Multiple periodic solutions of second order parameter-dependent equations via rotation numbers

We investigate the existence of multiple periodic solutions for a class of second order parameter-dependent equations of the form $x^{″}+f(t,x)=sp(t)$. We compare the behavior of its solutions with suitable linear and piecewise linear equations near positive infinity and infinity. Furthermore, in this context, the nonlinearity $f$ does not satisfy the usual sign condition, and the global existence of solutions for the Cauchy problem associated to the equation is not guaranteed. Our approach is based on the Poincaré-Birkhoff twist theorem, a rotation number approach and the phase-plane analysis. Our result generalizes the result in Fonda and Ghirardelli ^[1] for second order parameter-dependent equations.