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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.

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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