Upon a Theory of Infinite Systems of Non-Linear Implicit and Differential Equations

Abstract

Introduction. There has arisen, in recent years, a large literature upon infinite systems of non-linear implicit and differential equations.' The methods wrhich are employed have become classical in the treatment of finite systems. However these methods (employed in the treatment of finite systems) may be extended only to infinite systems of very special type, restricted by inequalities, which for many reasons are not fulfilled in possible applications of the method of infinitely many variables to various classical problems in analysis. Thus the theorems which have been proved by me a few years ago2 are the only ones to my knowledge which have found an application to concrete problems, in particular to the problems of Celestial Mechanics or the Calculus of Variations. I shall give here a comprehensive review of these questions without giving any essential extensions of my results as set forth in my previous papers and without assuming any previous knowledge of the theory of infinitely many variables. The applications will be excluded and the reader referred in this connection to some earlier papers appearing in the Mathemcbtische Annalen and in the Mathematische Zeitschrift. A characteristic application can be found in the paper of Martin appearing in this number of the Journal. The infinite system which I shall treat is composed exclusively of power series (and not of more general functions) of the infinite sequence of variables (as is indeed always the case in concrete applications). My problem was to introduce a method of treatment in which the power series are not subjected to inequalities which would not be fulfilled in concrete applications but which would be necessary in the classical modes of treatment. On page 252 there will be given a number of examples, of most important character, which will serve to illustratewhy the usual methods must fail. Inasmuch as the nature of the problems appears most clearly in its application to the complex space of Hilbert, the present paper will be restricted in its treatment to this space. It is of course possible by using the principles of General Analysis to further extend the methods here set forth. There is no difficulty in using other spaces ' than that of Hilbert. In the articles cited above also spaces other than that of Hilbert are treated; cf. the above mentioned paper of Martin. 241