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The Neumann problem for quasilinear differential equations

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*Tiziana Cardinali, Nikolaos S. Papageorgiou and Raffaella Servadei*

**Address.**University of Perugia,
Department of Mathematics and Computer Science,
via Vanvitelli 1, Perugia 06123, Italy

National Technical University,
Department of Mathematics,
Zografou Campus, Athens 15780, Greece

University of Roma `Tor Vergata',
Department of Mathematics,
via della Ricerca Scientifica, Roma 00133, Italy

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

**Abstract.**
In this note we prove the existence of extremal
solutions of the quasilinear Neumann problem $-(|x'(t)|^{p-2}x'(t))'
= f(t, x(t), x'(t))$, a.e. on $T$, $x'(0) =
x'(b) =0$, $2\leq p < \infty$ in the order interval
$[\psi,\varphi]$, where $\psi$ and $\varphi$ are respectively a
lower and an upper solution of the Neumann problem.

**AMSclassification. **35J60, 35J65.

**Keywords. ** Upper solution, lower solution,
order interval, truncation function, penalty function,
pseudomonotone operator, coercive operator, Leray-Schauder
principle, maximal solution, minimal solution.