2018
Том 70
№ 12

# Volume 13, № 3, 1961

Article (Russian)

### On periodic solutions of differential equations of the $n$-th order with a small parameter

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 3-12

An asymptotic method of constructing periodic solutions is developed for differential equations of the $n$-th order containing a small parameter. The method makes use of expansions in integral powers of the small parameter. A method of investigating the stability of the resulting solutions is presented. The convergence of the expansions is proved.

Article (Russian)

### On some transformations of integral equation kernels and their influence on the spectra of these equations

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 12-38

In recent publications by the authors (1, 2) and V. I. Matsayev (3,4) it was shown that the study of the abstract triangular representation of Volterra operators by the Brodsky integral naturally leads to a series of relations between the eigenvalues of Hermitian components of Volterra operators.
Though some specific properties of the triangular truncation transformation while deducing were used, these results admit in the main a generalization for any transformations (linear continuous operators acting in the Hilbert space of the Hilbert-Schmidt operators). By this generalization both the nature of relations under consideration and their proofs are simplified.
This generalization (§§ 2, 3, 4) is perhaps of interest as it leads to some new applications; it permits us, in particular, to obtain a number of precise estimations for the central stability zone for various Hamiltonian systems of linear differential equations with periodic coefficients (§ §5, 6, 7).

Article (Russian)

### On some Galois relations in the theory of groups

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 39-45

Article (Russian)

### Uber Reduktion des Systems der gewöhnlichen linearen Differentialgleichungen, die von einem Parameter abhängen

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 46-58

In dieser Arbeit sind die Fragen der Begründung des formalen Verfahrens der Reduktion des Systems der gewöhnlichen linearen Differential gleichungen, deren Koeffizienten von einem Parameter abhängen, zur Fastdiagonalform betrachtet.
Hier wird jener bei den Anwendungen am öftesten in Frage kommende Fall erforscht, wenn die unabhängige Veränderliche : und der Parametere reel sind. In diesem Fall ist der Beweis des Grundtheorems, das die Begründung der Methode gewährleistet, für das ganzen Anfangsgebiet der Veränderung von $t$ durchgeführt.

Article (Russian)

### Some generalizations of V. Markov's problem and his basic theorem corresponding to the P. L. Chebyshev — A. A. Markov criterion. І

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 59-74

The author considers the best — in the Chebyshev sense — approach of a continuous real-numerical function $f(x)$ given on a bicompact hausdorff space $G$, by means of a generalized polynomial $F(x) = \sum^n_{j=0}a_j\varphi_j(x)$ where continuous linearly independent functions $\{\varphi_j(x)\}^n_0$ form a system of Chebyshev functions ($T$-system) on the indicated space with $p \leq n$ linear links between the parameters of the polynomial.

Article (Russian)

### Boundary problems for certain self-conjoint differential equations of the second order degenerating on the border of the region

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 75-85

Problems of the Dirichlet and Neumann type in a half-space are solved in explicit form for equations (1,1) and (1,2).

Article (Russian)

### On the solution of a class of functional equations

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 86-94

The author considers functional equations of form (1). It is asserted that the process of solving (1) is determined by the kernel $\varphi(x)$. If the kernel consists of stationary points only, the solution of equation (1) is reduced to the solution of a system of ordinary equations; if the kernel has no stationary points, the method of steps is employed. If the kernel has no more than an even number of stationary points, then (see [4]) the real axis is divided into sets $M-i,\; i = 0, 1, 2, ...$ so that $\varphi(M_i) \subseteq M_i$. The method of steps is applied to each set except $M_0$. The idea of dividing the region of definition of the function into invariants in respect to the function of sets is then used once more for determining the nature of the solution obtained.

Brief Communications (Russian)

### On collision of elastic-plastic bodies

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 95-97

Brief Communications (Russian)

### Solution of the equations of motion of a balanced gyroscope by the acceleration method

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 97-100

Brief Communications (Russian)

### On dual dispersion relations

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 100-103

Brief Communications (Russian)

### Application of the averaging method for the investigation of oscillations, induced by instantaneous impulses, in self-oscillating systems of the second order with a small parameter

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 103-110

Brief Communications (Russian)

### On the numerical decomposition of a system of ordinary linear differential equations

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 109-113

Brief Communications (Russian)

### On the solution of a class of functional equations

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 113-115

BookReview (Russian)

### S. Z. Stokalo, Operational methods and their development in the theory of linear differential equations with variable coefficients

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 116-117