Integral Equations & Model Reduction For Fast Computation of Nonlinear Periodic Response

The paper’s Theorem 3 asserts existence, convergence, local uniqueness, and a quasi-best-approximation bound for collocation solutions of ζ = G ∘ T(ζ) with PN as in (22) and operators in (23)–(25), concluding ζN = PN ∘ G ∘ T(ζN) converges to ζ∗ and ηN = T(ζN) to η∗, with the local uniqueness and error bound under 1 ∉ σ(D(G ∘ T)(ζ∗)) . However, its proof relies on an imported result (Appendix A, Theorem 6) that requires the additional hypothesis that the limiting solution has nonzero index, an assumption not stated in Theorem 3(i) itself . This creates a gap in the stated assumptions. By contrast, the model’s solution supplies a complete argument: it establishes compactness and C1-regularity of F = G ∘ T, uses uniform PN-approximation on compact sets to make PN F map a small ball into itself and apply Brouwer for existence and convergence, and then uses collectively compact spectral stability and an implicit-function argument for (ii), matching the paper’s uniqueness and best-approximation claim under 1 ∉ σ(D(G ∘ T)(ζ∗)) .