Existence Of Positive Solutions To A Generalized Second-order Three-point Integral Boundary Value Problems

The researchers began to study the boundary value problems for the ordinarydifferential equations, since Picard began to explore the existence and uniqueness ofthe second-order two-point boundary value problems for the ordinary differentialequations by using iterative method in1983. Nearly half of this century, the boundaryvalue problems for the ordinary differential equations have been developing rapidly.Simultaneously, with the development of the functional theory and the need to solvepractical problems, the study on the existence of positive solutions for the boundaryvalue problems has been endowed with more significance in the theory research andpractical application. In this paper, by using the properties of the Green’s function andfixed point theorem, the writer studies the existence of positive solutions to thesecond-order three-point integral boundary value problems with different boundaryconditions.This paper consists of the following three parts.The first part is introduction. Mainly introduce the origin of the boundary valueproblems for the ordinary differential equations and briefly present its both researchstatus and research significance;In the second part, the writer introduces some properties of the Green’s functionand discusses the existence of positive solution boundary to the second-orderthree-point integral boundary value problems for ordinary differential equations whenthe boundary conditions are non-resonance. Some theorems of the existence forpositive solutions are received;In the third part, the writer discusses the same boundary value problems of thesecond part, but the boundary conditions are resonant. The writer studies the existenceof solutions for the boundary value problems at resonance. Some sufficient conditions for the existence of the solution for the equation are obtained. At last, an example isgiven to demonstrate the main results.