Local mapping relations and global implicit function theorems
Introduction.Consider the problem of solving a nonlinear equation (1) F(x,y) = zwith respect to y for given x and z.If, for example, x, y, z are elements of some Banach space, then under appropriate conditions about F, the well-known implicit function theorem ensures the "local" solvability of (1).The question arises when such a local result leads to "global" existence theorems.Several authors, for example, Cesari [2], Ehrmann [5], Hildebrandt and Graves [8], and Levy [9], have obtained results along this line.But these results are closely tied to the classical implicit function theorem, and no general theory appears to exist which permits the deduction of global solvability results once a local one is known.The problem has considerable interest in numerical analysis.In fact, when an operator equation(2) Fy = x is to be solved iteratively for y, the process converges usually only in a neighborhood of a solution.If for some other operator F0 the equation F0y = x has been solved, then one may try to "connect" F and F0 by an operator homotopy H(t,y) such that 77(0, y) = F0y and H(l,y) = Fy, and to "move" along the solution "curve" y(t) of H(t,y)=x, O^i^l, from the known solution y(0) to the unknown y(l), for instance, by using a locally convergent iterative process.This is the so-called "continuation method".For a review of earlier work about this method see, for example, the introduction of Ficken [6] who then proceeds to develop certain results about the global solvability of H(t, y) = x by using the local solvability provided by the implicit function theorem.Other results are due to Davidenko [3]; see also Yakovlev [14] and, more recently, Davis [4], and Meyer [10].In this paper we shall consider the problem of finding global existence theorems for (1) in the following setting.For fixed z the problem depends only on the (multivalued) relation between x and y given by (1).We therefore consider abstract