Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations.

*(English)*Zbl 0557.34036The 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 |

##### Keywords:

forced pendulum equation; Josephson equation; coincidence degree; lower solutions techniques; Degree theory; mountain pass theorem
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\textit{J. Mawhin} and \textit{M. Willem}, J. Differ. Equations 52, 264--287 (1984; Zbl 0557.34036)

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