Periodic solutions for nonlinear equations with mean curvature-like operators (Q2455426)

This theorem extends the Manásevich-Mawhin approach for periodic solutions of differential equations involving p-Laplacians to the case of equations of the form \[ (\phi(u'))' = f(t,u,u'), \] where \(\phi : \mathbb R \to (-1,1)\) is an increasing homeomorphism such that \(\phi(0) = 0.\) This class includes in particular the case of curvature operators for which \(\phi(s) = \frac{s}{\sqrt{1 + s^2}}\). Following Manásevich-Mawhin's approach, but taking in account the new difficulties introduced by the bounded range of \(\phi\), the authors reduce the problem to a fixed point problem in the space of periodic functions of class \(C^1\) and use topological degree to obtain the corresponding continuation theorem. An example is given. For similar results and other applications, see a recent series of papers by Bereanu and Mawhin.