Existence theorems of periodic solutions for a class of damped vibration problems (Q1004237)

The authors consider the damped vibration problem \[ \ddot{u} \left( t\right) +g\left( t\right) \dot{u}\left( t\right) =\nabla F\left( t,u\left( t\right) \right)\text{ a.e. on }\left[0,T\right] \] with periodic boundary conditions. Here, \(T>0\), \(g\in L^{\infty }\left( 0,T\right) \), \(F\) is a Caratheodory function subject to some standard growth. It is important that \(F\) is concave in \(u\), which means that the results obtained by the authors are new even for \(g=0\). The authors consider the variational formulation of the problem to which they apply three different critical points theorems (namely the direct method of the calculus of variations, the critical point theorem without compactness due to Yabri and Moussaoui and the mountain pass result) all of which allow to obtain at least one solution. They demonstrate that there are functions satisfying the growth conditions which are imposed in the paper.