PERIODIC SOLUTIONS OF LIÉNARD EQUATIONS WITH ASYMMETRIC NONLINEARITIES AT RESONANCE

Abstract

The existence of $2\pi$-periodic solutions of the second-order differential equation \[ x''+f(x)x'+ax^+-bx^-+g(x)=p(t), \qquad n\in \mathbb{N},\] where $a, b$ satisfy $1/\sqrt{a}+1/\sqrt{b}=2/n$ and $p(t)=p(t+2\pi)$, $t\in \mathbb{R}$, is examined. Assume that limits $\lim_{x\to\pm\infty}F(x)=F(\pm\infty)$ ($F(x)=\int_0^xf(u) du$) and $\lim_{x\to\pm\infty}g(x)=g(\pm\infty)$ exist and are finite. It is proved that the equation has at least one $2\pi$-periodic solution provided that the zeros of the function $\Sigma_1$ are simple and the zeros of the functions $\Sigma_1, \Sigma_2$ are different and the signs of $\Sigma_2$ at the zeros of $\Sigma_1$ in $[0,2\pi/n)$ do not change or change more than two times, where $\Sigma_1$ and $\Sigma_2$ are defined as follows: \[\Sigma_1(\theta)=\frac{n}{\pi}\left[\frac{g(+\infty)}{a}-\frac{g(-\infty)}{b} \right]-\frac {1}{2\pi}\int_0^{2\pi}p(t)\varphi(t+\theta)\,dt,\qquad \theta\in [0, 2\pi/n],\]\[\Sigma_2(\theta)=\frac{n}{\pi}[F(+\infty)-F(-\infty)]-\frac{1}{2\pi} \int_0^{2\pi}p(t)\varphi'(t+\theta)\,dt, \qquad\theta\in [0, 2\pi/n].\] Moreover, it is also proved that the given equation has at least one $2\pi$-periodic solution provided that the following conditions hold: \begin{eqnarray*} -\infty&<&\liminf_{x\to\pm\infty}\frac{F(x)}{|x|^{p-2}x}\le \limsup_ {x\to\pm\infty}\frac{F(x)}{|x|^{p-2}x}<+\infty,\\ 0&<&\liminf_{x\to\pm\infty}\frac{g(x)}{|x|^{q-2}x}\leq \limsup_{x\to\pm\infty} \frac{g(x)}{|x|^{q-2}x}<+\infty, \end{eqnarray*} with $1\leq p<q<2$.