# Volume 61, № 3, 2009

### On the completeness of algebraic polynomials in the spaces $L_p (ℝ, dμ)$

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 291-301

We prove that the theorem on the incompleteness of polynomials in the space $C^0_w$ established by de Branges in 1959 is not true for the space $L_p (ℝ, dμ)$) if the support of the measure μ is sufficiently dense

### Small oscillations of a viscous incompressible fluid with a large number of small interacting particles in the case of their surface distribution

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 302-321

We study the asymptotic behavior of solutions of the problem that describes small motions of a viscous incompressible fluid filling a domain Ω with a large number of suspended small solid interacting particles concentrated in a small neighborhood of a certain smooth surface Γ ⊂ Ω. We prove that, under certain conditions, the limit of these solutions satisfies the original equations in the domain Ω\Γ and some averaged boundary conditions (conjugation conditions) on Γ.

### Problem without initial conditions for linear and almost linear degenerate operator differential equations

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 322-332

We study the problem without initial conditions for linear and almost linear degenerate operator differential equations in Banach spaces. The uniqueness of a solution of this problem is proved in the classes of bounded functions and functions with exponential behavior as t → –∞. We also establish sufficient conditions for initial data under which there exists a solution of the considered problem in the class of functions with exponential behavior at infinity.

### Approximation of conjugate differentiable functions by biharmonic Poisson integrals

Kharkevych Yu. I., Zhyhallo K. M.

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 333-345

We determine the exact values of upper bounds of approximations by biharmonic Poisson integrals on classes of conjugate differentiable functions in uniform and integral metrics.

### p-Geodesic transformations and their groups in second-order tangent bundles induced by concircular transformations of bases

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 346-364

We investigate the flattening properties of the Lie group $G_r^{II}$ of transformations of a second-order tangent bundle $T^2(M)$ equipped with the lift $∇^{II}$ of an affine connection $∇$ and the lift $g^{II}$ of a metric $g$ on the base of $M$ induced by the Lie group $G_r$ of concircular transformations of the base of $M$. The obtained results reveal certain geometric features of the induced group $G_r^{II}$ within the framework of the theory of $p$-geodesic mappings.

### Girsanov theorem for stochastic flows with interaction

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 384-390

We prove an analog of the Girsanov theorem for the stochastic differential equations with interaction $$dz(u,t) = a(z(u,t),μt)dt + ∫R f(z(u,t)−p)W(dp,dt),$$ where $W$ is a Wiener sheet on $ℝ × [0; +∞)$ and $a(∙)$ is a function of special type.

### Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters

Molchanyuk I. V., Plotnikov A. V.

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 384-390

We study the problem of high-speed operation for linear control systems with fuzzy right-hand sides. For this problem, we introduce the notion of optimal solution and establish necessary and sufficient conditions of optimality in the form of the maximum principle.

### On elliptic systems in Hörmander spaces

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 391-399

We study a linear system of pseudodifferential equations uniformly elliptic in Petrovskii’s sense in the Hilbert scale of Hörmander functional spaces defined in $ℝ_n$. An a priori estimate is proved for the solution of the system and its interior smoothness in this scale of spaces is investigated. As an application, we establish a sufficient condition for the existence of continuous bounded derivatives of the solution.

### Boundedness of weak solutions of a nondiagonal parabolic system of two equations

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 400-407

We study the problem of boundedness of weak solutions of a general nondiagonal parabolic system of nonlinear differential equations whose matrix of coefficients satisfies special structural conditions. To do this, we use a procedure based on the estimation of a certain function of unknowns.

### Limiting absorbing belt for a quasiperiodically driven mapping of the shift of intervals

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 408-417

For a discontinuous dynamical system with discrete time on a two-dimensional cylinder generated by a quasiperiodically driven mapping of the shift of intervals with overlapping, we prove the existence and uniqueness of a limiting semiinvariant absorbing belt whose width lies within the same limits as the width of overlapping. In the case of overlapping of constant width, this belt is invariant, and the dynamics inside the belt is equivalent to a skew shift on a two-dimensional torus.

### Stability of a program manifold of control systems with locally quadratic relations

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 418-424

We establish sufficient conditions for the absolute stability of a program manifold of control systems. In the case where the Jacobi matrix is degenerate, sufficient conditions for the absolute stability of a program manifold is obtained by reduction to the central canonical form.

### Stationary distributions of fading evolutions

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 425-431

We study fading random walks on the line. We determine stationary distributions of the fading Markov evolution and investigate the special semi-Markov case where the sojourn times of the renewal process have Erlang distributions.