# Volume 61, № 9, 2009

### Direct and inverse problems for the Dirac operator with a spectral parameter linearly contained in a boundary condition

Amirov R. Kh., Keskin B., Özkan G.

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1155-1166

We investigate a problem for the Dirac differential operators in the case where an eigenparameter not only appears in the differential equation but is also linearly contained in a boundary condition. We prove uniqueness theorems for the inverse spectral problem with known collection of eigenvalues and normalizing constants or two spectra.

### Functor of weakly additive τ-smooth functionals and mappings

↓ Abstract

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1167-1173

We present a criterion for the τ-smoothness of weakly additive order-preserving functionals and establish that the functor of τ-smooth weakly additive functionals preserves the perfectness of mappings. We prove that if a continuous mapping between Tikhonov spaces is such that the mapping between the corresponding spaces of weakly additive functionals generated by it is open, then the indicated mapping is also open. It is shown that the converse statement is true under certain additional conditions.

### Degenerate nonlinear boundary-value problems

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1174-1188

We establish necessary and sufficient conditions for the existence of solutions of weakly nonlinear degenerate boundary-value problems for systems of ordinary differential equations with a Noetherian operator in the linear part. We propose a convergent iterative procedure for finding solutions and establish the relationship between necessary and sufficient conditions.

### Best $M$-Term trigonometric approximations of the classes $B^{Ω}_{p,θ}$ of periodic functions of many variables

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1189-1199

We obtain exact order estimates for the best $M$-term trigonometric approximations of the classes $B^{Ω}_{p,θ}$ of periodic functions of many variables in the space $L_q$.

### Conditions for the stability of an impulsive linear equation with pure delay

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1200-1207

We establish necessary and sufficient conditions for the stability of one class of impulsive linear differential equations with delay.

### Convergence of difference additive functionals under local conditions on their characteristics

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1208-1224

For additive functionals defined on a sequence of Markov chains that approximate a Markov process, we establish the convergence of functionals under the condition of local convergence of their characteristics (mathematical expectations).

### Hausdorff–Besicovitch dimension of the graph of one continuous nowhere-differentiable function

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1225-1239

We investigate fractal properties of the graph of the function $$y = f(x) = ∑^{∞}_{k−1}\frac{β_k}{2^k} ≡ Δ^2_{β_1β_2…β_k…},$$ where $$\beta_1 = \begin{cases} 0 & \mbox{if } \alpha_1(x) = 0,\\ 1 & \mbox{if } \alpha_1(x) \neq 0,\\ \end{cases}$$ $$\beta_k = \begin{cases} β_{k−1} & \mbox{if } \alpha_k(x) = \alpha_{k-1}(x),\\ 1 - β_{k−1} & \mbox{if } \alpha_k(x) \neq \alpha_{k-1}(x),\\ \end{cases}$$ and $α_k(x)$ is the kth ternary digit of $x$: In particular, we prove that this graph is a fractal set with Hausdorff–Besicovitch $α_0(Г_f) = \log_2(1 +2^{\log_32}$ dimension and cell dimension $α_K (Г_f) = 2-\log_32$.

### Structure of the semigroup $OT_n$

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1240-1246

We study the structure of the semigroup $OT_n$, which is a unique (up to an isomorphism) $R$-section of the semigroup $T_n$. For this semigroup, we describe Green relations, determine regular and nilpotent elements, describe maximal nilpotent subsemigroups, and determine the unique irreducible system of generatrices and maximal subsemigroups.

### Variational approach to the solution of linear multiparameter eigenvalue problems

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1247-1256

We associate a multiparameter spectral problem in a real Euclidean space with a variational problem of finding a minimum of a certain functional. We establish the equivalence of the spectralproblem and the variational problem. On the basis of the gradient procedure, we propose a numerical algorithm for the determination of its eigenvalues and eigenvectors. The local convergence of the algorithm is proved.

### Ideals of one-branch singularities of curves of the type $W$

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1257-1266

We establish necessary and sufficient conditions for a one-branch singularity of the type $W$ of a plane algebraic curve to have at most two-parameter families of ideals.

### On the absolute summability of Fourier series of almost-periodic besicovitch functions

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1267-1276

For almost-periodic Besicovitch functions whose spectrum has a limit point only at infinity, we establish criteria for the absolute Cesàro summability of their Fourier series of order greater than –1.

### On some systems of convolution-type first-order integrodifferential equations on the semiaxis

Khachatryan A. Kh., Khachatryan Kh. A.

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1277-1292

We study a class of vector convolution-type integrodifferential equations on the semiaxis used for the description of various applied problems of mathematical physics. By using a special three-factor decomposition of the original mathematical integrodifferential operator, we prove the solvability of these equations in certain functional spaces.

### On the boundedness of one recurrent sequence in a banach space

Gorodnii M. F., Vyatchaninov O. V.

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1293-1296

We establish necessary and sufficient conditions under which a sequence $x_0 = y_0,\; x_{n+1} = Ax_n + y_{n+1},\; n ≥ 0$, is bounded for each bounded sequence $\{y_n : n ⩾ 0\} ⊂ \left\{x ∈ ⋃^{∞}_{n=1} D(A_n)|\sup_{n ⩾ 0} ∥A^nx∥ < ∞\right\}$, where $A$ is a closed operator in a complex Banach space with domain of definition $D(A)$.