M. Hirsch's famous theorem on strongly monotone flows generated by autonomous systems u'(t) = f(u(t)) is generalized to the case where f depends also on t, satisfies Carathéodory hypotheses and is only locally Lipschitz continuous in u. The main result is a corresponding Comparison Theorem, where f(t,u) is quasimonotone increasing in u; it describes precisely for which components equality or strict inequality holds.
@article{bwmeta1.element.bwnjournal-article-apmv66z1p269bwm,
author = {Wolfgang Walter},
title = {On strongly monotone flows},
journal = {Annales Polonici Mathematici},
volume = {66},
year = {1997},
pages = {269-274},
zbl = {0870.34014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv66z1p269bwm}
}
Wolfgang Walter. On strongly monotone flows. Annales Polonici Mathematici, Tome 66 (1997) pp. 269-274. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv66z1p269bwm/
[000] [1] M. Hirsch, Systems of differential equations that are competitive or cooperative, II: convergence almost everywhere, SIAM J. Math. Anal. 16 (1985), 423-439. | Zbl 0658.34023
[001] [2] P. Volkmann, Gewöhnliche Differentialgleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorräumen, Math. Z. 127 (1972), 157-164. | Zbl 0226.34058
[002] [3] W. Walter, Gewöhnliche Differentialgleichungen, 5th ed., Springer, 1993.
[003] [4] T. Ważewski, Systèmes des équations et des inégalités différentielles ordinaires aux deuxièmes membres monotones et leurs applications, Ann. Soc. Polon. Math. 23 (1950), 112-166. | Zbl 0041.20705