A research has been made on the existence of extreme solutions to first-order impulsive differential equations with difference between two functions.By employing the approaches for upper and lower solutions and monotone iterative technique

the authors have obtained the sufficient conditions for the existence of coupled maximal solution and minimal solution.Then by using the theory of impulsive differential inequality

the authors have presented the unique solution to the boundary problem.The researches in this paper have offered solutions to initial value problem for ordinary differential equation involving difference of two monotone functions

ordinary differential equation with integral boundary conditions

and first-order impulsive ordinary differential equation with anti-periodic.