Existence Of Positive Solutions For Second Order Boundary Value Problems Of Ordinary Differential Equations With Parameters

The boundary value problems of differential equations are fundamental problemsin the research of the differential equations theories. With the great developmentof science and technology,all sorts of nonlinear problems have resulted from mathematics, physics, chemistry, biology, medicine, economics, engineering, cybernetics and so on. During the development of solving such problems, nonlinearfunctional analysis has become one of the most important research fields in modern mathematics. It mainly includes partial ordering method, topological degree theory, cone theory and the variational method. Also it provides a very effect theoretical tool for solving many nonlinear problems in the fields of the scienceand technology. And what is more, it is an important approach for studying nonlinear differential equations arising from many applied mathematics.Functional analysis have been the important theory basis gradually on the studies of boundary value problems of differential equations since 20th century. In fact, the common characteristic of ordinary differential operation and integral operation is that, after operations on a function we can get a new function. we call the two kinds of operations operators consistently. Functional analysis developson the basis of concept of operator. In the mid 1930s, French Mathematician J.Leray and J.Schauder established Leray-Schauder degree theory. Their methods are very successful on the research of linear differential, integral and functional equations. Especially, the theory applies on the boundary value problems of ordinarydifferential equations and becomes topological method or functional analysis method of ordinary differential equation. The core is the establishment and the application of the fixed point theorem.We have to talk about Green function while the functional methods are mentioned. Green function is an important tool on studying the boundary value problems of nonlinear ordinary differential equations. With the help of Green function,we can transform the existence of positive solutions of boundary value problems into the existence of the fixed points of the operators, and provide the condition to the existence of solutions,multiplicity solutions and uniqueness of boundary value problems. As discussing the existence of positive solutions for boundary value problems of ordinary differential equations, only their positive solutions are significant.Many authors have extensively studied the existence of positive solution of the following BVP ( Boundary Value Problem):where f∈C([0,1]×[0,+∞),[0,+∞)).In recent paper [5], Y.Li generalizes the above BVP to Sturm -Liouville boundary value problem and obtains results on its existence. In the special case, R.Ma and B.Thompson in [6]investigate the following BVP:and establish the results on its multiplicity solutions.In the references mentioned above, a common condition has been assumed: f is a nonnegative function. Since the nonnegativity on∫assure that corresponding integral operator maps the cone into the cone, the fixed point theorem on the cone can be applied.Inspired by [6],in the case of not requiring f(t,u) to be nonnegative in the paper, by transforming the boundary value problem into the integral equation system, and applying the fixed point index theory,the author studies the existence of positive solutions for some boundary value problems of ordinary differential equations with parameters. It is made up of three chapters and the main contents are as follows:Chapter 1 introduces some backgrounds and preliminaries about boundary value problems of differential equations.Chapter 2 discusses second order Robin problems with one parameter:and obtains the results on the existence of its positive solution. Chapter 3 discusses second order boundary value problems of ordinary differentialequations with two parameters:and obtains the results on the existence of its positive solution.