| Singular boundary value problems for nonlinear ordinary differential equations, which arise in a variety of areas such as mechanics, boundary layer theory, diffusion and reaction equations, biology, ect, become an important topic in ordinary equations fields.In this paper, existence theory for two positive solutions is presented to singular three-point boundry value problemThe function q(t) may be singular at t = 0 and the function f may be singular at y = 0. The existence of multiple positive solutions is obtained via the Leray-Shauder alternative and the fixed point theorem in cones.The present work is a direct extension of some results in [4] for the singular problem. Our technique of proof uses essentially the fixed point theorem in cones which is well adapted to this type of problem. We extended the results of two-point boundary value condition to three-point boundary value condition using the fixed point theorem in cones in [4].This thesis is composed of five parts. In the first part, we introduce the historical backgroud of problems which will be investigated and the main problem studied in this paper, namely nonlinear singular second-order three-point boundary value problem. There is a brief summerize of results of this problem in other literatures, and introdution of some basic knowledge and propositions which is neeeded in the proof of the theorem.In the second part, we proof a general existence principle for the Dirichlet boundary value problem, it will be used in the next part.The third part is our main results, we establish existence principles for nonlinear singular second-order three-point boundary value problem. In this part, we will use contents and methods from [4] and [17].The forth part is a particular condition of the third part, we establish existence principles for nonlinear singular second-order three-point boundary value problem which has two positive solutions. The last part is an example, the example is given to explain the main results.
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