Bound sets for a class of $\phi$-Laplacian operators

Abstract

We provide an extension of the Hartman-Knobloch theorem for periodic solutions of vector differential systems to a general class of $\phi$-Laplacian differential operators. Our main tool is a variant of the Man\'{a}sevich-Mawhin continuation theorem developed for this class of operator equations, together with the theory of bound sets. Our results concern the case of convex bound sets for which we show some new connections using a characterisation of sublevel sets due to Krantz and Parks. We also extend to the $\phi$-Laplacian vector case a classical theorem of Reissig for scalar periodically perturbed Li\'{e}nard equations.