We prove that the initial value problem x’(t) = f(t,x(t)), is uniquely solvable in certain ordered Banach spaces if f is quasimonotone increasing with respect to x and f satisfies a one-sided Lipschitz condition with respect to a certain convex functional.
@article{bwmeta1.element.bwnjournal-article-cmv76z1p19bwm, author = {Gerd Herzog}, title = {On a Theorem of Mierczy\'nski}, journal = {Colloquium Mathematicae}, volume = {78}, year = {1998}, pages = {19-29}, zbl = {0896.34055}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv76z1p19bwm} }
Herzog, Gerd. On a Theorem of Mierczyński. Colloquium Mathematicae, Tome 78 (1998) pp. 19-29. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv76z1p19bwm/
[000] [1] A. Chaljub-Simon, R. Lemmert, S. Schmidt and P. Volkmann, Gewöhnliche Differentialgleichungen mit quasimonoton wachsenden rechten Seiten in geordneten Banachräumen, in: General Inequalities 6 (Oberwolfach, 1990), Internat. Ser. Numer. Math. 103, Birkhäuser, Basel, 1992, 307-320. | Zbl 0763.34048
[001] [2] K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math. 296, Springer, Berlin, 1977.
[002] [3] G. Herzog, An existence and uniqueness theorem for ordinary differential equations in ordered Banach spaces, Demonstratio Math., to appear. | Zbl 0911.34055
[003] [4] G. Herzog, On ordinary differential equations with quasimonotone increasing right hand side, Arch. Math. (Basel), to appear. | Zbl 0896.34056
[004] [5] R. Lemmert, Existenzsätze für gewöhnliche Differentialgleichungen in geordneten Banachräumen, Funkcial. Ekvac. 32 (1989), 243-249. | Zbl 0721.34073
[005] [6] R. Lemmert, R. M. Redheffer and P. Volkmann, Ein Existenzsatz für gewöhnliche Differentialgleichungen in geordneten Banachräumen, in: General Inequalities 5 (Oberwolfach, 1986), Internat. Ser. Numer. Math. 80, Birkhäuser, Basel, 1987, 381-390. | Zbl 0625.34070
[006] [7] R. Lemmert, S. Schmidt and P. Volkmann, Ein Existenzsatz für gewöhnliche Differentialgleichungen mit quasimonoton wachsender rechter Seite, Math. Nachr. 153 (1991), 349-352.
[007] [8] R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Krieger, 1987.
[008] [9] J. Mierczyński, Strictly cooperative systems with a first integral, SIAM J. Math. Anal. 18 (1987), 642-646. | Zbl 0657.34033
[009] [10] J. Mierczyński, Uniqueness for a class of cooperative systems of ordinary differential equations, Colloq. Math. 67 (1994), 21-23. | Zbl 0831.34001
[010] [11] J. Mierczyński, Uniqueness for quasimonotone systems with strongly monotone first integral, in: Proc. Second World Congress of Nonlinear Analysts (WCNA-96), Athens, 1996, to appear. | Zbl 0896.34001
[011] [12] P. Volkmann, Gewöhnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorräumen, Math. Z. 127 (1972), 157-164. | Zbl 0226.34058
[012] [13] P. Volkmann, Cinq cours sur les équations différentielles dans les espaces de Banach, in: Topological Methods in Differential Equations and Inclusions (Montréal, 1994), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 472, Kluwer, Dordrecht, 1995, 501-520.