Numerically Implementable Existence Proofs

Abstract

Existence theorems are among the most frequently invoked theorems of mathematics since they assure that a solution to some equation exists. Some of the celebrated examples are the fundamental theorem of algebra, the fixed point theorems of Banach, Brouwer, Leray & Schauder, and Kakutani. With the exception of the Banach fixed point theorem, the classical statements of the above theorems merely assert the existence of a fixed point or a zero point of a map, but their traditional proofs in general do not offer any means of actually obtaining the fixed point or zero point. Many of the classical proofs of fixed point theorems can be given via the concept of the Brouwer degree. We will not need this concept in our subsequent discussions. However, for readers wishing to read up on degree theory we can suggest the books of Amann (1974), Berger (1977), Cronin (1964), Deimling (1974) or Schwartz (1969).