Resonance with Landesman-Lazer conditions for parameter-dependent equations: a multiplicity result via the Poincaré-Birkhoff theorem

Abstract

We investigate the resonance problem and prove the existence of multiple periodic solutions to a second order parameter-dependent equation $ x''+f(t, x) = sp(t) $. We weaken the usual requirement on the sublinearity of perturbations when $ |x| $ becomes large; and develop a more general method to investigate the rotational characterizations of the Landesman-Lazer conditions. Furthermore, $ f $ does not satisfy the common sign condition, and even the global existence of the solution is not guaranteed.

Keywords

• periodic solutions
• poincaré-birkhoff theorem
• resonance
• landesman-lazer conditions
• parameter-dependent