The pendulum equation: from periodic to almost periodic forcings

Abstract

where a > 0 is a parameter and p : R → R is an almost periodic function. Almost periodicity will be understood in the classical sense defined by Bohr [7]. The existence of almost periodic solutions has been already discussed in several papers by Blot [5], Mawhin [19, 20] and by Belley and Saadi Drissi [2]. Also the papers by Fink [10] and by Fournier, Szulkin and Willem [12] contain results applicable to (1). In all these works there is some restriction on the size of the forcing. This size is measured with respect to different norms of p, always with the intention of locating the solution on an interval where the sine function is decreasing, say ]π2 , 3π 2 [. The possible novelty of the present paper is that it searches for results valid for forcings of arbitrary size. There are many other papers on the forced pendulum equation but they deal with the periodic case. See [20] for a recent survey. A nice feature of the periodically forced pendulum is that most of the methods of Nonlinear Analysis can be applied and lead to interesting conclusions. In this sense the ∗Supported by D.G.I. MTM2005-03483, Ministerio de Educacion y Ciencia, Spain