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

TL;DR

This paper develops a new coincidence degree theory for $m$-accretive operators with weakly upper semicontinuous perturbations, and applies it to prove existence of solutions for certain nonlinear PDEs with discontinuities.

Contribution

It introduces a novel homotopy invariant coincidence degree for weakly upper semicontinuous perturbations of quasi-$m$-accretive operators, extending solution existence results.

Findings

Established a new coincidence degree for weakly upper semicontinuous perturbations.

Proved the existence of nontrivial positive solutions for nonlinear PDEs with discontinuities.

Applied the theory to specific second order PDEs to demonstrate its effectiveness.

Abstract

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 $A\colon D(A)\multimap E$ is an $m$-accretive operator in a Banach space $ 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.