Lyapunov stability for regular equations and applications to the Liebau phenomenon
Recently, some results on the existence and stability of positive periodic solutions for the following singular second-order differential equation \[ \ddot{u}+a\dot{u}=\frac{1}{u}(e(t)-b(t)\dot{u}^2)-c \] with friction have been presented. However, there are few works on the conservative case \(a = 0\), which is an idealization but it is interesting from both the mathematical and the physical point of view. By means of classical topological techniques like the Kolmogorov-Arnold-Moser (KAM) theory, the Moser twist theorem, the averaging method and the method of upper and lower solutions in the reversed order, the authors study the existence and stability of periodic solutions of two kinds of regular equations for this gap. As an application, they present some results on the existence and stability of T-periodic solutions of a Liebau-type equation.
scientific article; zbMATH DE number 6919436
scientific article