Systems of Differential Equations Subject to Mild Integral Conditions
William F. Trench · Proceedings of the American Mathematical Society · 1983
It is shown that solutions of a system $x’ = f(t,x)$ approach constant vectors as $t \to \infty$, under assumptions which do not require that $|| {f(t,x)} || \leqslant w(t,|| x ||)$, where $w$ is nondecreasing in $|| x ||$, and which permit some or all of the integral smallness conditions on $f$ to be stated in terms of ordinary—rather than absolute—convergence. Estimates of the order of convergence are given.