Aspects of Generalized Inverses in Analysis and Regularization

Abstract

This paper provides glimpses into some aspects of generalized inverses in analysis and some aspects of regularization of ill-posed linear operator equations. The paper consists of three parts. Part I, which deals with linear analysis, is devoted primarily to generalized inverses of integral operators of the first and second kinds and to related topics. Part II provides some uses of generalized inverses in nonlinear analysis (nonlinear alternative problems, inverse mapping and open mapping theorems, nonlinear operator equations and constrained optimization problems). Part III deals briefly with regularization methods in which generalized inverses play a role. Reference citations in this paper are to the Bibliography at the end of this volume. Also, the reader may refer to Section 2 of the paper on “Perturbations and Approximations…” in this volume for notation and results on generalized inverses that are used in the present paper.