An approximation theory for generalized Fredholm quadratic forms and integral-differential equations

Abstract

ABSTRACT. An approximation theory is given for a very general class of elliptic quadratic forms which includes the study of 2nth order (usually in integrated form), selfadjoint, integral-differential equations. These ideas follows in a broad sense from the quadratic form theory of Hestenes, applied to integraldifferential equations by Lopez, and extended with applications for approximation problems by Gregory.The application of this theory to a variety of approximation problem areas in this setting is given. These include focal point and focal interval problems in the calculus of variations/optimal control theory, oscillation problems for differential equations, eigenvalue problems for compact operators, numerical approximation problems, and finally the intersection of these problem areas.In the final part of our paper our ideas are specifically applied to the construction and counting of negative vectors in two important areas of current applied mathematics: In the first case we derive comparison theorems for generalized oscillation problems of differential equations. The reader may also observe the essential ideas for oscillation of many nonsymmetric (indeed odd order)ordinary differential equation problems which will not be pursued here. In the second case our methods are applied to obtain the "Euler-Lagrange equations"for symmetric tridiagonal matrices. In this significant new result (which will allow us to reexamine both the theory and applications of symmetric banded matrices) we can construct in a meaningful way, negative vectors, oscillation vectors, eigenvectors, and extremal solutions of classical problems as well as faster more efficient algorithms for the numerical solution of differential equations.In conclusion it appears that many physical problems which involve symmetric differential equations are more meaningful presented as integral differential equations (effects of friction on physical processes, etc.). It is hoped that this paper will provide the general theory and present examples and methods to study integral differential equations.Received by the editors August 27, 1973 and, in revised form, April 8, 1975 AMS (MOS) subject classifications (1970). Primary 45J05, 49A30, 34C10.Key words and phrases. Approximation theory, conjugate points, oscillations, calculus of variations, Fredholm integral differential equations, spline approximations. 1. Introduction. The main purpose of this paper is to present an approximation theory of quadratic forms which is applicable to a very general class of quadratic forms and to linear selfadjoint operators of a generalized Fredholm type. That is 2wth order, integral-differential systems such as T»(r) = vß~l(t), or as the generalized system: