Global implicit function theorems

Three results are given concerning relations of the form , in which and are variables in given spaces and , respectively, and is the zero element of a third space . 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 for in terms of , with the solution map continuous. Another theorem addresses the problem of determining conditions under which given any pair such that , there is a unique continuous map such that and for all , with independent of sufficiently small changes in . The third result gives, under similar reasonable hypotheses, necessary and sufficient conditions under which a relation is equivalent to for some homeomorphism of onto .