# Perestyuk N. A.

### Averaging of fuzzy systems

Perestyuk N. A., Skripnik N. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 412-428

We develop the ideas of the method of averaging for some classes of fuzzy systems (fuzzy differential equations with delay, fuzzy differential equations with pulsed action, fuzzy integral equations, fuzzy differential inclusions and differential inclusions with fuzzy right-hand sides without and with pulsed action).

### Anatolii Mykhailovych Samoilenko (on his 80th birthday)

Antoniouk A. Vict., Berezansky Yu. M., Boichuk О. A., Gutlyanskii V. Ya., Khruslov E. Ya., Kochubei A. N., Korolyuk V. S., Kushnir R. M., Lukovsky I. O., Makarov V. L., Marchenko V. O., Nikitin A. G., Parasyuk I. O., Pastur L. A., Perestyuk N. A., Portenko N. I., Ronto M. I., Sharkovsky O. M., Tkachenko V. I., Trofimchuk S. I.

Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 3-6

### Stability of global attractors of impulsive infinite-dimensional systems

Kapustyan O. V., Perestyuk N. A., Romanyuk I. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 29-39

The stability of global attractor is proved for an impulsive infinite-dimensional dynamical system. The obtained abstract results are applied to a weakly nonlinear parabolic equation whose solutions are subjected to impulsive perturbations at the times of intersection with a certain surface of the phase space.

### On the 100th birthday of outstanding mathematician and mechanic Yurii Oleksiiovych Mytropol’s’kyi (03.01.1917 – 14.06.2008)

Berezansky Yu. M., Boichuk О. A., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Parasyuk I. O., Perestyuk N. A., Samoilenko A. M., Sharkovsky O. M.

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 132-144

### Volodymyr Leonidovych Makarov (on his 75th birthday)

Korolyuk V. S., Lukovsky I. O., Nesterenko B. B., Nikitin A. G., Perestyuk N. A., Samoilenko A. M., Solodkii S. G., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1715-1717

### Global attractors of impulsive infinite-dimensional systems

Kapustyan O. V., Perestyuk N. A.

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 517-528

We study the existence of global attractors in discontinuous infinite-dimensional dynamical systems, which may have trajectories with infinitely many impulsive perturbations. We also select a class of impulsive systems for which the existence of a global attractor is proved for weakly nonlinear parabolic equations.

### On Preservation of the Invariant torus for Multifrequency Systems

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1498–1505

We establish new conditions for the preservation of an asymptotically stable invariant toroidal manifold of the linear extension of a dynamical system on a torus under small perturbations in a set of nonwandering points. The proposed approach is applied to the investigation of the existence and stability of the invariant tori of linear extensions of the dynamical systems with simple structures of limit sets and recurrent trajectories.

### Anatolii Mykhailovych Samoilenko (on his 75th birthday)

Berezansky Yu. M., Boichuk О. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Perestyuk N. A., Portenko N. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 3 - 6

### Averaging of set-valued impulsive systems

Perestyuk N. A., Skripnik N. V.

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 126-142

We give a review of the development of ideas of the averaging method for some classes of set-valued impulsive systems (impulsive differential inclusions, impulsive differential equations and inclusions with Hukuhara derivative, and impulsive fuzzy differential equations and inclusions).

### Mykola Ivanovych Shkil' (on his 80th birthday)

Korolyuk V. S., Lukovsky I. O., Perestyuk N. A., Pratsiovytyi M. V., Samoilenko A. M., Yakovets V. P.

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1720-1722

### Dmytro Ivanovych Martynyuk (on the 70th anniversary of his birthday)

Danilov V. Ya., Gorodnii M. F., Kirichenko V. V., Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 571-573

### Green–Samoilenko operator in the theory of invariant sets of nonlinear differential equations

Perestyuk N. A., Slyusarchuk V. Yu.

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 948-957

We establish conditions for the existence of an invariant set of the system of differential equations $$\frac{dφ}{dt} = a(φ),\quad \frac{dx}{dt} = P(φ)x + F(φ,x),$$ where $a: Φ → Φ, P: Φ → L(X, X)$, and $F: Φ × X→X$ are continuous mappings and $Φ$ and $X$ are finite-dimensional Banach spaces.

### Fifty years devoted to science (on the 70th birthday of Anatolii Mykhailovych Samoilenko)

Berezansky Yu. M., Dorogovtsev A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Mitropolskiy Yu. A., Perestyuk N. A., Rebenko A. L., Ronto A. M., Ronto M. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 3–7

### Some modern aspects of the theory of impulsive differential equations

Chernikova O. S., Perestyuk N. A.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 81–94

We give a brief survey of the main results obtained in recent years in the theory of impulsive differential equations.

### On the 90th birthday of Yurii Alekseevich Mitropol’skii

Berezansky Yu. M., Gorbachuk M. L., Korolyuk V. S., Koshlyakov V. N., Lukovsky I. O., Makarov V. L., Perestyuk N. A., Samoilenko A. M., Samoilenko Yu. I., Sharko V. V., Sharkovsky O. M., Stepanets O. I., Tamrazov P. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 147–151

### Once again on the Samoilenko numerical-analytic method of successive periodic approximations

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 472–488

A new numerical-analytic algorithm for the investigation of periodic solutions of nonlinear periodic systems of differential equations *dx/dt* = *A*(*t*) *x*+ *ƒ*(*t, x*) in the critical case is developed. The problem of the existence of solutions and their approximate construction is studied. Estimates for the convergence of successive periodic approximations are obtained.

### On the Solvability of Impulsive Differential-Algebraic Equations

Perestyuk N. A., Vlasenko L. A.

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 458–468

We establish theorems on the existence and uniqueness of a solution of the impulsive differential-algebraic equation $$\frac{d}{{dt}}[Au(t)] + Bu(t) = f(t,u(t)),$$ where the matrix A may be singular. The results are applied to the theory of electric circuits.

### Global Attractor of an Evolution Inclusion with Pulse Influence at Fixed Moments of Time

Kapustyan O. V., Perestyuk N. A.

Ukr. Mat. Zh. - 2003. - 55, № 8. - pp. 1058-1068

We consider an autonomous evolution inclusion with pulse perturbations at fixed moments of time. Under the conditions of global solvability, we prove the existence of a minimal compact set in the phase space that attracts all trajectories.

### Conditions for the Existence of Nonoscillating Solutions of Nonlinear Differential Equations with Delay and Pulse Influence in a Banach Space

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 790-798

For nonlinear differential second-order equations with delay and pulse influence in a Banach space, we establish necessary and sufficient conditions for the existence of their solutions nonoscillating with respect to a subspace.

### International Scientific Conference on the Theory of Evolution Equations (Fifth Bogolyubov Readings)

Konet I. M., Perestyuk N. A., Samoilenko A. M., Teplinsky Yu. V.

Ukr. Mat. Zh. - 2002. - 54, № 10. - pp. 1440

### Mykhailo Iosypovych Yadrenko (On His 70th Birthday)

Buldygin V. V., Korolyuk V. S., Kozachenko Yu. V., Mitropolskiy Yu. A., Perestyuk N. A., Portenko N. I., Samoilenko A. M., Skorokhod A. V.

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 435-438

### Dmytro Ivanovych Martynyuk (On the 60th Anniversary of His Birth)

Danilov V. Ya., Mitropolskiy Yu. A., Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2002. - 54, № 3. - pp. 291-292

### On Stability of Integral Sets of Impulsive Differential Systems

Chernikova O. S., Perestyuk N. A.

Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 249-257

We introduce the notion of stability of integral sets of impulsive differential systems of general form (with nonfixed times of impulse influence). We establish conditions sufficient for the stability of an integral set.

### On the Existence of Periodic Solutions for Certain Classes of Systems of Differential Equations with Random Pulse Influence

Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2001. - 53, № 8. - pp. 1061-1079

We establish conditions for the existence of periodic solutions for systems of differential equations with random right-hand side and random pulse influence at fixed times. We consider the case of small pulse perturbation and weakly nonlinear systems.

### On the Stability of Invariant Sets of Discontinuous Dynamical Systems

Chernikova O. S., Perestyuk N. A.

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 78-84

We establish sufficient conditions for the stability, asymptotic stability, and instability of invariant sets of discontinuous dynamical systems.

### Controlled Pulse Influence in Games with Fixed Termination Time

Ostapenko E. V., Perestyuk N. A.

Ukr. Mat. Zh. - 2000. - 52, № 8. - pp. 1112-1118

We construct optimal strategies for players and determine the sets of initial positions favorable for one player or another.

### On the stability of a trivial invariant torus of one class of impulsive systems

Dudzyanyi S. I., Perestyuk N. A.

Ukr. Mat. Zh. - 1998. - 50, № 3. - pp. 338–349

We consider the problem of asymptotic stability of the trivial invariant torus of one class of impulsive systems. Sufficient criteria of asymptotic stability are obtained by the method of freezing in one case, and by the direct Lyapunov method for the investigation of stability of solutions of impulsive systems in another case.

### Anatolii Mikhailovich Samoilenko (on his 60th birthday)

Berezansky Yu. M., Boichuk О. A., Korneichuk N. P., Korolyuk V. S., Koshlyakov V. N., Kulik V. L., Luchka A. Y., Mitropolskiy Yu. A., Pelyukh G. P., Perestyuk N. A., Skorokhod A. V., Skrypnik I. V., Tkachenko V. I., Trofimchuk S. I.

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 3–4

### On the stability of an invariant torus

Dudzyanyi S. I., Perestyuk N. A.

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1212-1222

We consider the problem of the asymptotic stability for trivial invariant torus of a linear extension of dynamical system on a torus. We formulate and prove sufficient criteria of asymptotic stability and study conditions for the existence and uniqueness of the Lyapunov functions of fixed sign.

### 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.

### Periodic solutions of a weakly nonlinear system of partial differential equations with pulse influence

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 601–605

We establish conditions of the existence of solutions periodic in *t* with period *T* for a weakly nonlinear system of partial differential equations with pulse influence.

### Stability of solutions of pulsed systems

Chernikova O. S., Perestyuk N. A.

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 98–111

We present the principal results in the theory of stability of pulse differential equations obtained by mathematicians of the Kiev scientific school of nonlinear mechanics. We also present some results of foreign authors.

### On a method for construction of successive approximations for investigation of multipoint boundary-value problems

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1243–1253

We suggest a new scheme of successive approximations. This scheme allows one to study the problem of existence and approximate construction of solutions of nonlinear ordinary differential equations with multipoint linear boundary conditions. This method enables one to study problems both with singular and nonsingular matrices in boundary conditions.

### Control over linear pulse systems

Akhmetov M. U., Perestyuk N. A., Tleubergenov M. I.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 307–314

Rank conditions for control of linear pulse systems are established. The Pontryagin maximum principle is obtained in sufficient form. An example of control synthesis in a problem for linear pulse systems is given.

### Periodic solutions of nonlinear differential equations with pulse influence in a banach space

Perestyuk N. A., Slyusarchuk V. Yu.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 370–380

Rank conditions for control of linear pulse systems are established. An example of control synthesis in a problem for linear pulse systems is given.

### On invariance of some properties of solutions under perturbation of a pulse system of differential equations

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1707–1713

Sufficient conditions for the invariance of boundedness and stability properties of solutions under perturbation of a pulse system of differential equations are established.

### Reducibility of nonlinear almost periodic systems of difference equations on an infinite-dimensional torus

Martynyuk D. I., Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1994. - 46, № 9. - pp. 1216–1223

### Green-Samoilenko function and existence of integral sets of linear extensions of nonautonomous equations

Asrorov F. A., Perestyuk N. A.

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1067–1071

An integral invariant set is constructed for systems of differential equations by using the Green-Samoilenko function. The problem of asymptotic stability of this set is studied.

### Reducibility of nonlinear almost periodic systems of difference equations given on a torus

Martynyuk D. I., Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 404–412

Sufficient conditions are established for the reducibility of a nonlinear system of difference equations $$x(x + 1) = x(1) + \omega + P(x(t), t) + \lambda,$$ where $P(x, t)$ is a function $2\pi$-periodic in $x_i(i = 1,..., n)$ and almost periodic in $t$ with a frequency basis $\alpha$, to the system $$y(t + 1) = y(t) + \omega.$$

### Integral sets of systems of difference equations

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1613–1621

The problem of existence of integral sets of systems of difference equations is studied. We establish sufficient conditions for the existence of these sets and their stability. For the system under consideration, the behavior of trajectories that originate in a sufficiently small neighborhood of integral sets is investigated.

### On a comparison method for pulse systems in the space $R^n$

Akhmetov M. U., Perestyuk N. A.

Ukr. Mat. Zh. - 1993. - 45, № 6. - pp. 753–762

A method for the study of differential equations with pulse influence on the surfaces, which was realized in [1] for a bounded domain in the phase space, is now extended to the entire space $R^n$. We prove theorems on the existence of integral surfaces in the critical case and justify the reduction principle for these equations.

### Integral sets of quasilinear pulse systems

Akhmetov M. U., Perestyuk N. A.

Ukr. Mat. Zh. - 1992. - 44, № 1. - pp. 5–11

Sufficient conditions for the existence of integral sets of weakly nonlinear systems of differential equations with pulse effect on a surface are presented. The asymptotic behavior of solutions originating on integral sets and in the vicinity of these sets is investigated.

### Asymptotic representation of solutions of regularly perturbed systems of differential equations with nonclassical right-hand side

Akhmetov M. U., Perestyuk N. A.

Ukr. Mat. Zh. - 1991. - 43, № 10. - pp. 1298–1304

### Method of freezing in systems with impulse action

Ashirov O. A., Perestyuk N. A.

Ukr. Mat. Zh. - 1991. - 43, № 6. - pp. 848–853

### Generalized solutions of impulse systems and the phenomenon of pulsations

Perestyuk N. A., Samoilenko A. M., Trofimchuk S. I.

Ukr. Mat. Zh. - 1991. - 43, № 5. - pp. 657–663

### Stability of periodic solutions of differential equations with impulse action on surfaces

Akhmetov M. U., Perestyuk N. A.

Ukr. Mat. Zh. - 1989. - 41, № 12. - pp. 1596–1601

### Differentiable dependence of the solutions of impulse systems on initial data

Akhmetov M. U., Perestyuk N. A.

Ukr. Mat. Zh. - 1989. - 41, № 8. - pp. 1028–1033

### Almost-periodic solutions of impulse systems

Akhmetov M. U., Perestyuk N. A.

Ukr. Mat. Zh. - 1987. - 39, № 1. - pp. 74-80

### Asymptotic integration of weakly nonlinear systems with impulses

Ishchuk V. V., Perestyuk N. A.

Ukr. Mat. Zh. - 1985. - 37, № 3. - pp. 361–363

### Averaging method in systems with impulses

Mitropolskiy Yu. A., Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1985. - 37, № 1. - pp. 56 – 64

### Almost-periodic solutions of one class of systems with impulses

Akhmetov M. U., Perestyuk N. A.

Ukr. Mat. Zh. - 1984. - 36, № 4. - pp. 486 – 490

### A contribution to the stability problem for solutions of systems of differential equations with impulses

Chernikova O. S., Perestyuk N. A.

Ukr. Mat. Zh. - 1984. - 36, № 2. - pp. 190 - 195

### Invariant sets of a class of discontinuous dynamical systems

Ukr. Mat. Zh. - 1984. - 36, № 1. - pp. 63 - 68

### Periodic and almost-periodic solutions of impulsive differential equations

Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1982. - 34, № 1. - pp. 66-73

### Periodic solutions of nonlinear differential equations with impulsive action

Perestyuk N. A., Shovkoplyas V. N.

Ukr. Mat. Zh. - 1979. - 31, № 5. - pp. 517–524

### The problem of justifying the averaging method for second-order equations with impulsive action

Mitropolskiy Yu. A., Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1977. - 29, № 6. - pp. 750–762

### The method of averaging in systems with an impulsive action

Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1974. - 26, № 3. - pp. 411–418

### Invariant sets of systems with instantaneous changes in standard form

Perestyuk N. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1973. - 25, № 1. - pp. 129-134