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In last thirty years, many nonlinear problems have resulted from mathematics, physics, chemistry, biology, medicine, economics, engineering, sybernetics and so on. With solving these problems, many important methods and theory such as partial ordering method, topological degree method, and the variational method have been developed gradually. They become very effective theoretical tools to solve many nonlinear problems in the fields of the science and technology.This paper mainly investigates the existence of positive solutions for boundary value problems of high-order differential system by using the theory of cone, the fixed point index theorems and the method of lower and upper solutions. The existence and uniqueness of positive solutions for differential equations have been considered extensively since twenty years ago. Here we discuss such problems on differential system.Chapter 1 investigates the existence of multiple positive solutions for boundary value problem of nonlinear high-order differential systemwhere f,g∈C[[0,1]×R+,R+],g（t,0）=0, R+ = [0,+∞). Papers [5] and [11]-[14] considered the existence of solutions for second-order coupled system, papers [6] and [7] considered the existence of solutions for third-order or fourth-order coupled system.On the base of these papers and the paper [2], this chapter improves their results. As far as we know, there is no paper to investigate the existence of solutions for coupled system as this paper does. The main tool used here is the fixed point theorem of cone expansion and compression. The result we get is the existence of multiple solutions.Chapter 2 investigates the existence of multiple solutions for third-order three point boundary value problem of singular semipositone differential systemwhereλ∈R+=[0,+∞),f∈C[（0,1）×（0,+∞）×R+,R],g∈C[（0,1）×R+×（0,+∞）,R].[19]-[22] considered positive solutions of singular boundary value problem for differential system, [23] considered positive solutions of singular semipositone boundary value problems. Using the fixed point index, we get the existence of multiple solutions. Finally, an example is worked out to demonstrate the applications.Chapter 3 investigates positive solutions for forth-order singular boundary value problem of differential systemwhere fi（t,x1,x2,x3,x4） （i = 1,2） may be singular at t = 0, t = 1,x1 = 0,x2 = 0,x3 =0,and x4 = 0. As we know, singular boundary value problems were considered many years ago, such as [1], [3]-[4], [19]-[20] and [22]-[28]. The nonlinear term in these papers may be singular at t = 0 and t = 1.In [29], f（t,x1,x2,x3） may be singular at t = 0,t =1,x1 = 0,x2 = 0 and x3 = 0, g（t,x1,x2） may be singular at t = 0, t = 1,x1 = 0 and x2 = 0. On the basis of the papers [24]-[29], we get the existence of the positive solutions, by using the method of lower and upper solutions and the comparison result. At last, an example is worked out to show the conditions are suitable.
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