## Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Liénard and Duffing equations.(English)Zbl 0536.34022

The paper considers the nonlinear oscillator $$x''+f(x)x'+g(t,x)=0$$, where f is continuous and g is 2$$\pi$$-periodic in t and satisfies Caratheodory conditions, and provides sufficient conditions in order that 2$$\pi$$- periodic solutions exist. The main assumptions are the following: $\beta(t)\leq \lim \inf_{| x| \to \infty}x^{-1}g(t,x)\leq \lim \sup_{| x| \to \infty}x^{-1}g(t,x)\leq \mu(t),$ where these inequalities hold a.e. in [0,2$$\pi]$$ and $$\beta$$ and $$\mu$$ satisfy (i) belongs to $$L^ 1(0,2\pi)$$ and $$\int_{[0,2\pi]}\beta(t)dt>0$$, (ii) $$\mu$$ (t)$$\leq 1$$, with strict inequality on a set of positive measure. To prove this result the authors employ the coincidence degree theory developed by the first author plus some new a priori estimates.
Reviewer: K.Schmitt

### MSC:

 34C25 Periodic solutions to ordinary differential equations 34L99 Ordinary differential operators

### Keywords:

nonlinear oscillator
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