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Volume 15
Issue 9
10.3390/axioms15090702
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Article

Mild Solutions and Exponential Stability of ψ-Caputo Neutral Stochastic Integro-Differential Delay Systems of Order σ∈(1,2) with Impulses and Poisson Jumps

by
Pradeepa Rajendran
1,
Mohammed Rabih
2,*,
Sivam Abhirami
3 and
Marappan Sathish Kumar
1,*
1
Department of Mathematics, Paavai Engineering College, Namakkal 637018, Tamil Nadu, India
2
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
3
Department of Mathematics, Sona College of Technology, Salem 636005, Tamil Nadu, India
*
Authors to whom correspondence should be addressed.
Axioms 2026, 15(9), 702; https://doi.org/10.3390/axioms15090702 (registering DOI)
Submission received: 4 August 2026 / Revised: 6 September 2026 / Accepted: 15 September 2026 / Published: 20 September 2026
(This article belongs to the Special Issue Existence of Solutions for Differential Equations: Theories, Methods and Applications)
Versions Notes

Abstract

We establish sufficient conditions for the existence and exponential stability of mild solutions of a class of fractional neutral stochastic integro-differential delay systems governed by the ψ-Caputo fractional derivative of order σ∈(1,2), which unifies the Caputo, Caputo–Hadamard and Caputo–Katugampola derivatives. The system carries dual integral memory kernels, α1 inside the neutral term and α2 inside the forcing term, and is subject to Poisson jump perturbations and to instantaneous impulses acting simultaneously on the state and on its ψ-derivative, in an infinite-dimensional Hilbert space. Existence is obtained by combining the Hausdorff measure of noncompactness with Mönch’s fixed-point theorem, which avoids compactness assumptions on the associated cosine family. Exponential stability is then derived from a ψ-weighted derivative impulsive integral inequality, yielding decay that is exponential in ψ(ν)−ψ(0) and hence exponential, algebraic or logarithmic in ν according to the growth of ψ. An illustrative example is presented to demonstrate the applicability of the theoretical results.
Keywords: ψ-Caputo; fractional derivative; fractional neutral stochastic delay systems; mild solutions; exponential stability; Mönch fixed point theorem; Poisson jumps; derivative impulses ψ-Caputo; fractional derivative; fractional neutral stochastic delay systems; mild solutions; exponential stability; Mönch fixed point theorem; Poisson jumps; derivative impulses

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MDPI and ACS Style

Rajendran, P.; Rabih, M.; Abhirami, S.; Kumar, M.S. Mild Solutions and Exponential Stability of ψ-Caputo Neutral Stochastic Integro-Differential Delay Systems of Order σ∈(1,2) with Impulses and Poisson Jumps. Axioms 2026, 15, 702. https://doi.org/10.3390/axioms15090702

AMA Style

Rajendran P, Rabih M, Abhirami S, Kumar MS. Mild Solutions and Exponential Stability of ψ-Caputo Neutral Stochastic Integro-Differential Delay Systems of Order σ∈(1,2) with Impulses and Poisson Jumps. Axioms. 2026; 15(9):702. https://doi.org/10.3390/axioms15090702

Chicago/Turabian Style

Rajendran, Pradeepa, Mohammed Rabih, Sivam Abhirami, and Marappan Sathish Kumar. 2026. "Mild Solutions and Exponential Stability of ψ-Caputo Neutral Stochastic Integro-Differential Delay Systems of Order σ∈(1,2) with Impulses and Poisson Jumps" Axioms 15, no. 9: 702. https://doi.org/10.3390/axioms15090702

APA Style

Rajendran, P., Rabih, M., Abhirami, S., & Kumar, M. S. (2026). Mild Solutions and Exponential Stability of ψ-Caputo Neutral Stochastic Integro-Differential Delay Systems of Order σ∈(1,2) with Impulses and Poisson Jumps. Axioms, 15(9), 702. https://doi.org/10.3390/axioms15090702

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