Linearized Oscillations in Autonomous Delay Impulsive Differential Equations

Abstract

In recent years, there have been intensive efforts to establish linearised oscillation results for onedimensional delay, neutral delay and advanced impulsive differential equations. An impressive number of these efforts have yielded fruitful results in many analytical and applied areas. This is particularly obvious in the areas of applied disciplines such as the linear delay impulsive differential equations. However, there still remains a lot more to be explored in this direction, especially, in the area of non-linear autonomous differential equations. In this paper, we are proposing the development of linearised oscillation techniques for some general non-linear autonomous impulsive differential equations with several delays.

Oscillation Conditions for a Type of Second Order Neutral Differential Equations with Impulses

“…The development of the theory of impulsive differential equations is one of the recent trends in the history of qualitative theory of differential equations [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. There are many monographs related to this subject [1,[29][30][31], etc.…”

“…They also appear in problems dealing with vibrating masses attached to an elastic as well as the Euler equation in some vibrational problems [2,3,[5][6][7]. Very recently, the unending list of results on oscillation and nonoscillation of solutions of first order neutral delay impulsive differential equations has increased [8,9,[11][12][13].…”

“…Since the continuity properties of the solutions play an important role in the analysis of the behaviour, the techniques used to handle the solutions of impulsive differential equations are fundamentally different including the definitions of some of the basic terms. In this section, we examine some of these changes [11].…”

Bounded Oscillation Theorem for Unstable-type Neutral Impulsive Differential Equations of the Second Order

“…Still more interesting, the theory of impulsive differential equations has brought in yet another dimension to the whole scenario and has helped to usher in a new body of knowledge for further considerations. The effects of these new inputs can be observed in the study of oscillatory properties of impulsive differential equations with deviating arguments as well as the investigation of neutral impulsive differential equations which have recently captured the attention of many applied mathematicians as well as other scientists around the world [7,8,9,10,11,12].…”

Criteria for Bounded (Unbounded) Oscillations of neutral Impulsive Differential Equations of the Second Order with Variable Coefficients

“…Among the numerous publications concerning the oscillation properties of IDEDA -with delayed or advanced arguments, we choose to refer to [2], [6] and [11]. Lately, the pioneering efforts of Isaac and Lipcsey ([7], [8], [9], [13]) in identifying some of the essential oscillatory and non-oscillatory conditions of neutral impulsive differential equations of the first order is also worth commending. However, relatively less attention has been given to oscillations of second-order neutral delay differential equations with impulses.…”