Existence Of Positive Solutions Of Multi-Point Boundary Value Problems For Singular Differential Equations

In this thesis, we discuss the existence of positive solutions of m-point boundaryvalue problems for singular differential equations (1) and (2) as followswhere 0 < a_i < 1, 0 <ξ_i < 1 ,i = 1, 2,…, m - 2, f may change sign and may be singular at x = 0 and x' = 0, andwhere f may be singular at x = 0 and x' = 0.M-point boundary value problems for differential equations are that boundary value conditions of differential equations depend on the value at interval endpoints and inside the interval,which arose in different fields of applicable mathematics and physics, including examples of the same cross-section and density of different sub-extensioncord vibration and elastic stability theory in many of the issues. Because the multi-point boundary value problems have widely applying background, it has an important research value.The study of multi-point BVPS (boundary value problems) for linear second-orderordinary differential equations was earlier studied by Il'in and Moiseev. Since then, many authors studied more general nonlinear multi-point BVPS, for examples [4-5], and references therein. Recently, using Leray-Schauder continuation theorem, R.Ma and D. O'Regan proved the existence of positive solutions of C~1[0,1) solutions for the BVPwhere f : [0,1] x R~2→R satisfies the Carath(?)odory's conditions (see[6]). However, up to now, there are a fewer results on the exisence of multiple solutions to equation (1) when the nonlinearity f depends on x' and may be singular. This thesis obtains the exisence of multiple solutions to equation (1) and (2) when the nonlinearity f may be singular at x = 0, x' = 0. So it is the improvement for existence of positive solutions for three-point boundary value problems.There are many results about above problems(see [5], [7-13], [23-32]). The method what we used is firstly to construct proper integral operators and to remove the part in which integral operators change sign because of f changing sign. Then, using Arzela-Ascoli theorem, we consider the set of the approximate solutions and obtain a convergent subsequence. The limit is a positive solution for equation (1). This thesis is divided into three chapters.In chapter 1, we present the existence of positive solutions of (1) when nonlinearitymay be singular at x = 0, x' = 0 and f may be change sign.In chapter 2, we mainly study the existence of multiple positive solutions of (1). Firstly, using the theory of fixed point index on a cone, we present the existence of multiple positive solutions to equation (1) when f may be singular at x' = 0 but not at x = 0. Secondly, under the condition f is singular at x' = 0 and x = 0, we present the existence of multiple positive solutions to equation (1). Finally, under the condition f is singular at x = 0 but not at x' = 0, we present the existence of multiple positive solutions to equation (1).In chapter 3, we present the existence of positive solution to equation (2) when the nonlinearity f(t,x,z) may be singular at x = 0 and z = 0,and the degree of singularity at x = 0 and z = 0 may be arbitrary.