Periodic oscillations of forced pendulum-like equations

Two-Point Boundary Value Problems for a Class of Second-Order Ordinary Differential Equations

“…The Dirichlet problem for the semilinear second-order ODE u g t, u, u 0 1.1 has been studied by many authors from the pioneering work of Picard 1 , who proved the existence of a solution by an application of the well-known method of successive approximations under a Lipschitz condition on g and a smallness condition on T . Sharper results were obtained by Hamel 2 in the special case of a forced pendulum equation see also 3,4 . The existence of periodic solutions for this case has been first considered by Duffing 5 in 1918.…”

Non–continuation of the periodic oscillations of a forced pendulum in the presence of friction

“…(6) This problem was already considered by Bates in [2]. He proved that if c = 0, there exists a continuum of solutions (see also [11,12]).…”

“…This result is a simple and elegant application of the variational method (see [3,6] for a proof and [8] for more information on this problem).…”

“…In this case variational methods do not seem applicable to the periodic problem and Mawhin asked in [6] and [7] whether or not it was possible to extend Theorem 1.1 to (3). In [10] it was found that the answer to this question is negative.…”

Forced oscillations with rapidly vanishing nonlinearities

“…Also \zx(t)\ = \zx(t) -zx(tx)\ < ß\t-tx\, from (9). Thus, for any Ô > 0, we have \zx(t)\<ô, provided \t-tx \ < 6/ß.…”