A new method for the boundedness of semilinear Duffing equations at resonance

Abstract

We introduce a new method for the boundedness problem of semilinear Duffing equations at resonance. In particular, it can be used to study a class of semilinear equations at resonance without the polynomial-like growth condition. As an application, we prove the boundedness of all the solutions for the equationẍ + n 2 x + g(x) + ψ(x) = p(t) under the Lazer-Leach condition on g and p, where n ∈ N + , p(t) and ψ(x) are periodic and g(x) is bounded.

Boundedness in Asymmetric Oscillations at Resonance in a Critical Situation

“…In 2016, Wang, Wang and Piao [13] showed that if ψ oscillates periodically in x, the Lazer-Leach condition (1.7) is sufficient and necessary for the boundedness of (1.8).…”