On the structure of periodic solutions of differential equations |
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Authors: | Giovanni Vidossich |
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Institution: | Departamento de Matemática, Universidade de Brasília, Brasília, Brazil |
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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]. |
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