By the use of a generalized version of the Poincare'–Birkhoff
fixed point theorem, we prove the existence of at least two periodic
solutions for a class of Hamiltonian systems in the plane, having in
mind the forced pendulum equation as a particular case. Our approach
is closely related to the one used by Franks in [14]. We thus provide a
new proof of a theorem by Mawhin and Willem [26], originally obtained
by the use of variational methods.
