The note presents a simple proof of a result due to F. Obersnel and P. Omari on the existence of periodic solutions with an arbitrary period of the first order scalar differential equation, provided equation has an n-periodic solution with the minimal period n > 1.
@article{BUMI_2009_9_2_2_445_0,
author = {Stanis\l aw S\c edziwy},
title = {Periodic Solutions of Scalar Differential Equations without Uniqueness},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {2},
year = {2009},
pages = {445-448},
zbl = {1178.34045},
mrnumber = {2537280},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2009_9_2_2_445_0}
}
Sȩdziwy, Stanisław. Periodic Solutions of Scalar Differential Equations without Uniqueness. Bollettino dell'Unione Matematica Italiana, Tome 2 (2009) pp. 445-448. http://gdmltest.u-ga.fr/item/BUMI_2009_9_2_2_445_0/
[1] - - , On a multivalued version of the Sharkovskii Theorem and its application to differential inclusions, Set-Valued Analysis, 10 (2002), 1-14. | MR 1888453
[2] - - , Period two implies all periods for a class of ODEs: A multivalued map approach, Proceedings of the AMS, 135 (2007), 3187-3191. | MR 2322749 | Zbl 1147.34031
[3] - , A version of Sharkovskii Theorem for differential equations, Proceedings of the AMS, 133 (2005), 449-453. | MR 2093067 | Zbl 1063.34030
[4] - , Theory of Ordinary Differential Equations, McGraw-Hill (New York, Toronto, London, 1955). | MR 69338 | Zbl 0064.33002
[5] - - , Differential Equations, Dynamical Systems, and an Introduction to Chaos, Ed. 2 Acad. Press (N. York, 2003). | MR 2144536
[6] - , Period three implies chaos, American Mathematical Monthly, 82 (1975), 985-992. | MR 385028 | Zbl 0351.92021
[7] - , Period two implies chaos for a class of ODEs, Proceedings of the AMS, 135 (2007), 2055-2058. | MR 2299480 | Zbl 1124.34335
[8] - , Old and new results for first order periodic ODEs without uniqueness: a comprehensive study by lower and upper solutions, Advanced Non-linear Studies, 4 (2004), 323-376. | MR 2079818 | Zbl 1072.34041