# Volume 60, № 12, 2008

### Method of accelerated convergence for the construction of solutions of a Noetherian boundary-value problem

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1587–1601

We study the problem of finding conditions for the existence of solutions of weakly nonlinear Noetherian boundary-value problems for systems of ordinary differential equations and the construction of these solutions. A new iterative procedure with accelerated convergence is proposed for the construction of solutions of a weakly nonlinear Noetherian boundary-value problem for a system of ordinary differential equations in the critical case.

### Logarithms of moduli of entire functions are nowhere dense in the space of plurisubharmonic functions

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1602 – 1609

We prove that the set of logarithms of moduli of entire functions of several complex variables is nowhere dense in the space of plurisubharmonic functions equipped with a topology that is a generalization of the topology of uniform convergence on compact sets. This topology is generated by a metric in which plurisubharmonic functions form a complete metric space. Thus, the logarithms of moduli of entire functions form a set of the first Baire category.

### Boundary-value problem for a parabolic system of integro-differential equations with integral conditions

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1610 – 1618

Using operators of fractional integration and differentiation, we prove a theorem on the wellposedness of a general parabolic boundary-value problem for a system of integro-differential equations with integral operators in boundary conditions.

### On the solvability of one class of parameterized operator inclusions

Kapustyan V. O., Kasyanov P. O., Kohut O. P.

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1619–1630

We consider a class of parameterized operator inclusions with set-valued mappings of \( {\bar S_k} \) type. Sufficient conditions for the solvability of these inclusions are obtained and the dependence of the sets of their solutions on functional parameters is investigated. Examples that illustrate the results obtained are given.

### Asymptotic relation for the density of a multidimensional random evolution with rare poisson switchings

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1631 – 1641

In the Euclidean space $\mathbb{R}^m,\quad m \geq 2,$ the symmetric random evolution
$\textbf{X}(t) = (X_1(t),...,X_m(t))$ controlled by a homogeneous Poisson process with parameter $\lambda > 0$ is considered.

An asymptotic formula for the transition density $p(\textbf{x},t),\quad t > 0,$ of the process $\textbf{X}(t)$ for $\lambda \rightarrow 0$ is obtained.
The behavior of $p(\textbf{x},t)$ near the boundary of the diffusion area in spaces of various dimensions is described.

### On sharp Kolmogorov-type inequalities taking into account the number of sign changes of derivatives

Kofanov V. A., Miropol'skii V. E.

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1642–1649

New sharp inequalities of the Kolmogorov type are established, in particular, the following sharp inequality for $2 \pi$-periodic functions $x \in L^r_{\infty}(T):$ $$||x^{(k)}||_l \leq \left(\frac{\nu(x')}{2} \right)^{\left(1 - \frac1p \right)\alpha} \frac{||\varphi_{r-k}||_l}{||\varphi_r||^{\alpha}_p} ||x||^{\alpha}_p \left|\left|x^{(r)}\right|\right|^{1-\alpha}_{\infty},$$ $k,\;r \in \mathbb{N},\quad k < r, \quad r \geq 3,\quad p \in [1, \infty],\quad \alpha = (r-k) / (r - 1 + 1/p), \quad \varphi_r$ is the perfect Euler spline of order $r,\quad \nu(x')$ is the number of sign changes of the derivative $x'$ on a period.

### Systems of equations of Kolmogorov type

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1650–1663

We consider one class of degenerate parabolic systems of equations of the type of diffusion equation with Kolmogorov inertia. For systems whose coefficients may depend only on the time variable, we construct a fundamental matrix of solutions of the Cauchy problem and obtain estimates for this matrix and all its derivatives.

### On some limit theorems for the maximum of sums of independent random processes

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1664–1674

We study conditions of weak convergence of maximum of sums of independent random processes in the space $L_p.$
We present a number of applications to asymptotic analysis of some $\omega^2$-type statistics.

### Heat equation and wave equation with general stochastic measures

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1675 – 1685

We consider the heat equation and wave equation with constant coefficients that contain a term given by an integral with respect to a random measure. Only the condition of sigma-additivity in probability is imposed on the random measure. Solutions of these equations are presented. For each equation, we prove that its solutions coincide under certain additional conditions.

### Classification of infinitely differentiable periodic functions

Serdyuk A. S., Shydlich A. L., Stepanets O. I.

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1686–1708

The set $\mathcal{D}^{\infty}$ of infinitely differentiable periodic functions is studied in terms of generalized
$\overline{\psi}$-derivatives defined by a pair $\overline{\psi} = (\psi_1, \psi_2)$ of sequences $\psi_1$ and $\psi_2$.
In particular, it is established that every function $f$ from the set $\mathcal{D}^{\infty}$ has at least one derivative whose parameters $\psi_1$ and $\psi_2$
decrease faster than any power function. At the same time, for an arbitrary function $f \in \mathcal{D}^{\infty}$ different from
a trigonometric polynomial, there exists a pair $\psi$ whose parameters $\psi_1$ and $\psi_2$ have the same rate of decrease
and for which the $\overline{\psi}$-derivative no longer exists.

We also obtain new criteria for $2 \pi$-periodic functions real-valued on the real axis to belong to the set of
functions analytic on the axis and to the set of entire functions.

### Littlewood - Paley theorem on $L^{p(t)}(\mathbb{R}^n)$ spaces

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1709 – 1715

We point out that when the Hardy - Littlewood maximal operator is bounded on the space $L^{p(t)}(\mathbb{R}^n),\quad 1 < a \leq p(t) \leq b < \infty,\quad t \in \mathbb{R}$, the well-known characterization of spaces $L^{p(t)}(\mathbb{R}^n),\quad 1 < p < \infty$, by the Littlewood - Paley theory extends to the space $L^{p(t)}(\mathbb{R}^n).$ We show that if $n > 1,$ the Littlewood -Paley operator is bounded on $L^{p(t)}(\mathbb{R}^n),\quad 1 < a \leq p(t) \leq b < \infty,\quad t \in \mathbb{R},$ if and only if $p(t) =$ const.

### Solvable subgroups in groups with self-normalizing subgroup

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1716–1721

We study the structure of some solvable finite subgroups in groups with self-normalizing subgroup.

### Bogolyubov Readings-2008. International Conference "Differential Equations, Theory of Functions and Applications" (on the occasion of the 70th anniversary of academician AM Samoilenko)

Romanyuk A. S., Samoilenko A. M.

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1722

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1723 - 1724

### Index of volume 60 of „Ukrainian Mathematical Journal”

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1725-1729