Existence and stability of periodic solutions for relativistic singular equations (Q507300)

The authors provide sufficient conditions for the existence and stability of periodic solutions of the equation \[ \left(\frac{x'}{\sqrt{1-(x')^2}}\right)'+ cx' + f(x) = e(t). \] This is a particular case of a scalar \(\phi\)-Laplacian equation, where the first term is the relativistic acceleration. The authors are interested in the case when the nonlinearity \(f(x)\) exhibits some type of singularity, i.e., it becomes unbounded at a given point. The existence result is based on the continuation method of coincidence degree theory, whereas the stability relies on a known relation between Brouwer degree and stability of periodic solutions due to \textit{R. Ortega} [Boll. Unione Mat. Ital., VII. Ser., B 3, No. 3, 533--546 (1989; Zbl 0686.34052)].