Volume 55, № 8, 2003

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

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.

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

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.

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.

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.

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 <...< y₁ < π using the equality yi = y_{i + 2s} + 2π furthermore, these points are zeros of a 2π-periodic continuous function f.

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.

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 $S_ϕ^p$.

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

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.

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 → ∞.