# Volume 49, № 8, 1997

### Optimal renewal of definite integrals of monotone functions from the class $H^{ω}$

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1011–1015

We obtain an exact estimate of the error of optimal renewal of an integral on a set of functions $f(t)$ monotone on $[a, b]$ with a convex majorant of the modulus of continuity, provided that $|f(b) − f(a)| = L > 0$.

### Tracing of pseudotrajectories of dynamical systems and stability of prolongations of orbits

Sharkovsky O. M., Vereikina M. B.

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1016–1024

We investigate properties of dynamical systems associated with the approximation of pseudotrajectories of a dynamical system by its trajectories. According to modern terminology, a property of this sort is called the “property of tracing pseudotrajectories” (also known in the English literature as the “shadowing property”). We prove that dynamical systems given by mappings of a compact set into itself and possessing this property are systems with stable prolongation of orbits. We construct examples of mappings of an interval into itself that prove that the inverse statement is not true, i.e., that dynamical systems with stable prolongation of orbits may not possess the property of tracing pseudotrajectories.

### Generalized factorized groups with dispersible subgroups

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1025–1031

We generalize the well-known nition of complementability of subgroups. We describe the structure of nondispersible generalized factorized groups all subgroups of which are dispersible.

### On absolute, perfect, and unconditional convergences of double series in Banach spaces

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1032–1041. August

We prove that, in the case of double series, perfect and unconditional convergences coincide, while absolute and perfect convergences do not coincide even for numerical series.

### Testing of numerous hypotheses with the use of an optimal extended nonrandomized procedure

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1042–1054

We propose a new procedure for testing hypotheses with the use of optimal statistical criteria.

### Aniterative method for the solution of some singularly perturbed cauchy problems

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1055–1060

We construct an iterative method for the solution of Cauchy problems for systems of singularly perturbed equations with fast time.

### Three problems related to the Kummer problem

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1061–1068

We give a brief survey of the principal results concerning the individual Kummer problem and present new results of the author concerning this problem.

### Rate of convergence of Fourier series on the classes of $\overline{\psi}$-integrals

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1069-1113

We introduce the notion of $\overline{\psi}$-integrals of 2π-periodic summable functions f, f ε L, on the basis of which the space L is decomposed into subsets (classes) $L^{\overline{\psi}}$. We obtain integral representations of deviations of the trigonometric polynomials $U_{n(f;x;Λ)}$ generated by a given Λ-method for summing the Fourier series of functions $f ε L^{\overline{\psi}}$. On the basis of these representations, the rate of convergence of the Fourier series is studied for functions belonging to the sets $L^{\overline{\psi}}$ in uniform and integral metrics. Within the framework of this approach, we find, in particular, asymptotic equalities for upper bounds of deviations of the Fourier sums on the sets $L^{\overline{\psi}}$, which give solutions of the Kolmogorov-Nikol'skii problem. We also obtain an analog of the well-known Lebesgue inequality.

### Contour-solid properties of finely hypoharmonic functions

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1114–1125

We prove contour-solid theorems for finely hypoharmonic functions defined in finely open sets of the complex plane.

### Vladimir Marchenko. '75

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1126

### On the existence of discontinuous limit cycles for one system of differential equations with pulse influence

Gorbachuk T. V., Perestyuk N. A.

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1127–1134

For one system of differential equations with pulse influence, we establish conditions under which a positive root of the equation for stationary amplitudes obtained from equations of the first approximation generates a discontinuous limit cycle. We construct improved first approximations for the system under consideration.

### Some inequalities for gradients of harmonic functions

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1135–1136

For a function *u(x, y)* harmonic in the upper half-plane *y*>0 and represented by the Poisson integral of a function *v(t) ∈ L* _{ 2 } *(−∞,∞)*, we prove that the inequality \(grad u (x, y)|^2 {\text{ }} \leqslant {\text{ }}\frac{1}{{4\pi ^3 }}{\text{ }}\int\limits_{ - \infty }^\infty {v^2 } (t)dt\) is true. A similar inequality is obtained for a function harmonic in a disk.

### On periodic solutions of linear degenerate second-order ordinary differential equations

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1137–1142

We consider a scalar linear second-order ordinary differential equation whose coefficient of the second derivative may change its sign when vanishing. For this equation, we obtain sufficient conditions for the existence of a periodic solution in the case of arbitrary periodic inhomogeneity.

### On the mean-value theorem for analytic functions

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1143–1147

A version of the mean-value theorem (formulas of finite increments) for analytic functions is proved.

### Structure of one class of groups with conditions of denseness of normality for subgroups

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1148–1151

We give a constructive description of locally graded groups *G* satisfying the following condition: For any pair of subgroups *A* and *B* such that *A , there exists a normal subgroup N that belongs to G and is such that A≦N≦B.*

*Chronicles (Ukrainian)
*

### Seminar-School “Mathematical Simulation”

Berezovsky A. A., Khomchenko A. N., Samoilenko A. M.

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1152