# Petryshyn R. I.

### Nonlocal multipoint (in time) problem for evolutionary pseudodifferential equations with analytic symbols in spaces of type $W$

Horodets’kyi V. V., Martynyuk O. V., Petryshyn R. I.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 9. - pp. 1208-1226

UDC 517.956

The correct solvability of a nonlocal multipoint (in time) problem for the evolution equations with differentiation operators of infinite order is established for an infinite time interval and an initial function, which is an element of the space of generalized functions of the type $ W'$.
The properties of the fundamental solution and the behavior of the solution as $ t \to + \infty $ are investigated.

### A problem for one class of pseudodifferential evolutionary equations multipoint in the time variable

Horodets’kyi V. V., Petryshyn R. I., Verezhak A. P.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 337-355

We establish the correct solvability of the multipoint (in the time variable) problem for the evolution equation with operator of differentiation of infinite order in generalized $S$-type spaces. The properties of the fundamental solution of this problem and the behavior of the solution $u(t, x)$ as $t \rightarrow +\infty$ are investigated.

### Correct Solvability of a Nonlocal Multipoint (in Time) Problem for One Class of Evolutionary Equations

Horodets’kyi V. V., Martynyuk O. V., Petryshyn R. I.

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 339-353

We study properties of a fundamental solution of a nonlocal multipoint (with respect to time) problem for evolution equations with pseudo-Bessel operators constructed on the basis of constant symbols. The correct solvability of this problem in the class of generalized functions of distribution type is proved.

### Averaging of initial-value and multipoint problems for oscillation systems with slowly varying frequencies and deviated argument

Danylyuk I. M., Petryshyn R. I., Samoilenko A. M.

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 412–430

We prove new theorems on the substantiation of the method of averaging over all fast variables on a segment and a semiaxis for multifrequency systems with deviated argument in slow and fast variables. An algorithm for the solution of a multipoint problem with parameters is studied, and an estimate for the difference of solutions of the original problem and the averaged problem is established.

### Reducibility of a nonlinear oscillation system with pulse influence in the neighborhood of an integral manifold

Dudnytskyi P. M., Petryshyn R. I., Samoilenko A. M.

Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1076–1094

In the neighborhood of an asymptotically stable integral manifold of a multifrequency system with pulse influence at fixed times, we perform a decomposition of the equations for angular and position variables.

### Construction of an Integral Manifold of a Multifrequency Oscillation System with Fixed Times of Pulse Action

Petryshyn R. I., Samoilenko A. M., Sopronyuk Т. M.

Ukr. Mat. Zh. - 2003. - 55, № 5. - pp. 641-662

We determine a class of multifrequency resonance systems with pulse action for which an integral manifold exists. We construct a function that determines a discontinuous integral manifold and investigate its properties.

### Error Estimates for the Averaging Method for Pulse Oscillation Systems

Lakusta L. M., Petryshyn R. I., Samoilenko A. M.

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 510-524

We prove new theorems on the justification of the averaging method on a segment and semiaxis in multifrequency oscillation systems with pulse action at fixed times.

### Justification of Averaging Method for Multifrequency Impulsive Systems

Petryshyn R. I., Sopronyuk Т. M.

Ukr. Mat. Zh. - 2003. - 55, № 1. - pp. 55-65

We prove new theorems on the justification of the averaging method for multifrequency oscillation systems with pulse influence at fixed times.

### Averaging of Boundary-Value Problems with Parameters for Multifrequency Impulsive Systems

Lakusta L. M., Petryshyn R. I., Samoilenko A. M.

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1237-1249

By using the averaging method, we prove the solvability of boundary-value problems with parameters for nonlinear oscillating systems with pulse influence at fixed times. We also obtain estimates for the deviation of solutions of the averaged problem from solutions of the original problem.

### Exponential Estimate for the Fundamental Matrix of a Linear Impulsive System

Petryshyn R. I., Sopronyuk Т. M.

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1101-1108

We investigate the properties of oscillating sums and integrals dependent on parameters. These properties are used for the estimation of the normal fundamental matrix of a linear system with rapidly oscillating coefficients and pulse influence at fixed moments of time.

### On integral manifolds of oscillating systems with slowly varying frequencies

Lakusta L. M., Petryshyn R. I.

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 87–93

By using averages of functions, we construct the integral manifold of an oscillating system that passes through resonances in the course of its evolution. We investigate the smoothness of the integral manifold and obtain estimates for its partial derivatives.

### Estimate of an error of the averaging method on a semiaxis for a multifrequency resonance system

Petryshyn R. I., Pokhyla O. M.

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 685–690

In this paper, we justify the averaging method for a multifrequency resonance system on a semiaxis under the assumption that the normal fundamental matrix of a variational system of averaged equations for slow variables exponentially tends to zero. We also study the quantitative dependence of the estimates on the magnitude of a small parameter.

### Averaging method in some problems of the theory of nonlinear oscillations

Ukr. Mat. Zh. - 1995. - 47, № 6. - pp. 801–810

By using the averaging method, we prove the solvability of multipoint problems for nonlinear oscillation systems. The deviation of the solutions of original and averaged problems is estimated.

### Method of averaging in multifrequency systems with slowly varying parameters

Petryshyn R. I., Samoilenko A. M.

Ukr. Mat. Zh. - 1988. - 40, № 4. - pp. 493-500

### Stability of certain two-particle systems

Petryshyn R. I., Samoilenko A. M.

Ukr. Mat. Zh. - 1986. - 38, № 4. - pp. 128–131

### Averaging with respect to a fast variable in triple-frequency systems of second approximation

Golets B. I., Golets V. L., Petryshyn R. I.

Ukr. Mat. Zh. - 1985. - 37, № 4. - pp. 437–443

### Computation of the multipliers of linear periodic systems of the second order

Ukr. Mat. Zh. - 1981. - 33, № 2. - pp. 262–267

### Averaging in oscillating systems passing through resonance

Golets B. I., Golets V. L., Petryshyn R. I.

Ukr. Mat. Zh. - 1980. - 32, № 4. - pp. 448–455