# Volume 63, № 8, 2011

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

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1011-1020

We consider a Cauchy-type boundary-value problem of, a problem with three boundary conditions, and the Dirichlet problem for a general fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain $\Omega \subset R^2$ with smooth boundary. Using the method of the Green formula, the theory of expansion of differential operators, and the theory of $L$-traces (i.e., traces associated with a differential operation $L$), we obtain necessary and sufficient (for elliptic operators) conditions for the solvability of each of the problems under consideration in the space $H^m(\Omega),\;\; m \geq 4$.

### Sojourn time of almost semicontinuous integral-valued processes in a fixed state

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1021-1029

Let $\xi(t)$ be an almost lower semicontinuous integer-valued process with the moment generating function of the negative part of jumps $\xi_k : \textbf{E}[z^{\xi_k} / \xi_k < 0] = \frac{1 − b}{z − b},\quad 0 ≤ b < 1.$ For the moment generating function of the sojourn time of $\xi(t)$ in a fixed state, we obtain relations in terms of the roots $z_s < 1 < \widehat{z}_s$ of the Lundberg equation. By passing to the limit $(s → 0)$ in the obtained relations, we determine the distributions of $l_r(\infty)$.

### On the asymptotic distribution of the Koenker?Bassett estimator for a parameter of the nonlinear model of regression with strongly dependent noise

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1030-1052

We prove that, under certain regularity conditions, the asymptotic distribution of the Koenker - Bassett estimator coincides with the asymptotic distribution of the integral of the indicator process generated by a random noise weighted by the gradient of the regression function.

### Cauchy problem for a differential equation in the Banach space with generalized strongly positive operator coefficient

Chaikovs'kyi A. V., Il'chenko Yu. V.

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1053-1070

The concept of strongly positive operator is generalized, and properties of the operators introduced are analyzed. The solutions of the Cauchy problem for a linear inhomogeneous differential equation with generalized strongly positive operator coefficient are found.

### Stability of smooth soHtary waves for the generahzed Korteweg - de Vries equation with combmed dispersion

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1071-1077

The orbital stability problem of the smooth solitary waves in the generalized Korteweg - de Vries equation with combined dispersion is considered. The results show that the smooth solitary waves are stable for an arbitrary speed of wave propagation.

### On the boundary behavior of solutions of the Beltrami equations

Kovtonyuk D. A., Petkov I. V., Ryazanov V. I.

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1078-1091

We show that every homeomorphic solution of the Beltrami equation $\overline{\partial} f = \mu \partial f$ in the Sobolev class $W^{1, 1}_{\text{loc}}$ is a so-called lower $Q$-homeomorphism with $Q(z) = K_{\mu}(z)$, where $K_{\mu}$ is a dilatation quotient of this equation. On this basis, we develop the theory of the boundary behavior and the removability of singularities of these solutions.

### Weyl's theorem for algebrascally $wF(p, r, q)$ operators with $p, q > 0$ and $q \geq 1$

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1092-1102

If $T$ or $T*$ is an algebraically $wF(p, r, q)$ operator with $p, r > 0$ and $q ≥ 1$ acting on an infinite-dimensional separable Hilbert space, then we prove that the Weyl theorem holds for $f(T)$, for every $f \in \text{Hol}(\sigma(T))$, where $ \text{Hol}(\sigma(T))$ denotes the set of all analytic functions in an open neighborhood of $\sigma(T)$. Moreover, if $T^*$ is a $wF(p, r, q)$ operator with $p, r > 0$ and $q ≥ 1$, then the $a$-Weyl theorem holds for $f(T)$. Also, if $T$ or $T^*$ is an algebraically $wF(p, r, q)$ operators with $p, r > 0$ and $q ≥ 1$, then we establish spectral mapping theorems for the Weyl spectrum and essential approximate point spectrum of T for every $f \in \text{Hol}(\sigma(T))$, respectively. Finally, we examine the stability of the Weyl theorem and $a$-Weyl theorem under commutative perturbation by finite-rank operators.

### On asymptotic equivalence of solutions of stochastic and ordinary equations

Novak I. H., Samoilenko A. M., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1103-1127

For a weakly nonlinear stochastic system, we construct a system of ordinary differential equations the behavior of solutions of which at infinity is similar to the behavior of solutions of the original stochastic system.

### On the openness and discreteness of mappings with unbounded characteristic of quasiconformality

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1128-1134

The paper is devoted to the investigation of the topological properties of space mappings. It is shown that sense-preserving mappings $f : D \rightarrow \overline{\mathbb{R}^n}$ in a domain $D \subset \mathbb{R}^n$, n ≥ 2, which are more general than mappings with bounded distortion, are open and discrete if a function Q corresponding to the control of the distortion of families of curves under these mappings has slow growth in the domain f(D), e.g., if Q has finite mean oscillation at an arbitrary point $y0 \in f(D)$.

### Volodymyr Leonidovych Makarov (on his 70th birthday)

Korolyuk V. S., Lukovsky I. O., Samoilenko A. M.

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1135-1136

### Yurii Serhiiovych Osypov (on his 75th birthday)

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1137-1139

### Strongly radical supplemented modules

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1140-1146

Zoschinger studied modules whose radicals have supplements and called these modules radical supplemented. Motivated by this, we call a module strongly radical supplemented (briefly srs) if every submodule containing the radical has a supplement. We prove that every (finitely generated) left module is an srs-module if and only if the ring is left (semi)perfect. Over a local Dedekind domain, srs-modules and radical supplemented modules coincide. Over a no-local Dedekind domain, an srs-module is the sum of its torsion submodule and the radical submodule.

### Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1147-1152

Polynomial matrices $A(x)$ and $B(x)$ of size $n \times n$ over a field $\mathbb{F}$ are called semiscalar equivalent if there exist a nonsingular $n \times n$ matrix $P$ over $\mathbb{F}$ and an invertible $n \times n$ matrix $Q(x)$ over $\mathbb{F}[x]$ such that $A(x) = PB(x)Q(x)$. We give a canonical form with respect to the semiscalar equivalence for a matrix pencil $A(x) = A_0x - A_1$, where $A_0$ and $A_1$ are $n \times n$ matrices over $\mathbb{F}$, and $A_0$ is nonsingular.