ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

# resolvent

## Mixed Problem for a Homogeneous Wave Equation with a Nonzero Initial Velocity and a Summable Potential

For a mixed problem defined by a wave equation with a summable potential equal-order boundary conditions with a derivative and a zero initial position, the properties of the formal solution by the Fourier method are investigated depending on the smoothness of the initial velocity u′t(x, 0) = ψ(x). The research is based on the idea of A. N. Krylov on accelerating the convergence of Fourier series and on the method of contour integrating the resolvent of the operator of the corresponding spectral problem.

## An Analogue of the Jordan–Dirichlet Theorem for the Integral Operator with Kernel Having Jumps on Broken Lines

In this paper the sufficient conditions (conditions such as Jordan–Dirichlet) expansion function f(x) in a uniformly convergent series of eigenfunctions and associated functions of the integral operator whose kernel is suffering jumps on the sides of the square, inscribed in the unit square. As is known, this expansion requires to f(x) is continuous and belong to the closure of the integral values operator. It turns out that if f(x) also is a function of bounded variation, these conditions are also sufficient.

## Integral operator with kernel having jumps on broken lines

In this paper we study equiconvergence expansions in trigonometric Fourier series, and in eigenfunctions and associated functions of an integral operator whose kernel suffers jumps at the sides of the square inscribed in the unit square.

## Riescz Basis Property of Eigen and Associated Functions of Integral Operators with Discontinuous Kernels, Containing Involution

For invertible integral operator which kernel is discontinuous on the diagonals of the unit square Riescz basis property of its eigen and associated functions in L2[0, 1] is proved.

## About the Classical Solution of the Mixed Problem for the Wave Equation

The classic solution of the mixed problem for a wave equation with a complex potential and minimal smoothness of initial data is established by the Fourier method. The resolvent approach consists of constructing formal solution with the help of the Cauchy – Poincaré method of integrating the resolvent of the corresponding spectral problem over spectral parameter. The method requires no information about eigen and associated functions and uses only the main part of eigenvalues asymptotics.

## Justification of Fourier Method in a Mixed Problem for Wave Equation with Non-zero Velocity

In the paper, using contour integration of the resolvent of the corresponding spectral problem operator, justification of Fourier method in two mixed problems for wave equation with trivial initial function and non-zero velocity is given. The boundary conditions of these problems, together with fixed endpoint conditions, embrace all cases of mixed problems with the same initial conditions for which the corresponding spectral operators in Fourier method have regular boundary conditions.

## On Riescz Bases of Eigenfunction of 2-nd Order Differential Operator with Involution and Integral Boundary Conditions

Riesz basisness with brackets of the eigen and associated function is proved for a 2-nd order differential operator with involution in the derivatives and with integral boundary conditions. To demonstrate this the spectral problem of the initial operator is reduced to the spectral problem of a 1-st order operator without involution in the 4-dimensional vector-function space.

## Resolvent Approach to Fourier Method in a Mixed Problem for Non-homogeneous Wave Equation

Fourier method of obtaining classic solution is being justified in a mixed problem for non-homogeneous wave equation with a complex potential and fixed boundary conditions under minimal conditions on initial data. The proof is based on resolvent approach which does not need any information on eigen and associated functions of the corresponding spectral problem.

## Solution of Integral Equations via Resolvents of Simplest Differential Operators

Families of operators for approximate solution of integral equations with unbounded inverse operators and right parts of equations speciﬁed by the root-mean-square approximations are constructed.

## Approximate Solution of an Optimal Control Problem with Linear Nonhomogeneous Control System in Hilbert Space

For an optimal control problem with a linear differential equation in Hilbert space and quadratic criteria necessary and sufficient conditions of control functions optimality and approximate formulas of the expansions of these functions in eigenfunctions of the control system operator have been obtained.