Positive Solutions for Second Order Differentiae Equations Boundary Value Problem on the Half-line
|School||Hebei University of Science and Technology|
|Keywords||Ordinary differential equation Fixed point theorem Positive solution Half-infinite interval Boundary value problem Cone|
Differential equations describe the relationship between unknown function and its derivative in the form of equation. In mathematics and applications, the significance of differential equations is that study on many physical and technical problems can be attributed to solve the differential equation problems. Study on differential equations boundary value problem of qualitative is one of the most important branch, many domestic and foreign workers in mathematics concern about the existence of positive solutions. Infinite interval boundary value problem can be used to reflect the developments in the future, help people to study development of prediction rules, so it has more significant value for research. In recent years, people pay more and more attention on the existence positive solutions of half-infinite interval boundary value problems, many scholars have studied the research with the different research tools and obtained a number of valuable research findings.This article studied several kinds of the existence of positive solutions for boundary value problems on the half-infinite interval by using the theory of cone, Leggett-William fixed point theorems and Avery-Peterson fixed point theorem. Infinite interval do not have the compactness, so we need to construct a special Banach space and a special cone, by using extension of Ascoli-Arzela theorem to complete the proof.The thesis is divided into five chapters according to contents. We mainly discuss the existence of positive solutions for the following three kinds of infinite interval boundary value problems by fixed point theorem.In the first chapter, we introduce the history, current situation of the theory of boundary value problems for ordinary differential equations, and the main content in this paper.In the second chapter, we consider the existence of positive solutions for boundary value problems on the half-line. And study on the nature and draw some inequalities by constructing a Green function.In the third chapter, Through the Leggett-William fixed point theorem, we obtain the existence of solutions for boundary value problems on the half-line with dependence on the first order derivative. The core of this part is the calculative process of Green Function, which guarantee that the functions defined on infinite interval have better properties.In the fourth chapter, we discuss triple positive solutions for multi-point boundary value problems on the half-line. The core of this part is the calculative process of Green Function, by using new fixed point theorem in the cone, we can prove that the existence of the fixed point in the cone.In the last chapter, we obtain triple positive solutions for multi-point p-Laplacian boundary value problems on the half-line. A special Banach space and a cone are introduced so that we can establish some similar inequalities, and then we can proceed with the Avery-Peterson fixed point theorem.