epsilon \in \left( 0,\epsilon _{1}\right] \) and \(\varsigma \in \left[ a,L^{\epsilon ^{-\beta }}\right] \) having
\(\square \)
4 Example
Let \(\psi \left( \varsigma \right) =\log \varsigma ,\) consider the FSDPEs
where \(a=1,{\mathcal {T}}=e,\lambda \in \left( 0,\frac{e-1}{e}\right) ,\alpha \in \left( \frac{1}{2},1\right) .\) The coefficients
and
fulfillment of conditions \((\Lambda 1),\) so there has a unique solution to FSDPEs (4.1).
Define
logically we see that \((\Lambda 2)\) achieve, so we get the expected form of (4.1) is
From Theorem 3.1, when \(\epsilon \rightarrow 0\), the solution \(X_{\epsilon }\left( \varsigma \right) \) and \(Z_{\epsilon }(\varsigma )\) to Eqs. (4.1) and (4.2) are similar in the sense of mean square.
5 Conclusion
Numerous studies have investigated the application of the averaging principle to fractional stochastic differential equations, with a focus on approximating solutions through mean square averaging. In our groundbreaking work (1.1), we specifically addressed a unique category of \(\psi \)-Caputo fractional stochastic differential equations (FSDPEs) driven by a Brownian motion. By adhering to the aforementioned two conditions, we successfully achieved the desired outcome. Additionally, we extended the classical Khasminskii approach to encompass \(\psi \)-Caputo FSDPEs. As we outline in our forthcoming research:
-
1.
Exploration of variable-order fractional differential equations and their modeling.
-
2.
Investigating the averaging principle with impulsive processes.
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Acknowledgements
The authors thank Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R14), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The authors Bahaaeldin Abdalla and Thabet Abdeljawad would like to thank Prince Sultan University for the support through the TAS research lab.
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MAA: Writing the original draft, editing, formal analysis, and investigations. HB: Writing the original draft, editing, formal analysis, validation, and investigations. BA: Editing, formal analysis, and investigations. TA: Editing, supervision, formal analysis, and investigations. All authors confrimed the last version
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Alqudah, M.A., Boulares, H., Abdalla, B. et al. Khasminskii Approach for \(\psi \)-Caputo Fractional Stochastic Pantograph Problem. Qual. Theory Dyn. Syst. 23, 100 (2024). https://doi.org/10.1007/s12346-023-00951-4
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DOI: https://doi.org/10.1007/s12346-023-00951-4