# Volume 48, № 11, 1996

### Yurii Dmitrievich Sokolov (on his 100th birthday)

Gorbachuk M. L., Luchka A. Y., Mitropolskiy Yu. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1443-1445

### Just people of the world

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1446-1447

### On the optimal rate of convergence of the projection-iterative method and some generalizations of it on a class of equations with smoothing operators

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1448-1456

For some classes of operator equations of the second kind with smoothing operators, we find the exact order of the optimal rate of convergence of generalized projection-iterative methods.

### On boundary-value problems for a second-order differential equation with complex coefficients in a plane domain

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1457-1467

We study boundary-value problems for a homogeneous partial differential equation of the second order with arbitrary constant complex coefficients and a homogeneous symbol in a bounded domain with smooth boundary. Necessary and sufficient conditions for the solvability of the Cauchy problem are obtained. These conditions are written in the form of a moment problem on the boundary of the domain and applied to the investigation of boundary-value problems. This moment problem is solved in the case of a disk.

### Multipoint problem for hyperbolic equations with variable coefficients

Klyus I. S., Ptashnik B. I., Vasylyshyn P. B.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1468-1476

By using the metric approach, we study the problem of classical well-posedness of a problem with multipoint conditions with respect to time in a tube domain for linear hyperbolic equations of order 2*n* (*n* ≥ 1) with coefficients depending on*x*. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of the problem.

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

Gorbachuk M. L., Yakymiv R. Ya.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1477-1483

For an equation*Au = f* where*A* is a closed densely defined operator in a Hilbert space*H, f* ε*H*, 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.

### On characteristic properties of singular operators

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1484-1493

For a linear operator*S* in a Hilbert space ℋ, the relationship between the following properties is investigated: (i)*S* is singular (= nowhere closable), (ii) the set ker*S* is dense in ℋ, and (iii)D(*S*)∩ℛ(*S*)={0}.

### On one variational criterion of stability of pseudoequilibrium forms

Lukovsky I. O., Mykhailyuk O. V., Timokha A. N.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1494-1500

We establish a variational criterion of stability for the problem of the vibrocapillary equilibrium state which appears in the theory of interaction of limited volumes of liquid with vibrational fields.

### Methods for the solution of equations with restrictions and the Sokolov projection-iterative method

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1501-1509

We establish consistency conditions for equations with additional restrictions in a Hilbert space, suggest and justify iterative methods for the construction of approximate solutions, and describe the relationship between these methods and the Sokolov projection-iterative method.

### Variational schemes for vector eigenvalue problems

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1510-1517

We construct and study exact and truncated self-adjoint three-point variational schemes of any degree of accuracy for self-adjoint eigenvalue problems for systems of second-order ordinary differential equations.

### Potential fields with axial symmetry and algebras of monogenic functions of a vector variable. I

Mel'nichenko I. P., Plaksa S. A.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1518-1529

We obtain a new representation of potential and flow functions for space potential solenoidal fields with axial symmetry. We study principal algebraic-analytical properties of monogenic functions of a vector variable with values in an infinite-dimensional Banach algebra of even Fourier series and describe the relationship between these functions and the axially symmetric potential and Stokes flow function. The suggested method for the description of the above-mentioned fields is an analog of the method of analytic functions in the complex plane for the description of plane potential fields.

### On the optimization of projection-iterative methods for the approximate solution of ill-posed problems

Pereverzev S. V., Solodkii S. G.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1530-1537

We consider a new version of the projection-iterative method for the solution of operator equations of the first kind. We show that it is more economical in the sense of amount of used discrete information.

### Moduli of continuity defined by zero continuation of functions and *K*-functionals with restrictions

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1537-1554

We consider the following*K*-functional: $$K(\delta ,f)_p : = \mathop {\sup }\limits_{g \in W_{p U}^r } \left\{ {\left\| {f - g} \right\|_{L_p } + \delta \sum\limits_{j = 0}^r {\left\| {g^{(j)} } \right\|_{L_p } } } \right\}, \delta \geqslant 0,$$ where ƒ ∈*L* _{ p }:=*L* _{ p }[0, 1] and*W* _{ p,U } ^{ r } is a subspace of the Sobolev space*W* _{ p } ^{ r } [0, 1], 1≤*p*≤∞, which consists of functions*g* such that \(\int_0^1 {g^{(l_j )} (\tau ) d\sigma _j (\tau ) = 0, j = 1, ... , n} \) . Assume that 0≤*l* _{ l }≤...≤*l* _{ n }≤*r*-1 and there is at least one point τ_{ j } of jump for each function σ_{ j }, and if τ_{ j }=τ_{ s } for*j* ≠*s*, then*l* _{ j } ≠*l* _{ s }. Let \(\hat f(t) = f(t)\) , 0≤*t*≤1, let \(\hat f(t) = 0\) ,*t*<0, and let the modulus of continuity of the function*f* be given by the equality $$\hat \omega _0^{[l]} (\delta ,f)_p : = \mathop {\sup }\limits_{0 \leqslant h \leqslant \delta } \left\| {\sum\limits_{j = 0}^l {( - 1)^j \left( \begin{gathered} l \hfill \\ j \hfill \\ \end{gathered} \right)\hat f( - hj)} } \right\|_{L_p } , \delta \geqslant 0.$$

We obtain the estimates \(K(\delta ^r ,f)_p \leqslant c\hat \omega _0^{[l_1 ]} (\delta ,f)_p \) and \(K(\delta ^r ,f)_p \leqslant c\hat \omega _0^{[l_1 + 1]} (\delta ^\beta ,f)_p \) , where β=(*pl* _{ l } + 1)/*p*(*l* _{1} + 1), and the constant*c*>0 does not depend on δ>0 and ƒ ∈*L* _{ p }. We also establish some other estimates for the considered*K*-functional.

### Sobolev problem in the complete scale of Banach Spaces

Roitberg Ya. A., Sklyarets A. V.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1555-1563

In a bounded domain*G* ⊂ ℝ^{ n }, whose boundary is the union of manifolds of different dimensions, we study the Sobolev problem for a properly elliptic expression of order 2*m*. The boundary conditions are given by linear differential expressions on manifolds of different dimensions. We study the Sobolev problem in the complete scale of Banach spaces. For this problem, we prove the theorem on a complete set of isomorphisms and indicate its applications.

### Coercive solvability of a generalized Cauchy-Riemann system in the Space $L_p (E)$

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1564-1569

For an inhomogeneous generalized Cauchy-Riemann system with nonsmooth coefficients separated from zero, we establish conditions for the solvability and estimation of a weighted solution and its first-order derivatives.

### Periodic solutions of Quasilinear Hyperbolic integro-differential equations of second order

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1572-1575

We study a periodic boundary-value problem for a quasilinear integro-differential equation with the d’Alembert operator on the left-hand side and a nonlinear integral operator on the right-hand side. We establish conditions under which the uniqueness theorems are true.

### On averaging of differential inclusions in the case where the average of the right-hand side does not exist

Plotnikov V. A., Savchenko V. M.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1572-1575

We consider the problem of application of the averaging method to the asymptotic approximation of solutions of differential inclusions of standard form in the case where the average of the right-hand side does not exist.

### Boundary-Value problems for systems of integro-differential equations with Degenerate Kernel

Boichuk О. A., Krivosheya S. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1576-1579

By using methods of the theory of generalized inverse matrices, we establish a criterion of solvability and study the structure of the set of solutions of a general linear Noether boundary-value problem for systems of integro-differential equations of Fredholm type with degenerate kernel.

### On the instability of lagrange solutions in the three-body problem

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1580-1585

We consider the relation between the Lyapunov instability of Lagrange equilateral triangle solutions and their orbital instability. We present a theorem on the orbital instability of Lagrange solutions. This theorem is extended to the planar*n*-body problem.