# Volume 65, № 3, 2013

### On a Nonlocal Boundary-Value Problem for Systems of Impulsive Hyperbolic Equations

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 315-328

We consider a nonlocal boundary-value problem for a system of impulsive hyperbolic equations. Conditions for the existence of a unique solution of the problem are established by the method of functional parameters, and an algorithm for its determination is proposed.

### Application of the ergodic theory to the investigation of a boundaryvalue problem with periodic operator coefficient

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 329-338

We establish necessary and sufficient conditions for the solvability of a family of differential equations with periodic operator coefficient and periodic boundary condition by using the notion of the relative spectrum of a linear bounded operator in a Banach space and the ergodic theorem. We show that if the existence condition is satisfied, then these periodic solutions can be constructed by using the formula for the generalized inverse of a linear bounded operator obtained in the present paper.

### Correct Solvability of a Nonlocal Multipoint (in Time) Problem for One Class of Evolutionary Equations

Gorodetskii V. V., Martynyuk O. V., Petryshyn R. I.

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 339-353

We study properties of a fundamental solution of a nonlocal multipoint (with respect to time) problem for evolution equations with pseudo-Bessel operators constructed on the basis of constant symbols. The correct solvability of this problem in the class of generalized functions of distribution type is proved.

### Asymptotic Representations for Some Classes of Solutions of Ordinary Differential Equations of Order $n$ with Regularly Varying Nonlinearities

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 354-380

Existence conditions and asymptotic (as $t \uparrow \omega (\omega \leq +\infty)$) representations are obtained for one class of monotone solutions of an $n$th-order differential equation whose right-hand side contains a sum of terms with regularly varying nonlinearities.

### Fredholm solvability of a periodic Neumann problem for a linear telegraph equation

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 381-391

We investigate a periodic problem for the linear telegraph equation $$u_{tt} - u_{xx} + 2\mu u_t = f (x, t)$$ with Neumann boundary conditions. We prove that the operator of the problem is modeled by a Fredholm operator of index zero in the scale of Sobolev spaces of periodic functions. This result is stable under small perturbations of the equation where p becomes variable and discontinuous or an additional zero-order term appears. We also show that the solutions of this problem possess smoothing properties.

### 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.

### Self-Affine Singular and Nowhere Monotone Functions Related to the *Q*-Representation of Real Numbers

Kalashnikov A. V., Pratsiovytyi M. V.

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 405-417

We study functional, differential, integral, self-affine, and fractal properties of continuous functions belonging to a finite-parameter family of functions with a continuum set of "peculiarities". Almost all functions of this family are singular (their derivative is equal to zero almost everywhere in the sense of Lebesgue) or nowhere monotone, in particular, nondifferentiable. We consider different approaches to the definition of these functions (using a system of functional equations, projectors of symbols of different representations, distribution of random variables, etc.).

### Multipoint Problem for *B*-Parabolic Equations

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 418-429

We establish conditions for the well-posedness of a problem for one class of parabolic equations with the Bessel operator in one of the space variables in a bounded domain with multipoint conditions in the time variable and some boundary conditions in the space coordinates. A solution of the problem is constructed in the form of a series in a system of orthogonal functions. We prove a metric theorem on lower bounds for the small denominators appearing in the solution of the problem.

### Asymptotic behavior of higher-order neutral difference equations with general arguments

Chatzarakis G. E., Khatibzadeh H., Miliaras G. N., Stavroulakis I. P.

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 430-450

We study the asymptotic behavior of solutions of the higher-order neutral difference equation $$Δm[x(n)+cx(τ(n))]+p(n)x(σ(n))=0,N∍m≥2,n≥0,$$ where $τ (n)$ is a general retarded argument, $σ(n)$ is a general deviated argument, $c ∈ R; (p(n)) n ≥ 0$ is a sequence of real numbers, $∆$ denotes the forward difference operator $∆x(n) = x(n+1) - x(n)$; and $∆^j$ denotes the jth forward difference operator $∆^j (x(n) = ∆ (∆^{j-1}(x(n)))$ for $j = 2, 3,…,m$. Examples illustrating the results are also given.

### 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