Some results on the forced pendulum equation

Abstract This paper is devoted to the study of the forced pendulum equation in the presence of friction, namely u ″ + a u ′ + sin u = f ( t ) with a ∈ R and f ∈ 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 α ∈ ( 0 , 1 ) by the Leray–Schauder theorem. Finally, we prove the existence of a bounded solution on [ 0 , + ∞ ) using a diagonal argument.