# Mikhailets V. A.

### Fredholm one-dimensional boundary-value problems with parameter in Sobolev spaces

Atlasiuk O. M., Mikhailets V. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1457-1465

For systems of linear differential equations on a compact interval, we investigate the dependence on a parameter $\varepsilon$ of the solutions to boundary-value problems in the Sobolev spaces $W^n_{\infty}$. We obtain a constructive criterion of the continuous dependence of the solutions of these problems on the parameter $\varepsilon$ for $\varepsilon = 0$. The degree of convergence of these solutions is established.

### Fredholm one-dimensional boundary-value problems in Sobolev spaces

Atlasiuk O. M., Mikhailets V. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1324-1333

For systems of ordinary differential equations on a compact interval, we investigate the character of solvability of the most general linear boundary-value problems in Sobolev spaces. We find the indexes of these problems and obtain a criterion of their well-posedness.

### Continuity in the parameter for the solutions of one-dimensional boundary-value problems for differential equations of higher orders in Slobodetsky spaces

Maslyuk H. O., Mikhailets V. A.

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 404-411

We introduce the most general class of linear boundary-value problems for systems of ordinary differential equations of order $r \geq 2$ whose solutions belong to the Slobodetsky space $^{Ws+r}_p\bigl( (a, b),C_m\bigr),$ where $m \in N,\; s > 0$ and $p \in (1,\infty )$. We also establish sufficient conditions under which the solutions of these problems are continuous functions of the parameter in the Slobodetsky space $W^{s+r}_p\bigl( (a, b),C_m\bigr)$.

### Limit theorems for the solutions of boundary-value problems

Mikhailets V. A., Pelekhata O. B., Reva N. V.

Ukr. Mat. Zh. - 2018. - 70, № 2. - pp. 216-223

We study the uniform limit with respect to a parameter for the solutions of a sequence of general boundary-value problems for systems of linear ordinary differential equations of any order on a finite interval. An essential generalization of the Kiguradze theorem (1987) for these problems is obtained.

### Fredholm Boundary-Value Problems with Parameter in Sobolev Spaces

Gnyp E. V., Kodlyuk T. I., Mikhailets V. A.

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 584-591

For systems of linear differential equations of order $r ∈ ℕ$, we study the most general class of inhomogeneous boundary-value problems whose solutions belong to the Sobolev space $W_p^{n + r} ([a, b],ℂ^m)$, where $m, n + 1 ∈ ℕ$ and $p ∈ [1,∞)$. We show that these problems are Fredholm problems and establish the conditions under which these problems have unique solutions continuous with respect to the parameter in the norm of this Sobolev space.

### Extended Sobolev Scale and Elliptic Operators

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 392-404

We obtain a constructive description of all Hilbert function spaces that are interpolation spaces with respect to a couple of Sobolev spaces $[H^{(s_0)}(\mathbb{R}^n), H^{(s_1)}(\mathbb{R}^n)]$ of some integer orders $s_0$ and $s_1$ and that form an extended Sobolev scale. We find equivalent definitions of these spaces with the use of uniformly elliptic pseudodifferential operators positive definite in $L_2(\mathbb{R}^n)$. Possible applications of the introduced scale of spaces are indicated.

### Myroslav L’vovych Horbachuk (on his 75 th birthday)

Berezansky Yu. M., Gerasimenko V. I., Khruslov E. Ya., Kochubei A. N., Mikhailets V. A., Nizhnik L. P., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 451-454

### Limit theorems for one-dimensional boundary-value problems

Kodlyuk T. I., Mikhailets V. A., Reva N. V.

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 70-81

We study the limit with respect to a parameter in the uniform norm for solutions of general boundary-value problems for systems of linear ordinary differential equations of the first order. A generalization of the Kiguradze theorem (1987) to these problems is obtained. The conditions on the asymptotic behavior of the coefficients of the systems are weakened as much as possible. Sufficient conditions for the Green matrices to converge uniformly to the Green matrix of the limit boundary-value problem are found as well.

### On the unconditional almost-everywhere convergence of general orthogonal series

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1360-1367

The Orlicz and Tandori theorems on the unconditional almost-everywhere convergence, with respect to Lebesgue measure, of real orthogonal series defined on the interval (0; 1) are extended to general complex orthogonal series defined on an arbitrary measure space.

### Regularization of two-term differential equations with singular coefficients by quasiderivatives

Goryunov A. S., Mikhailets V. A.

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1190-1205

We propose a regularization of the formal differential expression $$l(y) = i^m y^{(m)}(t) + q(t)y(t),\; t \in (a, b),$$ of order $m \geq 3$ by using quasiderivatives. It is assumed that the distribution coefficient $q$ has an antiderivative $Q \in L ([a, b]; \mathbb{C})$. In the symmetric case $(Q = \overline{Q})$, we describe self-adjoint and maximal dissipative/accumulative extensions of the minimal operator and its generalized resolvents. In the general (nonselfadjoint) case, we establish conditions for the convergence of the resolvents of the considered operators in norm. The case where $m = 2$ and $Q \in L_2 ([a, b]; \mathbb{C})$ was studied earlier.

### Myroslav L’vovych Horbachuk (on his 70th birthday)

Adamyan V. M., Berezansky Yu. M., Khruslov E. Ya., Kochubei A. N., Kuzhel' S. A., Marchenko V. O., Mikhailets V. A., Nizhnik L. P., Ptashnik B. I., Rofe-Beketov F. S., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 439–442

### Elliptic boundary-value problem in a two-sided improved scale of spaces

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 497–520

We study a regular elliptic boundary-value problem in a bounded domain with smooth boundary. We prove that the operator of this problem is a Fredholm one in a two-sided improved scale of functional Hilbert spaces and that it generates there a complete collection of isomorphisms. Elements of this scale are Hörmander-Volevich-Paneyakh isotropic spaces and some their modi.cations. An *a priori* estimate for a solution is obtained and its regularity is investigated.

### Singularly perturbed periodic and semiperiodic differential operators

Mikhailets V. A., Molyboga V. M.

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 785–797

Qualitative and spectral properties of the form sums $$S_{±}(V) := D^{2m}_{±} + V(x),\quad m ∈ N,$$ are studied in the Hilbert space $L_2(0, 1)$. Here, $(D_{+})$ is a periodic differential operator, $(D_{-})$ is a semiperiodic differential operator, $D_{±}: u ↦ −iu′$, and $V(x)$ is an arbitrary 1-periodic complex-valued distribution from the Sobolev spaces $H_{per}^{−mα},\; α ∈ [0, 1]$.

### 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.

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

### Improved scales of spaces and elliptic boundary-value problems. III

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 679–701

We study elliptic boundary-value problems in improved scales of functional Hilbert spaces on smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The local smoothness of a solution of an elliptic problem in an improved scale is investigated. We establish a sufficient condition under which this solution is classical. Elliptic boundary-value problems with parameter are also studied.

### Yaroslav Borisovich Lopatinsky (09.11.1906 - 03.10.1981)

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

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

### Regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1536–1555

We study a regular elliptic boundary-value problem for a homogeneous differential equation in a bounded domain. We prove that the operator of this problem is a Fredholm (Noether) operator in a two-sided improved scale of functional Hilbert spaces. The elements of this scale are Hörmander-Volevich-Paneyakh isotropic spaces. We establish an *a priori* estimate for a solution and investigate its regularity.

### Improved scales of spaces and elliptic boundary-value problems. II

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 352–370

We study improved scales of functional Hilbert spaces over $\mathbb{R}^n$ and smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The theory of elliptic boundary-value problems in these spaces is developed.

### Improved scales of spaces and elliptic boundary-value problems. I

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 217–235

We study improved scales of functional Hilbert spaces over *R ^{n}* and smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The theory of elliptic boundary-value problems in these spaces is developed.

### Elliptic Operators in a Refined Scale of Functional Spaces

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 689–696

We study the theory of elliptic boundary-value problems in the refined two-sided scale of the Hormander spaces $H^{s, \varphi}$, where $s \in R,\quad \varphi$ is a functional parameter slowly varying on $+\infty$. In the case of the Sobolev spaces $H^{s}$, the function $\varphi(|\xi|) \equiv 1$. We establish that the considered operators possess the properties of the Fredholm operators, and the solutions are globally and locally regular.

### Spectral analysis of elliptic differential equations in Hilbert space

Ukr. Mat. Zh. - 1986. - 38, № 1. - pp. 49–55

### On solvable and sectorial boundary problems for the Sturm-Liouville operator equations

Ukr. Mat. Zh. - 1974. - 26, № 4. - pp. 450–459