On the existence and stability of periodic solutions for pendulum-like equations with friction and \(\phi \)-Laplacian
The authors consider the differential equation \[ (\phi (x'))'+cx'+g(x)=e(t)+s \] and discuss the existence and stability properties of periodic solutions. The function \(e\) is periodic and has zero average, and the numbers \(c\geq 0\) and \(s\in \mathbb{R}\) act as parameters. The assumptions on the function \(\phi\) are typical of the so-called \(\phi\)-Laplacian operators, and they include the examples \(\phi (z)=\frac{z}{\sqrt{1\pm z^2}}\) corresponding to the mean curvature and relativistic operators. With respect to the nonlinearity \(g\), the main focus is on the function \(g(x)=k\sin x\) leading to some variants of the forced pendulum equation. In the study of stability properties, the authors observe that the linearized equation can be reduced to an equation of Hill's type via a change of the independent variable.
scientific article; zbMATH DE number 6146678