Ill-Posed Problems and Integral Equations

摘要

This document addresses 5 substantial problems in the theory and numerical analysis of ill-posed problems and integral equations: (1) Collocation, and Galerkin methods for Volterra and Abel equations of the first kind; (2) Galerkin and collocation methods for nonlinear Abel-Volterra integral equations on the half-line and on a finite interval; (3) a new approach to classification and regularization of ill-posed operator equations, and quantification of ill-posedness; (4) operator external problems in the theory of compensation and representation of control systems; (5) constrained least-squares solutions of linear inclusions and singular control problems in Hilbert space. New notions of bivariational and singular variational derivatives for functionals are also studies. They will be applied to extend the von Mises calculus for statistical functionals and its applications to robustness and approximation theorems. Keyboards: Multivalued linear mappings; Kernel functions.