Abstract
Eventual stability and eventual boundedness for nonlinear impulsive differential equations with supremums are studied. The impulses take place at fixed moments of time. Piecewise continuous Lyapunov functions have been applied. Method of Razumikhin as well as comparison method for scalar impulsive ordinary differential equations have been employed.
Highlights
The stability of solutions of differential equations via Lyapunov method has been intensively investigated in the past
It is obligatory to study the stability of such sets, which are not invariant with respect to a given system of differential equations
To the best of our knowledge, there are no results considering the stability of nonlinear impulsive differential equations with supremums, which is very important in theories and applications and is a very challenging problem
Summary
The stability of solutions of differential equations via Lyapunov method has been intensively investigated in the past. It is obligatory to study the stability of such sets, which are not invariant with respect to a given system of differential equations. This immediately excludes the stability in the sense of Lyapunov. Arisen in this situation, to be solved, a new notion is introduced – eventual stability [7, 17] In this case, the set under consideration, despite not being invariant in the usual sense, is invariant in the asymptotic sense. To the best of our knowledge, there are no results considering the stability of nonlinear impulsive differential equations with supremums, which is very important in theories and applications and is a very challenging problem. By employing a class of piecewise continuous functions which are generalization of the classical Lyapunov’s functions [6, 11] coupled with the Razumikhin technique [3, 5, 9, 10, 13, 14, 15] some sufficient conditions are found
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