Asymptotic properties of neutral type linear systems

The paper’s Theorem 3 states uniform exponential stability of ẋ(t) − A(t)ẋ(g(t)) = ∑k Bk(t)x(hk(t)) under µ(B(t)) ≤ −β, ‖A‖ < 1, and the smallness condition (3.11) involving ‖A/µ(B)‖ and τk‖Bk/µ(B)‖, deriving it from Theorems 1–2 (via a λ-perturbation and matrix-measure bounds) . The preliminaries define the fundamental matrix X(t,s), give the solution representation, and set the matrix-measure framework used throughout . The candidate solution proves the same stability condition and exponential bound by a different route: a direct variation-of-constants around ẏ = B(t)y with µ(B) ≤ −β and a weighted sup-norm argument that yields a fixed-point inequality ‖X‖α ≤ 1 + Q(α)‖X‖α, with Q(0) matching (3.11) and continuity ensuring α ∈ (0,β) with Q(α) < 1, hence ‖X(t,s)‖ ≤ Me^{−α(t−s)}. This aligns with the paper’s assumptions and conclusion and uses tools justified in the preliminaries (fundamental matrix, solution representation, and matrix-measure estimates), but the proof technique is distinct. Hence both are correct, with different proofs.