Ukr. Mat. Zh. - 2003νmber=2. - 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.
Ukr. Mat. Zh. - 2003νmber=2. - 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.
Ukr. Mat. Zh. - 2003νmber=2. - 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.
Ukr. Mat. Zh. - 2003νmber=2. - 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. - 2003νmber=2. - 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 $S^m$.
Ukr. Mat. Zh. - 2003νmber=2. - 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
Ukr. Mat. Zh. - 2003νmber=2. - 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
Ukr. Mat. Zh. - 2003νmber=2. - 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.
Ukr. Mat. Zh. - 2003νmber=2. - 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.
Ukr. Mat. Zh. - 2003νmber=2. - 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.
Ukr. Mat. Zh. - 2003νmber=2. - 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 : ℓ1), and the characterizations of some other matrix classes are obtained by means of a suitable relation.
Ukr. Mat. Zh. - 2003νmber=2. - 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.
Ukr. Mat. Zh. - 2003νmber=2. - 55, № 1. - pp. 126-131
We establish differential properties of generalized solutions of multipoint boundary-value problems for ordinary differential equations.
Ukr. Mat. Zh. - 2003νmber=2. - 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. - 2003νmber=2. - 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.