Existence Of Sign-changing Solutions For Multi-point Boundary Value Problems Of Second-order Differential Equations

The ordinary differential equation singular boundary value problem is one of the most important branches of ordinary differential equations. It is resulted from physics,chemistry,biology,medicine,economics,engineering,cybernetics and so on. Recently, as the development of the ordinary differential equation, multiple points boundary value problem is at present one of the most active fields , where the existence of sign-changing solutions for elliptic boundary value problems have been studied extensively. However, few paper can be found dealing with the sing-changing solutions for multi-point boundary value problems. So it is very meaningful for us to discuss the existence of sign-changing solutions to some multi-point boundary value problems.In this paper, by using the fixed-point index theory, we study the existence of sign-changing solutions to the following two kinds of three-point boundary value problems. Moreover, every multi-point boundary value problem also has at least one positive solution and one negative solution. In section 1, we give an introduction of the origin of the problems, the status of the research on related issues, and the trend of the future development. In section 2, we study the existence of sign-changing solutions of some three-point boundary value problem by the cone theory and the fixed-point index theory. Moreover, the three-point boundary value problem also has at least one positive solution and one negative solution. In section 3, we investigate the existence of sign-changing solutions to the second-order differential eqution with another boundary value problems using the method of section 2.Later, we can also combine it with some practical ecological models, to make them stand together, achieve better results.