In this paper, we obtain conditions of existence and uniqueness of periodic solutions for a nonlinear fourth-order differential equation utilizing explicit Green’s function and fixed point index theorem combining with operator spectral theorem. We discuss an iteration method for constant coefficient nonlinear differential equations and establish the theorem on the existence of positive solutions for a fourth-order boundary value problem with a variable parameter. Finally, we give an example to illustrate our results.

中文翻译：

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一类四阶非线性微分方程的正周期解

本文利用显式格林函数，不动点指数定理和算子谱定理，获得了非线性四阶微分方程周期解存在性和唯一性的条件。我们讨论了常系数非线性微分方程的迭代方法，并建立了带有可变参数的四阶边值问题正解存在性的定理。最后，我们举一个例子来说明我们的结果。