The Existence Of Solutions For Several Kinds Of Singular Boundary Value Problems On The Half-line

In recent years, the singular boundary value problems arise in the fields of nuclear physics, gas dynamics, the theory of boundary layer, nonlinear optics and so on. Because of the great value both in theory and application, such problems have received a great deal of attentions by many researchers since 1980s. Along with the problem study thoroughly, the method of upper and lower solutions, cone theory, fixed point theorem, partial ordering method and the variational method were gradually used to demonstrate the existence results of positive solution of singular boundary value problem.This paper deeply discusses the existence of solutions for several kinds of sin-gular boundary value problems on the half-line mainly by making use of upper and lower solutions method, the theory of Leray—Schauder degree, fixed point index and iteration theory. The dissertation contains four chapters:In chapter 1, we consider the existence of positive solutions of second order singular boundary value problem (BVP) of differential equation on the half-line where f:R+×R+→R is a continuous function; m:R0+→R+is a Lebesgue integrable function and may be singular at t= 0; p∈C C(R+)∩C1(R0+)with p> 0 on(0,+∞),∫0+∞1/p(s)ds<+∞；a,b,c,d≥0 withρ=bc+acB(0,+∞)+ad>0, in which B(t,s)=∫ts1/p(v)dv,R+=[0,+∞),R0+=(0,+∞).By using the method of upper and lower solutions and the theory of Leray-Schauder degree,we get the existence of one and three positive solutions,and the sign of f can be changed.In chapter 2,we investigate the boundary value problem whose f depends on p(t)u′(t) where f:R+×R+×R→R+is a continuous function；m:R0+→R+is a Lebesgue integrable function and may be singular at t=0；p∈C(R+)∩C1(R0+)with p>0 on (0,+∞),∫0+∞；a,b,c≥0,d≥c withρ=bc+acB(0,+∞)+ad>0, in which B(t,s)=∫ts1/p(v)dv,R+=[0,+∞),R0+=(0,+∞).The difference between this chapter and the chapter 1 is that f depends on p(t)u′(t).In this chapter,by using the fixed point index theory,we study the existence of at least one and two positive solutions to the bellow boundary value problem.In chapter 3,we consider the existence of positive solutions of second order multi-point singular boundary value problem on the half-line in a Banach space E where k>0,ai≥0,0<(?)ai<1,0<ξ1<ξ2<ξ3…<ξm-2<+∞,f∈C[J×P,P],m:J'→J is a continuous function and may be singular at t=θ, whereθis the zero element of E,J=[0,+∞),J'=(0,+∞).In this chapter,by constructing three convex open sets and using the fixed'point index theory in cone, we obtain the existence of positive solutions of the bellow boundary value problem.In chapter 4,we consider the existence of positive solutions of singular bound- ary value problem of impulsive diffrerential equations where{tk}satifies 0