Bounded Variation Solutions to Initial Value Problems for the First Order Linear Impulsive Differential Equations

In this paper,by using the generalized ordinary differential equation theory,the relationships between the first-order linear impulsive differential equation and the generalized linear ordinary differential equation are discussed.Necessary and suffcient conditions are given for the existence and uniqueness of bounded variation solutions to the initial value problem of the first-order linear impulsive differential equation,and the general solution formula are obtained by using the method of variation of constants.These results essentially generalize the existing continuous solution to the linear ordinary differential equation.