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

Abstract

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