Random periodic solutions of non-autonomous stochastic differential equations

The paper proves existence of a random periodic solution for the non-autonomous SDE via an input-to-state operator K and a gain operator Kh, showing Kh is a contraction on a carefully constructed measurable function space and that K(u) at the fixed point is random periodic; it also establishes pullback convergence and flow invariance. The candidate solution follows the same skeleton: defines the same space, proves well-posedness and positivity/monotonicity of K, shows Kh is a contraction under the same small-gain bound L d^2/(-λ) < 1, obtains the unique fixed point, and derives mild-solution, flow-invariance, pullback convergence, and random periodicity. The main differences are stylistic (the paper uses liminf/limsup monotone bounds to deduce the pullback limit, while the candidate uses a Volterra–Grönwall estimate). No substantive logical conflict was found.