Global implicit function theorems
Irwin W. Sandberg · IEEE Transactions on Circuits and Systems · 1981
Three results are given concerning relations of the formf(x,y)=\theta, in whichxandyare variables in given spacesUandV, respectively, and\thetais the zero element of a third spaceW. 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 solvef(x, y)= {\theta}forxin terms ofy, with the solution map continuous. Another theorem addresses the problem of determining conditions under which given any pair(x_0, y_0)such thatf(x_0, y_0)=\theta, there is a unique continuous mapgsuch thatx_0= g(y_0)andf(g(y), y)=\thetafor ally \in V, withgindependent of sufficiently small changes in(x_0, y_0). The third result gives, under similar reasonable hypotheses, necessary and sufficient conditions under which a relationf(x, y) =\thetais equivalent tox=g(y)for some homeomorphismgofVontoU.