Systems of differential equations subject to mild integral conditions

Trench, William F.

doi:10.1090/S0002-9939-1983-0681832-1

Systems of differential equations subject to mild integral conditions
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by William F. Trench PDF

Proc. Amer. Math. Soc. 87 (1983), 263-270 Request permission

Abstract:

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.

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