This paper is a continuous work of Liu et al. (2017, first order) and Xu et al. (2019, second order) for affine-periodic solutions to ordinary differential equations. It is a hard problem to obtain satisfied extremum principles. In this paper, we give several extremum principles (Theorem 2.1 and Lemma 2.2) for affine-periodic problems, especially for the case of higher order systems. By these extremum principles, we partly establish the existence of affine-periodic solutions for higher order ordinary differential equations.

中文翻译：

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高阶微分方程的仿射周期解

本文是Liu等人的连续工作。（2017年，一等）和Xu等。（2019年，二阶）用于常微分方程的仿射周期解。获得满意的极值原理是一个难题。在本文中，我们针对仿射周期问题，特别是对于高阶系统，给出了几种极值原理（定理2.1和引理2.2）。根据这些极值原理，我们部分建立了高阶常微分方程仿射周期解的存在性。