Random periodic solutions of nonautonomous stochastic feedback systems with multiplicative noise

The paper proves existence, uniqueness, periodicity, and pullback attraction of a positive random periodic solution for the nonautonomous stochastic feedback system by (i) establishing contractivity of the gain operator Kh on a complete metric space (Lemma 4.2) under (R) and (H1), with contraction factor L d^2 sup_{t≥0} E R(t)/λ < 1, and (ii) combining a monotone/anti-monotone squeezing argument with the Banach fixed-point theorem to obtain a unique fixed point u ∈ M and Y = K(u) that satisfies φ(t,s,ω)Y(s,ω)=Y(t,ω) and Y(t+T,ω)=Y(t,θ_T ω), plus a.s. pullback convergence along s = −nT (Theorem 4.3) . The candidate solution follows a streamlined but different route: it relies directly on Banach’s fixed point theorem for Kh, identifies Y = K(u) as a solution via the variation-of-constants formula, proves periodicity using the shift covariance of Φ, and derives pullback attraction via a Volterra-type inequality and Borel–Cantelli. These steps are logically sound and consistent with the paper’s assumptions and preliminaries, including the linear-flow estimate (L) and the variation-of-constants formula (2.5) for the flow ϕ (and Φ’s cocycle/positivity under (A)) . Minor issues: the model incorrectly states the multiplicative noise is “scalar (identity direction)”—the paper allows diagonal/noise-structured matrices—but this does not affect the argument’s validity; and the model omits the paper’s (anti-)monotonicity hypotheses, which are used in the paper’s squeezing lemmas but are not needed for the model’s direct contraction-based proof. Overall, both are correct; the proofs differ in technique.