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 Leray-Schauder 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.