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 ∈ F ( x ), x ∈ U , where A : D ( A ) ⊸ E is an m -accretive operator in a Banach space E , F : K ⊸ E is a weakly upper semicontinuous set-valued map constrained to an open subset U of a closed set K ⊂ E . Two different approaches are presented. The theory is applied to show the existence of non-trivial positive solutions of some nonlinear second-order partial differential equations with discontinuities. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.