Global implicit function theorems

Three results are given concerning relations of the form f (x, y) =, in which x and y are variables in given spaces U and V, respectively, and is the zero element of a third space W. Such relations often arise in applications. Under reasonable hypotheses, and in a general normed linear space setting, one of the theorems provides necessary and sufficient conditions under which it is possible to globally and uniquely solve f (x, y) = for x in terms of y, with the solution map continuous. Another theorem addresses the problem of determining conditions under which given any pair (x₀, y₀) such that f (x₀, y₀) =, there is a unique continuous map g such that x₀= g (y₀) and f (g (y), y) = for all y V, with g independent of sufficiently small changes in (x₀, y₀). The third result gives, under similar reasonable hypotheses, necessary and sufficient conditions under which a relation f (x, y) = is equivalent to x=g (y) for some homeomorphism g of V onto U.