Periodic solutions of some differential equations with nonlinear damping
Here, the author studies the existence and uniqueness of periodic solutions to the following equation \[ x'' + F(x, x') = p(t), \] where \(p\) is periodic in \(t\). Under some reasonable assumptions, he proves the existence of periodic solutions using the upper-and-lower solutions method. He gets a series of results when the function \(F\) satisfies one of the following conditions (1) \(F: \mathbb{R}^2\to \mathbb{R}\) is continuous in \(x\) and \(x^\prime\) (nonsingular case); (2) \(F: (0,+\infty)\times \mathbb{R}\to \mathbb{R}\) and \(\lim_{x\to 0^+}F(x,0) = +\infty\) (case of one singularity); (3) \(F: (0,1)\times \mathbb{R}\to \mathbb{R}\) and \(\lim_{x\to 0^+}F(x,0) = +\infty\), \(\lim_{x\to 1^-}F(x,0) = -\infty\) (the case of two attractive singularities). He also investigates the case of dry friction \(x'' + \mu\text{sgn}(x') + g(x) = p(t)\).
scientific article; zbMATH DE number 1836160