Global inverse function theorems

It is often of interest to determine whether it is possible to invert a given mapping of one subset of into another. This paper reports on a collection of pertinent results and related material. We first prove a theorem which is used in latter sections of the paper. It provides several characterizations of a homeomorphism between open subsets of . Corresponding results are then proved for -diffeomorphisms Another theorem, one of our main results, shows that a map between two spaces of a certain general type is a -diffeomorphism if and only if a certain steepest descent process can always be carried out, and for each converges globally to a unique solution of . A final section is concerned with specific results for mappings that take one open rectangular region in into another. These results are of direct interest, for example, in the area of economics, where they provide definitive invertibility conditions for a large class of price-demand relations.