Global Bifurcation for Second-Order Neumann Problem with a Set-Valued Term

Abstract

We study the global bifurcation of the differential inclusion of the form $-{\left(k{u}^{\prime }\right)}^{\prime }+g\left(\cdot ,u\right)\in \mu F\left(\cdot ,u\right),u^{\prime }\left(0\right)=0=u^{\prime }\left(1\right)$, where $F$ is a “set-valued representation” of a function with jump discontinuities along the line segment $\left[0,1\right]\times \left\{0\right\}$. The proof relies on a Sturm-Liouville version of Rabinowitz's bifurcation theorem and an approximation procedure.