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

Mawhin, J.; Ward, J. R.

Rocky Mt. J. Math. 12, 643-654 (1982).

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