MSC (1991): Primary 34D20; Secondary 34D05.

In this paper we study the instability of the semilinear ordinary differential equation x′(t) = Ax(t) + f(t, x), where f(t, 0) = 0 and |f(t, x) | ≤ γ(t)|x|α, 0 ≤ α ≤ 1. In the case 0 ≤ α < 1, we show that the existence of an eigenvalue λ of the constant matrix A satisfying Re λ> 0 implies the instability of the null solution, for a function γ(t) satisfying lim sup t→∞ eβtγ(t)> 0, β < 0. Key words and phrases: Liapounov instability, h-instability, di-chotomies