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

# дифференциальный оператор

## About Differential Operators and Matrices of the Second Order

Differential operators of the second order are studied. Conditions of their invertibility are obtained. The main results are obtained on the comparison of the operator matrix of the second order with the researching operator.

## On Riesz Basises of the Eigen and Associated Functions of the Functional-Differential Operator with a Variable Structure

For a functional-differential operator of a variable structure with integral boundary conditions the Riesz basisness of its eigen and associated functions in the space L32[0, 1] is proved.

## Expansions in Eigenfunctions of the n-th Order Differential Operator with Non-Regular Boundary Conditions

The paper deals with the expansions in eigenfunctions of the n-th order differential operator with non-regular boundary conditions of special type. Necessary and sufﬁcient conditions for existing of such expansions either on the interval [0, 1] or inside it are derived.

## Multipoint Differential Operators: „Splitting“ of the Multiple in Main Eigenvalues

We study the boundary value problem for the differential operator of the eighth order with a summable potential. The boundary conditions of the boundary value problem are multipoint. We derived the integral equation for solutions of differential equation which define the studied differential operator. The asymptotic formulas and estimates for the solutions of the corresponding differential equation for large values of the spectral parameter are obtained.

## The Il’in Spectral Method for Determination of the Properties of the Basis Property and the Uniform Convergence of Biorthogonal Expansions on a Finite Interval

The paper discusses the basics of the spectral method of V. A. Il’in on an example of a simple second order differential operator on a segment of the number line. The ﬁrst theorem of Il’in on the unconditional basis property is stated. Its detailed proof is given. A chain of generalizations of this theorem is traced. A recently established a theorem on the unconditional basis property for the differential operators with general integral boundary conditions is formulated.