# Volume 46, № 4, 1994

### On the 75th birthday of Vladislav Kirillovich Dzyadyk, Corresponding Member of the Ukrainian Academy of Sciences

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 315

### On the sixtieth birthday of Dmytro Yakovych Petrina, Corresponding Member of the Ukrainian Academy of Sciences

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 316

### On D. Ya. Petrina's works in contemporary mathematical physics

Gerasimenko V. I., Malyshev P. V., Rebenko A. L.

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 317–328

This is a brief survey of the works of Prof. D. Ya. Petrina in various branches of contemporary mathematical physics.

### New differentiability criteria for complex-valued functions

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 329–337

A theorem is proved which states that the existence of the asymptotic limit of $f_{\overline{z}}$ as $z \rightarrow z_0$ implies that the complex-valued function $f(z)$ is $\mathbb{R}$-differentiable at $z_0$.

### Classification of nonlocal boundary-value problems on a narrow strip

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 338–346

For a general linear partial differential equation with constant coefficients, we establish a well-posedness criterion for a boundary-value problem on a strip $Π_y = ℝ × [0,Y]$ with an integral in a boundary condition. A complete classification of such problems based on their asymptotic properties as $Y → 0$ is obtained.

### On oscillation of solutions of a nonautonomous quasilinear second-order equation

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 347–356

Sufficient conditions are obtained for the initial values of nontrivial oscillating (for $t = ω$) solutions of the nonautonomous quasilinear equation $y'' \pm \lambda (t)y = F(t,y,y'),$, where $t ∈ Δ = [a, ω[,-∞ < a < ω ≤ + ∞, λ(t) > 0, λ(t) ∈ C_Δ^{(1)},$ $ |F((t,x,y))| ≤ L(t)(|x|+|y|)^{1+α}, L(t) ≥ -0, α ∈ [0,+∞[,$ $ F: Δ × R^2 →R, F ∈ C_{Δ × R^2}, R$ is the set of real numbers, and $R^2$ is the two-dimensional real Euclidean space.

### Limit theorems for the number of crossings of a fixed plane by certain sequences of generalized dbffusion processes

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 357–371

We characterize the weak convergence of certain sequences of generalized diffusion processes by using a specific functional of a process, namely, the number of crossings of a fixed plane by this process.

### Boundary-value problems for parametric ordinary differential equations

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 372–377

By using semiinverse matrices and a generalized Green's matrix, we construct solutions of boundary-value problems for linear and weakly perturbed nonlinear systems of ordinary differential equations with a parameter in boundary conditions.

### Testing hypotheses by using optimal statistical criteria. I

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 378–388

We propose a method for constructing statistical criteria. It can be used for testing an arbitrary finite set of simple alternative hypotheses. A concept of an optimal statistical criterion is introduced, special cases of which are the Bayesian criterion and the minimax criterion. It is proved that any optimal statistical criterion can be constructed on the basis of the likelihood ratio.

### A remark about orthogonal polynomials

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 389–392

We suggest a renewal method for reconstructing a density in a special case by a system of polynomials orthogonal with respect to it.

### A numerical-analytic method for three-point boundary-value problems

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 393–403

We suggest a modification of the numerical-analytic iteration method. This method is used for studying the problem of existence of solutions and for constructing approximate solutions of nonlinear ordinary differential systems with linear three-point boundary conditions of general form.

### Reducibility of nonlinear almost periodic systems of difference equations given on a torus

Martynyuk D. I., Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 404–412

Sufficient conditions are established for the reducibility of a nonlinear system of difference equations $$x(x + 1) = x(1) + \omega + P(x(t), t) + \lambda,$$ where $P(x, t)$ is a function $2\pi$-periodic in $x_i(i = 1,..., n)$ and almost periodic in $t$ with a frequency basis $\alpha$, to the system $$y(t + 1) = y(t) + \omega.$$

### Collocation method for solving singularly perturbed boundary-value problems by using cubic splines

Blatov I. A., Pokornaya I. Yu., Strygin V. V.

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 411–417

We consider singularly perturbed boundary-value problems in the case of boundary layers. To find approximate solutions of these problems, we use a collocation method based on cubic splines of minimal defect on nonuniform meshes.

### On the exponential dichotomy of pulse evolution systems

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 418–424

The equivalence of regularity and exponential dichotomy is established for linear pulse differential equations with unbounded operators in a Banach space. The separatrix manifolds of a linear pulse system exponentially dichotomous on a semiaxis are studied in a finite-dimensional space. The conditions of weak regularity of this system are given.

### A generalization of operator stochastic integrals

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 425–429

We generalize the method for construction of operator stochastic integrals suggested by Berezanskii, Zhernakov, and Us. We extend the class of integrable commutative quantum processes and study properties of corresponding integrals.

### Asymptotic behavior of the coefficients of solutions of the Hill equation

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 430–432

A new method for finding the asymptotics of the coefficients of solutions of the Hill equation is given.

### Solution of volterra integral equations of the second kind with small nonlinearities by a spline-iteration method

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 433–437

We consider and justify a spline-iteration method for solving Volterra integral equations of the second kind with small nonlinearities.

### Characteristics of power growth of a multidimensional series of exponents

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 438–442

The behavior of sums of multidimensional series of exponents near the boundary of the region of absolute convergence is studied.

### On divergence of series of exponents representing functions regular in convex polygons

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 443–445

We prove that, on a convex polygon, there exist functions from the Smirnov class *E* whose series of exponents diverge in the metric of the space *E.* Similar facts are established for the convergence almost everywhere on the boundary of a polygon, for the uniform convergence on a closed polygon, and for the pointwise convergence at noncorner points of the boundary.

### On potentials of ergodic Markov chains

Moskal'tsova N. V., Shurenkov V. M.

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 446–449

Two theorems on the existence of the potential of an ergodic Markov chain in an arbitrary phase space are proved.

### Existence and asymptotic behavior of solutions of an $n$-th-order differential equation partially solved with respect to derivative

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 450–453

We consider a differential equation of then th order unsolved with respect to the derivative. Under certain assumptions, we investigate the existence and asymptotic behavior of solutions of this equation.

### On the problem without initial conditions for a nonlinear degenerating parabolic system

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 454–456

We indicate the classes for which a solution of the problem for a certain nonlinear degenerating parabolic system without initial conditions exists and is unique. The uniqueness conditions are established both in the case where restrictions are imposed on the behavior of solutions as $t \rightarrow -\infty$ and in the case where they are not imposed. The existence is proved for the arbitrary behavior of the right-hand side of the system as $t \rightarrow -\infty$.

### On one question of B. Amberg

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 457–461

In the case where a group $G$ is the product $G = AB$ of Abelian subgroups $A$ and $B$, one of which has і finite 0-rank, it is proved that the Fitting subgroup $F$ and the Hirsch - Plotkin radical $R$ admit the lecompositions $F = (F \bigcap A)(F \bigcap B)$ and $R = (R \bigcap A)(R \bigcap B)$, respectively. This gives the affinitive answer to B. Amberg's question.

### On harmonic functions satisfying nonlocal boundary conditions

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 462–467

We study functions that are harmonic on a strip and satisfy nonlocal boundary conditions, establish constraints that should be imposed on the coefficients in the boundary conditions to guarantee the uniqueness of nonnegative solutions, and present examples when the uniqueness theorems are not true.

### Application of existence theorems to asymptotic decompositions

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 468–470

By the example of a certain nonlinear boundary-value problem for a second-order hyperbolic equation we justify a new approach to the application of the Krylov-Bogolyubov-Mitropolskii asymptoti methods. For certain linear problems, we present the compatibility conditions and the relations enablinj one to construct exact solutions.

### The conference “Nonlinear problems of differential equations and mathematical physics. The second Bogolyubov readings”

Kolomiyets V. G., Mitropolskiy Yu. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 471