# Volume 51, № 3, 1999

### On Chernikov*p*-groups

Gudivok P. M., Shapochka I. V.

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 291–304

We investigate extensions of divisible Abelian*p*-groups with minimality condition by means of a finite*p*-group*H* and establish the conditions under which the problem of describing all nonisomorphic extensions of this sort is wild. All the nonisomorphic Chernikov*p*-groups are described whose factor-group with respect to the maximum divisible Abelian subgroup is a cyclic group of order*p* ^{ s },*s*≤2.

### To problems with continual derivative in boundary conditions for a parabolic equation

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 305–313

We reduce problems with continual derivatives in boundary conditions for a parabolic equation to a system of two singular integral Volterra equations of the second order.

### Information aspects in the theory of approximation and recovery of operators

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 314–327

We present a brief review of new directions in the theory of approximation which are associated with the information approach to the problems of optimum recovery of mathematical objects on the basis of discrete information. Within the framework of this approach, we formulate three problems of recovery of operators and their values. In the case of integral operator, we obtain some estimates of the error.

### On certain nonlinear pseudoparabolic variational inequalities without initial conditions

Lavrenyuk S. P., Ptashnik B. I.

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 328–337

We consider a nonlinear pseudoparabolic variational inequality in a tube domain semibounded in variable*t*. Under certain conditions imposed on coefficients of the inequality, we prove the theorems of existence and uniqueness of a solution without any restriction on its behavior as*t*→−∞.

### The correlation matrix of random solutions of a dynamical system with Markov coefficients

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 338–348

For dynamical systems which are described by systems of differential or difference equations dependent on a finite-valued Markov process, we suggest a new form of equations for moments of their random solution. We derive equations for a correlation matrix of random solutions.

### On conditions of technical stability of solutions of a nonlinear boundary-value problem describing processes under parametric excitation in a Hilbert space

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 349–363

Sufficient conditions for technical stability are obtained for solutions of a nonlinear boundary-value problem which describes distributed parametric processes in a Hilbert space.

### Interpolation of nonlinear functionals by integral continued fractions

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 364–375

We constructively prove the theorem of existence of an interpolation integral chain fraction for a nonlinear functional*F:Q*[0,1]→**R** ^{1}.

### Method of successive approximations for abstract volterra equations in a banach space

Mishura Yu. S., Tomilov Yu. V.

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 376–382

We apply the method of successive approximations to abstract Volterra equations of the form*x=f+a*Ax*, where*A* is a closed linear operator. The assumption is made that a kernel*a* is continuous but is not necessarily of bounded variation.

### Structure of locally graded CDN[)-groups

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 383–388

We introduce the notion of CDN[)-groups:*G* is a CDN[)-group if, for any pair of its subgroups*A* and*B* such that*A* is a proper nonmaximum subgroup, of*B*, there exists a normal subgroup*N* which belongs to*G* and satisfies the inequalities*A≤N . Fifteen types of nilpotent non-Dedekind groups and nine types of nonnilpotent locally graded groups of this kind are obtained.*

*Article (Ukrainian)
Article (Russian)
Brief Communications (English)
Brief Communications (Russian)
Brief Communications (Ukrainian)
Brief Communications (Russian)
Brief Communications (Russian)
*

### On instability of the equilibrium state of nonholonomic systems

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 389–397

We establish a criterion of instability for the equilibrium state of nonholonomic systems, in which gyroscopic forces may dominate over potential forces. We show that, similarly to the case of holonomic systems, the evident domination of gyroscopic forces over potential ones is not sufficient to ensure the equilibrium stability of nonholonomic systems.

### On the theory of groups with generalized minimality condition for closed subgroups

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 398–409

We prove that a topological Abelian locally compact group with generalized minimality condition for closed subgroups is a group of one of the following types: 1) a group with minimality condition for closed subgroups, 2) an additive group of the*J* _{ p }-ring of integer*p*-adic numbers, 3) an additive group*R* _{ p } of the field of*p*-adic numbers (*p* is a prime number).

### On groups factorized by finitely many subgroups

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 410–412

We prove that every group factorizable into a product of finitely many pairwise permutable central-by-finite minimax subgroups is a soluble-by-finite group.

### Decomposition of systems of quasidifferential equations with rapid and slow variables

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 413–417

We obtain the decomposition of systems of quasidifferential equations with rapid and slow variables.

### A modified projection-iterative method for solution of a singular integral equation with parameters and small nonlinearity

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 418–422

We suggest a modified version of the projection-iterative method of solving a singular integral equation with parameters and small nonlinearity.

### The riquier problem for a nonlinear equation unresolved with respect to the Lévy iterated laplacian

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 423–427

We present a method of solving for the nonlinear equation*f*(*U*(*x*),Δ _{ L } ^{2} *U*(*x*)) = Δ_{ L } *U*(*x*) (Δ_{ L } is an infinite-dimensional Laplacian) unresolved with respect to an iterated infinite-dimensional Laplacian and for the Riquier problem for this equation.

### Stability of difference rational systems

Khusainov D. Ya., Shevelenko E. E.

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 428–431

For systems of difference equations with rational functions on the right-hand sides represented in a unified vector matrix form, we obtain stability conditions and calculate a value of the radius of a disk for the domain of asymptotic stability on the basis of the second Lyapunov method.