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Existence of two solutions for quasilinear periiodic differential equations
with discontinuities

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*Nikolaos S. Papageorgiou and Francesca Papalini*

**Address**: N. S. Papageorgiou, National Technical University, Department
of Mathematics, Zografou Campus,

Athens 15780, Greece

F. Papalini, University of Ancona, Department of Mathematics, Via Brecce
Bianche, Ancona 60131, Italy

**E-mail:** npapg@math.ntua.gr

papalini@dipmat.unian.it

**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.

**AMSclassification. **34B15

**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.