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2003 Periodic solutions of Liénard equations at resonance
Anna Capietto, Zaihong Wang
Differential Integral Equations 16(5): 605-624 (2003).

## 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.

## Citation

Anna Capietto. Zaihong Wang. "Periodic solutions of Liénard equations at resonance." Differential Integral Equations 16 (5) 605 - 624, 2003.

## Information

Published: 2003
First available in Project Euclid: 21 December 2012

zbMATH: 1039.34034
MathSciNet: MR1973066

Subjects:
Primary: 34C25
Secondary: 34B15