Quadratic differential equations in Banach spaces and analytic functionals

Citation

Reed, Irving S. (1949) Quadratic differential equations in Banach spaces and analytic functionals. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-06242004-135659

Abstract

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In Chapter I the preliminary concepts are introduced and applied to the quadratic differential equations, [...], in a complete normed linear ring. In Chapters II and V the functional equation, [...] is studied under suitable assumptions and shown to be related to the Frechet differential of the above differential equation as a functional of [...]. In Chapter IV the more general functional equation, [...], where T is an endomorphism, is treated.

The existence of the Frechet differential with respect to [...] is shown in Chapter II by means of a power series for the solution of [...] where Q is a trilinear function. The solution of [...] is obtained in Chapter VI in the terminology of Chapter IV. Moreover, the solution is shown to satisfy unquely the differential system, [...], [...] and to possess a generalized Taylor series expansion. The above equation is generalized with similar results to [...], where T is a trilinear function.

Chapter VII is concerned with examples of Chapter VI. For instance, the solution of the matrix differential equation, [...], [...], is treated both as a function of [...] and as an analytic functional of the [...] continuous functions [...].

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