# On strongly monotone flows

Annales Polonici Mathematici (1997)

- Volume: 66, Issue: 1, page 269-274
- ISSN: 0066-2216

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topWolfgang Walter. "On strongly monotone flows." Annales Polonici Mathematici 66.1 (1997): 269-274. <http://eudml.org/doc/269959>.

@article{WolfgangWalter1997,

abstract = {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.},

author = {Wolfgang Walter},

journal = {Annales Polonici Mathematici},

keywords = {system of ordinary differential equations; initial value problem; comparison theorem; monotone flow; quasimonotonicity; strongly monotone flows; Carathéodory hypotheses; quasimonotone increasing},

language = {eng},

number = {1},

pages = {269-274},

title = {On strongly monotone flows},

url = {http://eudml.org/doc/269959},

volume = {66},

year = {1997},

}

TY - JOUR

AU - Wolfgang Walter

TI - On strongly monotone flows

JO - Annales Polonici Mathematici

PY - 1997

VL - 66

IS - 1

SP - 269

EP - 274

AB - 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.

LA - eng

KW - system of ordinary differential equations; initial value problem; comparison theorem; monotone flow; quasimonotonicity; strongly monotone flows; Carathéodory hypotheses; quasimonotone increasing

UR - http://eudml.org/doc/269959

ER -

## References

top- [1] M. Hirsch, Systems of differential equations that are competitive or cooperative, II: convergence almost everywhere, SIAM J. Math. Anal. 16 (1985), 423-439. Zbl0658.34023
- [2] P. Volkmann, Gewöhnliche Differentialgleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorräumen, Math. Z. 127 (1972), 157-164. Zbl0226.34058
- [3] W. Walter, Gewöhnliche Differentialgleichungen, 5th ed., Springer, 1993.
- [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. Zbl0041.20705

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