# Volume 55, № 8, 2003

### Construction of a Fundamental Solution of a Perturbed Parabolic Equation

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1011-1021

We propose a method for the construction of a solution of a parabolic equation in the case where the diffusion operator is perturbed.

### A Problem with Nonlocal Conditions for Partial Differential Equations Unsolved with Respect to the Leading Derivative

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1022-1034

In the domain that is the product of a segment and a *p*-dimensional torus, we investigate the well-posedness of a problem with nonlocal boundary conditions for a partial differential equation unsolved with respect to the leading derivative with respect to a selected variable. We establish conditions for the the classical well-posedness of the problem and prove metric theorems on the lower bounds of small denominators appearing in the course of its solution.

### On Necessary and Sufficient Conditions for the Asymptotic Stability of Impulsive Systems

Gladilina R. I., Ignat'ev A. O.

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1035-1043

We prove that the sufficient conditions for the asymptotic stability of impulsive systems obtained by Gurgula and Perestyuk are also necessary conditions.

### An Interpolation Problem in the Class $R[a,b]$

Choque Rivero A. E., Dyukarev Yu. M.

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1044-1057

We consider an interpolation problem for matrix functions of the class $R[a,b]$. In the nondegenerate case, we describe all solutions in terms of fractional linear transformations. An explicit formula for the resolvent matrix is obtained.

### Global Attractor of an Evolution Inclusion with Pulse Influence at Fixed Moments of Time

Kapustyan O. V., Perestyuk N. A.

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1058-1068

We consider an autonomous evolution inclusion with pulse perturbations at fixed moments of time. Under the conditions of global solvability, we prove the existence of a minimal compact set in the phase space that attracts all trajectories.

### Equilibrium and Nonequilibrium States of the Model Fröhlich–Peierls Hamiltonian

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1069-1086

The model Fröhlich–Peierls Hamiltonian for electrons interacting with phonons only in some infinite discrete modes is considered. It is shown that, in the equilibrium case, this model is thermodynamically equivalent to the model of electrons with periodic potential and free phonons. In the one-dimensional case, the potential is determined exactly in terms of the Weierstrass elliptic function, and the eigenvalue problem can also be solved exactly. Nonequilibrium states are described by the nonlinear Schrödinger and wave equations, which have exact soliton solutions in the one-dimensional case.

### Sign-Preserving Approximation of Periodic Functions

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1087-1098

We prove the Jackson theorem for a zero-preserving approximation of periodic functions (i.e., in the case where the approximating polynomial has the same zeros *y* _{i}) and for a sign-preserving approximation [i.e., in the case where the approximating polynomial is of the same sign as a function *f* on each interval (*y* _{i}, *y* _{ i − 1})]. Here, *y* _{i} are the points obtained from the initial points −π ≤ *y* _{2s } ≤_{y} _{2s}−1 <...< y_{1} < π using the equality yi = y_{i + 2s} + 2π furthermore, these points are zeros of a 2π-periodic continuous function f.

### On One Class of Divisors of Polynomial Matrices over Integral Domains

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1099-1106

We establish conditions for the existence of a unital divisor for a polynomial matrix over an integral domain of characteristic zero in the case where its eigenvalues are known.

### Best $n$-Term Approximations by Λ-Methods in the Spaces $S_ϕ^p$

Shydlich A. L., Stepanets O. I.

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1107

We determine the exact values of the upper bounds of $n$-term approximations of $q$-ellipsoids by Λ-methods in the spaces $S_ϕ^p$ in the metrics of the spaces $S_ϕ^p$.

### On the Cauchy Problem for Degenerate Difference Equations of the *m*th Order in a Banach Space

Semenyshyna I. V., Teplinsky Yu. V.

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1127-1137

We establish sufficient conditions for the solvability of the Cauchy problem for degenerate difference equations of the *m*th order in a Banach space.

### Estimate for the Best Approximation of Summable Functions of Several Variables with a Certain Symmetry of Fourier Coefficients

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1138-1142

An upper bound for the best approximation of summable functions of several variables by trigonometric polynomials in the metric of *L* is determined in terms of Fourier coefficients. We consider functions representable by trigonometric series with certain symmetry of coefficients satisfying a multiple analog of the Sidon–Telyakovskii conditions.

### On the Asymptotic Behavior of Solutions of the First Initial Boundary-Value Problems for Parabolic Equations

Nguyen Manh Hung, Tran Thi Loan

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1143-1152

We consider the first initial boundary-value problem for a strongly parabolic system on an infinite cylinder with nonsmooth boundary. We prove some results on the existence, uniqueness, and asymptotic behavior of solutions as *t* → ∞.