# Mathematics

## Linear Difference Equation of Second Order in a Banach Space and Operators Splitting

In differential and difference equations classical textbooks, the n-th order differential and difference equations reducing by standard substitution to first-order differential and difference equations system is described. Each of the cohering equations can be written in the operator form. Naturally there is a question of coincidence of a number of properties of differential and difference equations (operators) of the second order and the corresponding functional equations (operators) of first order.

## On Recovering Integro-Differential Operators from the Weyl Function

We study inverse problems of spectral analysis for second order integro-differential operators, which are a perturbation of the Sturm–Liouville operator by the integral Volterra operator. We pay the main attention to the nonlinear inverse problem of recovering the potential from the given Weyl function provided that the kernel of the integral operator is known a priori. We obtain properties of the spectral characteristics and the Weyl function, provide an algorithm for constructing the solution of the inverse problem and establish the uniqueness of the solution.

## On an Inner Estimate of a Convex Body by the Lebesgue Set of Convex Differentiable Function

A finite-dimentional problem of embedding the largest by the inclusion of lower Lebesgue set of given convex function f(x) in a given convex body D ⊂ R p is considered. This problem is the generalization of the problem of inscribed ball (function f(x) is some norm, and the Lebesgue sets are the corresponding balls). The function f(x) must be differentiable on R p possibly expending the point 0 p and 0 p is the uniqueness point of minimum. Mathematical formalization of this problem is proposed in the form of finding maximin of a function of the difference of arguments.

## Embeddings of Generalized Bounded Variation Function Spaces into Spaces of Functions with Given Majorant of Average Modulus of Continuity

In the present paper we study embeddings of some spaces of functions of generalized bounded variation into classes of functions with given majorant of average modulus of continuity introduced by B. Sendov and V. Popov. We consider the spaces ΛBV (p) of functions of bounded (Λ − p)-variation suggested by D. Waterman (for p = 1) and M. Shiba (for p > 1) and spaces V (v(n)) of functions with given majorant of its modulus of variation. The last quantity was introduced by Z. A. Chanturia. The necessary and sufficient conditions of such embeddings are proved.

## Well-posedness of the Dirichlet Problem for One Class of Degenerate Multi-dimensional Hyperbolic-parabolic Equations

It has been shown by Hadamard that one of the fundamental problems of mathematical physics, the analysis of the behavior of oscillating string is an ill-posed problem when the boundary-value conditions are imposed on the entire boudary of the domain. As noted by A. V. Bitsadze and A. M. Nakhushev, the Dirichlet problem is ill-posed not only for the wave equation but for hyperbolic PDEs in general.

## On Multiple Completeness of the Root Functions of the Pencils of Differential Operators with Constant Coefficients

A class of the pencils of ordinary differential operators of n-th order with constant coefficients is considered. The roots of the characteristic equation of the pencils from this class are supposed to lie on a straight line containing the origin, provided that one of the roots lies on one part from the origin, the rest lie on another part. The cases when the system of root functions is m-fold (3 ≤ m ≤ n − 1) complete in the space of square summable functions on main interval are described.

## Estimation of Operator Norms in Eigenvalue Problems for Equations with Discontinuous Operators

Existence of solutions of problems with a spectral parameter for the equations with discontinuous operators is considered. The estimations of the operator norms for these problems are received. Dirichlet problem for the higher-order elliptic equation with discontinuous nonlinearity is considered as an appendix.

## Finding of Accessory Parameters for Mixed Inverse Boundary Value Problem with Polygonal Known Part of Boundary

We consider a mixed inverse boundary value problem with respect to parameter x for the case when the known part of the boundary L1z is a polygonal line. Integral representation of solution to the problem depends on real parameters being the pre-images of the vertices of L1z under conformal mapping. By analogy with Schwartz – Christoffel integrals, we name them accessory parameters. It is suggested a new method of determining the accessory parameters.

## On Idempotent Elements of Semigroup of Increasing Monotonous Mappings

In some special classes of ordered topological spaces we characterize roundings as extreme points of set of non increasing isotonic mappings, and establish their stability in Hyers –Ulam sense.

## Solvability of Evolutionary Equations in Generalized Transmission Problems for Shallow Shells

We prove the solvability of the generalized transmission problem in the non-classical theory of shallow shells using the method of compactness and a new way of obtaining a priori estimates.