Subharmonic Solutions in Reversible Non-Autonomous Differential Equations

The paper correctly sets up a self-map F:E→E via L(u)=ü−u and A=Id−L^{-1}(N_A∘j−j), proves small- and large-radius G-admissible homotopies to A and Id respectively, and computes the annulus degree G-deg(F,Ω)=(G)−G-deg(A,B(E)), leading to Theorem 2.11’s parity-orbit conclusions. See the construction of F and L (equations (6)–(8)), the linearization A (equation (12)) and the non-resonance condition (A5) together with the near-zero homotopy (Lemma 2.3), the Nagumo condition (A6) and the large-radius homotopy (Lemmas 2.4–2.5), the abstract degree identity (Theorem 2.6), the eigenvalue sign test and the counting of β_i, η_i, ρ_i, and the main existence theorem (Theorem 2.11) with orbit-type conclusions. The candidate solution, while matching the paper’s final parity conclusions and multiplicity counting, replaces the paper’s self-map framework by F(u)=L u−N(u) with L:=−d^2/dt^2+A (mapping E→F, not E→E), and claims both small- and large-ball homotopies to the same linear L. This breaks the required self-map structure for Brouwer G-degree, and if taken literally would make the annulus degree trivial (deg_G(L) cancels), contradicting the nontrivial degree used for existence. The candidate also overlooks the j:E→F compact embedding that the paper crucially uses to produce F=Id−compact on E. Thus the paper’s argument is correct, while the model’s proof contains structural errors, even though its spectral counting and orbit-type conclusions agree with the paper.