ABSTRACT
Theorem 9.1. Suppose 1) X(t, x) is almost periodic in t uniformly with respect to x ∈ D; 2) there exists a derivative A(t) = Xx(t, 0), and, for t ∈ R, |x1|, |x2| ≤ r ≤ a
|X(t, x1)−X(t, x2)−A(t)(x1 − x2)| ≤ ω(r)|x1 − x2|,
where ω(r)→ 0 as r → 0; 3) the matrix
A = lim T→∞
1 T
T∫
A(s)ds
does not have eigenvalues with a zero real part; 4) and
lim T→∞
∣∣∣∣∣∣ 1 T
T∫
X(σ, 0)dσ
∣∣∣∣∣∣ = 0.
