STABILITY OF LINEAR IMPULSIVE DIFFERENTIAL EQUATIONS VIA t_∞-SIMILARITY

Abstract

In this paper we investigate h-stability for linear impulsive equations using the notion of t ∞-similarity and an impulsive integral inequality.

h-STABILITY OF NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS VIA t∞-SIMILARITY

“…The nonlinear variation of constants formula of Alekseev [1] has been used by several authors [2][3][4][6][7][8] to study h-stability of solutions of nonlinear differential systems. In this paper we use the nonlinear variation of constants formular of Alekseev [1] to study h-stability of solutions of nonlinear perturbed differential systems.…”

“…Choi and Ryu [3] investigated the important properties about hS for the various differential systems. Recently, Choi et al [4] and Goo [6] obtained results for hS of nonlinear differential systems via t ∞ -similarity. Furthermore, Goo and Ryu [7], Goo and Yang [8], and Goo [9] have investigated hS for the nonlinear differential systems with different forms of perturbed terms.…”

h-STABILITY AND BOUNDEDNESS IN FUNCTIONAL PERTURBED DIFFERENTIAL SYSTEMS

“…That is, Pinto extended the study of exponential asymptotic stability to a variety of reasonable systems called h-systems. Choi, Ryu [2] and Choi, Koo, and Ryu [3] investigated bounds of solutions for nonlinear perturbed systems. Also, Goo [7,8,9] and Goo et al [11] investigated boundedness of solutions for nonlinear perturbed systems.…”

A CONVERSE THEOREM ON h-STABILITY VIA IMPULSIVE VARIATIONAL SYSTEMS

“…Choi et al [5,6] studied h-stability for the linear impulsive differential equations by means of the notions of similarity, t ∞ -similarity, and impulsive integral inequality. In [4], we proved that two concepts of h-stability and h-stability in variation for nonlinear impulsive differential systems are equivalent via t ∞ -similarity of the associated variational impulsive systems and impulsive integral inequalities.…”

“…Also, we characterized h-stability for nonlinear impulsive differential systems by using the notions of piecewise continuous auxiliary functions and impulsive differential inequalities. Many authors [2,3,5,12,13,16,20] have studied the various types of stability of solutions for impulsive differential equations. However, to the best of our knowledge, there are no papers published on the converse h-stability theorem for nonlinear impulsive differential systems.…”