# Volume 55, № 1, 2003

### Irreducible System of Constraints for a General Polyhedron of Arrangements

Emets O. A., Nedobachii S. I., Roskladka О. V.

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 3-11

We construct a system of constraints for a general polyhedron of arrangements that does not contain superfluous inequalities. The derivation of an irreducible system enables one to substantially reduce the number of operations necessary for finding exact solutions of optimization problems on arrangements.

### Singularities of Solutions of One Class of Equations of Continuum Mechanics

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 12-21

We analyze one class of families of integral equations and describe the dependence of the singularities of solutions of integral equations on the dimensions of the families of kernels of equations. On the basis of these results, we propose procedures for the construction of approximate solutions for a small parameter.

### A Multipoint Problem for Pseudodifferential Equations

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 22-29

We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable *t* and periodic conditions with respect to coordinates *x* _{1},...,*x* _{p} for equations unsolved with respect to the leading derivative with respect to *t* and containing pseudodifferential operators. We establish conditions for the unique solvability of this problem and prove metric assertions related to lower bounds for small denominators appearing in the course of its solution.

### Approximation Properties of Two-Dimensional Continued Fractions

Kuchmins’ka Kh. Yo., Sus' О. M., Vozna S. M.

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 30-44

By using the difference formula for approximations of two-dimensional continued fractions, the method of fundamental inequalities, the Stieltjes–Vitali theorem, and generalizations of divided and inverse differences, we estimate the accuracy of approximations of two-dimensional continued fractions with complex elements by their convergents and obtain estimates for the real and imaginary parts of remainders of two-dimensional continued fractions. We also prove an analog of the van Vleck theorem and construct an interpolation formula of the Newton–Thiele type.

### Characterization of the Points of $ϕ$-Strong Summability of Fourier–Laplace Series for Functions of the Class $L_p(S^m),\; p > 1$

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 45-54

We consider the behavior of the ϕ-strong means of Fourier–Laplace series for functions that belong to $L_p(S^m),\; p > 1$, on a set of points of full measure on an $m$-dimensional sphere $S^m$.

### Justification of Averaging Method for Multifrequency Impulsive Systems

Petryshyn R. I., Sopronyuk Т. M.

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 55-65

We prove new theorems on the justification of the averaging method for multifrequency oscillation systems with pulse influence at fixed times.

### Lyapunov–Schmidt Approach to Studying Homoclinic Splitting in Weakly Perturbed Lagrangian and Hamiltonian Systems

Prykarpatsky A. K., Samoilenko A. M., Samoilenko V. G.

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 66-74

We analyze the geometric structure of the Lyapunov–Schmidt approach to studying critical manifolds of weakly perturbed Lagrangian and Hamiltonian systems.

### On the Fréchet Differentiability of Invariant Tori of Countable Systems of Difference Equations Defined on Infinite-Dimensional Tori

Marchuk N. A., Teplinsky Yu. V.

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 75-90

By using the method of Green–Samoilenko functions, in the space of bounded number sequences we construct invariant tori of linear and nonlinear systems of discrete equations defined on infinite-dimensional tori. We establish sufficient conditions for the Fréchet differentiability of invariant tori.

### Boundedness of the *l*-Index of Laguerre–Pólya Entire Functions

Bordulyak M. T., Sheremeta M. M.

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 91-99

We investigate conditions on zeros of an entire function *f* of the Laguerre–Pólya class under which *f* is a function of bounded *l*-index.

### Darboux–Protter Spectral Problems for One Class of Multidimensional Hyperbolic Equations

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 100-107

For a multidimensional hyperbolic equation with a wave operator in the principal part, we show that the Darboux–Protter spectral problem has the countable set of eigenfunctions, and its dual problem is the Volterra problem.

### On the Space of Sequences of *p*-Bounded Variation and Related Matrix Mappings

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 108-118

The difference sequence spaces ℓ_{∞}(▵), *c*(▵), and *c* _{0}(▵) were studied by Kızmaz. The main purpose of the present paper is to introduce the space *bv* _{p} consisting of all sequences whose differences are in the space ℓ_{ p }, and to fill up the gap in the existing literature. Moreover, it is proved that the space *bv* _{p} is the BK-space including the space ℓ_{ p }. We also show that the spaces *bv* _{p} and ℓ_{ p } are linearly isomorphic for 1 ≤ *p* ≤ ∞. Furthermore, the basis and the α-, β-, and γ-duals of the space *bv* _{p} are determined and some inclusion relations are given. The last section of the paper is devoted to theorems on the characterization of the matrix classes (*bv* _{p} : ℓ_{∞}), (bv_{∞} : ℓ_{ p }), and (*bv* _{p} : ℓ_{1}), and the characterizations of some other matrix classes are obtained by means of a suitable relation.

### Determination of the Free Term and Leading Coefficient in a Parabolic Equation

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 119-125

We consider the inverse problem of finding the unknown time-dependent leading coefficient and the free term in a parabolic equation. Boundary conditions and overdetermination conditions are local. We find conditions for the uniqueness and local existence.

### Multipoint Boundary Conditions for Differential Operators

Golets B. I., Valitskii Yu. N., Zelenyak T. I.

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 126-131

We establish differential properties of generalized solutions of multipoint boundary-value problems for ordinary differential equations.

### Solvability of a Three-Point Boundary-Value Problem for a Second-Order Differential Inclusion

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 132-137

We investigate the problem of the existence of solutions of a three-point boundary-value problem for a second order differential inclusion.

### Expansion of a Self-Adjoint Absolutely Continuous Singular Integral Operator in Generalized Eigenvectors and Its Diagonalization

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 138-145

We describe the relationship between the expansion of a self-adjoint operator in generalized eigenvectors and the direct integral of Hilbert spaces. We perform the explicit diagonalization of a self-adjoint absolutely continuous singular integral operator *Y* using an Hermitian nonnegative kernel consisting of boundary values of the determining function of the operator *T* = *X* + *iY* with respect to the resolvent of the imaginary part of *Y*.