Periodic Solutions of Duffing Equation with an Asymmetric Nonlinearity and a Deviating Argument

Abstract

We study the existence of periodic solutions of the second-order differential equation $x^{\mathrm{\prime \prime }}+a{x}^{+}-b{x}^{-}+g(x(t-\tau ))=p(t)$, where $a,b$ are two constants satisfying $\mathrm{1}/\sqrt{a}+\mathrm{1}/\sqrt{b}=\mathrm{2}/n$, $n\in \mathrm{N}$, $\tau$ is a constant satisfying , $g,p:\mathrm{R}\to \mathrm{R}$ are continuous, and $p$ is $\mathrm{2}\pi$-periodic. When the limits ${\text{lim}}_{x\to \pm \mathrm{\infty }}g(x)=g(\pm \mathrm{\infty })$ exist and are finite, we give some sufficient conditions for the existence of $\mathrm{2}\pi$-periodic solutions of the given equation.