Bulletin of the Belgian Mathematical Society - Simon Stevin

Poincaré's problem in the class of almost periodic type functions

Pablo Figueroa and Manuel Pinto

Full-text: Open access


We consider the Poincaré's classical problem of approximation for second order linear differential equations in the class of almost periodic type functions. We obtain an explicit form for solutions of these equations by studying a Riccati equation associated with the logarithmic derivative of a solution. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions of the Riccati equation. A decomposition property of the perturbations induces a decomposition on the Riccati equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also $p$-almost periodic solutions to the Riccati equation.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 22, Number 2 (2015), 177-198.

First available in Project Euclid: 28 May 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34A05: Explicit solutions and reductions 34A30: Linear equations and systems, general 34E10: Perturbations, asymptotics 34E05: Asymptotic expansions

Perturbed second order linear differential equation Almost periodic solutions Riccati equation


Figueroa, Pablo; Pinto, Manuel. Poincaré's problem in the class of almost periodic type functions. Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 2, 177--198. doi:10.36045/bbms/1432840857. https://projecteuclid.org/euclid.bbms/1432840857

Export citation