Positive Solutions Of Singular Nonlinear Boundary Value Problems For Differential Equations

Recently, the existence of positive solutions of boundary value problems for singular ordinary differential equations and systems draws close attention, in which the nonlinear function is endowed with the conditions of some different kinds in much literature. In chapter 1 of this thesis, we explain the present condition of this question. In chapter 2 of this thesis, we consider the abstract existence theorems of positive solutions for singular nonlinear boundary value problems, and in chapter 3 we study the positive solutions of n-order singular nonlinear periodic boundary value problems.In chapter 2 we consider the abstract existence theorems of boundary value problems (L(p)(x)=h(x)f((φ(x)), (φ∈D(L), (1) where L:D(L)(?)C2(Ω)→L2(Ω) is a unbounded linear operator,Ω(?)Rn is bounded domain, h∈L2(Ω),h(x)≥0 and∫Ωh(x)dx>0.In this thesis, we allow h(x) to be discontinuous in (?)(Ω). The existence results of positive solutions are given by means of the cone theory and the fixed point index. If the superlinear or sublinear condition is satisfied, the problems (1) has at least one positive solution. The conclusions essentially extend and improve the main results of Yongxiang Li.In chapter 3 we consider the n-order singular nonlinear periodic boundary value problems. is continuous and f(t,u) is not identically vanishing, f(t,u) is uniformly continuous for u, and f(t,x)≤h(t)m(x), h:(0,1)→[0,+∞) is continuous, m: [0,+∞)→[0,∞) is continuous too.We draw into the control functionsThe existence results of positive solutions are given by means of the cone theory and the fixed point index, the problems (2) has at least one positive solution. The conclusions essentially extend and improve the main results of Yongxiang Li and Qingliu Yao.