Singular points of functional equations

Abstract

Hildebrandt and Graves [9](2) showed that if the partial differential dxb(0, 0; h), considered as a linear transformation of X, has a continuous everywhere-defined inverse, then there exists a unique continuous singlevalued function q defined on a neighborhood of the origin in 2) with values in X such that 4(0) =0 and b [k(y), y] =0 for all y in this neighborhood. Graves [8; 2, p. 408] showed that if dx4(0, 0; h) maps onto X, then there will be at least one solution corresponding to sufficiently small y. Cronin [3] recently considered a case in which dA need not map onto X and obtained, under suitable restrictions, theorems concerning the existence of solutions in terms of the topological degree theory. While our methods are closely related to hers, we focus our attention on the problem of studying the branching of the solutions that this situation allows. In a particular case we are able to apply Dieudonne's modification [5] of the Newton polygon method to obtain results exactly parallel to some for ordinary algebraic functions over the real or complex field. It is also seen that the work of E. Schmidt [17], L. Lichtenstein [14], and R. Iglisch [12] for a class of nonlinear integral equations hold valid for a general class of functions defined on Banach spaces. Also, in both their work and that of T. Shimizu [18], the assumption of analyticity can be replaced by that of the existence of a few continuous derivatives. Further, because of the simpler form for the equations we derive, it is possible to study particular cases in terms of initially given data. Our final part indicates briefly how these results can be applied to nonlinear differential equations with fixed end point boundary conditions. It is possible to treat questions of existence and uniqueness of solutions in the neighborhood of a given solution for a very general type of equation.