# Article

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Keywords:
upper and lower solutions; weak solution; evolution triple; compact embedding; distributional derivative; operator of type $(S)_{+}$; operator of type $L-(S)_{+}$; $L$-pseudomonotone operator; multivalued problem; extremal solutions; Zorn’s lemma
Summary:
We consider nonlinear parabolic boundary value problems. First we assume that the right hand side term is discontinuous and nonmonotone and in order to have an existence theory we pass to a multivalued version by filling in the gaps at the discontinuity points. Assuming the existence of an upper solution $\phi$ and of a lower solution $\psi$ such that $\psi \le \phi$, and using the theory of nonlinear operators of monotone type, we show that there exists a solution $x \in [\psi ,\phi ]$ and that the set of all such solutions is compact in $W_{pq}(T)$. For the problem with a Caratheodory right hand side we show the existence of extremal solutions in $[\psi ,\phi ]$.
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