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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.14444 |
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| _version_ | 1866910551598170112 |
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| author | Bustos, Harold Figueroa, Pablo Pinto, Manuel |
| author_facet | Bustos, Harold Figueroa, Pablo Pinto, Manuel |
| contents | We address the Poincaré-Perron's classical problem of approximation for high order linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in [23]. We obtain explicit formulae for solutions of these equations, for any fixed order $n\ge 3$, by studying a Riccati type equation associated with the logarithmic derivative of a solution. Moreover, we provide sufficient conditions to ensure the existence of a fundamental system of solutions. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions to this Riccati type equation. A decomposition property of the perturbations induces a decomposition on the Riccati type equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also $p$-almost periodic solutions to the Riccati type equation. We illustrate our results for a third order linear differential equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14444 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Poincaré-Perron problem for high order differential equations in the class of almost periodic type functions Bustos, Harold Figueroa, Pablo Pinto, Manuel Classical Analysis and ODEs We address the Poincaré-Perron's classical problem of approximation for high order linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in [23]. We obtain explicit formulae for solutions of these equations, for any fixed order $n\ge 3$, by studying a Riccati type equation associated with the logarithmic derivative of a solution. Moreover, we provide sufficient conditions to ensure the existence of a fundamental system of solutions. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions to this Riccati type equation. A decomposition property of the perturbations induces a decomposition on the Riccati type equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also $p$-almost periodic solutions to the Riccati type equation. We illustrate our results for a third order linear differential equation. |
| title | Poincaré-Perron problem for high order differential equations in the class of almost periodic type functions |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2407.14444 |