On a general iterative method for the approximate solution of linear operator equations

Abstract

1. Introduction. A number of iterative procedures have been developed for the approximate solution of a linear operator equation of the form Au = f, where f is a given element in some suitably normed linear space and A is either a matrix, an integral, or an abstract operator in this space. The purpose of this paper is to unify and extend investigations of finite algebraic systems by von Mises and Polaczek [10], Cesari and Picone [4], Quade [12], Keller [8, 9] and others [19]; integral equations of the Fredholm type by Neumann [5], Wiarda [18], Bilckner [2, 3], Wagner [17], Samuelson [14], and Fridman [7]; and operator equations in abstract Hilbert or Banach spaces by Sch6nberg [15], Rail [13], Bialy [1], and Petryshyn [11]. This generalization and unification of various methods in terms of conditions for convergence and error estimates is accomplished by studying a rather general iteration procedure of which the above methods are special cases. It is hoped that the procedure presented here can be used as a basis for possible discovery of new iterative methods when applied to concrete problems.