# Volume 57, № 2, 2005

### Some Problems of Simultaneous Approximation of Functions of Two Variables and Their Derivatives by Interpolation Bilinear Splines

Myskin K. Yu., Vakarchuk S. B.

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 147–157

Exact estimates for the errors of approximation of functions of two variables and their derivatives by interpolation bilinear splines are obtained on certain classes.

### Estimate for a Rearrangement of a Function Satisfying the “Reverse Jensen Inequality”

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 158–169

We show that an equimeasurable rearrangement of any function satisfying the “reverse Jensen inequality” with respect to various multidimensional segments also satisfies the “reverse Jensen inequality” with the same constant.

### Markov Uniqueness and Rademacher Theorem for Smooth Measures on an Infinite-Dimensional Space under Successful-Filtration Condition

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 170–186

For a smooth measure on an infinite-dimensional space, a “successful-filtration” condition is introduced and the Markov uniqueness and Rademacher theorem for measures satisfying this condition are proved. Some sufficient conditions, such as the well-known Hoegh-Krohn condition, are also considered. Examples demonstrating connections between these conditions and applications to convex measures are given.

### Strong Summability of Faber Series and Estimates for the Rate of Convergence of a Group of Deviations in a Closed Domain with Piecewise-Smooth Boundary

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 187–197

We establish estimates for groups of deviations of Faber series in closed domains with piecewise-smooth boundary.

### Stability and Comparison of States of Dynamical Systems with Respect to a Time-Varying Cone

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 198–213

We investigate classes of dynamical systems in a partially ordered space with properties of monotonicity type with respect to specified cones. We propose new methods for the stability analysis and comparison of solutions of differential systems using time-varying cones. To illustrate the results obtained, we present examples using typical cones in vector and matrix spaces.

### A Limit Theorem for Integral Functionals of an Extremum of Independent Random Processes

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 214–221

We prove a theorem on the convergence of integral functionals of an extremum of independent stochastic processes to a degenerate law of distributions.

### Differentiation of Singular Integrals with Piecewise-Continuous Density and Boundary Values of Derivatives of a Cauchy-Type Integral

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 222–229

We establish sufficient conditions for the differentiability of a singular Cauchy integral with piecewise-continuous density. Formulas for the *n*th-order derivatives of a singular Cauchy integral and for the boundary values of the *n*th-order derivatives of a Cauchy-type integral are obtained.

### Approximation of Continuous Functions by de La Vallee-Poussin Operators

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 230–238

For $\sigma \rightarrow \infty$, we study the asymptotic behavior of upper bounds of deviations of functions blonding to the classes
$\widehat{C}_{\infty}^{\overline{\Psi}}$ and $\widehat{C}^{\overline{\Psi}} H_{\omega}$ from the so-called Vallee Poussin operators.
We find asymptotic equalities that, in some important cases, guarantee the solution of the Kolmogorov - Nikol's'kyi problem for the Vallee Poussin operators on the classes
$\widehat{C}_{\infty}^{\overline{\Psi}}$ and $\widehat{C}^{\overline{\Psi}} H_{\omega}$.

### Phragmen-Lindelof Principle for Some Quasilinear Evolution Equations of the Second Order

Shishkov A. E., Sleptsova I. P.

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 239–249

We consider the equation $u_{tt} + A (u_t) + B(u) = 0$, where $A$ and $B$ are quasilinear operators with respect to the variable x of the second order and the fourth order, respectively. In a cylindrical domain unbounded with respect to the space variables, we obtain estimates that characterize the minimum growth of any nonzero solution of the mixed problem at infinity.

### Some Results on Asymptotic Stability of Order α

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 250–257

Quasi-equiasymptotic stability of order $α (α ∈ ℝ_{+} * )$ with respect to a part of variables is considered. Some sufficient conditions, a converse theorem, and a theorem on multistability are proved.

### A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 258–264

We obtain a differential analog of the main lemma in the theory of Markov branding processes $\mu(t),\quad t \geq 0$, of continuous time. We show that the results obtained can be applied in the proofs of limit theorems in the theory of branching processes by the well-known Stein - Tikhomirov method. In contrast to the classical condition of nondegeneracy of the branching process $\{\mu(t) > 0\}$, we consider the condition of nondegeneracy of the process in distant $\{\mu(\infty) > 0\}$ and justify in terms of generating functions. Under this condition, we study the asymptotic behavior of trajectory of the considered process.

### On Groups with Minimality Condition for Noninvariant Abelian pd-Subgroups

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 265-270

We study properties and the structure of non-Abelian groups with minimality condition for non-invariant Abelian pd-subgroups in the case where they do not satisfy the minimality condition for Abelian pd-subgroups. We prove the solvability of these groups and establish relations with non-Abelian groups in which all infinite Abelian pd-subgroups are invariant.

### Newton-Kantorovich Iterative Regularization for Nonlinear Ill-Posed Equations Involving Accretive Operators

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 271–276

The Newton-Kantorovich iterative regularization for nonlinear ill-posed equations involving monotone operators in Hilbert spaces is developed for the case of accretive operators in Banach spaces. An estimate for the convergence rates of the method is established.

### Shape-Preserving Smoothing of 3-Convex Splines of Degree 4

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 277–283

For every 3-convex piecewise-polynomial function s of degree ≤ 4 with n equidistant knots on [0, 1] we construct a 3-convex spline $s_1 (s_1 ∈ C (3))$ of degree ≤ 4 with the same knots that satisfies the inequality $$\left\| {S - S_1 } \right\|_{C_{[0,1]} } \leqslant c\omega _5 (s;1/n),$$ where $c$ is an absolute constant and $ω_5$ is the modulus of smoothness of the fifth order.

### Two-Sided Approximation of Solutions of Boundary-Value Problems

Mentynskyi S. M., Shuvar B. A.

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 284–288

We propose a general scheme for the two-sided approximation of solutions of boundary-value problems for ordinary differential equations. This scheme involves a number of known and new two-sided methods. In our investigation, we use constructions of the Samoilenko numerical-analytic method together with the procedure of the construction of two-sided methods proposed by Kurpel’ and Shuvar.