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

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.