Nonlinear Singular Differential Equations Boundary Value Problems

With the development of modern physics and applied mathematics, it de-mands the mathematical ability of analyzing and controling the objective phe-nomena toward to the overall high and precision level, which made the results of the nonlinear analysis was accumulated, and gradually formed an important branch subject of the present analysis mathematics-Nonlinear functional analy-sis. Nonlinear functional analysis is a research discipline in analysis mathematics both to have the profound theory and to have the widespread application. It takes the nonlinear problems appearing in mathematics and the natural sciences as background to establish some general theories and methods to handle nonlinear problem. Because it can commendably explain all kinds of natural phenomenal, coupled with the widely application in the realistic production and life, it has received highly attention of the domestic and foreign mathematics and natural science field in recent years.The boundary value problem of nonlinear stems from the applied mathe-matics, the physics, the cybernetics and each kind of application discipline. It is an important kind of question in the differential equations, it is one of most active domains of functional analysis studiesin at present.The singular nonlinear differential equation boundary value problem is also the hot spot which has been discussed in recent years. So it attachs more and more attention.In this paper, we use the cone theory, the fixed point theory, the topological degree theory as well as the fixed point index theory, to study the existence of positive solutions for several kinds of boundary value problems for nonlinear singular differential equation.The thesis is divided into three chapters according to contents.In chapter 1, by using the fixed point theorem in cone and combining with the relevent knowledge of cone theorey, we discuss the existence of positive solu-tions to the following nonlinear singular third-order three-point boundary value problem whereλ>0 is a positive parameter,α,βandηare constants withα>β≥0,0<η<1,and a:(0,1)�'R+is continuous and may be singular at t=0 and／or 1,f:[0,1]×R+×R+×R+�'R+is continuous,in which R+=[0,+∞). The problem we investigated is more general than that is considered in[8],and our results generalize and extend previous results in the field.In chapter 2,we study the existence of positive solutions for singular second-order boundary value problems for the differential systems on the half-line where f1:R+×(0,+∞)×R+�'+R+and f2:R+×R+×(0,+∞)�'R+ are continuous functions and f1(t,x,y)may be singular at x=0 while f2(t,x,y) may be singular at y=0；m1,m2:(0,+∞)�'R+ are continuous and may be singular at t=0,t=1；For i=1 or i=2,pi∈C(R+,R+)∩C'(0,+∞) with pi>0 on(0,+∞),∫0+∞1/(pi(s))ds＜+∞;αi,βi,γi,δi≥0 withρiγiβi+αiγiBi(0,+∞)+αiδi>0,in which Bi(t,s)=∫ts 1/(pi(τ))dτ.To obtain the results we use the fixed point theorey.In chapter 3,by employing a well-known fixed point index theorem and combining with the relevant knowledge of operator theorey, we study the existence of positive solutions for the nonlinear singular third-orderboundary value problem with integral boundary conditions where f∈C((0,1)×(0,+∞),R+) and f(t,x) may be singular at t= 0, t= 1, x= 0; a(t), b(t), c(t) are continuous and positive on [0,1].