Abstract and Applied Analysis

Lazer-Leach Type Conditions on Periodic Solutions of Liénard Equation with a Deviating Argument at Resonance

Zaihong Wang

Abstract

We study the existence of periodic solutions of Liénard equation with a deviating argument $x\mathrm{\prime \prime }+f\left(x\right)x\mathrm{\text{'}}+{n}^{\mathrm{2}}x+g\left(x\left(t-\tau \right)\right)=p\left(t\right),$ where $f,g,p:\mathrm{R}\to \mathrm{R}$ are continuous and $p$ is $\mathrm{2}\pi$-periodic, $\mathrm{0}\le \tau <2\pi$ is a constant, and $n$ is a positive integer. Assume that the limits ${\mathrm{\text{l}}\mathrm{\text{i}}\mathrm{\text{m}}}_{x\to ±\infty }g\left(x\right)=g\left(±\infty \right)$ and ${\mathrm{\text{l}}\mathrm{\text{i}}\mathrm{\text{m}}}_{x\to ±\infty }F\left(x\right)=F\left(±\infty \right)$ exist and are finite, where $F\left(x\right)={\int }_{\mathrm{0}}^{x}\mathrm{‍}f\left(u\right)du$. We prove that the given equation has at least one $\mathrm{2}\pi$-periodic solution provided that one of the following conditions holds: $\mathrm{2}\mathrm{\text{c}}\mathrm{\text{o}}\mathrm{\text{s}}\left(n\tau \right)\left[g\left(+\infty \right)-g\left(-\infty \right)\right]\ne {\int }_{\mathrm{0}}^{\mathrm{2}\pi }\mathrm{‍}p\left(t\right)\mathrm{\text{s}}\mathrm{\text{i}}\mathrm{\text{n}}\left(\theta +nt\right)dt$, for all $\theta \in \left[\mathrm{0,2}\pi \right],$$\mathrm{2}n\mathrm{\text{c}}\mathrm{\text{o}}\mathrm{\text{s}}\left(n\tau \right)\left[F\left(+\infty \right)-F\left(-\infty \right)\right]\ne$${\int }_{\mathrm{0}}^{\mathrm{2}\pi }\mathrm{‍}p\left(t\right)\mathrm{\text{s}}\mathrm{\text{i}}\mathrm{\text{n}}\left(\theta +nt\right)dt$, for all $\theta \in \left[\mathrm{0,2}\pi \right],$$\mathrm{2}\left[g\left(+\infty \right)-g\left(-\infty \right)\right]-\mathrm{2}n\mathrm{\text{s}}\mathrm{\text{i}}\mathrm{\text{n}}\left(n\tau \right)\left[F\left(+\infty \right)-F\left(-\infty \right)\right]\ne$${\int }_{\mathrm{0}}^{\mathrm{2}\pi }\mathrm{‍}p\left(t\right)\mathrm{\text{s}}\mathrm{\text{i}}\mathrm{\text{n}}\left(\theta +nt\right)dt$, for all $\theta \in \left[\mathrm{0,2}\pi \right],$$\mathrm{2}n\left[F\left(+\infty \right)-F\left(-\infty \right)\right]-\mathrm{2}\mathrm{\text{s}}\mathrm{\text{i}}\mathrm{\text{n}}\left(n\tau \right)\left[g\left(+\infty \right)-g\left(-\infty \right)\right]\ne$${\int }_{\mathrm{0}}^{\mathrm{2}\pi }\mathrm{‍}p\left(t\right)\mathrm{\text{s}}\mathrm{\text{i}}\mathrm{\text{n}}\left(\theta +nt\right)dt$, for all $\theta \in \left[\mathrm{0,2}\pi \right].$

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 906972, 10 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511880

Digital Object Identifier
doi:10.1155/2013/906972

Mathematical Reviews number (MathSciNet)
MR3055862

Zentralblatt MATH identifier
1277.34102

Citation

Wang, Zaihong. Lazer-Leach Type Conditions on Periodic Solutions of Liénard Equation with a Deviating Argument at Resonance. Abstr. Appl. Anal. 2013 (2013), Article ID 906972, 10 pages. doi:10.1155/2013/906972. https://projecteuclid.org/euclid.aaa/1393511880

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