# Volume 67, № 5, 2015

### On the 90th Birthday of Yurii Makarovych Berezans’kyi

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 579-583

### 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.

### 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.

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 $U(t)x$.

### Convergence and Approximation of the Sturm–Liouville Operators with Potentials-Distributions

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 602–610

We study the operators $L_n y = −(p_n y′)′+q_n y, n ∈ ℤ_{+}$, given on a finite interval with various boundary conditions. It is assumed that the function $q_n$ is a derivative (in a sense of distributions) of $Q_n$ and $1/p_n , Q_n /p_n$, and $Q^2_n/p_n $ are integrable complex-valued functions. The sufficient conditions for the uniform convergence of Green functions $G_n$ of the operators $L_n$ on the square as $n → ∞$ to $G_0$ are established. It is proved that every $G_0$ is the limit of Green functions of the operators $L_n$ with smooth coefficients. If $p_0 > 0$ and $Q_0(t) ∈ ℝ$, then they can be chosen so that $p_n > 0$ and $q_n$ are real-valued and have compact supports.

### Solvability of the Nonlocal Boundary-Value Problem for a System of Differential-Operator Equations in the Sobolev Scale of Spaces and in a Refined Scale

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 611-624

We study the solvability of the nonlocal boundary-value problem with one parameter for a system of differential-operator equations in the Sobolev scale of spaces of functions of many complex variables and in the scale of Hörmander spaces which form a refined Sobolev scale. By using the metric approach, we prove the theorems on lower estimates of small denominators appearing in the construction of solutions of the analyzed problem. They imply the unique solvability of the problem for almost all vectors formed by the coefficients of the equation and the parameter of nonlocal conditions.

### Two-Term Differential Equations with Matrix Distributional Coefficients

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 625–634

We propose a regularization of the formal differential expression $$\begin{array}{cc}\hfill l(y)={i}^m{y}^{(m)}(t)+q(t)y(t),\hfill & \hfill t\in \left(a,b\right)\hfill \end{array},$$ of order $m ≥ 2$ with matrix distribution $q$. It is assumed that $q = Q^{([m/2])}$, where $Q = (Q_{i,j})_{i,j = 1}^s$ is a matrix function with entries $Q_{i,j} ϵ L_2[a, b]$ if $m$ is even and $Q_{i,j} ϵ L_1[a, b]$, otherwise. In the case of a Hermitian matrix $q$, we describe self-adjoint, maximal dissipative, and maximal accumulative extensions of the associated minimal operator and its generalized resolvents.

### A Problem with Condition Containing an Integral Term for a Parabolic-Hyperbolic Equation

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 635-644

In a layer obtained as the Cartesian product of an interval $[−T_1 ,T_2], T_1 ,T_2 > 0$, and a space $ℝ_p, p ≥ 1$, we study a problem with nonlocal condition in the time variable containing an integral term for a mixed parabolic-hyperbolic equation in the class of functions almost periodic in the space variables. For this problem, we establish a criterion of uniqueness and sufficient conditions for the existence of solutions. To solve the problem of small denominators encountered in the construction of the solution, we use the metric approach.

### Mixed Problems for the Two-Dimensional Heat-Conduction Equation in Anisotropic Hörmander Spaces

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 645-656

For some anisotropic inner-product Hörmander spaces, we prove the theorems on well-posedness of initial-boundary-value problems for the two-dimensional heat-conduction equation with Dirichlet or Neumann boundary conditions. The regularity of the functions from these spaces is characterized by a couple of numerical parameters and a function parameter regularly varying at infinity in Karamata’s sense and characterizing the regularity of functions more precisely than in the Sobolev scale.

### Schrödinger Operators with Distributional Matrix Potentials

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 657–671

We study $1D$ Schrödinger operators $L(q)$ with distributional matrix potentials from the negative space $H_{unif}^{− 1} (ℝ, ℂ^{m × m})$. In particular, the class $H_{unif}^{− 1} (ℝ, ℂ^{m × m})$ contains periodic and almost periodic generalized functions. We establish the equivalence of different definitions of the operators $L(q)$, investigate their approximation by operators with smooth potentials $q ∈ L_{unif}^{− 1} (ℝ, ℂ^{m × m})$, and also prove that the spectra of operators $L(q)$ belong to the interior of a certain parabola.

### Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces

Chepurukhina I. S., Murach A. A.

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 672–691

We study elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems were introduced by Lawruk. We prove that the operator corresponding to a problem of this kind is bounded and Fredholm in appropriate couples of the inner product isotropic Hörmander spaces $H^{s,φ}$, which form the refined Sobolev scale. The order of differentiation for these spaces is given by a real number $s$ and a positive function $φ$ slowly varying at infinity in Karamata’s sense. We consider this problem for an arbitrary elliptic equation $Au = f$ in a bounded Euclidean domain $Ω$ under the condition that $u ϵ H^{s,φ} (Ω),\; s < \text{ord} A$, and $f ϵ L_2 (Ω)$. We prove theorems on the a priori estimate and regularity of the generalized solutions to this problem.

### On the Continuity in a Parameter for the Solutions of Boundary-Value Problems Total with Respect to the Spaces $C^{(n+r)}[a, b]$

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 692–700

We study a broad class of linear boundary-value problems for systems of ordinary differential equations, namely, the problems total with respect to the space $C^{(n+r)}[a, b]$, where $n ∈ ℕ$ and $r$ is the order of the equations. For their solutions, we prove the theorem of existence, uniqueness, and continuous dependence on the parameter in this space.

### On Decompositions of a Scalar Operator into a Sum of Self-Adjoint Operators with Finite Spectrum

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 701–716

We consider the problem of classification of nonequivalent representations of a scalar operator $λI$ in the form of a sum of $k$ self-adjoint operators with at most $n_1 , ...,n_k$ points in their spectra, respectively. It is shown that this problem is *-wild for some sets of spectra if $(n_1 , ...,n_k)$ coincides with one of the following $k$ -tuples: $(2, ..., 2)$ for $k ≥ 5,\; (2, 2, 2, 3),\; (2, 11, 11),\; (5, 5, 5)$, or $(4, 6, 6)$. It is demonstrated that, for the operators with points 0 and 1 in the spectra and $k ≥ 5$, the classification problems are *-wild for every rational $λ ϵ 2 [2, 3]$.