Periodic solutions of dynamical systems with bounded potential (Q1103767)

This paper deals with the existence of T-periodic solutions for systems of ordinary differential equations of the form \(-\ddot x=\nabla_ xV(t,x),\) where \(V\in C^ 2(R\times R^ N,R)\), \(V(t+T,x)=V(t,x)\) for any \((t,x)\in {\mathbb{R}}\times {\mathbb{R}}^ N\), V is bounded and \(\nabla_ xV(t,x)\to 0\) as \(| x| \to +\infty\). The author uses Morse theory to prove the existence of periodic solutions for the given equation. The use of Morse theory is done through defining on a suitable function space E a functional f whose critical points are the T-periodic solutions of the equation. Then he shows that the Palais-Smale condition holds on \(\{x\in E:f(x)\geq c+\epsilon \}\) and estimates the homology groups \(H_ q(\{x\in E;f(x)\leq c+\epsilon \})\).