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ArticleOriginal scientific text
Title
On strongly monotone flows
Authors 1
Affiliations
- Mathematisches Institut I, Universität Karlsruhe, D-76128 Karlsruhe, Germany
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.
Keywords
system of ordinary differential equations, initial value problem, comparison theorem, monotone flow, quasimonotonicity
Bibliography
- M. Hirsch, Systems of differential equations that are competitive or cooperative, II: convergence almost everywhere, SIAM J. Math. Anal. 16 (1985), 423-439.
- P. Volkmann, Gewöhnliche Differentialgleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorräumen, Math. Z. 127 (1972), 157-164.
- W. Walter, Gewöhnliche Differentialgleichungen, 5th ed., Springer, 1993.
- 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.
Pages:
269-274
Main language of publication
English
Received
1995-08-21
Published
1997
Exact and natural sciences