# Pokutnyi О. О.

### 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.

### Bounded solutions of evolutionary equations

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

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.

### 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.

### 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.