Generalized Solutions of Operator Equations and Extreme Elements [Hardcover]

Abstract models for many problems in science and engineering take the form of an operator equation. The resolution of these problems often requires determining the existence and uniqueness of solutions to these equations. Generalized Solutions of Operator Equations and Extreme Elements presents recently obtained results in the study of the generalized solutions of operator equations and extreme elements in linear topological spaces. The presented results offer new methods of identifying these solutions and studying their properties. These new methods involve the application of a priori estimations and a general topological approach to construct generalized solutions of linear and nonlinear operator equations. The monograph is intended for mathematicians, graduate students and researchers studying functional analysis, operator theory, and the theory of optimal control.This book examines recent results in the study of the generalized solutions of operator equations and extreme elements in linear topological spaces. The material presented here offers new methods of identifying these solutions and studying their properties.Preface1. Fundamental notions, general and auxiliary facts2. Simplest schemes of generalized solution of linear operator equations2.1. Strong generalized solution2.2. Strong almost solution2.3. Weak generalized solution2.4. Weak almost solution2.5. Unique existence of weak generalized solution2.6. Relationship between weak and strong generalized solutions3. A priori estimations for linear continuous operator3.1. A priori inequalities3.2. Generalized solution of operator equation in Banach spaces3.3. Generalized solution of operator equation in locally convex topological spaces3.4. Relationship between generalized solutions in Banach and locally convex topological spaces4. Applications of the theory of generalized solvability of linear equations4.1. Equations with Hilbert-Schmidt operator in Hilbert space L2(-p,p)4.2. Generalized solution of infinitelC,