Global inverse function theorems

It is often of interest to determine whether it is possible to invert a given mappingfof one subset ofR^ninto 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 ofR^n. Corresponding results are then proved forC^k-diffeomorphisms Another theorem, one of our main results, shows that aC^kmapfbetween two spaces of a certain general type is aC^k-diffeomorphism if and only if a certain steepest descent process can always be carried out, and for eachyconverges globally to a unique solutionxoff(x)=y. A final section is concerned with specific results for mappings that take one open rectangular region inR^ninto 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.