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On the existence of harmonic solutions of Liénard systems. (English) Zbl 0735.34033

The author considers Liénard systems of the form (*) $\stackrel{¨}{x}+\left(d/dt\right)\text{grad}F\left(x\right)+\text{grad}G\left(x\right)=p\left(t\right)$, $F\in {C}^{2}\left({R}^{n},R\right)$, $G\in {C}^{1}\left({R}^{n},R\right)$, $p\in {C}^{0}\left(R,{R}^{n}\right)$ and $p\left(t+T\right)=p\left(t\right)$ for a constant $T>0$, and proves two theorems for the existence of harmonic solutions, in which the conditions are somewhat weaker than those assumed in other papers quoted by the author. Denoting by $\left(·,·\right)$ the inner product in ${R}^{n}$ and assuming $|x|=\sqrt{\left(x,x\right)}$ for $x\in {R}^{n}$, Theorem 2 is:

Suppose that ${\int }_{0}^{T}p\left(s\right)ds=0$, and that there is a nonsingular matrix A and a constant ${R}_{0}>0$ such that $\left(Ax,\text{grad}G\left(x\right)\right)>0$ for $x\in {R}^{n}$ with $|x|>{R}_{0}$,

$\underset{|x|\to \infty }{lim}|\left(\text{grad}F\left(x\right),\text{grad}G\left(x\right)\right)|/|\text{grad}G\left(x\right)|>M=\underset{\left[0,T\right]}{max}|P\left(t\right)|·$

Then system (*) has at least one harmonic solution.

##### MSC:
 34C25 Periodic solutions of ODE
##### Keywords:
Liénard systems; existence of harmonic solutions