# Volume 55, № 10, 2003

### On Schur–Nevanlinna–Pick Indefinite Interpolation Problem

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1299-1315

The Schur–Nevanlinna–Pick interpolation problem is considered in the class of generalized Schur functions and reduced to the problem of the extension of a certain isometric operator *V* that acts in the Pontryagin space. The description of the solutions of this problem is based on the theory of the resolvent matrix developed by Krein.

### Equilibrium Problems for Potentials with External Fields

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1315-1339

We investigate the problem on the minimum of energy over fairly general (generally speaking, noncompact) classes of real-valued Radon measures associated with a system of sets in a locally compact space in the presence of external fields. The classes of admissible measures are determined by a certain normalization or by a normalization and a certain majorant measure σ. In both cases, we establish sufficient conditions for the existence of minimizing measures and prove that, under fairly general assumptions, these conditions are also necessary. We show that, for sufficiently large σ, there is a close correlation between the facts of unsolvability (or solvability) of both variational problems considered.

### Lie Symmetries, *Q*-Conditional Symmetries, and Exact Solutions of Nonlinear Systems of Diffusion-Convection Equations

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1340-1355

A complete description of Lie symmetries is obtained for a class of nonlinear diffusion-convection systems containing two Burgers-type equations with two arbitrary functions. A nonlinear diffusion-convection system with unique symmetry properties that is simultaneously invariant with respect to the generalized Galilei algebra and the operators of *Q*-conditional symmetries with cubic nonlinearities relative to dependent variables is found. For systems of evolution equations, operators of this sort are found for the first time. For the nonlinear system obtained, a system of Lie and non-Lie ansätze is constructed. Exact solutions, which can be used in solving relevant boundary-value problems, are also found.

### On Polymer Expansion for Gibbsian States of Nonequilibrium Systems of Interacting Brownian Oscillators

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1356-1377

The convergence of polymer cluster expansions for correlation functions of general Gibbs oscillator-type systems and related nonequilibrium systems of Brownian oscillators is established. The initial states for the latter are Gibbsian. It is proved that the sequence of the constructed correlation functions of the nonequilibrium system is a generalized solution of a diffusion BBGKY-type hierarchy.

### Extremal Problems of Approximation Theory in Linear Spaces

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1378-1409

We propose an approach that enables one to pose and completely solve main extremal problems in approximation theory in abstract linear spaces. This approach coincides with the traditional one in the case of approximation of sets of functions defined and square integrable with respect to a given σ-additive measure on manifolds in *R* ^{m}, *m* ≥ 1.

### On the Boundedness of a Recurrence Sequence in a Banach Space

Gomilko A. M., Gorodnii M. F., Lagoda O. A.

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1410-1418

We investigate the problem of the boundedness of the following recurrence sequence in a Banach space *B*: \(x_n = \sum\limits_{k = 1}^\infty {A_k x_{n - k} + y_n } ,{ }n \geqslant 1,{ }x_n = {\alpha}_n ,{ }n \leqslant 0,\) where |*y* _{n}} and |α_{ n }} are sequences bounded in *B*, and *A* _{k}, *k* ≥ 1, are linear bounded operators. We prove that if, for any ε > 0, the condition \(\sum\limits_{k = 1}^\infty {k^{1 + {\varepsilon}} \left\| {A_k } \right\| < \infty } \) is satisfied, then the sequence |*x* _{n}} is bounded for arbitrary bounded sequences |*y* _{n}} and |α_{ n }} if and only if the operator \(I - \sum {_{k = 1}^\infty {\text{ }}z^k A_k } \) has the continuous inverse for every *z* ∈ *C*, | *z* | ≤ 1.

### Qualitative Investigation of the Singular Cauchy Problem $\sum\limits_{k = 1}^n {(a_{k1} t + a_{k2} x)(x')^k = b_1 t + b_2 x + f(t,x,x'),x(0) = 0}$

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1419-1424

We prove the existence of continuously differentiable solutions $x:(0,ρ] → R$ with required asymptotic properties as $t → +0$ and determine the number of these solutions.

### Aggregate-Iterative Methods for the Approximation of Solutions of Boundary-Value Problems

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1425-1431

We construct a special aggregate-iterative algorithm (a two-parameter method) for the iterative integration of a differential equation with two-point boundary conditions. We establish conditions for the convergence of this method and present partial cases of the two-parameter aggregate-iterative algorithm.

### Singular Nonlinear Eigenvalue Problem for One Class of Second-Order Differential Equations

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1431-1438

The eigenvalue problem for a singular nonlinear differential equation of the second order is considered on a semiaxis. For this problem, we establish sufficient conditions for the existence of a solution with given number of zeros monotonically decreasing to zero at infinity.

### Aleksei Vasil'evich Pogorelov

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1439-1440