Asymptotic behavior of solutions of the cauchy problem x′ =f(t,x,x′), x(0) = 0

We prove the existence of continuously differentiable solutions x : (0, ρ]→ ℝn such that ∥x(t)-ξ(t)∥ = O(η(t)), ∥x′(t)-ξ′(t)∥ = O(η(t)/t), t→+0 or ∥x(t)-SN(t)∥ = O(tN+1), ∥x′(t)-S N′(t)∥ = O(tN), t→+0, where ξ: (0, τ) → ℝn, η:(0,τ)→(0, +∞), ∥ξ(t)∥ = o(1), η(t) = o(t), η(t) = o(∥ξ(t)∥), t→+0, SN(t) = Σk=2Nc ktk, ck ∈ ℝn, k ∈ {2, ... , N}, 0 < ρ < τ, ρ is sufficiently small. © 2002 Plenum Publishing Corporation.