Exponential Dichotomy for Noninvertible Linear Difference Equations

The uploaded paper’s Theorem 5.2 proves robustness of exponential dichotomy under multiplicative perturbations for noninvertible linear difference equations, with the same smallness condition ρδ = K(1−e^{−α})^{-1}(1+e^{−α})δ < 1, the same perturbed exponent β = −log(cosh α − √(sinh^2 α − 2Kδ sinh α)), the same bound constant L, and the same continuity estimate |Q(k)−P(k)| ≤ KL(1−e^{−(α+β)})^{-1}(1+e^{−(α+β)})δ as stated in the candidate solution. See Theorem 5.2 and the proof via a variation-of-constants fixed-point mapping T on bounded sequences, which establishes contraction when ρδ < 1 and then derives the explicit constants and β, as well as the extension from Z to half-axes via Lemma 5.3 (Theorem statement and proof details: ; fixed-point contraction bounds and construction: ; final continuity estimate for projections: ; extension to [a,∞) or (−∞,b] via embedding: ; equivalence of dichotomy definitions used in the argument: ). The model’s solution reaches the same quantitative conclusions but via a different route: a graph-transform/Riccati equation for the invariant bundle, Lyapunov–Perron/Green operator estimates in adapted norms (K=1), and a weighted contraction yielding the identical β. Minor omissions in the model’s write-up (e.g., justifying the resolvent for the operator DX and the adapted norms in the noninvertible setting) are standard and do not affect correctness. Hence both are correct, with different proofs.