Some results on the forced pendulum equation

This paper is devoted to the study of the forced pendulum equation in the presence of friction, namely u '' + au' + sin u = f(t) with a epsilon R and f epsilon L-2(0, T). Using a shooting type argument, we prove the existence of at least two essentially different T-periodic solutions under appropriate conditions on T and f. We also prove the existence of solutions decaying with a fixed rate alpha epsilon (0, 1) by the Leray-Schauder theorem. Finally, we prove the existence of a bounded solution on [0, +infinity) using a diagonal argument. (c) 2007 Elsevier Ltd. All rights reserved.