## Abstract and Applied Analysis

### A degree theory for compact perturbations of proper $C^{1}$ Fredholm mappings of index $0$

#### Abstract

We construct a degree for mappings of the form $F+K$ between Banach spaces, where $F$ is $C^{1}$ Fredholm of index $0$ and $K$ is compact. This degree generalizes both the Leray-Schauder degree when $F=I$ and the degree for $C^{1}$ 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 $F(\lambda,x)+K(\lambda,x)=0$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2005, Number 7 (2005), 707-731.

Dates
First available in Project Euclid: 3 October 2005

https://projecteuclid.org/euclid.aaa/1128345977

Digital Object Identifier
doi:10.1155/AAA.2005.707

Mathematical Reviews number (MathSciNet)
MR2202179

Zentralblatt MATH identifier
1117.47049

#### Citation

Rabier, Patrick J.; Salter, Mary F. A degree theory for compact perturbations of proper $C^{1}$ Fredholm mappings of index $0$. Abstr. Appl. Anal. 2005 (2005), no. 7, 707--731. doi:10.1155/AAA.2005.707. https://projecteuclid.org/euclid.aaa/1128345977

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