Degree for weakly upper semicontinuous perturbations of quasi-$m$-accretive operators

In the paper we provide the construction of a coincidence degree being a homotopy invariant detecting the existence of solutions of equations or inclusions of the form $Ax\in F(x)$, $x\in U$, where $A\colon D(A)\multimap E$isan$m$-accretiveoperatorinaBanachspace$ E$,$F\colon K\multimap E$ is a weakly upper semicontinuous set-valued map constrained to an open subset $U$ of a closed set $K\subset E$. Two different approaches will be presented. The theory is applied to show the existence of nontrivial positive solutions of some nonlinear second order partial differential equations with discontinuities.