Positive Solutions Of Boundary Value Problems For Nonlinear Second Order Ordinary Equations

The aim of this thesis is to study the existence of positive solutions of boundary value problem for nonlinear second order ordinary differential equations , mainly using the fixed point index theory in a cone. The conditions of the existence of positive solutions of the boundary value problem are given in the thesis. We improve the result of the existence of positive solutions either in the sublinear or in the superlinear conditons. So we get an essential existenc result because of its involving the first positive eigenvalue of the equation.In Chapter 1, we are devoted to introducing the development and the achievement of the existence of positive solutions of ordinary differential equations, also presenting the problems that will be studied.In Chapter 2, we introduce some essential definitions, preliminary theorems related to this thesis, involving the two important theorems of fixed point index theory, also presenting the corresponding Green function of the boundary value problem for the linear sencond order odinary differential equation. Furthermore, we introduce the eigenvalue problem of ordinary differential equations.In Chapter 3, using the fixed point index theory in a cone, we prove the existence of positive solutions of boundary value problem of nonlinear second order ordinary differential equations. We also present the conditions of existence of positive solutions. Thus, we get an essential existence result because of its involving the first positive eigenvalue of the equation. Meanwhile, we obtain some properties of the linear operator from the transformed second order ordinary differential equation. Finally, We obtain the sublinear and superlinear conditions under the particular case of the result .