Multiple critical points for a class of periodic lower semicontinuous functionals

We deal with a class of functionals I on a Banach space $X,$having the structure I=Ψ+ G, with Ψ : X → (-∞ , + ∞ ] proper, convex, lower semicontinuous andG: X → R of class C¹. Also, I isG-invariant with respect to a discrete subgroup G⊂ Xwith dim (span\ G)=N. Under some appropriate additionalassumptions we prove that I has at least $N+1$ critical orbits.As a consequence, we obtain that the periodically perturbedN-dimensional relativistic pendulum equation has at least $N+1$geometrically distinct periodic solutions.