On bounded solutions of differential systems

Abstract

The question of the existence of bounded solutions on an infinite interval of a linear inhomogeneous system of ordinary differential equations in a finite-dimensional space is considered. The study of bounded solutions of systems of ordinary differential equations is one of the most important problems in the qualitative theory of differential equations. In the study of the asymptotic behavior of solutions to differential systems, the works of A. Poincare and A.M. Lyapunov. Various conditions for the existence of bounded solutions of a linear system of ordinary differential equations have been obtained by many authors. Note the works of O. Perron, A. Walter, H. Shpet, D. Caligo, N.I. Gavrilova, M. Hukukara, M. Nagumo, M. Caratheodori, U. Barbouti, N. Ya. Lyashchenko,  B.P. Demidovich, A. Wintner, R. Bellman, Yu.S. Bogdanov, Z. Vazhevsky,          N. Levinson, M. Markus, L. Cesari and others. In this paper, we establish sufficient conditions for the boundedness of all solutions of a linear inhomogeneous system of differential equations on an infinite interval. A coefficient criterion for the boundedness of all solutions on an infinite interval of a linear inhomogeneous system of differential equations in a certain class of differential systems is given. Applied methods of differential equations and function theory. The results obtained are used in applications of differential equations and are of practical value.