Periodic solutions of planar Hamiltonian systems with asymmetric nonlinearities

In this paper, we look for periodic solutions of planar Hamiltonian systems $$\left \{ \textstyle\begin{array}{l} x'=f(y)+p_{1}(t,y),\\ y'=-g(x)+p_{2}(t,x). \end{array}\displaystyle \right . $$ By using the Poincare-Birkhoff twist theorem, we prove the existence and multiplicity of periodic solutions of the given system when f satisfies an asymmetric condition and the related time map satisfies an oscillating condition.