# Volume 63, № 9, 2011

### Estimates for the norms of fractional derivatives in terms of integral moduli of continuity and their applications

Babenko V. F., Churilova M. S.

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1155-1168

For functions defined on the real line or a half-line, we obtain Kolmogorov-type inequalities that estimate the $L_p$-norms $(1 \leq p < \infty)$ of fractional derivatives in terms of the Lp-norms of functions (or the $L_p$-norms of their truncated derivatives) and their $L_p$-moduli of continuity and establish their sharpness for $p = 1$. Applications of the obtained inequalities are given.

### Laplacian with respect to a measure on a Hilbert space and an *L *_{2}-version of the Dirichlet problem for the Poisson equation

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1169-1178

We propose a version of the Laplace operator for functions on a Hilbert space with measure. In terms of this operator, we investigate the Dirichlet problem for the Poisson equation.

### Submanifolds of compact operators with fixed multiplicities of eigenvalues

Bondar A. A., Dymarskii Ya. M.

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1179-1189

The manifold of symmetric real matrices with fixed multiplicities of eigenvalues was considered for the first time by V. Arnold. In the case of compact real self-adjoint operators, analogous results were obtained by Japanese mathematicians D. Fujiwara, M. Tanikawa, and S. Yukita. They introduced a special local diffeomorphism that maps Arnold's submanifold to a flat subspace. The properties of the indicated diffeomorphism were further studied by Ya. Dymarskii. In the present paper, we describe the smooth structure of submanifolds of finite-dimensional and compact operators of the general form in which a selected eigenvalue is associated with a single Jordan block.

### Regularization of two-term differential equations with singular coefficients by quasiderivatives

Goryunov A. S., Mikhailets V. A.

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1190-1205

We propose a regularization of the formal differential expression $$l(y) = i^m y^{(m)}(t) + q(t)y(t),\; t \in (a, b),$$ of order $m \geq 3$ by using quasiderivatives. It is assumed that the distribution coefficient $q$ has an antiderivative $Q \in L ([a, b]; \mathbb{C})$. In the symmetric case $(Q = \overline{Q})$, we describe self-adjoint and maximal dissipative/accumulative extensions of the minimal operator and its generalized resolvents. In the general (nonselfadjoint) case, we establish conditions for the convergence of the resolvents of the considered operators in norm. The case where $m = 2$ and $Q \in L_2 ([a, b]; \mathbb{C})$ was studied earlier.

### On modules over integer-valued group rings of locally soluble groups with rank restrictions imposed on subgroups

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1206-1217

We study the $ZG$-module $A$ such that $Z$ is the ring of integers, the group $G$ has infinite section $ p$-rank (or infinite 0-rank), $C_G(A) = 1$, $A$ is not a minimax $Z$-module, and, for every proper subgroup $H$ of infinite section $p$-rank (or infinite 0-rank, respectively), the quotient module $A/C_A(H)$ is a minimax $Z$-module. It is proved that if the group $G$ under consideration is locally solvable, then $G$ is a solvable group. Some properties of a solvable group of this type are obtained.

### Structure of a finite commutative inverse semigroup and a finite bundle for which the inverse monoid of local automorphisms is permutable

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1218-1226

For a semigroup $S$, the set of all isomorphisms between subsemigroups of $S$ is an inverse monoid with respect to composition, which is denoted by $P A(S)$ and is called the monoid of local automorphisms of $S$. A semigroup $S$ is called permutable if, for any pair of congruences $p, \sigma$ on $S$, one has $p \circ \sigma = \sigma \circ p$. We describe the structure of a finite commutative inverse semigroup and a finite band whose monoids of local automorphisms are permutable.

### On the theory of convergence and compactness for Beltrami equations with constraints of set-theoretic type

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1227-1240

We prove theorems on convergence and compactness for classes of regular solutions of degenerate Beltrami equations with set-theoretic constraints imposed on the complex coefficient and construct variations for these classes.

### On the Skorokhod mapping for equations with reflection and possible jump-like exit from a boundary

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1241-1256

For a solution of a reflection problem on a half-line similar to the Skorokhod reflection problem but with possible jump-like exit from zero, we obtain an explicit formula and study its properties. We also construct a Wiener process on a half-line with Wentzell boundary condition as a strong solution of a certain stochastic differential equation.

### Systems of essentially infinite-dimensional differential equations

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1257-1262

We investigate systems of differential equations with essentially infinite-dimensional elliptic operators (of the Laplace - Levy type). For nonlinear systems, we prove theorems on the existence and uniqueness of solutions. For a linear system, we give an explicit formula for the solution.

### Solvability of inhomogeneous boundary-value problems for fourth-order differential equations

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1263-1278

Some new oscillation criteria are established for the nonlinear damped differential equation $$(r(t)k_1 (x, x'))' + p (t) k_2 (x, x') x' + q (t) f (x (t)) = 0,\quad t \geq t_0.$$ The results obtained extend and improve some existing results in the literature.

### On minimal non- *MSP* -groups

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1279-1278

A finite group $G$ is called an $MSP$-group if all maximal subgroups of the Sylow subgroups of $G$ are Squasinormal in $G$. In this paper, wc give a complete classification of those groups which are not $MSP$-groups but whose proper subgroups are all $MSP$-groups.

### On the polynomial approximation of a conformal mapping of a domain with nonzero corner

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1285-1289

Let $G$ be a bounded domain with a Jordan boundary that is smooth at all points except a single point at which it forms a nonzero angle. We prove Korevaar’s conjecture on the order of polynomial approximation of a conformal mapping of this domain into a disk. We also obtain a pointwise estimate for the error of approximation.

### On solutions defined on an axis for differential equations with shifts of the argument

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1290-1296

We consider linear first-order differential equations with shifts of arguments with respect to functions with values in a Banach space. Sufficient conditions for the existence of nontrivial solutions of homogeneous equations are obtained. Ordinary differential equations are constructed for which all solutions defined on an axis are solutions of a given equation with shifts of the argument.