Localised time-periodic solutions of discrete nonlinear Klein-Gordon systems with convex on-site potentials

Both the paper and the model use essentially the same Schauder fixed-point scheme on X0 = L^2-per([0,T]; l^2(Z)) with zero time mean, factor the equation as q = M^{-1}∘N(q), and rely on the non-resonance Ω^2 > 1+4κ. However, the compactness step is not justified: the paper asserts X2 ⋐ X0 and that S maps into a compact subset Y0 ⊂ X0, but on an infinite lattice neither claim holds as stated due to spatial translation invariance; the model’s alternative compactness claim via “uniform tail control” is also unproven. In addition, neither argument excludes the trivial fixed point q ≡ 0—nontriviality is argued physically rather than by a rigorous degree/index or exclusion argument. Hence both proofs are incomplete.