Periodic solutions of Lagrangian inclusions (Q1180335)

The paper deals with the Lagrangian inclusion (1) \({d\over dt}{\partial{\mathcal L}\over\partial\xi}(t,q,\dot q)\in\partial_ q{\mathcal L}(t,q,\dot q)\), where \({\mathcal L}(t,q,\xi)={1\over 2}\sum^ N_{i,j=1}a_{ij}(q)\xi_ i\xi_ j-V(t,q)\), \(t\in\mathbb{R}\), \(q,\xi\in\mathbb{R}^ N\), \(V\in C(\mathbb{R}^{N+1},\mathbb{R})\) is \(T\)-periodic in \(t\) and locally Lipschitz continuous in \(q\) (uniformly with respect to \(t\)) and \(a_{ij}\) are locally Lipschitz continuous functions. The symbol \(\partial_ q\) denotes the generalized gradient introduced by F. H. Clarke. Solutions of (1) are found in the set of \(T\)-periodic absolutely continuous functions. Using variational methods the authors prove the existence of at least one solution when \(V\) is subquadratic in \(q\) at infinity and a multiplicity result when \(V\) does not depend on \(t\). The proofs are based on the Rabinowitz saddle point theorem for locally Lipschitz continuous functions.