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′+sinu=f(t)u″+au′+sinu=f(t) with a∈Ra∈R and f∈L2(0,T)f∈L2(0,T).Using a shooting type argument, we prove the existence of at least two essentially different TT-periodic solutions under appropriate conditions on TT and ff. We also prove the existence of solutions decaying with a fixed rate α∈(0,1)α∈(0,1) by the Leray–Schauder theorem. Finally, we prove the existence of a bounded solution on [0,+∞)[0,+∞) using a diagonal argument.