### Synopses & Reviews

The 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 a general functional analytic approach to solving operator equations in a general form. This unique and valuable monograph presents recently obtained results in the study of the generalized solutions of operator equations and extreme elements in linear topological spaces. The results are clearly and thoroughly presented and offer new methods of identifying these solutions and studying their properties. These methods are based on a priori estimations and a general topological approach to construct generalized solutions of linear and nonlinear operator equations. This volume is intended for mathematicians, graduate students and researchers studying functional analysis, operator theory, and the theory of optimal control. Prerequisites include knowledge of basic functional analysis.

#### Review

From the reviews: "This monograph principally deals with a unified approach to the notion of generalized solutions for linear and nonlinear operator equations between Banach spaces X and Y and, to a smaller degree, to the notion of generalized solutions to extremal problems for real functions defined on a set M of a Banach space. ... this book is written sufficiently well and is easy readable. Undoubtedly, it will be interesting for all specialists in Functional Analysis and Partial Differential Equations." (P. P. Zabreĭko, Mathematical Reviews, Issue 2012 j) "This is a well-written, self-contained and easily readable monograph, with a reasonable number of references to English written papers and books. It can be recommended to everyone interested in the topic and having some background in functional analysis and topology." (Robert Plato, Zentralblatt MATH, Vol. 1240, 2012)

#### Review

From the reviews:

"This monograph principally deals with a unified approach to the notion of generalized solutions for linear and nonlinear operator equations between Banach spaces X and Y and, to a smaller degree, to the notion of generalized solutions to extremal problems for real functions defined on a set M of a Banach space. ... this book is written sufficiently well and is easy readable. Undoubtedly, it will be interesting for all specialists in Functional Analysis and Partial Differential Equations." (P. P. Zabreĭko, Mathematical Reviews, Issue 2012 j)

"This is a well-written, self-contained and easily readable monograph, with a reasonable number of references to English written papers and books. It can be recommended to everyone interested in the topic and having some background in functional analysis and topology." (Robert Plato, Zentralblatt MATH, Vol. 1240, 2012)

#### Synopsis

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.

#### Synopsis

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.

### Table of Contents

Preface 1. Fundamental notions, general and auxiliary facts 2. Simplest schemes of generalized solution of linear operator equations 2.1. Strong generalized solution 2.2. Strong almost solution 2.3. Weak generalized solution 2.4. Weak almost solution 2.5. Unique existence of weak generalized solution 2.6. Relationship between weak and strong generalized solutions 3. A priori estimations for linear continuous operator 3.1. A priori inequalities 3.2. Generalized solution of operator equation in Banach spaces 3.3. Generalized solution of operator equation in locally convex topological spaces 3.4. Relationship between generalized solutions in Banach and locally convex topological spaces 4. Applications of the theory of generalized solvability of linear equations 4.1. Equations with Hilbert-Schmidt operator in Hilbert space L2(-,) 4.2. Generalized solution of infinite system of linear algebraic equations 4.3. Volterra Integral Equation of the First Kind 4.4. Statistics of random processes 4.5. Parabolic PDE in a connected domain 4.6. Parabolic PDE in a disconnected domain 5. Scheme of generalized solutions of linear operator equations 5.1. Generalized solution of linear operator equations in locally convex linear topological spaces 5.2. Examples of generalized solutions 5.3. Properties of generalized solutions in spaces E1, E2 6. Scheme of generalized solutions of nonlinear operator equations 6.1. Generalized solution of nonlinear operator equation 6.2. Almost solution of nonlinear operator equations 6.3. Unique existence of generalized solution 6.4. Correctness of generalized solutions 6.5. Pseudo-generalized and essentially generalized solutions 6.6. Embedding of space of pseudo-generalized solutions into space of generalized solutions 6.7. Examples of operators 6.8. Computation of generalized solution 7. Generalized extreme elements 7.1. Examples of generalized extreme elements 7.2. Generalized extreme elements for linear and positively homogeneous convex functional 7.3. Generalized extreme elements for general convex functional 7.4. Some remarks Reference