Discussion On The Boundary Value Prblem Of Nonlinear Differential Equations With Caratheodory-Function Term

In this paper, by using the theory of cone, the fixed point index theory and the fixed point theorems of cone expansion and compression, we study the existence of positive solutions for several classes of boundary value problem of second-order nonlinear differential equations with Caratheodory-function term. We establish some theorems of the existence of positive solutions to these problems, some relevant results are extended and improved. The structure of this paper as follows:In chapter 1, we introduce the background of the discussed problems of this thesis briefly and state the main results of this thesis concretely.In chapter 2, we consider second-order three-point boundary value problem (BVP) where 0<η,aη<1,and h> 0. Nonlinear term f is a Caratheodory-function.By using the fixed point theorems of norm cone expansion and compression, we obtain some sufficient conditions for the existence of multiple positive solutions of BVP (2.1.1).As application,we give an example to illustrate our result.In chapter 3,we discuss three-point boundary value problem of second-order impulsive differential equation whereh∈L1[0,1],f:J×R+×R+→R+ is a Quasi-Caratheodory function,η∈(0,1), 0<α<(?),Ik,Ik∈C[R+,R+],J=[0,1],J'=J\{t1,t2,…,tm},00(i=1,2,…,m-2),0<ξ1<ξ2<·<ξm-2<1,andξ≠tk,i=1,2,…,m-2；k=1,2,…,n,by using fixed-point theorem of cone expansion and compression, we get the existence of positive solutions of BVP(4.1.1). As application, we give an example to apply our results at the end of this thesis.