Bulletin of the Belgian Mathematical Society - Simon Stevin

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

Pablo Figueroa and Manuel Pinto

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Abstract

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

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

Dates
First available in Project Euclid: 28 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1432840857

Digital Object Identifier
doi:10.36045/bbms/1432840857

Mathematical Reviews number (MathSciNet)
MR3351035

Zentralblatt MATH identifier
1333.34019

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

Keywords
Perturbed second order linear differential equation Almost periodic solutions Riccati equation

Citation

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


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