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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

abstract and applied analysis

A degree theory for compact perturbations of proper<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$C^{1}$"><mml:msup><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math>Fredholm mappings of index<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext=" $0$"><mml:mn>0</mml:mn></mml:math>

A degree theory for compact perturbations of properC1Fredholm mappings of index0