D. A. Klyushin

Generalized Solutions of Operator Equations and Extreme Elements

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(-i degree,i degree) 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

Concept of Generalized Solution of Nonlinear Operator Equation

The purpose of Chap. 7 is to give a natural definition of generalized solution of non-solvable non-linear equation. In Sect. 7.1 we give the definition of generalized solution of equation with operator which acts in metric spaces. In Sect. 7.2 we give the definition of new concept of near-solution of non-linear operator equation. In Sect. 7.3 and 7.4 the existence, uniqueness, and correctness of generalized solution are proved. In Sect. 7.5 and 7.6 we investigate issues of embedding of metric spaces and extension of operators which are important for the theory of generalized solutions. Sections 7.7 and 7.8 contain examples of operators and describe numerical aspects of computation of generalized solutions. Section 7.9 is devoted to the development of the theory of generalized solutions of equations in uniform spaces and proximity spaces.

The Simplest Schemes of Generalized Solution of Linear Operator Equation

The purpose of Chap. 2 is to give a natural definition of generalized solution of non-solvable linear operator equation in Banach spaces. In Sects. 2.1–2.4 two concepts of generalized solutions are proposed: strong generalized solution and weak generalized solution based on the duality. In Sect. 2.5 the theorem on existence of a weak generalized solution is proved. The relation between these two approaches is studied in Sect. 2.6 in the case of equations with linear injective operator.

General Scheme of the Construction of Generalized Solutions of Operator Equations

In Chap. 6 we propose the general approach to development of the theory of generalized solvability of linear operator equations. In Sect. 6.1 we prove the theorems on existence and uniqueness of generalized solutions of linear operator equations in locally convex linear spaces. In Sect. 6.2 we show that the proposed construction generalizes the definitions of generalized solutions given in Chaps. 2 and 3. We study the a priory inequalities method for proving theorems on existence of generalized solutions in abstract cases.

A Priori Estimates for Linear Continuous Operators

In Chap. 3 the operator equations with continuous linear injective operator which acts in linear normed spaces are considered in greater detail. The spaces of strong and weak generalized solutions defined in previous chapter do not allow any constructive description in general case. In Sects. 3.1 and 3.2 we develop a variant of the theory, using the duality techniques, that is more useful for concrete applications. In Sects. 3.3 and 3.4 we study the alternative constructive approach to general solvability, based on the duality techniques.

The Major Definitions, Concepts and Auxiliary Facts

In Chap. 1 we give major definitions, concepts and auxiliary facts concerning the theory of generalized solutions, namely: linear vector space, dual pair, linear functional, continuous functional, conjugate space, algebraically conjugate space, second conjugate space, bilinear form, linear operator, injective operator, bijective operator, surjective operator, adjoint operator, algebraically adjoint operator, intermediate space, topological space, linear normed space, embedding, dense embedding, natural embedding, strong topology, weak topology, bounded set, subdifferential, Banach space, Sobolev space, neighborhood.

Generalized Extreme Elements

Chapter is devoted to the concept of generalized solution of extreme problems. In Sect. 8.1 we give necessary motivations and the definition of generalized extreme element of bounded continuous functional defined of a bounded closed subset of a Banach space. In Sect. 8.2 we prove the theorem on existence of the generalized extreme element of a linear continuous functional. In Sect. 8.3 we study an interest issue concerning the possibility of compact dense embedding of linear normed space in Banach one. The purpose of Sects. 8.4 and 8.5 is to develop the theory of generalized solvability of convex minimization problems in infinite-dimensional Banach spaces.

Computation of Near-Solutions of Operator Equations

Chapter 5 is devoted to numerical aspects of developed theory of generalized solvability. In Sect. 5.2–5.6 we discuss conceptual aspects of reliability and effectiveness of precise and iterative method for solving system of linear algebraic equations obtained as a result of finite-dimensional approximation of initial operator equations. In Sect. 5.7 we prove the criteria of classic solvability of linear operator equations in Banach and Hilbert spaces associated with the Neumann series.

Applications of the Theory of Generalized Solvability of Linear Equations

Chapter 4 is the largest one. It is in a sense “barycenter” of the book. Here we consider typical examples of applications of generalized solutions in various branches of pure and applied analysis. In Sect. 4.1, 4.3, and 4.4 we prove theorems on generalized solvability of equations with Hilbert–Schmidt operators and Volterra equations of the first kind and describe their applications in the random processes estimation theory. In Sect. 4.2 the generalized solvability of linear operator equations in classic spaces of sequences is studied. In Sect. 4.5, 4.6, and 4.7 the theorems on generalized solvability of boundary value problems for parabolic and generalized wave equations are proved. Section 4.7 is devoted to discussion of theorems of Lax-Milgram kind in Banach and locally convex spaces.

A structural approach to solving the 6th Hilbert problem

The paper deals with an approach to solving the 6th Hilbert problem based on interpreting the field of random events as a partially ordered set endowed with a natural order of random events obtained by formalization and modification of the frequency definition of probability. It is shown that the field of events forms an atomic generated, complete, and completely distributive Boolean algebra. The probability distribution of the field of events generated by random variables is studied. It is proved that the probability distribution generated by random variables is not a measure but only a finitely additive function of events in the case of continuous random variables (both rationaland real-valued).