Planar differential systems at resonance

Abstract

We consider the system $$ J\dot u=\nabla { \mathcal {H}}(u)+f(u)+p(t)\,, $$ where ${ \mathcal {H}}:{{\mathbb R}} ^2\to{{\mathbb R}} $ is of class $C^1$ with locally Lipschitz continuous gradient, $f:{{\mathbb R}} ^2\to{{\mathbb R}} ^2$ is locally Lipschitz continuous and bounded, and $p:{{\mathbb R}} \to{{\mathbb R}} ^2$ is measurable, bounded and $T-$periodic. Here, $J= \begin{pmatrix} \scriptstyle 0 & \!\!\!\!\scriptstyle -1 \cr \scriptstyle 1 & \scriptstyle \!0 \end{pmatrix} $ is the standard symplectic matrix. For some classes of functions $f,$ we give new existence theorems for periodic solutions and for unbounded solutions. Applications are given to forced second-order differential equations with separated nonlinearities.