On asymptotic solutions of systems of differential equations with deviating argument in certain critical cases

Sufficient conditions are found for the existence of particular solutions of the system
$$ \dot x(t) =f\bigl(x(t),x(t+t_1),\dots,x(t+t_k)\bigr),\qquad x\in \mathbb R^n, $$
that converge to the critical point $x=0$ as $t\to+\infty$ or as $t\to-\infty$, but not exponentially.