Convergence of asymptotic systems in Cohen–Grossberg neural network models with unbounded delays

The paper’s Theorem 3.3 proves lim_{t→∞}|x(t)−x̂(t)|=0 for solutions of (1.2) and its asymptotic system (2.4) under H1–H7 via a fluctuation-lemma argument with the Lyapunov-like quantity y_i(t)=sign(x_i−x̂_i)d_i^{-1}∫_{x̂_i}^{x_i}1/a_i(t,v)dv, plus boundedness/global existence (Lemmas 3.1–3.2) and a negativity condition equivalent to H7 (eqs. (2.5)–(2.7), (3.5), (3.16)) . The candidate solution proves the same limit using a weighted Halanay-type inequality on a sup-norm V(t), obtaining D^+V ≤ r(t)V+ρ(t) with limsup r<0 and ρ→0, then V→0. This is a sound alternative route given H1–H7 and the asymptotic closeness (2.6), which yields the vanishing perturbation terms ρ(t)→0 once boundedness and uniform continuity are known . Two minor issues in the candidate writeup: (i) well-posedness for infinite-delay systems is assumed rather than explicitly handled (the paper uses the proper phase-space framework and continuation lemma); (ii) the last step should explicitly observe that the −ā_iA_i(t) term included in r(t) cancels the growth from the weight w_i(t), ensuring |e_i(t)|→0, rather than appealing to a “finite prefactor.” These are fixable. Overall, both arguments establish the same claim under the stated hypotheses.