Positive Solutions And Sign-changing Solutions For Several Classes Of Ordinary Differential Equation Boundary Value Problems

The boundary value problem for ordinary differential equation is one of the most important branches of ordinary differential equations. Many problems of engineerings physics and biology can be handled by the boundary value problem for ordinary differential equation. Recently, the research on the boundary value problem for ordinary differential equation is very active,so the reader can see [20]-[26] for the results. And most of them are the existence of positive solutions. However, there were few papers that considered the sign-changing solutions for boundary value problem. So it is very meaningful to discuss the existence of positive solutions and sign-positive solutions to some boundary value problem for ordinary differential equation.In this paper, by using fixed point theorem, the fixed-point index method, we get the existence of positive solutions and sign-changing solutions for boundary value problem for several classes of second-order differential equation boundary value problems. The dissertation contains four chapters.Chapter I investigates Sturm-Liouville boundary value problem By using the fixed-point index method, we obtain the existence of one positive solution、one negative solution and one sign-changing solution. Meanwhile,one example is worked out to demonstrate the main result. In chapter2, we consider the second-order of three-point boundary value prob-lemwhere0<α, η <1.By using the properties of asymptotically linear operator andthe fixed-point index theory, we obtain the existence of two positive solutions!twonegative solutions and one sign-changing solution. Meanwhile, we give one exampleto demonstrate the main result.In chapter3, we study the second-order boundary value problemBy using the properties of asymptotically linear operator along cone and the fixed-point index theory, we obtain the existence of two positive solutions!two nega-tive solutions and one sign-changing solution. Meanwhile, we give one example todemonstrate the main result.In chapter4, we study Sturm-Liouville boundary value problemBy using the fixed-point theorem, we obtain the existence of two positive solutions.