Lipschitz lower sigma-exponents of linear differential systems
N. A. Izobov · Differential Equations · 2013
For the lower sigma-exponent of the linear differential system ẋ = A(t)x, x ∈ R n , t ≥ 0, defined by the formula Δσ(A) ≡ infλ[Q]≤-σ λ 1(A + Q), σ > 0, on the basis of the lower characteristic exponents λ 1(A+Q) of perturbed linear systems with Lyapunov exponents λ[Q] ≤ −σ 0. For any nondecreasing bounded function f(σ) of the parameter σ ∈ (0,+∞) that coincides with a constant on some infinite interval (σ 0,+∞), σ 0 ≥ 0, and satisfies the Lipschitz condition on the complementary interval (0, σ 0], we prove the existence of a linear system with coefficient matrix A f (t) bounded on the half-line [0,+∞) whose lower sigma-exponent Δσ(A f ) coincides with the function f(σ) on the entire interval (0,+∞).