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The boundary value problems for ordinary differential equations are very important. With the development of science and technology, all kinds of natural and marginal prob-lems in the fields of engineering, mechanics, astronomy, economics, cybernetics, biology etc can be described into the ordinary differential equations boundary value problems. As we all know, it is quite difficult to find out the solutions of the differential equations. So, the theoretical study of the existence of the solutions and their character from the theory attracts a great attention. With the constant development of the ordinary differ-ential equations, the study of multi point boundary value problems becomes increasingly active.The ordinary differential equations with multi point boundary value problems mean that the boundary conditions not only depend on the solution in the interval endpoints, but also depend on the solution in the range of interval points. For the reason, it can describe many important physics phenomenas more clearly, on the other hand, it can put many classical two point boundary value problems into the same framework. Therefore, it has important theoretical significance and application value and gets a great attention of many domestic and external mathematics workers. The concrete examples include the density of the same cross-section of different sub-branch of cable vibration problem in engineering and lots of problems in the theory of elastic stability. It is precisely because multi point boundary value problems have broad application background, and it pos-sesses important research value.The earliest research document of multi point boundary value problems was D.Barr, T.Sherman [9]spaper in 1973. For three point boundary value problems, Gupta, O’Regan, Ma and so on published many research achievements [3,14,15,16,21]. With more broad research of the three point boundary value problems, four point boundary value problems are the enthuse research problems. But so far, the research of four point boundary value problems is less. This provides vast space for the research of four point boundary value problems.Four point boundary value problems for singular differential equations also have broad actual significance. It has broad application in the turbulent medium flow hole theory of temperament, the elastic beam vibration theory, the topological transversal-ity theory, the physical universe, the plasma problem and so on. Early in 1992, Irena Rachunkova published a paper which researched the four point singular boundary value problem using the upper and lower solutions method in Nonlinear Analysis,in 1997,Junyu wang and Daqing jiang published a paper which researched the four point singular bound-ary value problem using the upper and lower solutions method in Journal of Mathematical Analysis and Applications. We can see, the research for the existence and multi-solution of four point boundary value problems for singular differential equations has important theoretical significance and application value.In 2008, Rahmat.Ali.Khan [4] studied the four point boundary value problem This is called Duffing equation.Duffing equation is a well-known nonlinear equation and has many applications in applied sciences, for example, it is used as a tool to discuss mechanical oscillators, periodic orbit extractio, etc. Another important application of Duffing equation is in the field of prediction of diseases,see [5]. Where 0<η≤δ<1,λx’(t) is the damping and the nonlinearity f:(0,1)xR\{0}→R is continuous and may be singular at x=0,t=0 or t=1.In this paper, author develop the method of upper and lower solutions for the existence of positive solution for the four-point singular BVP and the generalized quasilinearization to approximate the solution.In 2003, Ruyun Ma and Haiyan Wang [6] studied the three point boundary value problem This paper show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones. In first chapter of this paper,we use this method in the literature of the fixed point theorem on cone to discuss the ex-istence of at least one positive solution if f is superlinear or sublinear for our boundary value problems, this simplifies the complex degree in [4]of finding the upper and lower solutions,but also the appropriate example to prove theorem is applied.In 2009, Rahmat Ali Khan and Naseer Ahmad Asif [7] studied the two point bound-ary value problem where, the nonlinearity f∈C((0,1) x (0,∞) x (-∞,∞), (-∞,∞)) is allowed to change sign and is singular at t=0,t=1. Under much weaker hypothesis on f, where, 0≤f(t,x(t),x’(t))≤k(t)F(x(t)),∫10t(1-t)k(t)dt<∞,∫∞0du/F(u)=∞.In 2003, Guangchong Yang [8] studied the two point boundary value problem In handling the singularity, the author uses the diagonal sequence method to prove con-vergence of sequences, thus solved the difficult problem of singular. This paper in chapter 2 discuss Duffing equation four boundary value problem using above two ideas of refer-ences, thus increase the singular scope of nonlinear term f, expand system applied.In 2005, Zhongli Wei [10] studied the four point boundary value problem For nonlinear term f satisfying that there is a constantκ,μ(1<κ≤μ<+∞) for t∈(0,1),u∈(0,+∞), we have Cμf(t,u)≤f(t,Cu)≤Cκf(t,u),0≤C≤1, Cκf(t,u)≤f(t,Cu)≤Cμf(t,u), C≥1, where f satisfies the superlinear conditions. Author uses on cone fixed point theorem to prove the existence of C2[0,1] and C3[0,1] positive solutions of fourth-order super linear differential equation singular boundary value problems And we give necessary and sufficient conditions for the existence of solutions By this method, in this paper the third chapter uses the above superlinear conditions and gives the existence of solutions for Duffing equation four boundary value problems , and also gives necessary and sufficient conditions for the existence of positive solutions of Duffing equation four boundary value problem.
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