Mapping Degree for A-Proper Operators

Abstract

In Part I we demonstrated the fundamental importance of the Leray-Schauder mapping degree for operator equations involving compact operators. In this chapter we will generalize the Leray-Schauder mapping degree deg(I − C, G, b) for compact operators $$ to a mapping degree

$$$$

for A-proper operators $$. In this connection, we use the same approximation process as in Chapter 12. However, in contrast to Chapter 12, now the mapping degree of the equations being approximated need not tend to a unique limit value; hence, DEG(A, G, b) is, in the general case, a set of integers.

The concept of degree of mapping, in all its different forms, is one of the most effective tools for studying the properties of existence and multiplicity of solutions of nonlinear equations.

Felix E. Browder (1983)