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Keywords:
pseudomonotone operator; $L$-pseudomonotonicity; operator of type $(S)_{+}$; operator of type $L$-$(S)_{+}$; coercive operator; surjective operator; evolution triple; compact embedding; multifunction; upper solution; lower solution; extremal solution; $R_{\delta }$-set
pseudomonotone operator; $L$-pseudomonotonicity; operator of type $(S)_{+}$; operator of type $L$-$(S)_{+}$; coercive operator; surjective operator; evolution triple; compact embedding; multifunction; upper solution; lower solution; extremal solution; $R_{\delta }$-set
Summary:
In this paper we study nonlinear parabolic equations using the method of upper and lower solutions. Using truncation and penalization techniques and results from the theory of operators of monotone type, we prove the existence of a periodic solution between an upper and a lower solution. Then with some monotonicity conditions we prove the existence of extremal solutions in the order interval defined by an upper and a lower solution. Finally we consider problems with discontinuities and we show that their solution set is a compact $R_{\delta }$-set in $(CT,L^2(Z))$.
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