On a theory of linear differential equations in general analysis

Abstract

The fundamental principles of generalization have been well formulated by E. H. Moore in the statement: The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features. In applying this principle to the analogies between the theory of linear integral equations and the theory of systems of linear algebraic equations and biquadratic forms, Moore was led to the construction of a body of General Analysis, consisting of a theory of properties of classes of functions on a general range, developed in his Introduction to general analysis;t a theory of functional operations, sketched in the papers, On a form of general analysis with applications to linear differential and integral equations, t and On the fundamental functional operation of a general theory of linear integral equations;? a general theory of linear integral equations, outlined in the paper on Foundations of the theory of linear integral equations|| and the paper on Functional operations; and an existence theorem for general differential equations. ? It stands to reason that these general theories, in addition to the wide range of results obtainable as a result of specialization, may also serve as a basis for the consideration of generalizations in analogous fields. The present paper is an application of some of the concepts of the General Analysis and some of the results of the general theory of integral equations to the theory of general linear differential equations. The possibility of such an application