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
HTML articles powered by AMS MathViewer

by Roland Uhl

Proc. Amer. Math. Soc. 126 (1998), 1999-2003

DOI: https://doi.org/10.1090/S0002-9939-98-04311-1

PDF | 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).

References

E. Kamke, Zur Theorie der Systeme gewöhnlicher Differentialgleichungen II, Acta Math. 58 (1932), 57–85.
Roland Lemmert, Existenzsätze für gewöhnliche Differentialgleichungen in geordneten Banachräumen, Funkcial. Ekvac. 32 (1989), no. 2, 243–249 (German). MR 1019432
M. Müller, Über das Fundamentaltheorem in der Theorie der gewöhnlichen Differentialgleichungen, Math. Z. 26 (1927), 619–645.
Sabina Schmidt, Fixed points for discontinuous quasi-monotone maps in sequence spaces, Proc. Amer. Math. Soc. 115 (1992), no. 2, 361–363. MR 1081098, DOI 10.1090/S0002-9939-1992-1081098-2
Alice Simon and Peter Volkmann, Remark on quasimonotonicity, Inequalities and applications, World Sci. Ser. Appl. Anal., vol. 3, World Sci. Publ., River Edge, NJ, 1994, pp. 543–548. MR 1299583
Alice Simon and Peter Volkmann, Parabolic inequalities in ordered topological vector spaces, Nonlinear Anal. 25 (1995), no. 9-10, 1051–1054. MR 1350727, DOI 10.1016/0362-546X(95)00099-H
R. Uhl, An extension of Max Müller’s theorem to differential equations in ordered Banach spaces, Funkcial. Ekvac. 39 (1996), 203–216.
Peter Volkmann, Gewöhnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorräumen, Math. Z. 127 (1972), 157–164 (German). MR 308547, DOI 10.1007/BF01112607
Peter Volkmann, Über die Invarianz konvexer Mengen und Differentialungleichungen in einem normierten Raume, Math. Ann. 203 (1973), 201–210 (German). MR 322305, DOI 10.1007/BF01629254
Peter Volkmann, Cinq cours sur les équations différentielles dans les espaces de Banach, Topological methods in differential equations and inclusions (Montreal, PQ, 1994) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 472, Kluwer Acad. Publ., Dordrecht, 1995, pp. 501–520 (French, with English and French summaries). MR 1368679, DOI 10.1007/978-94-011-0339-8_{1}1
Wolfgang Walter, Gewöhnliche Differential-Ungleichungen im Banachraum, Arch. Math. (Basel) 20 (1969), 36–47. MR 244594, DOI 10.1007/BF01898989
Wolfgang Walter, Differential and integral inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 55, Springer-Verlag, New York-Berlin, 1970. Translated from the German by Lisa Rosenblatt and Lawrence Shampine. MR 0271508, DOI 10.1007/978-3-642-86405-6