The Optimal Interval Of Green’s Function Get Negative Definite For A Third-order Two Point Disconjugate Boundary Value Problem

The ordinary differential equation boundary value problems were both old andyoung subjects. To say they were old, it could date back to200-300years ago, forexample, there were some very classical instances called the problem of brachistochrone,and the problem of Sturm-Liouville eigenvalue. These examples had a long history,ofcourse many mathematicians paid much attention to them. If for the ordinarydifferential equation initial value problem, we had the theorem called the uniqueness ofsolution of the initial value. But for boundary value problems, since the boundarycondition consisted of two endpoints, thus the solution were more complex than the oneinvolved in initial value condition. There may exist the unique solution, multiplesolutions or no solution. It would be not convenient when we discussed about them. Butin recent years, many authors used some approaches to solve boundary value problems.For instance, the fixed point method, variational ways, the obtention of a priori-boundsfor the possible solutions and then the application of topological degree arguments, thetheory of upper and lower solutions, disconjugate theory, fixed point index, Morse indexand rotation and Lie group approaches. These techniques had proved to be very strongand fruitful.We treat the problem as a young one, because in the field of differential equation itappeared some fractional order case.(include fractional ordinary differential equationand fractional partial differential equation). So that some theories would be consideredagain to come up with new method.The ordinary differential equation boundary value problem essentially came fromphysics, including mechanics, electricity, thermodynamics phenomenons. The originalequation may be a partial differential equation with initial and boundary value problem,via some transformations we could acquire an ordinary differential equation boundaryvalue problem, so the significance was important.In this paper, the existence of the optimal interval where the Green’s function wouldget negative definite is proved, at the same time the theoretical result is checked bybackward searching method. Then the left and right endpoints are found, and newprinciples of comparision are established, at last the solvability for a class of the thirdorder differential equations is proved as application.