| In recent years, because of the high practical value in the fields of gas dynamics fluid mechanics, the theory of boundary layer, nonlinear optics and so on, singular periodic boundary value problem in Banach space becomes one of the important prob-lems that attract the attention of mathematicians and other technicians gradually Along with the problem study thoroughly, the method of upper and lower solutions, topological degree and cone theory or method of approximation were gradually used to demonstrate the existence results of positive solution of singular periodic boundary value promblem. This paper attemps to make use of fixed point index, cone compres-sion and expansion fixed point theorem to discuss such problems more generally on the basis of above references.In chapter 1, we consider the existence of positive solutions of second order semi-positive singular periodic boundary value problem in Banach space wheref(t,u(t))is singularity at t= 0, t=2π, u=0. There exist M＞0 satisfied that f(t,u)+M> 0. We mainly use the fixed point index, discuss the existence of positive solutions for singular boundary value problems in Banach space.What is more,we also consider the existence of positive solutions under super-linear and sequence-linear condition.In chapter 2, we consider the equation of third order singular periodic boundary value problem of differential equations in Bananch space. Mainly through definition (Au)(t)=∫02πG(t, s)f(s, Ju(s))ds, the solution of the equation is equivalent to the solution of boundary value problems By using cone compression and expansion fixed point theorem, we presents the exis-tence of positive solutions of third order periodic boundary value problems.In chapter 3, we study the third order semipositive periodic boundary value prob-lems in Banach space By using cone compression and expansion fixed point theorem, this chapter presents the existence of positive solutions of third order semipositive periodic boundary value problems.
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