Existence Of Positive Solutions To The Multi Point Boundary Value Problem Of Ordinary Differential Equations In Banach Spaces

Differential equations date from all sorts of applied subjects, for example: nuclear physics, gas kinetics, hydrodynamics, nonlinear optics and so on. The study of multi-point boundary value problems for linear second order ordinary differential equations was initiated by Il'in and Moiseev, the existence of positive solutions to the multi-point boundary value problems of ordinary differential equations had been extensively studied in the last several years, and there has been some results about it; recently, on the other hand, the theory of ordinary differential equations in Ba-nach spaces has become a new important branch of mathematics, it combines the theory of ordinary differential equations with the theory of functional analysis, using the method of functional analysis, studies ordinary differential equations in Banach spaces. However, to the author's knowledge, few papers can be found in the literature for the existence of positive solutions for multi-point boundary value problem in Banach space. So it is very meaningful for us to discuss the existence of positive solutions for multi-point boundary value problems in Banach spaces.In this paper, we consider these problems from the following aspects: first of all, by using the fixed points of strict-set-contractions, we study the existence of positive solutions to a nonlinear four-point boundary value problem in Banach spaces, and several sufficient conditions about the existence of at least one or two positive solutions to this kind of equations are obtained respectively under two different situations. Then we investigate the existence of positive solutions to a generalized Sturm-Liouville four-point boundary value problem in Banach spaces, using the same theorem as above, we obtain some similar conditions of the existence of at least one or two positive solutions to this kind of equations respectively.There is great searching space about nonlinear multi-point boundary value problems in Banach spaces. By changing the conditions or eqution of the problem, I hope we will get better results in future.