A NONUNIFORM MARKUS–YAMABE CONJECTURE: TRIANGULAR CASE VIA UNIFORMIZATION

The paper proves global nonuniform asymptotic stability for triangular nonautonomous systems under hypotheses (a)–(b) by (i) establishing a nonuniform exponential dichotomy for upper block triangular linear systems via a Uniformization Lemma and a boundedness condition on upper-diagonal blocks, and (ii) using diagonal significance of the nonuniform spectrum to verify (G3) and then a recursive scalar argument to conclude stability (Theorem 4) . The candidate solution gives a direct contraction estimate for the difference of any two solutions of the triangular system, obtaining |x(t)−y(t)| ≤ K e^{E s} e^{−γ(t−s)} |x(s)−y(s)| and deducing global nonuniform asymptotic stability, Lyapunov stability (with δ(t0,η)=η/(K e^{E t0})), and uniqueness of the equilibrium. Its key steps mirror the paper’s structural assumptions: (a) yields nonuniform decay on each diagonal transition, and (b) provides uniform bounds on the super-diagonal columns in the time-dependent norm family that satisfies L1|v| ≤ |v|_t ≤ L2 e^{θ t}|v| (cf. (4.3)) . While the paper proceeds through spectral/Uniformization machinery and a triangular-to-scalar recursion, the model’s proof is a self-contained variation-of-constants and convolution estimate on the induced upper-triangular linear system for differences. Under the same hypotheses (including the existence of the norm family used in (b)), both arguments are logically sound and reach the same conclusion.