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# Article

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Keywords:
upper solution; lower solution; order interval; truncation function; penalty function; pseudomonotone operator; coercive operator; Leray-Schauder principle; maximal solution; minimal solution
Summary:
In this note we prove the existence of extremal solutions of the quasilinear Neumann problem $-( \vert x^{^{\prime }}(t) \vert ^{p-2}x^{^{\prime }}(t))^{^{\prime }} = f(t,x(t),x ^{^{\prime }}(t))$, a.e. on $T$, $x^{^{\prime }}(0) = x^{^{\prime }}(b) =0$, $2\le p < \infty$ in the order interval $[\psi ,\varphi ]$, where $\psi$ and $\varphi$ are respectively a lower and an upper solution of the Neumann problem.
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