V. V. Semenov

A hybrid method without extrapolation step for solving variational inequality problems

In this paper, we introduce a new method for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. The iterative process is based on well-known projection method and the hybrid (or outer approximation) method. However we do not use an extrapolation step in the projection method. The absence of one projection in our method is explained by slightly different choice of sets in the hybrid method. We prove a strong convergence of the sequences generated by our method.

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.