Continuous-Time and Discrete-Time Quasilinear Systems with Asymptotically Unpredictable Solutions

The paper’s Theorem 3.1 establishes, under (A1)–(A3), the existence, uniqueness, and global exponential stability of a unique bounded solution Φ to x'(t)=Ax(t)+f(x(t−τ))+φ(t), and proves that Φ is asymptotically unpredictable when φ is asymptotically unpredictable. It does so by splitting φ=ψ+θ (with ψ unpredictable and θ→0), invoking standard DDE theory for a unique globally exponentially stable solution, and then applying a contraction argument on a tailored operator T to show ‖Φ−Ψ‖→0, where Ψ is the unpredictable solution corresponding to ψ . The candidate solution uses variation-of-constants and a fixed-point on BC(R) for existence/uniqueness with contraction ratio q=NLf/λ, a weighted sup estimate yielding stability with α=(2NLfe^{λτ/2})/λ<1, and then shows asymptotic unpredictability by combining an external unpredictability result with a direct decay estimate for the difference. The small-gain constants match the paper’s A3 (λ−2NLfe^{λτ/2}>0), and the logic is sound. While the paper’s proof of asymptotic closeness uses a bespoke contraction mapping T on a bounded set S, the model’s proof uses a Halanay-type weighted norm; these are methodologically different but substantively consistent. Hence both are correct, with different proof presentations.