New results for Brillouin electron beam focusing system
In this paper, the authors study the following singular equation \[ x''(t)+a(t)x(t)=\frac{1}{x(t)}, \] where \(a(t)\in C(\mathbb{R}, [0,+\infty))\) and \(a(t+T)=a(t)\;\forall t\in \mathbb{R}\). Under the assumption \[ |a|_\infty:=\max_{t\in[0,T]}|a(t)|<\frac{4\pi^2}{T^2} \] the existence of positive \(T\)-periodic solutions is proved. The proof is based on the Manasevich-Mawhin's theorem. As an application, they obtain the existence of positive \(2\pi\)-periodic solutions for the Brillouin electron beam focusing equation \[ x''+a(1+\cos 2t)x=\frac{1}{x}, \] where \(a\in(0,\frac{1}{2})\).
scientific article; zbMATH DE number 6806918