Definition 1 It is said that a function x defined on the set {− 1 } ∪ [ − τ, ∞ ) is a solution of Eq. (1)-(2) if it satisfies the following conditions: ( D$_{1}$ ) x ( t ) is continuous on [ − τ, ∞ ) with the possible exception of the points t$_{i}$, i = 1 , 2 , ... ( D$_{3}$ ) x ( t ) differentiable and satisfies (1) for any t ∈ R $^{+}$, with the possible exception of the points t$_{i}$, i = 1 , 2 , ..., and [ t ] ∈ [0 , ∞ ) , where one-sided derivatives exist, Definition 2. A function x ( t ) is called oscillatory if it is neither positive nor negative for t ≥ T where T is sufficiently large. Otherwise, the solution is called nonoscillatory. Remark 1. In this paper we assume that −∞ < b$_{i}$ < 1 for all i = 1 , 2 , ... Otherwise, from the impulse conditions (2) it is obtained that the solutions are already oscillatory. Remark 2. We assume that b ( t ) ̸≡ 0 or c ( t ) ̸≡ 0 . If b ( t ) ≡ 0 and c ( t ) ≡ 0 , then Eq. (1)-(2) reduces an ordinary differential equation with impulses. The results on the oscillation of impulsive ordinary differential equations can be found in the survey paper [15]. In this paper we also consider following differential inequalities. 3 x $^{′}$( t ) + a ( t ) x ( t ) + b ( t ) x ( t − τ ) + c ( t ) x ([ t − 1]) ≤ 0 , t = t$_{i}$, t ≥ t$_{0}$ > 0 , (3) Δ x ( t$_{i}$ ) = b$_{i}$x ( t$_{i}$ ) , i = 1 , 2 , ..., ( D$_{2}$ ) x ( t ) is right continuous and has left-hand limit at the points t$_{i}$, i = 1 , 2 , ... ( D$_{4}$ ) x ( t ) satisfies (2) at the points t$_{i}$, i = 1 , 2 , ... 2. Main Results