Volume 48, № 8, 1996
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 obtaina posteriori error estimates.
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 obtaina 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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.