Abstract and Applied Analysis 
Volume 2005 (2005), Issue 7, Pages 707-731
doi:10.1155/AAA.2005.707

A degree theory for compact perturbations of proper C1 Fredholm mappings of index 0

Patrick J. Rabier1 and Mary F. Salter2

 1Department of Mathematics, University of Pittsburgh, Pittsburgh 15260, PA, USA
2Department of Mathematics and Computer Science, Franciscan University of Steubenville, Steubenville 43952, OH, USA
Received 14 June 2004

Abstract

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.