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

TL;DR

This paper extends the Hartman-Knobloch theorem and classical results to a broad class of $\,phi$-Laplacian operators, using bound sets and continuation methods to establish periodic solutions.

Contribution

It introduces a novel extension of the Hartman-Knobloch theorem for $\,phi$-Laplacian systems and develops new connections involving convex bound sets and sublevel set characterizations.

Findings

Extended Hartman-Knobloch theorem to $\,phi$-Laplacian operators

Developed a variant of the Manásevich-Mawhin continuation theorem for this class

Established new links between convex bound sets and sublevel set characterizations

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.