# Volume 64, № 4, 2012

### A companion of Dragomir's generalization of Ostrowski's inequality and applications in numerical integration

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 435-450

\lambda) f(x) - \int^b_a f(t)dt\right]\right| \leq$$ $$\leq\left[\frac{(b-a)^2}{4}(\lambda^2 + (1 - \lambda)^2) + \left(x - \frac{a + b}{2}\right)^2\right] ||f'||_{\infty}$$ are established. Some sharp inequalities are proved. An application to a composite quadrature rule is provided.

### Neumann problem and one oblique-derivative problem for an improperly elliptic equation

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 451-462

We investigate the solvability of an inhomogeneous Neumann problem and oblique-derivative problem for an improperly elliptic scalar differential equation with complex coefficients in a bounded domain. The model case where the domain is the unit disk and the equation does not have lower-order terms is studied. It is proved that the classes of boundary data for which the problems have unique solutions in a Sobolev space are the spaces of functions with exponentially decreasing Fourier coefficients.

### Estimation of the number of ultrasubharmonics for a two-dimensional almost autonomous Hamiltonian system periodic in time

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 463-489

Using the Arnold method of detection of fixed points of symplectic diffeomorphisms, we find lower estimates for the number of ultrasubharmonics in a Hamiltonian system on a two-dimensional symplectic manifold with almost autonomous time-periodic Hamiltonian. We show that the asymptotic behavior of these estimates as the perturbation parameter tends to zero depends on which of the four zones of a ring domain foliated by closed level curves of the unperturbed Hamiltonian the generating unperturbed ultrasubharmonics belong to.

### On the summability of double Walsh - Fourier series of functions of bounded generalized variation

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 490-507

The convergence of Cesaro means of negative order of double Walsh-Fourier series of functions of bounded generalized variation is investigated.

### On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 508-524

We study the following modification of the Landau-Kolmogorov problem: Let $k, r \in \mathbb{N}, \quad 1 \leq k \leq r -1$ and $p, q, s \in [1, \infty]$. Also let $MM^m,\; m \in \mathbb{N}$, be the class of nonnegative functions defined on the segment $[0,1]$ whose derivatives of orders $1, 2,... , m$ are nonnegative almost everywhere on $[0,1]$. For every $\delta > 0$, find the exact value of the quantity $$w^{k, r}_{p, q, s}(\delta; MM^m) := \sup \left\{ ||x^{(k)}||_q : \; x \in MM^m,\; ||x||_p \leq \delta, \;\; ||x^{(r)}||_s \leq 1\right\}$$ We determine the quantity $w^{k, r}_{p, q, s}(\delta; MM^m)$ in the case where $s = \infty$ and $m \in \{r,\; r — 1,\; r — 2\}$. In addition, we consider certain generalizations of the above-stated modification of the Landau-Kolmogorov problem.

### Boundary controllability problems for the equation of oscillation of an inhomogeneous string on a semiaxis

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 525-541

We consider a wave equation on a semiaxis, namely, $w_{tt}(x,t) = w_{xx}(x,t) — q(x)w(x,t), x > 0$. The equation is controlled by one of the following two boundary conditions: $w(0,t) = u_0(t)$ and $w_x(0,t) = u_1(t), t \in (0,T)$, where $u_0, u_1$ are controls. In both cases, the potential q satisfies the condition $q \in C[0, \infty)$, the controls belong to the class $L^{\infty}$ and the time $T >$ 0 is fixed. These control systems are considered in Sobolev spaces. Using the operators adjoint to the transformation operators for the Sturm - Liouville problem, we obtain necessary and sufficient conditions for the null-controllability and approximate null-controllability of these systems. The controls that solve these problems are found in explicit form.

### Local time at zero for arratia flow

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 542-556

We study the Arratia flow $x(u,t)$. We prove that $x(\cdot,t)$ is a Markov process whose phase space is a certain subset $K$ of the Skorokhod space. We introduce the notion of total local time at zero for the Arratia flow. We prove that it is an additive, nonnegative, continuous functional of the flow and calculate its characteristic.

### Conformal isoparametric spacelike hypersurfaces in conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 557-570

We study the conformal geometry of conformal spacelike hypersurfaces in the conformal spaces $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$. We obtain a complete classification of conformal isoparametric spacelike hypersurfaces in $\mathbb{Q}^4_1$ and $\mathbb{Q}^5_1$.

### Dmytro Ivanovych Martynyuk (on the 70th anniversary of his birthday)

Danilov V. Ya., Gorodnii M. F., Kirichenko V. V., Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 571-573

### Yurii Ivanovych Samoilenko (on the 80th anniversary of his birthday)

Bakhtin A. K., Gerasimenko V. I., Plaksa S. A., Samoilenko A. M., Sharko V. V., Trohimchuk Yu. Yu, Yacenko V. O., Zelinskii Yu. B.

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 574-576