Existence of periodic solutions in the systems of the billiard type (Q1981912)

The paper analyses a billiard-type system describing the free motion of a particle in a bounded domain \(\Omega\), under dissipation and periodic potential, and with elastic reflection when a collision with the boundary occurs. More precisely, the authors consider the second-order differential equation \[ \ddot x + \nabla V(t,x) + f(x,\dot x) = 0, \quad (t,x) \in \mathbb{R}\times \Omega, \] where \(V\) is a smooth potential \(T\)-periodic in the \(t\)-variable, and \(f\) is the dissipation term. The approximation scheme in [\textit{V. Benci} and \textit{F. Giannoni}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6, No. 1, 73--93 (1989; Zbl 0667.34054)] is extended to prove the existence of a \(T\)-periodic bounce solution. Moreover, if the potential \(V\) is coercive, the existence is valid also when the domain \(\Omega\) is unbounded. Some final applications are discussed: periodic bounce orbits are provided for the Fermi model, the bouncing ball model with dissipation, and a class of multi-degrees-of-freedom impact systems.