# Volume 63, № 7, 2011

### Generalized (*n, d*)-ray systems of points and inequalities for nonoverlapping domains and open sets

Bakhtin A. K., Targonskii A. L.

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 867-879

We solve the extremal problem of finding the maximum of the functional.

### Structure of nodal algebras

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 880-879

The structure of nodal algebras over a complete discrete valuation ring with algebraically closed residue field is described.

### Relative Chebyshev point of a system of continuously varying bounded closed sets

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 889-903

For the problem of finding a relative Chebyshev point of a system of continuously varying (in the sense of the Hausdorff metric) bounded closed sets of a normed space linear over the field of complex numbers, we establish some existence and uniqueness theorems, necessary and sufficient conditions, and criteria for a relative Chebyshev point and describe properties of the extremal functional and the extremal operator.

### On the stability of abstract monotone impulsive differential equations in terms of two measures

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 904-923

We consider differential equations in a Banach space subjected to pulse influence at fixed times. It is assumed that a partial order is introduced in the Banach space with the use of a certain normal cone and that the differential equations are monotone with respect to initial data. We propose a new approach to the construction of comparison systems in a finite-dimensional space that does not involve auxiliary Lyapunov type functions. On the basis of this approach, we establish sufficient conditions for the stability of this class of differential equations in terms of two measures, choosing a certain Birkhoff measure as the measure of initial displacements, and the norm in the given Banach space as the measure of current displacements. We give some examples of investigation of impulsive systems of differential equations in critical cases and linear impulsive systems of partial differential equations.

### Existence criteria and asymptotics for some classes of solutions of essentially nonlinear second-order differential equations

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 924-938

We establish existence theorems and asymptotic representations for some classes of solutions of second-order differential equations whose right-hand sides contain nonlinearities of a more general form than nonlinearities of the Emden - Fowler type.

### Approximation of functions from the classes $C^{\psi}_{\beta, \infty}$ by biharmonic Poisson integrals

Kharkevych Yu. I., Zhyhallo K. M.

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 939-959

Asymptotic equalities are obtained for upper bounds of deviations of biharmonic Poisson integrals on the classes of $(\psi, \beta)$-differentiable periodic functions in the uniform metric.

### On necessary conditions for the convergence of Fourier series

Ivashchuk O. V., Zaderei P. V.

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 960-968

We obtain necessary conditions for the convergence of multiple Fourier series of integrable functions in the mean.

### Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 969-984

For arbitrary $[\alpha, \beta] \subset \textbf{R}$ and $p > 0$, we solve the extremal problem $$\int_{\alpha}^{\beta}|x^{(k)}(t)|^q dt \rightarrow \sup, \quad q \geq p, \quad k = 0, \quad \text{or} \quad q \geq 1, \quad k \geq 1,$$ on the set of functions $S^k_{\varphi}$ such that$\varphi ^{(i)}$ is the comparison function for $x^{(i)},\; i = 0, 1, . . . , k$, and (in the case $k = 0$) $L(x)_p \leq L(\varphi)_p$, where $$L(x)_p := \sup \left\{\left(\int^b_a|x(t)|^p dt \right)^{1/p}\; :\; a, b \in \textbf{R},\; |x(t)| > 0,\; t \in (a, b) \right\}$$ In particular, we solve this extremal problem for Sobolev classes and for bounded sets of the spaces of trigonometric polynomials and splines.

### Volterra quadratic stochastic operators of a two-sex population

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 985-998

We introduce the notion of Volterra quadratic stochastic operators of a bisexual population. The description of the fixed points of Volterra quadratic stochastic operators of a bisexual population is reduced to the description of the fixed points of Volterra-type operators. Several Lyapunov functions are constructed for the Volterra quadratic stochastic operators of a bisexual population. By using these functions, we obtain an upper bound for the ω-limit set of trajectories. It is shown that the set of all Volterra quadratic stochastic operators of a bisexual population is a convex compact set, and the extreme points of this set are found. Volterra quadratic stochastic operators of a bisexual population that have a 2-periodic orbit (trajectory) are constructed.

### Estimates for weighted eigenvalues of fourth-order elliptic operator with variable coefficients

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 999-1008

We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in $R^n$. We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the $(k + 1)$-th eigenvalue in terms of the first $k$ eigenvalues. Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang -Xia inequality (J. Funct. Anal. - 2007. - 245) for the clamped plate problem to a fourth-order elliptic operator with variable coefficients.