# Volume 54, № 8, 2002

### Topological Properties of Periodic Components of *A*-Diffeomorphisms

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1031-1041

We consider periodic components of *A*-diffeomorphisms on two-dimensional manifolds. We study properties of these components and give a topological description of their boundaries.

### Manifolds of Eigenfunctions and Potentials of a Family of Periodic Sturm–Liouville Problems

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1042-1052

We consider a family of boundary-value problems in which the role of a parameter is played by a potential. We investigate the smooth structure and homotopic properties of the manifolds of eigenfunctions and degenerate potentials corresponding to double eigenvalues.

### Mixed Problem for an Ultraparabolic Equation in Unbounded Domain

Lavrenyuk S. P., Protsakh N. P.

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1053-1066

We investigate a mixed problem for a nonlinear ultraparabolic equation in a certain domain *Q* unbounded in the space variables. This equation degenerates on a part of the lateral surface on which boundary conditions are given. We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation; these conditions do not depend on the behavior of the solution at infinity. The problem is investigated in generalized Lebesgue spaces.

### On the Correct Solvability of One Cauchy Problem

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1067-1076

We establish a criterion for convolutors in certain *S*-type spaces. Using this criterion, we prove the correct solvability (in both directions) of one Cauchy problem in these spaces.

### Reconstruction of a Pair Integral Operator of the Convolution Type

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1077-1088

For an arbitrary operator, we pose a general reconstruction problem inverse to the problem of finding solutions. For the pair operator considered, this problem is reduced to the equivalent problem of reconstruction of the kernels of the pair integral equation of the convolution type that generates this operator. In the cases investigated, we prove theorems that characterize the reconstruction of the corresponding kernels, which are constructed in terms of two functions from different Banach algebras of the type *L* _{1}(−∞, ∞) with weight.

### Singularly Perturbed Equations with Impulse Action

Kaplun Yu. I., Samoilenko A. M., Samoilenko V. G.

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1089-1009

We propose and justify an algorithm for the construction of asymptotic solutions of singularly perturbed differential equations with impulse action.

### Approximation Properties of the de la Vallée-Poussin Method

Rukasov V. I., Stepanets O. I.

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1100-1125

We present a survey of results concerning the approximation of classes of periodic functions by the de la Vallée-Poussin sums obtained by various authors in the 20th century.

### Stochastic Lyapunov Functions for a System of Nonlinear Difference Equations

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1126

We study problems related to the stability of solutions of nonlinear difference equations with random perturbations of semi-Markov type. We construct Lyapunov functions for different classes of nonlinear difference equations with semi-Markov right-hand side and establish conditions for their existence.

### Well-Posed and Regular Nonlocal Boundary-Value Problems for Partial Differential Equations

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1135-1142

The present paper deals with the well-posedness and regularity of one class of one-dimensional time-dependent boundary-value problems with global boundary conditions on the entire time interval. We establish conditions for the well-posedness of boundary-value problems for partial differential equations in the class of bounded differentiable functions. A criterion for the regularity of the problem under consideration is also obtained.

### Structure of Matrices and Their Divisors over the Domain of Principal Ideals

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1143-1148

We investigate the structure of matrices and their divisors over the domain of principal ideals.

### On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1149-1153

Let *M* _{f}(*r*) and μ_{f}(*r*) be, respectively, the maximum of the modulus and the maximum term of an entire function *f* and let Φ be a continuously differentiable function convex on (−∞, +∞) and such that *x* = *o*(Φ(*x*)) as *x* → +∞. We establish that, in order that the equality \(\lim \inf \limits_{r \to + \infty} \frac{\ln M_f (r)}{\Phi (\ln r)} = \lim \inf \limits_{r \to + \infty} \frac{\ln \mu_f (r)}{\Phi (\ln r)}\) be true for any entire function *f*, it is necessary and sufficient that ln Φ′(*x*) = *o*(Φ(*x*)) as *x* → +∞.