Nonoscillatory and Oscillatory Criteria for First Order Nonlinear Neutral Impulsive Differential Equations

Abstract

A survey of recent studies in neutral impulsive differential equations reveals that most of such works revolve around the quest for oscillatory conditions for linear impulsive differential equations. The development of oscillatory and nonoscillatory criteria for nonlinear impulsive differential equations has so far attracted very little attention. In this paper, we obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions for nonlinear first order neutral impulsive differential equations with constant delays

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

“…This could be due to the fact that neutral impulsive differential equations play fundamental roles in the present drive to further develop information technology. Indeed, neutral differential equations appear in networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits) [8]. 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].…”

“…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].…”

“…In effect, the solution ( ) Without further mentioning, we will assume throughout this paper that every solution ( ) y t of equation (1) ( ) y t is defined and is strictly positive (negative) for t T ≥ [9];…”

“…ii) non-oscillatory, if it is either finally positive or finally negative; and iii) oscillatory, if it is neither finally positive nor finally negative [1,8]. Remark 1.1: All functional inequalities considered in this paper are assumed to hold finally, that is, they are satisfied for all t large enough.…”

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].…”