This paper studies the "internal structure" of the periodic solutions of differential equations with the aim of stating when they are constant functions. Yorke [21] and Lasota and Yorke [10] are the first works which show the existence, uńder certain conditions, of a lower bound for the period of non-constant solutions. As applications of the general results proved in Section 1 we obtain a negative solution to an open problem of Browder, the discovery that the periodic solutions ensured by Vidossich [17, Theorem 3.16], are constant functions, and conditions under which the periodic solutions of hyperbolic and parabolic equations are constant functions. Finally, we note that Li [11] applies the results of Section 1 to differential equations with delay. Various result of this paper point out a strong connection between the existence of periodic solutions of small period of x' = f( x) and the fact that the origin belongs to the range of f. This situation is explored in [19].

On the structure of periodic solutions of differential equations / Vidossich, Giovanni. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 21:2(1976), pp. 263-278. [10.1016/0022-0396(76)90122-4]

On the structure of periodic solutions of differential equations

Vidossich, Giovanni
1976-01-01

Abstract

This paper studies the "internal structure" of the periodic solutions of differential equations with the aim of stating when they are constant functions. Yorke [21] and Lasota and Yorke [10] are the first works which show the existence, uńder certain conditions, of a lower bound for the period of non-constant solutions. As applications of the general results proved in Section 1 we obtain a negative solution to an open problem of Browder, the discovery that the periodic solutions ensured by Vidossich [17, Theorem 3.16], are constant functions, and conditions under which the periodic solutions of hyperbolic and parabolic equations are constant functions. Finally, we note that Li [11] applies the results of Section 1 to differential equations with delay. Various result of this paper point out a strong connection between the existence of periodic solutions of small period of x' = f( x) and the fact that the origin belongs to the range of f. This situation is explored in [19].
1976
21
2
263
278
https://doi.org/10.1016/0022-0396(76)90122-4
https://www.sciencedirect.com/science/article/pii/0022039676901224?via%3Dihub
Vidossich, Giovanni
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/86308
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