Integral Contraction Approach to Coupled Integro-Fractional Stochastic Differential Systems
- 1Department of Mathematics, Faculty of Sciences, Sakarya University, Sakarya 54050, Türkiye
- 2Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, Pakistan
- 3Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia
Abstract
We consider a coupled system of Caputo–Katugampola integro–fractional stochastic differential equations with Volterra memory, nonlocal integral boundary conditions, Wiener perturbations, and compensated Poisson jumps. The main purpose is to give a mathematically explicit fixed-point treatment in which the two components are solved simultaneously on a product space. A Green representation is derived directly from the two boundary conditions; this derivation fixes the boundary denominator and the complete Green kernel used in all subsequent estimates. Integral contractors are formulated with a correction operator and a residual modulus. In particular, a cubic nonlinearity is used to show that a coefficient need not be globally Lipschitz on the whole state space in order to satisfy the contractor hypothesis on an invariant ball. The moment order p≥2 is fixed throughout each theorem, so all contractor and contraction constants are explicitly allowed to depend on p. The coupled fixed-point problem is represented by a nonnegative 2×2 matrix Qp and the condition ρ(Qp)<1 is used as the actual contraction condition. Ulam–Hyers stability is obtained from the same matrix by an explicit Neumann-series estimate. For trajectory control we distinguish ordinary predictable finite-energy controls from generalized semimartingale controls; this distinction is necessary because a two-point stochastic boundary-value problem is generally noncausal and an ordinary drift control cannot cancel independent martingale terms pathwise. A numerical experiment based on the same Green representation verifies the contraction criterion, the non-Lipschitz contractor construction, the Ulam–Hyers estimate, and the generalized trajectory-control mechanism. The presentation also records the precise scope of the adaptedness assumption for the stochastic integrals.
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BibTeX
@article{liaqat2026integral,
title = {{Integral Contraction Approach to Coupled Integro-Fractional Stochastic Differential Systems}},
author = {Muhammad Imran Liaqat and Abdulaziz Khalid Alsharidi},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100737},
url = {https://www.mdpi.com/2075-1680/15/10/737},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Integral Contraction Approach to Coupled Integro-Fractional Stochastic Differential Systems
AU - Muhammad Imran Liaqat
AU - Abdulaziz Khalid Alsharidi
PY - 2026
DA - 2026/10/01
JO - Axioms
DO - 10.3390/axioms15100737
UR - https://www.mdpi.com/2075-1680/15/10/737
AB - We consider a coupled system of Caputo–Katugampola integro–fractional stochastic differential equations with Volterra memory, nonlocal integral boundary conditions, Wiener perturbations, and compensated Poisson jumps. The main purpose is to give a mathematically explicit fixed-point treatment in which the two components are solved simultaneously on a product space. A Green representation is derived directly from the two boundary conditions; this derivation fixes the boundary denominator and the complete Green kernel used in all subsequent estimates. Integral contractors are formulated with a correction operator and a residual modulus. In particular, a cubic nonlinearity is used to show that a coefficient need not be globally Lipschitz on the whole state space in order to satisfy the contractor hypothesis on an invariant ball. The moment order p≥2 is fixed throughout each theorem, so all contractor and contraction constants are explicitly allowed to depend on p. The coupled fixed-point problem is represented by a nonnegative 2×2 matrix Qp and the condition ρ(Qp)<1 is used as the actual contraction condition. Ulam–Hyers stability is obtained from the same matrix by an explicit Neumann-series estimate. For trajectory control we distinguish ordinary predictable finite-energy controls from generalized semimartingale controls; this distinction is necessary because a two-point stochastic boundary-value problem is generally noncausal and an ordinary drift control cannot cancel independent martingale terms pathwise. A numerical experiment based on the same Green representation verifies the contraction criterion, the non-Lipschitz contractor construction, the Ulam–Hyers estimate, and the generalized trajectory-control mechanism. The presentation also records the precise scope of the adaptedness assumption for the stochastic integrals.
SN - 2075-1680
ER -