Existence Of Positive Solutions For Singular Boundary Value Problems Of Several Kinds Of Differetial Equations

It is well known that singular boundary value problem has been one of the most important problems that attract the attention of mathematicians and other technicians. It arises in the fields of nuclear physics,gas dynamics,Newtonian fluid mechanics,the theory of boundary layer,nonlinear optics and so on.Along with the problem study thoroughly,the method of upper and lower solutions,topological degree and cone theory or method of approximation were gradually used to demonstrate the existence results of positive solutions of singular boundary value problem.This paper mainly attemps to make use of fixed point index,cone compression and expansion fixed point theorem and the first eigenvalue of the relevant linear operator to discuss such problems more generally on the basis of above references.Chapter 1 investigates the existence of positive solutions for second order two point singular boundary value problem where f(t.u)is required to be nonnegetive and continuous,and f(t.u)may be singular at t = 0,t = 1 mostand x = 0.By constructing a special cone,using the fixed point index theory on cone and method of approximation,we get the existence of at least two positive solutions.Chapter 2 considers positive solutions of the three-point boundary value problem for second order differential equations with an advanced argument whereη∈(0,1),α≥0,β≥0,0 < k <(α+β)/(β+αη),a(t)may be singular at t = 0,t = 1.Firstly,by using the fixed point index method,we establish the existence of at least one positive solution to BVP(2.1.1).Secondly,by using cone compression and expansion fixed point theorem to get the existence of at least two positive solutions to BVP(2.1.1)when the parameterλis small enough.Chapter 3 investigats positive solutions of generalized Sturm-Liouville four-point boundary value problems whereα,β.γ.δ≥0,0<μ₁,μ₂<1,0<η≤ξ<1,h(t)may be singular at t = 0.t = 1. This paper is considered under some conditions concerning the first eigenvalue of the relevant linear operator.At the same time,the fixed point index theory is used to get the existence of at least one positive solution to BVP(3.1.1).Chapter 4 investigates the existence of two positive solutions for singular fourthorder boundary value problems in Banach space where J=[0,1],f∈C[(0,1)×P×P×P,P],f(t,x,y,z)may be singular at t = 0,t = 1. P is a cone of real Banach space E.This paper gets two positive solutions in Banach space by using the fixed point index theory of a strict set contraction operator.