# Volume 56, № 4, 2004

### Generalized two-parameter Lebesgue-Stieltjes integrals and their applications to fractional Brownian fields

Il'chenko S. A., Mishura Yu. S.

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 435–450

We consider two-parameter fractional integrals and Weyl, Liouville, and Marchaut derivatives and substantiate some of their properties. We introduce the notion of generalized two-parameter Lebesgue-Stieltjes integral and present its properties and computational formulas for the case of differentiable functions. The main properties of two-parameter fractional integrals and derivatives of Hölder functions are considered. As a separate case, we study generalized two-parameter Lebesgue-Stieltjes integrals for an integrator of bounded variation. We prove that, for Hölder functions, the integrals indicated can be calculated as the limits of integral sums. As an example, generalized two-parameter integrals of fractional Brownian fields are considered.

### Estimates of groups of deviations of faber sums and strong summability of faber series on classes of ψ-integrals

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 451–461

We establish upper bounds for a group of ϕ^{*}-deviations of Faber sums on the classes of ψ-integrals in a complex plane introduced by Stepanets.

### Stability of positive and monotone systems in a partially ordered space

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 462–475

We investigate properties of positive and monotone dynamical systems with respect to given cones in the phase space. Stability conditions for linear and nonlinear differential systems in a partially ordered space are formulated. Conditions for the positivity of dynamical systems with respect to the Minkowski cone are established. By using the comparison method, we solve the problem of the robust stability of a family of systems.

### Malmquist theorem for solutions of differential equations in a neighborhood of a logarithmic singular point

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 476–483

The Malmquist theorem (1913) on the growth of meromorphic solutions of the differential equation *f ′ = P(z,f) / Q(z,f)*, where *P(z,f)* and *Q(z,f)* are polynomials in all variables, is proved for the case of meromorphic solutions with logarithmic singularity at infinity.

### On some properties of bundles of trajectories of a controlled bilinear inclusion

Komleva T. A., Plotnikov A. V.

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 484–494

We consider a differential bilinear inclusion with control and present conditions under which the reachability set for this inclusion is compact.

### Approximation of infinitely differentiable periodic functions by interpolation trigonometric polynomials

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 495–505

We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities on the classes of periodic infinitely differentiable functions *C* _{Ψ} ^{β} *C* whose elements can be represented in the form of convolutions with fixed generating kernels. We obtain asymptotic equalities for upper bounds of approximations by interpolation trigonometric polynomials on the classes *C* _{Ψ} ^{β,∞} and *C* _{Ψ} ^{β} *H* _{ω}.

### Regularity of a boundary point for singular parabolic equations with measurable coefficients

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 506–516

We investigate the continuity of solutions of quasilinear parabolic equations near the nonsmooth boundary of a cylindrical domain. We prove a sufficient condition for the regularity of a boundary point, which coincides with the Wiener condition for the Laplace *p*-operator. The model case of the equations considered is the equation \(\frac{{\partial u}}{{\partial t}} - \Delta _p u = 0\) with the Laplace *p*-operator Δ _{ p } for 2*n* */* (*n* + 1) < *p* < 2.

### On the relation between fourier and leont’ev coefficients with respect to smirnov spaces

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 517–526

Yu. Mel’nik showed that the Leont’ev coefficients Κ_{ f }(λ) in the Dirichlet series \({{2n} \mathord{\left/ {\vphantom {{2n} {\left( {n + 1} \right) < p < 2}}} \right. \kern-0em} {\left( {n + 1} \right) < p > 2}}\) of a function *f* ∈*E* ^{ p } *(D)*, 1 < *p* < ∞, are the Fourier coefficients of some function *F* ∈*L* ^{ p }, ([0, 2π]) and that the first modulus of continuity of *F* can be estimated by the first moduli and majorants in *f.* In the present paper, we extend his results to moduli of arbitrary order.

### A goodness-of-fit test for a polynomial errors-in-variables model

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 527–543

Polynomial regression models with errors in variables are considered. A goodness-of-fit test is constructed, which is based on an adjusted least-squares estimator and modifies the test introduced by Zhu et al. for a linear structural model with normal distributions. In the present paper, the distributions of errors are not necessarily normal. The proposed test is based on residuals, and it is asymptotically chi-squared under null hypothesis. We discuss the power of the test and the choice of an exponent in the exponential weight function involved in test statistics.

### Congruences on ternary semigroups

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 544–559

We study ternary semigroups as universal algebras with one associative operation. We investigate their algebraic structure and associated representations. Results for congruences of ternary semigroups generated by binary relations are presented.

### V. G. Georgii Mykolaiovych Polozhyi (on his 90th birthday)

Glushchenko A. A., Lyashko I. I., Mitropolskiy Yu. A., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G.

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 560-561

### Periodic solutions of systems of hyperbolic equations bounded on a plane

Assanova A. T., Dzhumabaev D. S.

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 562-572

For a linear system of hyperbolic equations of the second order with two independent variables, we investigate the problem of the existence and uniqueness of a solution periodic in both variables and a solution periodic in one of the variables and bounded on a plane. By using the method of introduction of functional parameters, we obtain sufficient conditions for the unique solvability of the problems under consideration.

### Separately $Fσ$-measurable functions are close to functions of the first baire class

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 573–576

We prove that a Borel separately $Fσ$-measurable function $f: X \times Y → R$ on the product of Polish spaces is a function of the first Baire class on the complement $X × Y \backslash M$ of a certain projectively meager set $M ⊂ X × Y$.