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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
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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