Bounded solutions for a forced bounded oscillator without friction

TL;DR

This paper proves the existence of bounded solutions and their derivatives for a nonlinear second-order differential equation with a bounded reaction term, using variational methods and a dual Nehari approach.

Contribution

It introduces a variational proof for bounded solutions of a nonlinear oscillator without friction under Landesman-Lazer conditions, employing a dual Nehari method.

Findings

Existence of solutions bounded on the real line

Solutions have bounded first derivatives

Applicable to equations with bounded reaction terms

Abstract

Under the validity of a Landesman-Lazer type condition, we prove the existence of solutions bounded on the real line, together with their first derivatives, for some second order nonlinear differential equation of the form $\ddot u + g(u) = p(t)$, where the reaction term $g$ is bounded. The proof is variational, and relies on a dual version of the Nehari method for the existence of oscillating solutions to superlinear equations.