Using closed orbits to bifurcate many periodic solutions for pendulum-type equations
The paper deals with the boundary value problem for a pendulum-like equation of the type \[ u^{\prime\prime}+ cu^\prime+g(u)=\bar e+\tilde e(t)\,, \quad u(T)-u(0)=0\,,\quad u^\prime(T)-u^\prime(0)=0\,. \] Here, \(T>0\) is a real constant, \(g\in C(R/2\pi Z)\) is a continuous \(2\pi\)-periodic function with zero mean (i.e., \(\int_0^{2\pi} g(u)\,du=0\)), and the external force \(e=\bar e+\tilde e\in L_1[0,\,T]\) is decomposed as a sum of a constant \(\bar e\) and a function \(\tilde e\) with zero mean. A result of \textit{F. Donati} [C. R. Acad. Sci. Paris Sér. I Math. 317, 667--672 (1993; Zbl 0783.34033)] says that in the periodic problem for the conservative forced pendulum equation (i.e., with \(g(u)=\Lambda \sin(u)\)), it is always possible to find forcing terms \(\tilde e\) such that the problem has at least four geometrically different solutions (two solutions are called geometrically different if their difference is not a multiple of \(2\pi\)). This result has been extended by \textit{R. Ortega} [Rocky Mt. J. Math. 27, 861--876 (1997; Zbl 0899.34031)], who has shown that within the same framework it is possible to change \(4\) by any number. In the present paper it is shown how Ortega's theorem remains valid in the presence of friction, and for a bigger class of oscillating functions \(g\), namely, those which are restrictions to the real line of entire functions. The proof is based on topological tools.
scientific article; zbMATH DE number 2136650