Differential and Integral Equations

Periodic solutions of Liénard equations at resonance

Anna Capietto and Zaihong Wang

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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

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

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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