# Boichuk О. A.

### Bounded solutions of the nonlinear Lyapunov equation and homoclinic chaos

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 761-773

UDC 517.9

The paper is devoted to the investigation of bounded solutions of a nonlinear Lyapunov-type problem in Banach and Hilbert
spaces. Necessary and sufficient conditions for the existence of bounded solutions are obtained under the assumption that
the homogeneous equation admits exponential dichotomy on the semiaxes. Conditions for the existence of homoclinic
chaos in nonlinear evolution equations are presented.

### 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

### Bounded solutions of evolutionary equations

Boichuk О. A., Pokutnyi О. О., Zhuravlyov V. Р.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 7-28

We study the problems of existence and representations of the solutions bounded on the entire axis for both linear and nonlinear differential equations with unbounded operator coefficients in the Fr´echet and Banach spaces under the condition of exponential dichotomy on the semiaxes of the corresponding homogeneous equation.

### 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

### Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Boichuk О. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144

### Perturbation Theory of Operator Equations in the FréChet and Hilbert Spaces

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1181-1188

The perturbation theory is constructed in the Fréchet and Hilbert spaces. An iterative process is proposed for finding branching solutions.

### Exponential Dichotomy and Bounded Solutions of Differential Equations in the Fréchet Space

Ukr. Mat. Zh. - 2014. - 66, № 12. - pp. 1587-1597

We establish necessary and sufficient conditions for the existence of bounded solutions of linear differential equations in the Fréchet space. The solutions are constructed with the use of a strong generalized inverse operator.

### Yurii Stephanovych Samoilenko (on his 70th 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., Nizhnik L. P., Samoilenko A. M., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1408-1409

### Application of the ergodic theory to the investigation of a boundaryvalue problem with periodic operator coefficient

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 329-338

We establish necessary and sufficient conditions for the solvability of a family of differential equations with periodic operator coefficient and periodic boundary condition by using the notion of the relative spectrum of a linear bounded operator in a Banach space and the ergodic theorem. We show that if the existence condition is satisfied, then these periodic solutions can be constructed by using the formula for the generalized inverse of a linear bounded operator obtained in the present paper.

### Normally solvable operator equations in a Banach space

Boichuk О. A., Pokutnyi О. О., Zhuravlev V. F.

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 163-174

On the basis of a generalization of the well-known Schmidt lemma to the case of linear, bounded, normally solvable operators in Banach spaces, we propose a procedure for the construction of a generalized inverse for a linear, bounded, normally solvable operator whose kernel and image are complementable in the indicated spaces. This construction allows one to obtain a solvability criterion for linear normally solvable operator equations and a formula for finding their general solutions.

### 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

### Degenerate nonlinear boundary-value problems

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1174-1188

We establish necessary and sufficient conditions for the existence of solutions of weakly nonlinear degenerate boundary-value problems for systems of ordinary differential equations with a Noetherian operator in the linear part. We propose a convergent iterative procedure for finding solutions and establish the relationship between necessary and sufficient conditions.

### Method of accelerated convergence for the construction of solutions of a Noetherian boundary-value problem

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1587–1601

We study the problem of finding conditions for the existence of solutions of weakly nonlinear Noetherian boundary-value problems for systems of ordinary differential equations and the construction of these solutions. A new iterative procedure with accelerated convergence is proposed for the construction of solutions of a weakly nonlinear Noetherian boundary-value problem for a system of ordinary differential equations in the critical case.

### A criterion for the existence of the unique invariant torus of a linear extension of dynamical systems

Ukr. Mat. Zh. - 2007. - 59, № 1. - pp. 3–13

Under the assumption that a linear homogeneous system defined on the direct product of a torus and the Euclidean space is exponentially dichotomous on semiaxes, we obtain a necessary and sufficient condition for the existence of the unique invariant torus of the corresponding inhomogeneous linear system.

### Solutions of Weakly-Perturbed Linear Systems Bounded on the Entire Axis

Boichuk А. A., Boichuk О. A., Samoilenko A. M.

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1517-1530

We establish conditions under which solutions of weakly-perturbed systems of linear ordinary differential equations bounded on the entire axis *R* emerge from the point ε = 0 in the case where the corresponding unperturbed homogeneous linear differential system is exponentially dichotomous on the semiaxes *R* _{+} and *R* _{−}.

### A Condition for the Existence of a Unique Green–Samoilenko Function for the Problem of Invariant Torus

Ukr. Mat. Zh. - 2001. - 53, № 4. - pp. 556-559

Under the assumption that a linear homogeneous system defined on the direct product of a torus and a Euclidean space is exponentially dichotomous on the semiaxes, we obtain a condition for the existence of a unique Green–Samoilenko function for the problem of invariant torus. We find an expression for this function in terms of projectors that determine the dichotomy on the semiaxes.

### Criterion of the solvability of matrix equations of the Lyapunov type

Boichuk О. A., Krivosheya S. A.

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1021–1026

By using the theory of generalized inverse operators, we establish a criterion of the solvability of the Lyapunov-type matrix equations *AX - XB = D* and *X - AXB = D* and investigate the structure of the set of their solutions.

### Nonlinear boundary-value problems for systems of ordinary differential equations

Ukr. Mat. Zh. - 1998. - 50, № 2. - pp. 162–171

We consider nonlinear boundary-value problems (with Noetherian operator in the linear part) for systems of ordinary differential equations in the neighborhood of generating solutions. By using the Lyapunov — Schmidt method, we establish conditions for the existence of solutions of these boundary-value problems and propose iteration algorithms for their construction.

### 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

### Boundary-value problems for systems of difference equations

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 832–835

Boundary-value problems for systems of difference equations with discrete argument whose linear part is the Noetherian operator are considered. The necessary and sufficient conditions of the solvability of difference boundary-value problems of this sort are obtained.

### Weakly nonlinear boundary-value problems for operator equations with pulse influence

Boichuk О. A., Samoilenko A. M., Zhuravlev V. F.

Ukr. Mat. Zh. - 1997. - 49, № 2. - pp. 272–288

We consider the problem of finding conditions of solvability and algorithms for construction of solutions of weakly nonlinear boundary-value problems for operator equations (with the Noetherian linear part) with pulse influence at fixed times. The method of investigation is based on passing by methods of the Lyapunov—Schmidt type from a pulse boundary-value problem to an equivalent operator system that can be solved by iteration procedures based on the fixed-point principle.

### Boundary-Value problems for systems of integro-differential equations with Degenerate Kernel

Boichuk О. A., Krivosheya S. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1576-1579

By using methods of the theory of generalized inverse matrices, we establish a criterion of solvability and study the structure of the set of solutions of a general linear Noether boundary-value problem for systems of integro-differential equations of Fredholm type with degenerate kernel.

### Generalized Green operator of a boundary-value problem with degenerate pulse influence

Boichuk О. A., Chuiko E. V., Chuiko S. M.

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 588-594

We find necessary and sufficient conditions for solvability of nonhomogeneous linear boundary-value problems for systems of ordinary differential equations with impulsive force in a general case where the number of boundary-value conditions in not equal to the order of the differential systems (Noetherian problems). We construct a generalized Greens's operator for boundary-value problems, not every solution of which can be extended from the left end point to the right end point of the interval where the solution is defined.

### Periodic solutions of autonomous systems with pulse influence in critical cases

Ukr. Mat. Zh. - 1995. - 47, № 11. - pp. 1478–1484

We obtain existence conditions and iterative schemes for constructing periodic solution of weakly nonlinear autonomous systems with pulse influence in critical cases.

### Construction of the solutions of linear operator equations in Banach spaces

Boichuk О. A., Zhuravlev V. F.

Ukr. Mat. Zh. - 1991. - 43, № 10. - pp. 1343–1350

### Construction of solutions of two-point boundary problems for weakly perturbed nonlinear systems in critical cases

Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1416–1420

### Construction of periodic solutions of nonlinear systems in critical cases

Ukr. Mat. Zh. - 1988. - 40, № 1. - pp. 62-69