Amin Benaissa Cherif

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1. Introduction and Position of Problem 1.1. Results and Discussion Nowadays, mathematics contains many references related to impulsive differential equations. We mention here the development of some of them in this area. Impulsive differential equations is considered in [1], where the authors obtained results related to oscillation and the behaviour of solutions of the system v ′ ( s ) + r ( s ) v ( s − τ ) = 0 , s ≠ ϕ k , s ≥ s 0 , v ( ϕ k + ) − v ( ϕ k − ) = I k ( v ( ϕ k + ) ) , k ∈ N .

1. Introduction and Some Historical Review Differential equations with impulse action are an important subclass of hybrid systems, i.e., systems of differential equations that merge continuous and discrete time dynamics. They are a new direction in differential equations theory, which has many applications to models of mathematical problems in mechanics, biology, and engineering [1,2]. The very important problem for this type of system is the question of the qualitative properties of solutions.