# Chaikovs'kyi A. V.

### On the Third Moduli of Continuity

Bezkryla S. I., Chaikovs'kyi A. V., Nesterenko A. N.

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1420-1424

An inequality for the third uniform moduli of continuity is proved. This inequality implies that an arbitrary 3-majorant is not necessarily a modulus of continuity of order 3.

### On solutions defined on an axis for differential equations with shifts of the argument

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1290-1296

We consider linear first-order differential equations with shifts of arguments with respect to functions with values in a Banach space. Sufficient conditions for the existence of nontrivial solutions of homogeneous equations are obtained. Ordinary differential equations are constructed for which all solutions defined on an axis are solutions of a given equation with shifts of the argument.

### Cauchy problem for a differential equation in the Banach space with generalized strongly positive operator coefficient

Chaikovs'kyi A. V., Il'chenko Yu. V.

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1053-1070

The concept of strongly positive operator is generalized, and properties of the operators introduced are analyzed. The solutions of the Cauchy problem for a linear inhomogeneous differential equation with generalized strongly positive operator coefficient are found.

### Functions of shift operator and their applications to difference equations

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1408–1419

We study the representation for functions of shift operator acting upon bounded sequences of elements of a Banach space. An estimate is obtained for the bounded solution of a linear difference equation in the Banach space. For two types of differential equations in Banach spaces, we present sufficient conditions for their bounded solutions to be limits of bounded solutions of the corresponding difference equations and establish estimates for the rate of convergence.

### Improvement of one inequality for algebraic polynomials

Chaikovs'kyi A. V., Nesterenko A. N., Tymoshkevych T. D.

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 231-242

We prove that the inequality $||g(⋅/n)||_{L_1[−1,1]}||P_{n+k}||_{L_1[−1,1]} ≤ 2||gP_{n+k}||_{L_1[−1,1]}$, where $g : [-1, 1]→ℝ$ is a monotone odd function and $P_{n+k}$ is an algebraic polynomial of degree not higher than $n + k$, is true for all natural $n$ for $k = 0$ and all natural $n ≥ 2$ for $k = 1$. We also propose some other new pairs $(n, k)$ for which this inequality holds. Some conditions on the polynomial $P_{n+k}$ under which this inequality turns into the equality are established. Some generalizations of this inequality are proposed.

### Investigation of One Linear Differential Equation by Using Generalized Functions with Values in a Banach Space

Ukr. Mat. Zh. - 2001. - 53, № 5. - pp. 688-693

We present a generalization of some facts of the theory of generalized functions of slow growth to the case of operator-valued test functions. We propose a construction of regular generalized functions with values in a Banach space. The results obtained are used for the description of slowly increasing solutions of linear homogeneous differential equations with shifted arguments and bounded operator coefficients in a Banach space.