Abstract. In this paper we examine a quasilinear
periodic problem driven by the one- dimensional $p$-Laplacian and
with discontinuous forcing term $f$. By filling in the gaps at the
discontinuity points of $f$ we pass to a multivalued periodic
problem. For this second order nonlinear periodic differential
inclusion, using variational arguments, techniques from the theory
of nonlinear operators of monotone type and the method of upper
and lower solutions, we prove the existence of at least two non
trivial solutions, one positive, the other negative.
Keywords. One dimensional $p$-Laplacian, maximal
monotone operator, pseudomonotone operator, generalized
pseudomonotonicity, coercive operator, first nonzero
eigenvalue, upper solution, lower solution, truncation map,
penalty function, multiplicity result.