A degree theory for compact perturbations of proper $C^{1}$ Fredholm mappings of index $0$

Abstract

We construct a degree for mappings of the form $F+K$ between Banach spaces, where $F$ is $C^1$ Fredholm of index $0$ and $K$ is compact. This degree generalizes both the Leray-Schauder degree when $F=I$ and the degree for $C^1$ Fredholm mappings of index $0$ when $K=0$. To exemplify the use of this degree, we prove the “invariance-of-domain” property when $F+K$ is one-to-one and a generalization of Rabinowitz's global bifurcation theorem for equations $F\left(\lambda ,x\right)+K\left(\lambda ,x\right)=0$.