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Stochastic Clairaut Equation on Algebra of Generalized Functions

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Hafedh Rguigui at Umm al-Qura University
Hafedh Rguigui
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Abstract

Based on an infinite dimensional distributions space, we study the solution of the generalized stochastic Clairaut equation using a suitable convolution calculus. The solution of such equation is shown to be positive and its integral representation with respect to the Radon measure is given. Moreover, the contractivity property is studied. Finally, the system is shown to be finite-time stochastically stable.
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Complex Analysis and Operator Theory (2024) 18:19
https://doi.org/10.1007/s11785-023-01466-1
Complex Analysis
and Operator Theory
Stochastic Clairaut Equation on Algebra of Generalized
Functions
Hafedh Rguigui 1,2
Received: 9 November 2023 / Accepted: 26 November 2023 / Published online: 3 January 2024
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024
Abstract
Based on an infinite dimensional distributions space, we study the solution of the
generalized stochastic Clairaut equation using a suitable convolution calculus. The
solution of such equation is shown to be positive and its integral representation with
respect to the Radon measure is given. Moreover, the contractivity property is studied.
Finally, the system is shown to be finite-time stochastically stable.
Keywords Finite-time stable ·Convolution product ·Generalized stochastic Clairaut
equation ·Space of entire functions with θ-exponential growth condition of minimal
type
1 Introduction
In mathematics and more specifically in mathematical analysis, Clairaut equation
(named after Alexis Clairaut (1713–1765)) is a differential equation of the following
form
y(x)=xdy
dx +fdy
dx,(1)
Communicated by Palle Jorgensen.
This article is part of the topical collection “Infinite-dimensional Analysis and Non-commutative Theory”
edited by Marek Bozejko, Palle Jorgensen and Yuri Kondratiev.
BHafedh Rguigui
hmrguigui@uqu.edu.sa
1Department of Mathematics, Al-Qunfudhah University College, Umm Al-Qura University,
Mecca, Kingdom of Saudi Arabia
2High School of Sciences and Technology of Hammam Sousse, University of Sousse, Rue Lamine
Abassi, 4011 Hammam Sousse, Tunisia
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