A degree theory for compact perturbations of properC1 Fredholm mappings of index 0

We construct a degree for mappings of the form F +K between Banach spaces, where F is C1 Fredholm of index 0 and K is compact. This degree generalizes both the LeraySchauder degree when F = I and the degree for C1 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(λ,x) +K(λ,x)= 0.