# Volume 58, № 4, 2006

### On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 435–450

In a plane domain bounded by a biquadratic curve, we consider the problem of the uniqueness of a solution of the Dirichlet problem for the string equation. We show that this problem is equivalent to the classical Poncelet problem in projective geometry for two appropriate ellipses and also to the problem of the solvability of the Pell-Abel algebraic equation; some other related problems are also considered.

### Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 451–471

Let $\mathcal{M}_{(n)},\quad n = 1, 2,...,$ be the supercritical branching random walk in which the family sizes may be infinite with positive probability. Assume that a natural martingale related to $\mathcal{M}_{(n)},$ converges almost surely and in the mean to a random variable $W$. For a large subclass of nonnegative and concave functions $f$ , we provide a criterion for the finiteness of $\mathbb{E}W f(W)$. The main assertions of the present paper generalize some results obtained recently in Kuhlbusch’s Ph.D. thesis as well as previously known results for the Galton-Watson processes. In the process of the proof, we study the existence of the $f$-moments of perpetuities.

### Once again on the Samoilenko numerical-analytic method of successive periodic approximations

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 472–488

A new numerical-analytic algorithm for the investigation of periodic solutions of nonlinear periodic systems of differential equations *dx/dt* = *A*(*t*) *x*+ *ƒ*(*t, x*) in the critical case is developed. The problem of the existence of solutions and their approximate construction is studied. Estimates for the convergence of successive periodic approximations are obtained.

### Properties of a wiener process with coalescence

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 489–504

A Wiener process with coalescence and its analog are discussed. We prove the existence of an initial distribution with preset final probabilities for this analog and investigate the problem of the existence of such distributions concentrated at a single point or absolutely continuous with respect to the Lebesgue measure. The behavior of a semigroup of a Wiener process with coalescence in the two-dimensional case and properties of a Wiener flow with coalescence are studied.

### Topological methods in the theory of operator inclusions in Banach spaces. II

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 505–521

We develop topological methods for the investigation of operator inclusions in Banach spaces, prove the generalized Ky Fan inequality, and study the critical points of many-valued mappings in topological spaces.

### Representation of holomorphic functions of many variables by Cauchy-Stieltjes-type integrals

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 522–542

We consider functions of many complex variables that are holomorphic in a polydisk or in the upper half-plane. We give necessary and sufficient conditions under which a holomorphic function is a Cauchy-Stieltjes-type integral of a complex charge. We present several applications of this criterion to integral representations of certain classes of holomorphic functions.

### Investigation of exponential dichotomy of linear Itô stochastic systems with random initial data by using quadratic forms

Krenevych A. P., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 543–553

We study conditions for the mean-square exponential dichotomy of linear Itô stochastic systems. We prove that a sufficient condition for exponential dichotomy is the existence of a quadratic form whose derivative along the solutions of a system is negative definite. The converse theorem is also proved.

### On interpolation approximation of differentiable operators in a Hilbert space

Khlobystov V. V., Popovicheva T. N.

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 554–563

In a Hilbert space, we construct an interpolation approximation of the Taylor polynomial for differentiable operators. By using this approximation, we obtain estimates of accuracy for analytic operators that strengthen previously known results and for operators containing finitely many Fréchet derivatives.

### Olexiy Bogolyubov (03.25.1911 - 01.11.2004)

Dobrovol'skii V. A., Lykova O. B., Mitropolskiy Yu. A., Pustovoytov M. O., Samoilenko A. M., Urbansky V. M.

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 564–567

### Functions of the first Baire class with values in metrizable spaces

Karlova O. O., Mykhailyuk V. V.

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 568–572

We show that every mapping of the first functional Lebesgue class that acts from a topological space into a separable metrizable space that is linearly connected and locally linearly connected belongs to the first Baire class. We prove that the uniform limit of functions of the first Baire class $f_n : \; X \rightarrow Y$ belongs to the first Baire class if $X$ is a topological space and $Y$ is a metric space that is linearly connected and locally linearly connected.

### Generalization of the Prokhorov multidimensional analog of the Chebyshev inequality

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 573–576

We prove two theorems on upper and lower bounds for probabilities in the multidimensional case. We generalize and improve the Prokhorov multidimensional analog of the Chebyshev inequality and establish a multidimensional analog of the generalized Kolmogorov probability estimate.