A degree theory for compact perturbations of properC1Fredholm mappings of index0

We construct a degree for mappings of the form F+K betweenBanach spaces, where F is C1 Fredholm of index0 and K is compact. This degree generalizesboth the Leray-Schauder degree when F=I and the degree forC1 Fredholm mappings of index 0 when K=0. To exemplifythe use of this degree, we prove the “invariance-of-domain”property when F+K is one-to-one and a generalization ofRabinowitz's global bifurcation theorem for equationsF(λ,x)+K(λ,x)=0