Estimate of error of an approximated solution by the method of moments of an operator equation

For an equationAu = f whereA is a closed densely defined operator in a Hilbert spaceH, f εH, we estimate the deviation of its approximated solution obtained by the moment method from the exact solution. All presented theorems are of direct and inverse character. The paper refers to direct methods of mathematical physics, the development of which was promoted by Yu. D. Sokolov, the well-known Ukrainian mathematician and mechanic, a great humanitarian and righteous man. We dedicate this paper to his blessed memory.     

Existence of a solution and averaging of nonlinear multifrequency problems with delay

The method of averaging over the fast variables is applied to the investigation of multifrequency systems with linearly transformed argument. We prove the existence of solutions of the initial-and boundary-value problems in a small neighborhood of the solution of the averaged problem and estimate the error of the method of averaging for slow variables. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 435–446, April, 2007.     

Justification of the averaging method for a system of singularly perturbed differential equations with impulses

The averaging method is justified for a system of singularly perturbed differential equations of the form $$\dot x(t) = X\left( {t,\frac{t}{\varepsilon },x(t),y(t),\varepsilon } \right),\varepsilon \dot y(t) = Y(t,x(t),y(t))$$ , in the presence of impulses.     

Boundary-value problems with parameters for a multifrequency oscillation system

By using the averaging method, we prove the solvability of boundary-value problems with parameters for nonlinear oscillation systems. We obtain estimates for the deviation of solutions of averaged problems from solutions of original problems.     

Embedding theorems with an exponential weight on the real semiaxis

We state embedding theorems between spaces of functions defined on the real semi-axis, which can grow exponentially both at 0 and at +∞.     

Justification of the averaging method for a system of functional-differential equations with variable structure and impulses

This paper presents a justification of the averaging method for a system of functional-differential equations with variable structure, impulses, and transformed argument, depending on the time and on the unknown function.     

On Kolmogorov-type inequalities for functions defined on a semiaxis

We established necessary and sufficient conditions for the existence of a function from the class S ⁻ with given integral norms of three successive derivatives (generally speaking, of fractional order). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1299–1312, October, 2007.     

Polynomial approximation with an exponential weight on the real semiaxis

We consider the polynomial approximation on (0,+∞), with the weight $u(x)= x^{\gamma}e^{-x^{-\alpha}-x^{\beta}}$ , α>0, β>1 and γ≧0. We introduce new moduli of smoothness and related K-functionals for functions defined on the real semiaxis, which can grow exponentially both at 0 and at +∞. Then we prove the Jackson theorem, also in its weaker form, and the Stechkin inequality. Moreover, we study the behavior of the derivatives of polynomials of best approximation.     

Exact inequalities for derivatives of functions of low smoothness defined on an axis and a semiaxis

We obtain new exact inequalities of the form

Inverse nonstationary problem of scattering for a hyperbolic system of n equations of the first order on a semiaxis

In connection with a joint study of n-1 problems with diverse boundary conditions we construct a scattering operator for a hyperbolic system on a semiaxis for the case of n-1 incident and one scattered wave. The possibility of the unique recovery of the coefficients of the system with respect to the scattering operator is shown.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1638–1646, December, 1991.     

On a modification of the averaging method for seeking higher approximations

Systems in the N.M. Bogolyubov standard form as well as systems with rapid phases are considered. It is proposed to seek the solution in the form of an asymptotic series in a small parameter with coefficients representable in the form of the sum of two functions. The first depends on slow time and is found as the solution of a simpler equation in a finite segment. The second is a trigonometric polynomial of the time (or the angular displacements) with coefficients which depend on the slow time (it is found in an explicit manner). It is convenient to use the results in solving certain problems in celestial mechanics.