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Zbl 0603.34036
Critical point theory and a theorem of Amaral and Pera.
(English)
[J] Boll. Unione Mat. Ital., VI. Ser., B 3, 583-598 (1984). ISSN 0392-4041

For the differential equation $u''(t)+g(t,u(t))=0$ where g is $2\pi$- periodic in t, {\it L. Amaral} and {\it 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 $H\sb 1$ and $\nu$, with $\nu <1$, such that for large x, $$H\sb 1\le g(t,x/x)\le \nu \quad and\quad \int\sp{2\pi}\sb{0}\lim\sb{\vert x\vert \to \infty}\inf \frac{g(t,x)}{x} dt>0.$$ 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 $$\int\sp{2\pi}\sb{0}\lim\sb{\vert x\vert \to \infty}\inf (sgn x) g(t,x) dt>0.$$ This theorem and the theorem of Amaral and Pera are derived using the variational methods introduced by {\it P. Rabinowitz} [Nonlinear analysis, Collect. Pap. Honor H. Rothe, 161-177 (1978; Zbl 0466.58015)].
[C.Chicone]
MSC 2000:
*34C25 Periodic solutions of ODE
34C05 Qualitative theory of some special solutions of ODE
34C15 Nonlinear oscillations of solutions of ODE

Keywords: second order differential equation; variational methods

Citations: Zbl 0472.34028; Zbl 0466.58015

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