Ordinary differential inequalities and quasimonotonicity in ordered topological vector spaces

Uhl, Roland

doi:10.1090/S0002-9939-98-04311-1

Ordinary differential inequalities and quasimonotonicity in ordered topological vector spaces
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Proc. Amer. Math. Soc. 126 (1998), 1999-2003 Request permission

Abstract:

A well known comparison theorem on ordinary differential inequalities with quasimonotone right-hand side $f(t,x)$ was carried over by Volkmann (1972) to (pre)ordered topological vector spaces. We prove that the quasimonotonicity of $f$ is a necessary condition here if $f$ is continuous. Then it is shown that quasimonotonicity can be verified by considering only a few positive continuous linear functionals in the definition (for instance in $\ell _{\infty }$ by taking coordinate functionals).

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