# Mathematics

## 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.

## On Classical Solvability of One-Dimensional Mixed Problem for Fourth Order Semilinear Biparabolic Equations

Existence and uniqueness of classical solution of one-dimensional mixed problem with Riquier type homogenous boundary conditions for one class of fourth order semilinear biparabolic equations are studied. A priori estimates method is used to prove the existence in large theorem for classical solution of mixed problem under consideration..

## On the Number of Solutions of Nonlinearity Boundary Value Problems with a Stieltjes Integral

In this paper we obtain sufficient conditions for the existence of multiple solutions for nonlinear boundary value problem with a Stieltjes integral.

## Substantiation of Fourier Method in Mixed Problem with Involution

In this paper the mixed problem for the first order differential equation with involution is investigated. Using the received specified asymptotic formulas for eigenvalues and eigenfunctions of the corresponding spectral problem, the application of the Fourier method is substantiated. We used techniques, which allow to transform a series representing the formal solution on Fourier method, and to prove the possibility of its term by term differentiation. At the same time on the initial problem data minimum requirements are imposed.

## Applicathion the Pontryagin‘s Maximum Principle to Optimal Economics Models

In this paper three models of firm are considerd as the discrete optimal control problems. The algorithm for solution is based on Pontryagin‘s Maximum Principle. The paper contains numerical examples.

## On Congruences of Partial n-ary Groupoids

Ri-congruence is defined for partial n-ary groupoids as a generalization of right congruence of a full binary groupoid. It is proved that for any i the Ri-congruences of a partial n-ary groupoid G form a lattice, where the congruence lattice of G is not necessary a sublattice. An example is given, demonstrating that the congruence lattice of a partial n-ary groupoid is not always a sublattice of the equivalence relations lattice of G. The partial n-ary groupoids G are characterized such that for some i, all the equivalence relations on G are its Ri-congruences.

## Cramer’s Formulas for Systems of Linear Equations and Inequalities Over Boolean Algebra

There obtained analogies of classical Cramer’s formulas for systems of linear equations and inequalities with square matrix of coefficients from Boolean algebra.