Stability of solutions of one class of nonlinear dynamic equations

Abstract

We study the stability of the zero solution of a nonlinear dynamic equation on a time scale under certain assumptions on the right-hand side of this equation. In addition to conditions for the existence and uniqueness of a solution of the Cauchy problem, we also assume that the exponential function of the linear approximation is bounded, and the norms of the nonlinear part and its derivatives with respect to the components of the space variable are majorized by power functions of the norm of the space variable. Using the generalized method of Lyapunov functions, we obtain sufficient conditions for the stability of the zero solution of the nonlinear equation under consideration.

On the connected stability of the large-scale systems of dynamical equations on the time scale in critical cases

“…We note that the solution of Eq. (3.1) depends continuously on the initial data (by Lemma 2 in [3]). Then, choosing y 0 from a sufficiently small η 1 -neighborhood of zero (η 1 < η), it is possible to obtain that the solution y(t; t 0 , y 0 ) on the integration intervals in formula (3.27) is in a neighborhood of the point y = 0 with a beforehand given radius.…”

“…satisfy all conditions of the lemma of differentiability with respect to a parameter of the solutions of dynamical equations on the time scale [3]. Therefore, by differentiating both sides of identity (3.13) with respect to y i (i = 1, 2, .…”