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Multiple solutions for nonlinear periodic problems with discontinuities

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*Nikolaos S. Papageorgiou and Nikolaos Yannakakis*

**Address.** National Technical University, Department of Mathematics,
Zografou Campus\newline Athens 157 80, GREECE
**E-mail:** npapg@math.ntua.gr

**Abstract.** In this paper we consider a periodic problem driven
by the one dimensional $p$-Laplacian and with a discontinuous right hand
side. We pass to a multivalued problem, by filling in the gaps at the discontinuity
points. Then for the multivalued problem, using the nonsmooth critical
point theory, we establish the existence of at least three distinct periodic
solutions.

**AMSclassification.** 34C25.

**Keywords.** Multiple solutions, periodic problem, one-dimensional
$p$-Laplacian, discontinuous vector field, nonsmooth Palais-Smale condition,
locally Lipschitz function, generalized subdifferential, critical point,
Saddle Point Theorem, Ekeland variational principle.