Implicit Functions and Their Differentials in General Analysis

Implicit function theorems occur in analysis in many different forms, and have a fundamental importance.Besides the classical theorems as given for example in Goursat's Cours d'Analyse and Bliss's Princeton Colloquium Lectures, and the classical theorems on linear integral equations, implicit function theorems in the domain of infinitely many variables have been developed by Volterra, Evans, Levy, W. L. Hart and others.ÎThe existence and imbedding theorems for solutions of differential equations, as treated for example by Bliss, § have also received extensions to domains of infinitely many variables by Moulton, Hart, Barnett, Bliss and others.If Special properties in the case of linear differential equations in an infinitude of variables have been treated by Hart and Hildebrandt.|| On the other hand, Hahn and Carathéodory** have made important generalizations of the notion of differential equation by removing continuity restrictions on the derivatives and by writing the equations in the form of integral equations.