## Differential and Integral Equations

### Periodic solutions of Liénard equations at resonance

#### Abstract

In this paper we study the existence of periodic solutions of the second order differential equation $$x''+f(x)x'+n^2x+\varphi(x)=p(t), \quad n\in {{\bf N}} .$$ We assume that the limits $$\lim_{x\to\pm\infty}\varphi(x)=\varphi(\pm\infty),\quad \lim_{x\to\pm\infty}F(x)=F(\pm\infty)\quad \Big( F(x) =\int_0^xf(u)du \Big)$$ exist and are finite. We prove that the given equation has at least one $2\pi$-periodic solution provided that (for $A=\int_0^{2\pi}p(t)\sin nt dt, B=\int_0^{2\pi}p(t)\cos nt dt$) one of the following conditions is satisfied: $$2(\varphi(+\infty)-\varphi(-\infty))>\sqrt{A^2+B^2}$$ $$2n(F(+\infty)-F(-\infty))>\sqrt{A^2+B^2}$$ $$2(\varphi(+\infty)-\varphi(-\infty))=\sqrt{A^2+B^2}, \quad F(+\infty)\not=F(- \infty)$$ $$2n(F(+\infty)-F(-\infty))=\sqrt{A^2+B^2}, \quad \varphi(+\infty)\not=\varphi(- \infty).$$ On the other hand, we prove the non-existence of $2\pi$-periodic solutions provided that the inequality $$2(\varphi(+\infty)-\varphi(-\infty))+2n(F(+\infty)-F(-\infty))\leq\sqrt {A^2+B^2}$$ and other conditions hold. We also deal with the existence of $2\pi$-periodic solutions of the equation when $\varphi$ satisfies a one-sided sublinear condition and $F$ is bounded.

#### Article information

Source
Differential Integral Equations, Volume 16, Number 5 (2003), 605-624.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356060630

Mathematical Reviews number (MathSciNet)
MR1973066

Zentralblatt MATH identifier
1039.34034

Subjects
Primary: 34C25: Periodic solutions
Secondary: 34B15: Nonlinear boundary value problems

#### Citation

Capietto, Anna; Wang, Zaihong. Periodic solutions of Liénard equations at resonance. Differential Integral Equations 16 (2003), no. 5, 605--624. https://projecteuclid.org/euclid.die/1356060630