Numerical methods for impulsive differential equation

In this paper, the asymptotical stability of the numerical methods with the constant stepsize for impulsive differential equation x ̇ ( t ) = α x , t ≠ k , t > 0 , △ x = β x , t = k , x ( 0 + 0 ) = x 0 , where α ≠ 0 , β , x 0 ∈ R , 1 + β ≠ 0 , k ∈ N , are investigated. The asymptotical stability conditions of the analytic solution of this equation and the numerical solutions are obtained. Finally, some experiments are given.