Inverse Functions and Existence Principles

Abstract

Each one of six general existence principles of compactness (the extreme value theorem), completeness (the Newton method or the modified Newton method), topology (Brouwer’s fixed point theorem), homotopy (on contractions of a sphere to its center), variational analysis (Ekeland’s principle), and monotonicity (the Minty–Browder theorem)) is shown to lead to the inverse function theorem, each one giving some novel insight. There are differences in assumptions and algorithmic properties; some of the propositions have been constructed specially for this paper. Simple proofs of the last two principles are included. The proof by compactness is shorter and simpler than the shortest and simplest known proof, that by completion. This gives a very short self-contained proof of the Lagrange multiplier rule, which depends only on optimization methods. The proofs are of independent interest and are intended as well to be useful in the context of the ongoing efforts to obtain new variants of methods that are based on the inverse function theorem, such as comparative statics methods.