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

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].
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
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/86308
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 13
  • ???jsp.display-item.citation.isi??? 10
social impact