Multiple periodic solutions of infinite-dimensional pendulum-like equations

We prove the multiplicity of periodic solutions for an equation in a separable Hilbert space $H$, with $T$-periodic dependence in time, of the type $$ ddot x+{cal A}x+ abla_xV(t,x)=e(t),. $$ Here, ${cal A}$ is a semi-negative definite bounded selfadjoint operator, with nontrivial null-space ${cal N}({cal A})$, the function $V(t,x)$ is bounded above, periodic in $x$ along a basis of ${cal N}({cal A})$, with $ abla_xV$ having its image in a compact set, and $e(t)$ has mean value in ${cal N}({cal A})^perp$. Our results generalize several well-known theorems in the finite-dimensional setting, as well as a recent existence result by Boscaggin, Fonda and Garrione.