Asymptotic Behavior of Solutions of the Cauchy Problem x′ = f(t, x, x′), x(0) = 0 - Zernov, A. E.; Kuzina, Yu. V. (Ukrains’kyi Matematychnyi Zhurnal, 2002)
We prove the existence of continuously differentiable solutions \(x:(0,{ρ ]} → \mathbb{R}^n\) such that $$\left\| {x\left( t \right) - {ξ }\left( t \right)} \right\| = O\left( {{η }\left( t \right)} \right),{ }\left\| {x'\left( t \right) - {ξ '}\left( t \right)} \right\| = O\left( {{η }\left( t \right)/t} \right),{ }t → + 0$$ or $$\left\| {x\left( t \right) - S_N \left( t \right)} \right\| = O\left( {t^{N + 1} } \right),{ }\left\| {x'\left( t \right) - S'_N \left( t \right)} \right\| = O\left( {t^N } \right),{ }t → + 0,$$ where $${ξ }:\left( {0,{τ }} \right) → \mathbb{R}^n ,{ η }:\left( {0,{τ }} \right) → \left( {0, + ∞ } \right),{ }\left\| {{ξ }\left( t \right)} \right\| = o\left( 1 \right),$$ $${η }\left( t \right) = o\left( t \right),{ η }\left( t \right) = o\left( {\left\| {{ξ }\left( t \right)} \right\|} \right),{ }t → + 0,{ }S_N \left( t \right) = \sum\limits_{k = 2}^N {c_k t^k ,}$$ $$c_k \in \mathbb{R}^n ,k \in \left\{ {2,...,N} \right\},{ }0 < {ρ } < {τ },{ ρ is sufficiently small}{.}$$