Sigmoidal approximations of a nonautonomous neural network with infinite delay and Heaviside function

The paper’s Theorem 5.1 proves lim_{ε→0+} Dist_{Cγ}(A_{Φ^ε}(t), A_{Φ^0}(t)) = 0 via (i) the monotonicity A_{Φ^0}(t) ⊂ A_{Φ^ε}(t), and (ii) a compactness/limit argument using Theorem 3.12 to pass from solutions of (3.6)^ε to a solution of (3.6)^0, plus a representation of attractors by complete bounded solutions, yielding a contradiction if the distance does not vanish. The candidate solution gives the same high-level structure but passes to the limit directly in the Aumann integrals using graph convergence of χ_ε to χ, and it establishes uniform dissipativity and asymptotic compactness independent of ε to obtain precompactness. These approaches are consistent; the paper relies on its internal Lemmas/Thms (Lemmas 4.2–4.6, Theorems 3.12, 4.7) and a contradiction argument (Theorem 5.1), whereas the candidate uses a direct limsup/outer limit argument and standard measurable selection results. No substantive logical conflict was found. The candidate omits some technical details (measurability/selection for Aumann integrals and the exact uniformity in ε), but these are standard under the paper’s hypotheses (D), (M), (I), (A).