Periodic solutions of \((2n+1)\)th order ordinary differential equations
The author uses the Leray-Schauder degree theory to prove that the equation \[ x^{(2n+1)}+f(t,x,x',...,x^{(2n)})=0 \] has a unique \(2\pi\)-periodic solution provided that \(f\) is \(C^1\), \(2\pi\)-periodic in time and the following conditions hold: (H1) \(|f(t,x)-f(t,y)|\leq \sum_{i=0}^{2n} c_{i} |x_{i}-y_{i}|\) for some \(c_0>0\) and \(c_i\geq 0\), (H2) there exists \(m>0\) such that \(m\leq |f_{x_{0}}|\leq c_{0}\) on \(\mathbb{R}\times \mathbb{R}^{2n+1}\) and \[ \left({c_{0} \over m} +1\right) \sum_{i=1}^{2n} (2\pi)^{2n-i+1} c_{i}<1. \]
scientific article; zbMATH DE number 1836790