We study a system of first-order partial differential equations with generalized functions as coefficients that describes the heat and mass transfer process in domains with thin inclusions. It is proved that the operator of the problem is continuous and injective, and a theorem on the existence and uniqueness of a generalized solution is also established.

A new two-stage method is proposed for the approximate solution of variational inequalities with pseudo-monotone and Lipschitz-continuous operators acting in a finite-dimensional linear normed space. This method is a modification of several well-known two-stage algorithms using the Bregman divergence instead of the Euclidean distance. Like other schemes using Bregman divergence, the proposed method can sometimes efficiently take into account the structure of the feasible set of the problem. A theorem on the convergence of the method is proved and, in the case of a monotone operator and convex compact feasible set, non-asymptotic estimates of the efficiency of the method are obtained.

We study a linear parabolic system in a domain with a thin low-permeable inclusion. A new formulation of the problem is obtained with unknowns (u, ω). Under this approach, the main second-order parabolic equation is converted to a set of first-order partial differential equations with distributional coefficients. The relations of the approach to the classical and weak formulations of the problem are analyzed.

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.

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.

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 solution

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.

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.

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.

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.

We consider the following linear parabolic system in a domain with a thin low-permeable insertion (“imperfect interface”):∂u∂t+q(ξ)u+∇⋅ω→=f(t,ξ),ω→=−K∇u,(t,ξ)∈Q1∪Q2⊂Rn,u|t=0=0,u|ξ∈∂Ω=0,K={kij(ξ)}i,j=1n,[(ω→,n→)Rn]=0,α[u]+limξ→ξ0(ω→,n→)Rn=0,(t,ξ0)∈Q3=Q¯1∩Q¯2. We consider a new formulation of the problem where the unknowns are (u,ω→), and the parabolic problem is converted to a first-order system of partial differential equations with distributional coefficients. We also prove inequalities for negative norms for the parabolic operator with the distributional coefficients and theorems of existence and uniqueness. For optimization problems for the processes we show existence of optimal controls, investigate smoothness of a performance criterion and give a simple condition for controllability of the system. In addition, we consider applications of the obtained results to a pulse control problem and prove convergence of a control mapping regularization procedure.

A linear parabolic equation in a disconnected domain with inhomogeneous transmission conditions of the nonideal contact type is studied. A generalized formulation of the problem is considered. An analogue of the Galerkin method is proposed for solving the problem, and the stability of the method is investigated. This makes it possible to prove existence and uniqueness theorems for the equation under different assumptions on the data smoothness.

In this paper, we consider the solvability of the Cauchy problem for pseudohyperbolic equations (partial differential equations of third order). For the case in which the right-hand side is a generalized function (distribution) of finite order, we establish a theorem on the unique solvability for a sufficiently general pseudohyperbolic operator. The method of proof is based on a specially constructed “scale” of a priori inequalities for the direct and adjoint operators.

We consider the problem of solvability and optimization for a pseudohyperbolic operator of the general form. We prove theorems on existence and uniqueness for various right-hand sides of the equation. The results obtained are applied to the problem of trajectory-final controllability.

For a sufficiently broad class of partial differential operators, we prove a theorem on homeomorphisms. Applications of this theorem to some classical operators are considered.