Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

The paper proves existence of a T-periodic weak solution in the invariant subspace Mo under asymptotic nonresonance conditions by: (i) constructing the Mo/Me decomposition when p is even and establishing good spectral separation and Fredholm properties for Lo; (ii) using a Leray–Schauder continuation theorem relative to the Fredholm operator Lo; and (iii) deriving a uniform a priori bound along the homotopy via a carefully built truncation g(t,x,y) and Lemma 3.1, which ensures a spectral gap estimate for Lo−γI when γ stays in the window [α−ε, β+ε] (Theorem 4.1 and its proof sketch are explicit in the PDF). This argument is coherent and complete in the manuscript’s framework . By contrast, the model’s solution uses a different (potentially viable) degree setup based on K(y)=(Lo−aI)^{-1}(f̂−ay) and a blow-up/limit argument, but it contains critical flaws: (1) it omits the paper’s necessary hypothesis p is even, which underpins the Mo/Me reduction and spectral isolation for odd temporal modes ; (2) it asserts compact resolvent of Lo and uses Leray–Schauder degree for a compact perturbation of identity without justifying Lo’s resolvent compactness or the continuity/compactness needed for K on L2(Ω) under only (H2), whereas the paper carefully works with Lo-compactness and a Fredholm degree framework; and (3) it defines a homotopy Kτ(y)=(Lo−aI)^{-1}(τ f̂−ay) but then incorrectly claims “K0≡0,” which is false for that homotopy. These issues render the model’s written proof incorrect in its current form, even though parts of its strategy resemble a standard alternative route via choosing a in the spectral gap and projecting onto Eλ⊕Eλ̄⊕W.