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The real difference-dynamic system is considered. The case of unlimited continuity of solutions is investigated for this difference-dynamic system. Unlimited continuity to the right of solutions of the differencedynamic system is a necessary condition for Lyapunov stability of solutions of this system. Using functions similar to Lyapunov functions, sufficient conditions for the unlimited continuity of solutions of the difference-dynamical system are obtained. The concepts of Lagrange stability and Poisson stability are introduced. Relations between Lyapunov functions and Lagrange stability are investigated. Using the discrete analogue of the second Lyapunov method, the necessary and sufficient conditions for Lagrange stability of the solution of the difference-dynamical system are obtained. A dynamic-difference system is considered for which a discrete operator is constructed, with the help of which the Poisson stability of the system solutions is investigated.

izvestiâ nacionalʹnoj akademii nauk respubliki kazahstan. seriâ fiziko-matematičeskaâ/izvestiâ nacionalʹnoj akademii nauk respubliki kazahstan. seriâ fiziko-matematičeskaâ

ON LAGRANGE STABILITY AND POISSON STABILITY OF THE DIFFERENTIAL-DYNAMIC SYSTEMS