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Boundedness, periodicity, and convergence of solutions in a retarded Liénard equation. (English) Zbl 0803.34064

From authors’ introduction: “The paper deals with the retarded Liénard equation

${x}^{\text{'}\text{'}}+f\left(x\right){x}^{\text{'}}+g\left(x\left(t-h\right)\right)=e\left(t\right),\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $h\ge 0$ is the delay and $e$ is a bounded function. With appropriate assumptions on $f$ and $g$ the authors obtain necessary and sufficient conditions for solutions of $\left(*\right)$ to be uniformly ultimately bounded. Thus by a fixed point theorem, those conditions imply that $\left(*\right)$ has a $T$-periodic solution whenever $e$ is $T$-periodic. Also are given conditions under which all solutions of $\left(*\right)$ converge.

##### MSC:
 34K99 Functional-differential equations 34C11 Qualitative theory of solutions of ODE: growth, boundedness 34C25 Periodic solutions of ODE 34D40 Ultimate boundedness (MSC2000)
##### References:
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