# Volume 67, № 11, 2015

### Conditional Symmetry of a System of Nonlinear Reaction-Diffusion Equations

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1443-1449

The conditional symmetry of a system of nonlinear reaction-diffusion equations is investigated. It is shown that the operators of conditional symmetry exist for the systems of nonlinear reaction-diffusion equations with an arbitrary number of independent variables. Moreover, these operators are found in the explicit form.

### Boundary Trace Operator in a Domain of Hilbert Space and the Characteristic Property of its Kernel

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1450-1460

We prove an infinite-dimensional analog of the classical theorem on density of the set $C_0^1 (G)$ of finite smooth functions in the kernel of the boundary trace operator $γ: H_1(G) → L_2(∂G)$.

### Conditions of Convergence Almost Everywhere for the Convolution of a Function with Delta-Shaped Kernel to this Function

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1461-1476

We establish sufficient conditions for the convergence of the convolution of a function with delta-shaped kernel to this function. These conditions are used for the construction of the subspaces of solutions of differential equations and systems of these equations isometric to the spaces of real functions.

### Weakly Nonlinear Fredholm Integral Equations with Degenerate Kernel in Banach Spaces

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1477-1490

We consider weakly nonlinear Fredholm integral equations with degenerate kernel in Banach spaces and establish a necessary condition and sufficient conditions for the existence of solutions of equations of this kind. The convergent iterative procedures are proposed for the construction either of a single solution or of at least one possible solution.

### t-Generalized Supplemented Modules

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1491-1497

In the present paper, $t$-generalized supplemented modules are defined starting from the generalized ⨁-supplemented modules. In addition, we present examples separating the $t$-generalized supplemented modules, supplemented modules, and generalized ⨁-supplemented modules and also show the equality of these modules for projective and finitely generated modules. Moreover, we define cofinitely $t$-generalized supplemented modules and give the characterization of these modules. Furthermore, for any ring $R$, we show that any finite direct sum of $t$-generalized supplemented $R$-modules is $t$-generalized supplemented and that any direct sum of cofinitely $t$-generalized supplemented $R$-modules is a cofinitely $t$-generalized supplemented module.

### Application of the Laplace Transform of Tempered Distributions to the Construction of Functional Calculus

Lopushanskyi A. O., Sharyn S. V.

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1498-1511

We use the generalized *n*-dimensional Laplace transform of tempered distributions whose supports are located in a positive *n*-dimensional cone to construct functional calculus for the commutative collections of injective generators of *n*-parameter analytic semigroups of operators acting in a Banach space.

### On Simple-Layer Potentials for One Class of Pseudodifferential Equations

Osipchuk M. M., Portenko N. I.

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1512-1524

We construct single-layer potentials for a class of pseudodifferential equations connected with symmetric stable stochastic processes. An operator similar to the operator of gradient in the classical potential theory is selected and an analog of the classical theorem on the jump of (co)normal derivative of single-layer potential is established. This result allows us to construct solutions of some initial-boundary-value problems for pseudodifferential equations of the indicated kind.

### Generalizations of Steffensen’s Inequality by Lidstone’s Polynomials

Pečarić J. E., Perušić A., Smoljak K.

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1525-1539

We obtain generalizations of Steffensen’s inequality by using Lidstone’s polynomials. Furthermore, the functionals associated with the obtained generalizations are used to generate *n*-exponentially and exponentially convex functions, as well as the new Stolarsky-type means.

### Estimation of the Entropy Numbers and Kolmogorov Widths for the Nikol’skii–Besov Classes of Periodic Functions of Many Variables

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1540-1556

We establish order estimates for the entropy numbers of the Nikol’skii–Besov classes $B_{p,θ}^r$ of periodic functions of many variables in the space $L_q$ with certain relations between the parameters $p$ and $q$. By using the obtained lower estimates of the entropy numbers, we establish the exact-order estimates for the Kolmogorov widths of the same classes of functions in the space $L_1$.

### Conditions of Invertibility for Functional Operators with Shift in Weighted Hölder Spaces

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1557-1568

We consider functional operators with shift in weighted Hölder spaces. The main result of the work is the proof of the conditions of invertibility for these operators. We also indicate the forms of the inverse operators. As an application, we propose to use these results for the solution of equations with shift encountered in the study of cyclic models for natural systems with renewable resources.

### Bounds for the Periods of Periodic Solutions of Ordinary Differential Equations

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1569-1573

We consider a nonautonomous system of ordinary differential equations. It is supposed that this system has a periodic solution. We establish the lower bound for the period of this solution.

### Variations on Giuga Numbers and Giuga’s Congruence

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1573-1578

A $k$ -strong Giuga number is a composite integer such that $∑_{j = 1}^{n − 1} j^{n − 1} ≡ − 1 (mod n)$. We consider the congruence $∑_{j = 1}^{n − 1} j^{k(n − 1)} ≡ − 1 (mod n)$ for each $k ϵ ℕ$ (thus extending Giuga’s ideas for $k = 1$). In particular, it is proved that a pair $(n, k)$ with composite n satisfies this congruence if and only if $n$ is a Giuga number and $⋋(n) | k(n − 1)$. In passing, we establish some new characterizations of Giuga numbers and study some properties of the numbers n satisfying $⋋(n) | k(n − 1)$.

### Kolmogorov Widths for Analogs of the Nikol’skii–Besov Classes with Logarithmic Smoothness

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1579-1584

We establish the exact-order estimates of Kolmogorov widths and entropy numbers for analogs of the Nikol’skii–Besov classes with logarithmic smoothness.