Existence Of Solutions Of Three-point Boundary Value Problems For Impulsive Differential Equations

In this paper, we discuss the existence of solutions of three-point boundary value problems for impulsive differential equations as followsusing monotone iterative method, consider the existence of positive solutions of three-point boundary value problems for singular impulsive differential equationsas followsusing upper and lower solution technique and study the existence of multiple positive solutions of three-point boundary value problems for impulsive differential equations as followsusing fixed point theorem of cone expansion and compression.Multi-point boundary value problems arise in different fields of applicable mathematics and physics. In [1], Barr and Sherman firstly studied the multipoint boundary value problems and after then there were lots of results onthis aspect since 199l'2 5' because multi-point boundary value problems have wide background. In recent years, people begin paying attention to the study of multi-point boundary value problems for impulsive differential equations as the development of the theory for impulsive differential equations. The solutions of two-point boundary value problems for impulsive differential equations have been investigated extensively'14"21] , But the existence of solutions of three-point boundary value problems for impulsive differential equations are seldom studied'8"9' . So we study the existence of solutions of three-point boundary value problems (1), (2^) and (3). This paper is divided into three chapters.In chapter 1, first we establish one comparison theorem of three-point boundary value problem (1), then we get the existence of the minimal and the maximal solutions to (1) by an improved monotone iterative technique on the base of the above theorem. Also we give an estimate to x'. Finally, an example is given to show the application of the theorems. The comparison theorem of [10] isn't applicable for the case impulse and x' occur at the same time. In the special case, we get corresponding results when the nonlinear term / doesn't include x'. Particularly, L, = I* = 0, i = 1, 2, ? ? ?, m, the monotone iterative technique of [10] hasn't been effective for the case that / is dependent on x'. Therefore, the results are new even if there is no impulse.In chapter 2, first we establish the upper and lower solution technique for impulsive three-point boundary value problem (2A) . Second, using above technique, we prove that there is a A* such that problem (2\) has at least one positive solution if 0 < A < A* and problem (2>) has no positive solution if A > A*. Finally, an example is given to show the effectiveness of the theorems. We can notice that the upper and lower solution method of [10] is not suitable to impulsive three-point boundary value problems.In chapter 3, first we construct a special cone, and then we discuss the existence of multiple positive solutions of boundary value problems (3) by the fixed point theory. The results on this aspect are seldom seen. Finally, two examples are given to show the applications of the theorems. In this chapter, we also show that the equation which has a unique solution can have multiple solutions with impulsive effect.