# Volume 56, № 2, 2004

### Anatolii Mykhailovych Samoilenko is a member of the European Academy of Sciences

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 147

### Perturbation Method for a Parabolic Equation with Drift on a Riemannian Manifold

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 148-159

We construct a fundamental solution of a parabolic equation with drift on a Riemannian manifold of nonpositive curvature by the perturbation method on the basis of a solution of an equation without drift. We establish conditions for the drift field under which this method is applicable.

### Finite-Dimensional Nonlocal Reductions of the Inverse Korteweg–de Vries Dynamical System

Pritula N. N., Vorobyova O. V.

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 160-168

We study finite-dimensional Moser-type reductions for the inverse nonlinear Korteweg–de Vries dynamical system and the Liouville integrability of these reductions in quadratures.

### Asymptotically Well-Posed Boundary-Value Problems

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 169-184

In a domain that is the Cartesian product of an interval and a straight line, we investigate a two-point boundary-value problem for partial differential equations. We establish conditions under which this problem is asymptotically well posed in the class of bounded differentiable functions.

### Correct Solvability of the Cauchy Problem for One Equation of Integral Form

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 185-197

We describe spaces of test functions that generalize *S*-type and *W*-type spaces. In these spaces, we establish the complete solvability of the Cauchy problem for one equation of integral form with Bessel fractional integro-differential operator.

### Bernstein-Type Theorems and Uniqueness Theorems

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 198-213

Let \(f\) be an entire function of finite type with respect to finite order \(\rho {\text{ in }}\mathbb{C}^n \) and let \(\mathbb{E}\) be a subset of an open cone in a certain *n*-dimensional subspace \(\mathbb{R}^{2n} {\text{ ( = }}\mathbb{C}^n {\text{)}}\) (the smaller \(\rho \) , the sparser \(\mathbb{E}\) ). We assume that this cone contains a ray \(\left\{ {z = tz^0 \in \mathbb{C}^n :t > 0} \right\}\) . It is shown that the radial indicator \(h_f (z^0 )\) of \(f\) at any point \(z^0 \in \mathbb{C}^n \backslash \{ 0\} \) may be evaluated in terms of function values at points of the discrete subset \(\mathbb{E}\) . Moreover, if \(f\) tends to zero fast enough as \(z \to \infty \) over \(\mathbb{E}\) , then this function vanishes identically. To prove these results, a special approximation technique is developed. In the last part of the paper, it is proved that, under certain conditions on \(\rho \) and \(\mathbb{E}\) , which are close to exact conditions, the function \(f\) bounded on \(\mathbb{E}\) is bounded on the ray.

### Essentially Infinite-Dimensional Evolution Equations

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 214-220

We investigate the Cauchy problem for evolution equations with essentially infinite-dimensional elliptic operators.

### Scattering Problem for a Wave Equation with Absorption

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 221-227

We prove the solvability of the scattering problem for a wave equation with absorption and study the structure of the scattering operator.

### On Starting Control of Vibrations of a String

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 228-238

Within the framework of the theory of games, we consider the problem of starting control of oscillations of points of a string according to a given law. As control parameters for players, the initial position and the starting velocity of the string are taken. We determine the optimal control for players in both discrete case and continuous case.

### Additive Functions and Chain Complexes of Projective Modules

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 239-246

We study additive functions given on a category of finitely generated projective modules. Using these functions, we define *p*-minimal epimorphisms and give a necessary and sufficient condition for their existence. We prove results concerning the structure of *p*-minimal chains of projective modules.

### Boundedness of the *l*-Index of the Naftalevich–Tsuji Product

Sheremeta M. M., Trukhan Yu.S.

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 247-256

We investigate conditions for zeros under which the Naftalevich–Tsuji product is a function of a bounded *l*-index analytic in the unit disk.

### Averaging of Oscillation Systems with Delay and Integral Boundary Conditions

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 257-263

We prove the existence of a solution and obtain an estimate for the error of the averaging method for a multifrequency system with linearly transformed argument and integral boundary conditions.

### Critical Cases of the π-Stability of a Nonautonomous Quasilinear Equation of the *n*th Order

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 264-270

We establish sufficient conditions for the π-stability of the trivial solution of a quasilinear equation of the *n*th order.

### Structure of Binary Darboux-Type Transformations for Hermitian Adjoint Differential Operators

Prykarpatsky A. K., Samoilenko V. G.

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 271-275

For Hermitian adjoint differential operators, we consider the structure of Darboux–Bäcklund-type transformations in the class of parametrically dependent Hilbert spaces. By using the proposed new method, we obtain the corresponding integro-differential symbols of the operators of transformations in explicit form and consider the problem of their application to the construction of two-dimensional Lax-integrable nonlinear evolution equations and their Darboux–Bäcklund-type transformations.

### Plane Closed Trajectories on Certain Manifolds with Rotation Metric

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 276-283

Trajectories on spherical and toroidal manifolds are studied by methods of infinitesimal and global geometry.

### On Some Spectral Properties of the Energy Operator for an Infinite System in a Magnetic Field

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 284-289

For systems in a magnetic field, we investigate the form sum of an infinite-dimensional energy operator perturbed by a potential. We also investigate changes in the spectrum of the energy operator in the case of its perturbation by a potential.