Eigenvalue Criteria For Existence Of Multiple Positive Solutions Of Boundary Value Problems For Ordinary Differetial Equations

Ordinary differential equations are an important and irreplaceable subjects in mathematics and gradually become a powerful instrument to analyze and solve problems in the modern science and technology. It is important to study the boundary value problems of differential equations in contemporary practice of production and life. Theory of boundary value problem in ordinary differential equations is an active research field and there are a lot of productions in their field especially in nonlinear differential equations of the multi-point boundary value problems. Many authors have achieved fruitful results.Existence of solutions of ordinary differential equations is one of the important problems. The study in the existence of solutions of ordinary differential equations can be converted into the existence problem of the fixed point in the cone of Banach space with the corresponding integral operator. There are a lot of fixed point theorems to prove existence of solutions. For example, Schauder fixed point theorem, Krasnosel'skii fixed point theorem, Leggett-Williams fixed point theorem and the five functions fixed point theorem, and so on. In addition, there have been some new ways to study existence of solutions, such as the use the eigenvalue of integral operator to research existence of solutions is one of them. In this article, by using the eigenvalue of operator methods we have solved the boundary value problem for differential equations.The paper is divided into three main parts, as follows:First of all, through the studing of nth-order m-point boundary value problems for differential equations, we apply the new method of eigenvalue to discuss the existence of solutions of boundary value problem. The nonlinear term in the equation is discontinuities with a singularity in [0, 1] . We have proved that the nth–order m- point nonlinear boundary value problems have at least three positive solutions .Secondly, we discuss the existence of positive solutions of third order differential equations with integral boundary conditions. We study the eigenvalue of integral operator which corresponding to the equation, according to the fixed point index theorem, and we determine the existence of multiple positive solutions of third-order differential equations.Finally, we study the boundary value problem witch with integral boundary condition and it's nonlinear terms containing the second derivative. We define the norm that containing derivative norm in Banach spaces, to prove the existence of positive solution.