Razumikhin-type theorems for impulsive differential equations with piecewise constant argument of generalized type

Abstract

In this paper, we focus on developing Razumikhin technique for stability analysis of impulsive differential equations with piecewise constant argument. Based on the Lyapunov-Razumikhin method and impulsive control theory, we obtain some Razumikhin-type theorems on uniform stability, uniform asymptotic stability, and global exponential stability, which are rarely reported in the literature. The significance and novelty of the results lie in that the stability criteria admit the existence of piecewise constant argument and impulses, which may be either slight at infinity or persistently large. Examples are given to illustrate the effectiveness and advantage of the theoretical results.

Unpredictable solutions of quasilinear differential equations with generalized piecewise constant arguments of mixed type

“…Differential equations with generalized piecewise constant functions as arguments (EPCAG) have been introduced and developed in papers [1,2,3,4,5,6,10]. The ideas suggested in these papers became very useful not only in modeling but also in methodological sense, since the construction of equivalent integral equations for EPCAG has opened the research gate for methods of operator theory and functional analysis [7,9,24,26,28,29,32,43,48,49,50,52,53]. This was also confirmed with applications in neuroscience [8,9,11,25,41,42,44,47,50,51].…”