2017
Том 69
№ 9

All Issues

Gorbachuk M. L.

Articles: 44
Article (Ukrainian)

Spaces of smooth and generalized vectors of the generator of an analytic semigroup and their applications

Gorbachuk M. L., Gorbachuk V. M.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 4. - pp. 478-509

For a strongly continuous analytic semigroup $\{ e^{tA}\}_{t\geq 0}$ of linear operators in a Banach space $B$ we investigate some locally convex spaces of smooth and generalized vectors of its generator $A$, as well as the extensions and restrictions of this semigroup to these spaces. We extend Lagrange’s result on the representation of a translation group in the form of exponential series to the case of these semigroups and solve the Hille problem on description of the set of all vectors $x \in B$ for which there exists $$\mathrm{l}\mathrm{i}\mathrm{m}_{n\rightarrow \infty }\biggl( I + \frac{tA}n \biggr)^n x$$ and this limit coincides with etAx. Moreover, we present a short survey of particular problems whose solutions are necessary for the introduction of the above-mentioned spaces, namely, the description of all maximal dissipative (self-adjoint) extensions of a dissipative (symmetric) operator; the representation of solutions to operator-differential equations on an open interval and the analysis of their boundary values, and the existence of solutions to an abstract Cauchy problem in various classes of analytic vector-valued functions.

Anniversaries (Ukrainian)

Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Boichuk A. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.

Full text (.pdf)

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144

Article (Ukrainian)

Representations of a Group of Linear Operators in a Banach Space on the Set of Entire Vectors of its Generator

Gorbachuk M. L., Gorbachuk V. M.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 592-601

For a strongly continuous one-parameter group $\{U(t)\} t ∈(−∞,∞)$ of linear operators in a Banach space $\mathfrak{B}$ with generator $A$, we prove the existence of a set $\mathfrak{B}_1$ dense in $\mathfrak{B}_1$ on the elements $x$ of which the function $U(t)x$ admits an extension to an entire B$\mathfrak{B}$-valued vector function. The description of the vectors from $\mathfrak{B}_1$ for which this extension has a finite order of growth and a finite type is presented. It is also established that the inclusion $x ∈ \mathfrak{B}_1$ is a necessary and sufficient condition for the existence of the limit ${ \lim}_{n\to 1}{\left(I+\frac{tA}{n}\right)}^nx$ and this limit is equal to $U(t)x$.

Article (Ukrainian)

History of the Appearance of Infinite-Dimensional Analysis and its Development in Ukraine

Gorbachuk M. L.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2014. - 66, № 8. - pp. 1058–1073

We present a brief survey of the development of functional analysis in Ukraine and the problems of infinite-dimensional analysis posed and solved for thousands of years, which laid the foundations of this branch of mathematics.

Anniversaries (Ukrainian)

Yurii Stephanovych Samoilenko (on his 70th birthday)

Berezansky Yu. M., Boichuk A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Nizhnik L. P., Samoilenko A. M., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

Full text (.pdf)

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1408-1409

Anniversaries (Ukrainian)

Anatolii Mykhailovych Samoilenko (on his 75th birthday)

Berezansky Yu. M., Boichuk A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Perestyuk N. A., Portenko N. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

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Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 3 - 6

Anniversaries (Ukrainian)

Oleksandr Ivanovych Stepanets’ (on the 70 th anniversary of his birthday)

Gorbachuk M. L., Lukovsky I. O., Makarov V. L., Motornyi V. P., Romanyuk A. S., Samoilenko A. M., Serdyuk A. S., Sharko V. V., Zaderei P. V.

Full text (.pdf)

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 579-581

Anniversaries (Ukrainian)

Yuri Yurievich Trokhimchuk (on his 80th birthday)

Berezansky Yu. M., Bojarski B., Gorbachuk M. L., Kopilov A. P., Korolyuk V. S., Lukovsky I. O., Mitropolskiy Yu. A., Portenko N. I., Reshetnyak Yu. G., Samoilenko A. M., Sharko V. V., Shevchuk I. A., Skorokhod A. V., Tamrazov P. M., Zelinskii Yu. B.

Full text (.pdf)

Ukr. Mat. Zh. - 2008. - 60, № 5. - pp. 701 – 703

Anniversaries (Ukrainian)

Fifty years devoted to science (on the 70th birthday of Anatolii Mykhailovych Samoilenko)

Berezansky Yu. M., Dorogovtsev A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Mitropolskiy Yu. A., Perestyuk N. A., Rebenko A. L., Ronto A. M., Ronto M. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M.

Full text (.pdf)

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 3–7

Obituaries (Ukrainian)

Alexander Ivanovich Stepanets

Gorbachuk M. L., Lukovsky I. O., Mitropolskiy Yu. A., Romanyuk A. S., Rukasov V. I., Samoilenko A. M., Serdyuk A. S., Shevchuk I. A., Zaderei P. V.

Full text (.pdf)

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1722-1724

Anniversaries (Ukrainian)

Mark Grigorievich Krein (to the centenary of his birth)

Adamyan V. M., Arov D. Z., Berezansky Yu. M., Gorbachuk M. L., Gorbachuk V. I., Mikhailets V. A., Samoilenko A. M.

Full text (.pdf)

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 579-587

Article (Ukrainian)

Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space

Gorbachuk M. L.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 650–657

For a second-order elliptic differential equation considered on a semiaxis in a Banach space, we show that if the order of growth of its solution at infinity is not higher than the exponential order, then this solution tends exponentially to zero at infinity.

Anniversaries (Ukrainian)

Evgen Yakovich Khruslov (on his 75 th birthday)

Berezansky Yu. M., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Marchenko V. O., Mitropolskiy Yu. A., Nizhnik L. P., Pastur L. A., Samoilenko A. M., Sharko V. V.

Full text (.pdf)

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 549-550

Anniversaries (Ukrainian)

On the 90th birthday of Yurii Alekseevich Mitropol’skii

Berezansky Yu. M., Gorbachuk M. L., Korolyuk V. S., Koshlyakov V. N., Lukovsky I. O., Makarov V. L., Perestyuk N. A., Samoilenko A. M., Samoilenko Yu. I., Sharko V. V., Sharkovsky O. M., Stepanets O. I., Tamrazov P. M., Trohimchuk Yu. Yu

Full text (.pdf)

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 147–151

Anniversaries (Ukrainian)

Yaroslav Borisovich Lopatinsky (09.11.1906 - 03.10.1981)

Gorbachuk M. L., Lyantse V. É., Markovskii A. I., Mikhailets V. A., Samoilenko A. M.

Full text (.pdf)

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1443-1445

Article (Ukrainian)

On the correct solvability of the Dirichlet problem for operator differential equations in a Banach space

Gorbachuk M. L., Gorbachuk V. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1462–1476

We investigate the structure of solutions of an equation $y″(t) = By(t)$, where $B$ is a weakly positive operator in a Banach space B, on the interval $(0, \infty)$ and establish the existence of their limit values as $t → 0$ in a broader locally convex space containing $B$ as a dense set. The analyticity of these solutions on $(0, \infty)$ is proved and their behavior at infinity is studied. We give conditions for the correct solvability of the Dirichlet problem for this equation and substantiate the applicability of power series to the determination of its approximate solutions.

Article (Ukrainian)

On the behavior of orbits of uniformly stable semigroups at infinity

Gorbachuk M. L., Gorbachuk V. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 148–159

For uniformly stable bounded analytic $C_0$-semigroups $\{T(t)\} t ≥ 0$ of linear operators in a Banach space $B$, we study the behavior of their orbits $T (t)x, x ∈ B$, at infinity. We also analyze the relationship between the order of approaching the orbit $T (t)x$ to zero as $t → ∞$ and the degree of smoothness of the vector $x$ with respect to the operator $A^{-1}$ inverse to the generator A of the semigroup $\{T(t)\}_{t \geq 0}$. In particular, it is shown that, for this semigroup, there exist orbits approaching zero at infinity not slower than $e^{-at^{\alpha}}$, where $a > 0,\; 0 < \alpha < \pi/(2 (\pi - 0 )),\; \theta$ is the angle of analyticity of $\{T(t)\}_{t \geq 0}$, and the collection of these orbits is dense in the set of all orbits.

Anniversaries (Ukrainian)

Leonid Pavlovych Nyzhnyk (on his 70-th birthday)

Berezansky Yu. M., Gorbachuk M. L., Gorbachuk V. I., Khruslov E. Ya., Kostyuchenko A. G., Kuzhel' S. A., Marchenko V. O., Samoilenko A. M., Samoilenko Yu. S.

Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1120-1122

Anniversaries (Ukrainian)

Yurij Makarovich Berezansky (the 80th anniversary of his birth)

Gorbachuk M. L., Gorbachuk V. I., Kondratiev Yu. G., Kostyuchenko A. G., Marchenko V. O., Mitropolskiy Yu. A., Nizhnik L. P., Rofe-Beketov F. S., Samoilenko A. M., Samoilenko Yu. S.

Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 3-11

Article (Ukrainian)

Functional Analysis in the Institute of Mathematics of the National Academy of Sciences of Ukraine

Berezansky Yu. M., Gorbachuk M. L., Gorbachuk V. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 582–600

We give a brief survey of results on functional analysis obtained at the Institute of Mathematics of the Ukrainian National Academy of Sciences from the day of its foundation.

Article (Ukrainian)

Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method

Gorbachuk M. L., Hrushka Ya. I., Torba S. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 633–643

For an arbitrary self-adjoint operator B in a Hilbert space \(\mathfrak{H}\), we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector \(x \in \mathfrak{H}\) with respect to the operator B, the rate of convergence to zero of its best approximation by exponential-type entire vectors of the operator B, and the k-modulus of continuity of the vector x with respect to the operator B. The results are used for finding a priori estimates for the Ritz approximate solutions of operator equations in a Hilbert space.

Anniversaries (Ukrainian)

Dmytro Yakovych Petryna (on his 70 th birthday)

Gorbachuk M. L., Khruslov E. Ya., Lukovsky I. O., Marchenko V. O., Mitropolskiy Yu. A., Pastur L. A., Samoilenko A. M., Skrypnik I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 291-292

Anniversaries (Ukrainian)

Bohdan Iosypovych Ptashnyk

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

Full text (.pdf)

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1155–1156

Article (Ukrainian)

Behavior of the extended spectral measure of a self-adjoint operator at infinity

Gorbachuk M. L., Gorbachuk V. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 510–516

We study the dependence of the rate of growth of the extended spectral measure of a self-adjoint operator at infinity on the order of singularity of the vector on which this measure is considered.

Article (Ukrainian)

Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems

Gorbachuk M. L., Gorbachuk V. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 55–62

Article (Ukrainian)

Inverse problem for a sturm-liouville elliptic equation in a Hilbert space

Gorbachuk M. L., Knyukh B. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1991. - 43, № 11. - pp. 1537–1540

Anniversaries (Russian)

Samuil Davidovich Eidelman (On his sixtieth birthday)

Berezansky Yu. M., Gorbachuk M. L., Ivasyshen S. D., Korolyuk V. S., Mitropolskiy Yu. A., Skrypnik I. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1991. - 43, № 5. - pp. 578

Article (Ukrainian)

Extension theory for symmetric operators and boundary value problems for differential equations

Gorbachuk M. L., Gorbachuk V. I., Kochubei A. N.

Full text (.pdf)

Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1299–1313

Article (Ukrainian)

Convergence of series of independent Gaussian operators

Gorbachuk M. L., Gorodetskii V. V.

Full text (.pdf)

Ukr. Mat. Zh. - 1984. - 36, № 4. - pp. 500 – 502

Article (Ukrainian)

Behavior at infinity of solutions of operator differential equations

Gorbachuk M. L., Vynnyshyn Ya. F.

Full text (.pdf)

Ukr. Mat. Zh. - 1983. - 35, № 4. - pp. 489—493

Article (Ukrainian)

Weak solutions of differential equations in Hilbert space

Gorbachuk M. L., Kashpirovskii A. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1981. - 33, № 4. - pp. 513–518

Article (Ukrainian)

Scattering problem for first-order differential equations with operator coefficients

Gorbachuk M. L., Gorbachuk V. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1978. - 30, № 4. - pp. 452–461

Article (Ukrainian)

Some questions of the spectral theory of differential equations of elliptic type in the space of vector-functions

Gorbachuk M. L., Gorbachuk V. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1976. - 28, № 3. - pp. 313–324

Article (Ukrainian)

Some questions of the spectral theory of differential equations of elliptic type in the space of vector functions on a finite interval

Gorbachuk M. L., Gorbachuk V. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1976. - 28, № 1. - pp. 12–26

Article (Ukrainian)

The spectrum of self-adjoint extensions of the minimal operator generated by a Sturm-Liouville equation with operator potential

Gorbachuk M. L., Gorbachuk V. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1972. - 24, № 6. - pp. 726—734

Article (Ukrainian)

Classes of boundary-value problems for the sturm — Liouville equation with an operator potential

Gorbachuk M. L., Gorbachuk V. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1972. - 24, № 3. - pp. 291—304

Article (Ukrainian)

Problems of the spectral theory of the second order linear differential equation with unbounded operator coefficients

Gorbachuk M. L., Gorbachuk V. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1971. - 23, № 1. - pp. 3–14

Article (Ukrainian)

On self-adjoint semibounded abstract differential operators

Gorbachuk M. L., Vainerman L. I.

Full text (.pdf)

Ukr. Mat. Zh. - 1970. - 22, № 6. - pp. 806—808

Article (Ukrainian)

The stability of motion of a solid body containing a liquid-filled cavity suspended by a cord

Gorbachuk M. L., Sleptsova G. P., Temchenko M. Ye.

Full text (.pdf)

Ukr. Mat. Zh. - 1968. - 20, № 5. - pp. 586–602

Article (Ukrainian)

Basic results of studies at the mathematical-analysis department of the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR

Berezansky Yu. M., Gorbachuk M. L.

Full text (.pdf)

Ukr. Mat. Zh. - 1967. - 19, № 6. - pp. 73–92

Brief Communications (Russian)

On the continuation of positively definite functions of two variables

Berezansky Yu. M., Gorbachuk M. L.

Full text (.pdf)

Ukr. Mat. Zh. - 1965. - 17, № 5. - pp. 96-102

Brief Communications (Russian)

On the description of continuations of a positively definite operator function

Gorbachuk M. L.

Full text (.pdf)

Ukr. Mat. Zh. - 1965. - 17, № 5. - pp. 102-110

Article (Russian)

On the representation of positively determined operator functions

Gorbachuk M. L.

Full text (.pdf)

Ukr. Mat. Zh. - 1965. - 17, № 2. - pp. 29-46

Brief Communications (Russian)

Spectral expansions of Infinitesimal operators of the fourth dimensional group of translations

Gorbachuk M. L., Hachok V. P.

Full text (.pdf)

Ukr. Mat. Zh. - 1964. - 16, № 4. - pp. 528-534