First-Order Impulses for an Impulsive Stochastic Differential Equation System

Tayeb Blouhi, Safa M. Mirgani (), Fatima Zohra Ladrani, Amin Benaissa Cherif, Khaled Zennir and Keltoum Bouhali
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Tayeb Blouhi: Department of Mathematics, Faculty of Mathematics and Informatics, University of Science and Technology of Oran Mohamed-Boudiaf (USTOMB), El Mnaouar, BP 1505, Oran 31000, Algeria
Safa M. Mirgani: Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia
Fatima Zohra Ladrani: Department of Exact Sciences, Higher Training Teacherś School of Oran Ammour Ahmed (ENSO), Oran 31000, Algeria
Amin Benaissa Cherif: Department of Mathematics, Faculty of Mathematics and Informatics, University of Science and Technology of Oran Mohamed-Boudiaf (USTOMB), El Mnaouar, BP 1505, Oran 31000, Algeria
Khaled Zennir: Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
Keltoum Bouhali: Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia

Mathematics, 2025, vol. 13, issue 19, 1-21

Abstract: We consider first-order impulses for impulsive stochastic differential equations driven by fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) involving a nonlinear ϕ -Laplacian operator. The system incorporates both state and derivative impulses at fixed time instants. First, we establish the existence of at least one mild solution under appropriate conditions in terms of nonlinearities, impulses, and diffusion coefficients. We achieve this by applying a nonlinear alternative of the Leray–Schauder fixed-point theorem in a generalized Banach space setting. The topological structure of the solution set is established, showing that the set of all solutions is compact, closed, and convex in the function space considered. Our results extend existing impulsive differential equation frameworks to include fractional stochastic perturbations (via fBm) and general ϕ -Laplacian dynamics, which have not been addressed previously in tandem. These contributions provide a new existence framework for impulsive systems with memory and hereditary properties, modeled in stochastic environments with long-range dependence.

Keywords: stochastic differential equation; energy and industry; fractional Brownian motion; impulsive differential equations; matrix; generalized Banach space; iterative methods (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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