Topological Methods in Nonlinear Analysis

Massera's theorem for quasi-periodic differential equations

Rafael Ortega and Massimo Tarallo

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For a scalar, first order ordinary differential equation which depends periodically on time, Massera's Theorem says that the existence of a bounded solution implies the existence of a periodic solution. Though the statement is false when periodicity is replaced by quasi-periodicity, solutions with some kind of recurrence are anyway expected when the equation is quasi-periodic in time. Indeed we first prove that the existence of a bounded solution implies the existence of a solution which is quasi-periodic in a weak sense. The partial differential equation, having our original equation as its equation of characteristics, plays a key role in the introduction of this notion of weak quasi-periodicity. Then we compare our approach with others already known in the literature. Finally, we give an explicit example of the weak case, and an extension to higher dimension for a special class of equations.

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Topol. Methods Nonlinear Anal., Volume 19, Number 1 (2002), 39-61.

First available in Project Euclid: 2 August 2016

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Ortega, Rafael; Tarallo, Massimo. Massera's theorem for quasi-periodic differential equations. Topol. Methods Nonlinear Anal. 19 (2002), no. 1, 39--61.

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