Higher order analysis on the existence of periodic solutions in continuous differential equations via coincidence degree

The paper states and proves a higher-order averaging existence theorem for continuous (non-Lipschitz) systems using coincidence degree. Hypothesis (H) and the boundary-gap condition lead to Theorem A (via the full averaged map f) and then to Theorem B (via the truncated average F_k), with the key reduction d((L, N̂_ε), Ω) = d_B(J Q N_ε|_{Ω∩Ker L}, Ω∩Ker L, 0) justified by Proposition 5 and Lemma 6 together with the homotopy Ñ_ε(x, λ) = QN_ε(x) + λε(I − Q)N_ε(x) and Condition (C) implied by (H) and f ≠ 0 on ∂V (see Hypothesis (H) and Theorems A–B; proofs via Proposition 5, Lemma 6, and Condition (C) checks) . The model’s approach mirrors this plan but makes a critical error: it asserts that the coincidence degree D(L − ε λ N, W) is well-defined and constant in λ for all λ ∈ [0,1], and then evaluates it at λ = 0 by treating Lx = 0 directly. At λ = 0 the equation Lx = 0 has a continuum of boundary solutions (constant functions on ∂V), so the coincidence degree is not defined there; one must instead use the paper’s equivalent formulation (via N̂_ε and the JQN_ε reduction) to compute the Brouwer degree at λ = 0. This technical step is essential and is handled correctly in the paper but is missing/misstated in the model’s proof .