Abstract
Research in impulsive delay differential equations has been undergoing some exciting growth in recent times. This to a large extent can be attributed to the quest by mathematicians in particular and the science community as a whole to unveil nature the way it truly is. The realization that differential equations, in general, and indeed impulsive delay differential equations are very important models for describing the true state of several real-life processes/phenomena may have been the tunic. One can attest that in most human processes or natural phenomena, the present state is most often affected significantly by their past state and those that were thought of as continuous may indeed undergo abrupt change at several points or even be stochastic. In this study, a special strictly ascending continuous delay is constructed for a class of system of impulsive differential equations. It is demonstrated that even though the dynamics of the system and the delay have ideal continuity properties, the right side may not even have limits at some points due to the impact of past impulses in the present. The integral equivalence of the formulated system of equations is also obtained via a scheme similar to that of Perron by making use of certain assumptions.
Highlights
Introduction and Statement of ProblemThe theory of impulsive delay differential equations (IDE) is based on the behaviour of processes or phenomena which undergo abrupt changes in their state and past events affect the current behaviour
Interest is on the increase largely due to the fact that a lot of everyday phenomena in Sciences, Economics, Engineering, Space sciences, and control systems are modeled by impulsive delay differential equations [1, 3,4,5]
Ballinger’s Ph.D thesis and his subsequent work provide a good working tool for further research work in this area, especially, as it relates to existence, uniqueness, boundedness, continuation, and stability of solutions of Impulsive Delay Differential Equations (IDDE) [3]
Summary
The theory of impulsive delay differential equations (IDE) is based on the behaviour of processes or phenomena which undergo abrupt changes in their state and past events affect the current behaviour (delay). Ballinger’s Ph.D thesis and his subsequent work provide a good working tool for further research work in this area, especially, as it relates to existence, uniqueness, boundedness, continuation, and stability of solutions of Impulsive Delay Differential Equations (IDDE) [3]. This happens to be a fusion of two areas – Delay Differential Equations, and Impulsive Differential Equations. Time-Dependent Continuous Delay story may become significantly different This surge in the number of discontinuous points creates several problems as it pertains to existence, stability/instability of solutions, just to mention a few. Some recent results in impulsive delay differential equations with constant impulsive jumps can be seen in [7,8,9,10,11,12,13,14]
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