# Volume 48, № 8, 1996

### On the separation of isolated solutions of nonlinear integral equations

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1011-1020

We present the theoretical justification and a method for practical realization of the process of separation of solutions isolated in a bounded domain for some classes of nonlinear integral equations. We study the problem of construction of a sequence of approximation equations by the method of mechanical quadratures and the problem of existence of solutions of these equations. We also present methods for approximate solution of these equations and obtain*a posteriori* error estimates.

### Theorem on the central manifold of a nonlinear parabolic equation

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1021-1036

Under certain assumptions, we prove the existence of an *m*-parameter family of solutions that form the central invariant manifold of a nonlinear parabolic equation. For this purpose, we use an abstract scheme that corresponds to energy methods for strongly parabolic equations of arbitrary order.

### Local estimates of solutions of the stationary two-dimensional first boundary-value problem of magnetohydrodynamics

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1037-1046

For solutions of a two-dimensional first boundary-value problem of magnetohydrodynamics, we obtain*a priori* asymptotic (for high Hartmann numbers) estimates of components of the velocity of a liquid and the stream function in the interior of the flow in spaces of continuous functions.

### Some characteristics of sequences of iterations with random perturbations

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1047-1063

For a sequence of random iterations, we study the set of partial limits and the frequency of visiting their neighborhoods.

### Asymptotic stability of integral sets

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1064-1073

We prove theorems on asymptotic, equiasymptotic, and uniform asymptotic stability of the integral sel of a nonautonomous system of ordinary differential equations.

### Localization of spectrum and stability of certain classes of dynamical systems

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1074-1079

We develop a method for the localization of spectra of multiparameter matrix pencils and matrix functions, which reduces the problem to the solution of linear matrix equations and inequalities. We formulate algebraic conditions for the stability of linear systems of differential, difference, and difference-differential equations.

### On the existence of a measurable function with given values of the best approximations in $L_0$

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1080-1085

In the space of convergence in measure, we study the Bernstein problem of existence of a function with given values of the best approximations by a system of finite-dimensional subspaces strictly imbedded in one another.

### Singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1086-1095

The structure of the distribution of a random variable for which elements of the corresponding elementary continued fraction are independent random variables is completely studied. We prove that the distribution is pure and the absolute continuity is impossible, give a criterion of singularity, and study the properties of the spectrum. For the distribution of a random variable for which elements of the corresponding continued fraction form a uniform Markov chain, we describe the spectrum, obtain formulas for the distribution function and density, give a criterion of the Cantor property, and prove that an absolutely continuous component is absent.

### Averaging method in multipoint problems of the theory of nonlinear oscillations

Petryshyn Ya. R., Samoilenko A. M.

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1096-1103

By using the averaging method, we prove the solvability of multipoint problems for nonlinear oscillation systems and estimate the deviation of solutions of original and averaged problems.

### Theorems on instability of systems with respect to linear approximation

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1104-1113

We study the problem of instability of solutions of differential equations with a stationary linear part and a nonstationary nonlinear compact part in a Banach space.

### Complexity of projective methods for the solution of ill-posed problems

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1114-1124

We consider the problem of finite-dimensional approximation for solutions of equations of the first kind and propose a modification of the projective scheme for solving ill-posed problems. We show that this modification allows one to obtain, for many classes of equations of the first kind, the best possible order of accuracy for the Tikhonov regularization by using an amount of information which is far less than for the standard projective technique.

### On invariant tori of stochastic systems on a plane

Kopas' I. N., Stanzhitskii A. N.

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1125-1129

We study invariant tori of stochastic systems of the ltd type on a plane and present conditions for stability of such sets in probability.

### $J$-fractional regularization of linear ill-posed equations

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1130-1143

We present a method for solution of linear ill-posed equations in function spaces based on the use of continuous $J$-fractions. We obtain a meromorphic solution of regularized equations and indicate some cases where a solution can be represented in terms of rational functions.

### On periodic solutions of countable systems of linear and quasilinear difference equations with periodic coefficients

Samoilenko M. V., Teplinsky Yu. V.

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1144-1152

We present conditions for the existence of periodic solutions of linear difference equations with periodic coefficients in spaces of bounded number sequences. In the case where the generating linear equation has a unique periodic solution, we indicate sufficient conditions for the existence of a periodic solution of a quasilinear difference equation.