Monotone Iterative Technique For Impulsive Differential Equations

In recent years, the theory of impulsive differential equations is an important area of investigation since it is much richer than the corresponding theory of differential equations. An interesting and fruitful technique for proving existence results for nonlinear problems is the method of upper and lower solutions. This paper consists of three parts. In part I, using the method of upper and lower solutions conbined with monotone iterative technique, we discuss boundary value problem for the following impulsive differential equation with a parameterThe existence of extremal solutions is considered. In this part, we weaken the limit to functions f and G, and we needn't the condition D(b) < 1. So we improved the results of paper [1].In part II, by the same way, we consider first-order impulsive differential equations with integral boundary value problems. First of all, we discuss the existence of solutions for first-order impulsive differential equationsWe extended the corresponding conclutions of paper [3]. Moreover, a comparision theorem is given, which extended the results of paper [2].Secondly, we consider integral boundary value problem for impulsive differential equations involving the difference of two functionsIn this part, the problems which we considered includes periodic boundary value problems, initial value problems, anti-periodic boundary value problems. Our results containsthe corresponding results of paper [3,4,5], if Ik=Gk = 0, different numerical value is given to i, i = 1,2,3, theorems in this paper is the corresponding theorems of paper [3,4,5].In part III, firstly, we get a comparison result of a class of first-order impulsive differential equations. First-order impulsive differential inequalityis considered. We get the sufficient conditions of x(t) 0, x(t) 0, and Mx(t) 0. Note that if Lk = 0, |M| > N, the theorem is the case of paper [6]. If W = 0, the theorem is the corresponding results of paper [1], where rm = 0.Secondly, with the method of decreasing order, we consider periodic boundary value problem for second-order impulsive differential equations]the existence of extremal solutions is given, which extended the corresponding results of paper [6].