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Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. (English) Zbl 0557.34036
The motivation of this paper is the study of the existence of multiple solutions for the problem $(1)\quad \ddot x+f(x)\dot x+a \sin x=e(t),\quad x(0)-x(2\pi)=\dot x(0)-\dot x(2\pi)=0.$ In section 1, using upper and lower solutions techniques, it is proved that, for a $$2\pi$$-periodic f, if we write $$e=\tilde e+\bar e$$ with $$\bar e=(1/2\pi)\int^{2\pi}_{0}e(t)dt,$$ $$\tilde e=e-\bar e$$ then, for each $$\tilde e,$$ the set $$R(\tilde e)$$ of $$\bar e\in {\mathbb{R}}$$ for which (1) is solvable is a non-empty closed interval contained in [-a,a]. In section 2 it is proved that if $$| f(x)| \geq c>\sqrt{2/3}| \tilde e|_{L^ 2}$$ then 0 is an interior point of $$R(\tilde e)$$. In section 3 we deduce the existence of two solutions when $$| e|_{\infty}<a$$ and of one solution if equality holds. Degree theory is the basic ingredient of sections 2 and 3. In sections 4 and 5 we consider the conservative case (f$$\equiv 0)$$. The existence of two solutions is obtained by a mountain pass theorem when $$\bar e=0$$. In particular $$0\in R(\tilde e)$$ for every $$\tilde e.$$ In section 5 it is shown that the set of $$\tilde e$$ for which $$R(\tilde e)$$ is a neighborhood of 0 in $${\mathbb{R}}$$ is open and dense in the space of continuous functions with mean value 0.

##### MSC:
 34C25 Periodic solutions to ordinary differential equations 37G99 Local and nonlocal bifurcation theory for dynamical systems 58C30 Fixed-point theorems on manifolds 47H10 Fixed-point theorems 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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