Basic Analytic Degree Theory of a Mapping

Abstract

The problem of finding the number of solutions of a given equation has engaged a number of mathematicians. Brouwer, Bohl, Cauchy, Descartes, Gauss, Hadamard, Hermite, Jacobi, Kronecker, Ostrowski, Picard, Sturm and Sylvester had contributed to this topic. The Argument principle propounded by Cauchy on the zeros of a function inside a domain and Sturm’s theorem on the number of zeros of a real polynomial in a closed bounded interval have evolved into Degree theory of mappings. Even as the degree of a nonconstant polynomial gives the number of zeros of a polynomial, the degree of a mapping provides the number of zeros of nonlinear mapping in a domain. In this chapter, an elementary degree theory of mappings is described from an analytic point of view proposed by Heinz [3]. For more elaborate treatment, Cronin [1], Deimling [2], Lloyd [4] Outerelo and Ruiz [6] and Rothe [7] may be referred. It should be mentioned that Ortega and Rheinboldt [5] had provided a more accessible version of Heinz’s treatment.