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Critical point theory and a theorem of Amaral and Pera. (English) Zbl 0603.34036

For the differential equation ${u}^{\text{'}\text{'}}\left(t\right)+g\left(t,u\left(t\right)\right)=0$ where g is $2\pi$- periodic in t, L. Amaral and M. P. Pera [Boll. Unione Mat. Ital., V. Ser., C, Anal. Funz. Appl. 18, 107-117 (1981; Zbl 0472.34028)] showed that there is a $2\pi$-periodic solution if there are constants ${H}_{1}$ and $\nu$, with $\nu <1$, such that for large x,

${H}_{1}\le g\left(t,x/x\right)\le \nu \phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}{\int }_{0}^{2\pi }\underset{|x|\to \infty }{lim}inf\frac{g\left(t,x\right)}{x}dt>0·$

In the paper under review it is shown that if (sgn x) g(t,x) is bounded below, then the integral condition just stated can be replaced by

${\int }_{0}^{2\pi }\underset{|x|\to \infty }{lim}inf\left(sgnx\right)g\left(t,x\right)dt>0·$

This theorem and the theorem of Amaral and Pera are derived using the variational methods introduced by P. Rabinowitz [Nonlinear analysis, Collect. Pap. Honor H. Rothe, 161-177 (1978; Zbl 0466.58015)].

Reviewer: C.Chicone
##### MSC:
 34C25 Periodic solutions of ODE 34C05 Location of integral curves, singular points, limit cycles (ODE) 34C15 Nonlinear oscillations, coupled oscillators (ODE)