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In this paper cone theory , fixed point index theory and cone expansion and compression fixed point theorem for several nonlinear term is Caratheodory- function Positive Solutions of Second Order Differential Equations existence, extend and improve some of the literature findings , obtained positive solutions to these problems the existence theorem . Text is structured as follows : The first chapter is the introduction , this paper briefly describes the background and current status of research questions , and the main results of this paper were elaborated specific second chapter , which discusses the second-order three-point boundary value problem : 0 lt; η, αη lt; 1, and h gt; 0. nonlinear term f is a Caratheodory- function norm in this chapter use the form cone expansion and compression fixed point theorem, we obtain boundary value problem ( 2.1 . 1 ) the existence of multiple positive solutions sufficient condition as an application , the article finally gives an example to prove the results of the third chapter discusses the following point boundary value problem of second order impulsive differential equations where h ∈ L1 [0 index theory to get the problem ( 3.1.1 ) there is a sufficient condition for positive solutions , and gives as an example of application of the results . , m-2), 0 lt; ξ1 lt; ξ2 lt; ... lt; (?) m-2 lt; 1, and ξi ≠ tk, i = 1,2, ..., m-2; k = 1,2 , ..., n, the use of cone expansion and compression fixed point theorem problem ( 4.1.1 ) the existence of positive solutions . As an application , the article finally gives examples.
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