Well-posedness and asymptotic behavior of difference equations with a time-dependent delay and applications

The paper’s Theorem 5.14 proves the desired exponential estimate by (i) using the representation formula x(t)=A^{n(t)}x0(σ_{n(t)}(t)) from Theorem 4.11, and (ii) working in a norm with |A|<1 (guaranteed by ρ(A)<1) so that |x(t)|≤|A|^{n(t)}sup|x0|; then (H11) yields n(t)≥αt+β and the exponential bound follows . The candidate solution also unfolds to x(t)=A^{n(t)}x0(σ_{n(t)}(t)) (Step 1, matching Theorem 4.11) and uses (H11) the same way, but obtains the decay of A^k via a standard polynomial-times-exponential estimate for powers of a matrix with ρ(A)<1 (via Schur/Jordan), then absorbs the polynomial term into a slightly larger base q<1. This is a valid alternative to the paper’s “renorming” step. No missing hypotheses were used beyond those in the paper, and both arguments are sound.