On the Asymptotic Behavior of Solutions of Differential Systems

Abstract

There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let $ϕ (t)$ be a scalar continuous monotonically increasing positive function tending to ∞ as $t → ∞$. It is established that if all solutions of a differential system satisfy the inequality $$\left\| {x(t;t_0 ,\;x_0 )} \right\| \leqslant M\frac{{\varphi (t_0 )}}{{\varphi (t)}}\quad \operatorname{for} \;all\quad t \geqslant t_0 ,\quad x_0 \in \left\{ {x:\left\| x \right\| \leqslant \alpha } \right\},$$ then the solution $x(t; t_0, x_0)$ of this differential system tends to 0 faster than $M\frac{{\varphi (t_0 )}}{{\varphi (t)}}$