Positive Solutions For Singular Differential Equation With Sign Changing Nonlinearities

The theory of singular differential equations is an important branch of dif ferential equation. In recent years, the existence of positive solutions for singular initial and boundary value problems have been widely studied. For instance, Yang Guangchong and Ge Weigao at home, Donal O'Regan, Ravi P. Agarwal and Stanek abroad have obtained many results under different conditions. These results are mostly on nonlinearity f＞0, but the case f can change sign is studied very little. Now in the thesis we will discuss the case latter.The thesis contains three chapters. We will study the positive solution for the initial and boundary value problems of the second order and first order singular differential equations using the fixed point index theory on a cone. At the same time we consider the affects of the singularity and the changing sign on differential equation.In the first chapter, we consider the initial singular question:(1)where f(t, y, y') can change sign and be singular at y=0 or(and) y'=0.In the second chapter,we consider the boundary singular question:(2)where f(t, y, py') can change sign and be singular at y =0 or (and) py'=0.Donal O'Regan and Ravi P. Agarwal discussed the existence of positive solution for (1)and (2). They used the conditions: (a)f(t,u,p)≤h(u)[g(p)+r(p)]; (b)f(t,u,p)≥Ψ_H,L(t)u^γ, (t,u,p)∈[0,1]×[0, H]×(0, L] and then construct the approximate equation without singularity. In these two chapters we improve their work and consider the existence of positive solutions when f can change sign and be singular. We construct special operator and use the condition: f(t, u, z)≥β(t), |z|≤δto overcome the difficulty from the singularity and sign changing.In the third chapter, we consider the first order singular initial question:(3)where f(t, y) can change sign and be singular at y=0.Donal O'Regan and Ravi P.Agarwal have discussed this question in 1998.They used two special conditions to overcome the difficulty from the singularity and sign changing: (a)there is a unincreasing constant sequenceρ_n and lim_{n→∞}ρ_n=0, f(t,ρ)≥0, T/n≤t≤T; (b)there is a functionα∈C[0, T]∩C¹(0, T],α(0)=0,α(t)＞0, t∈(0, T], q(t).f(t,y)≥α'(t), (t,y)∈(0,T]×{y∈(0,∞):y＜α(t)}.In this chapter we also construct a special operator and use the condition simalar to (a) and f(t. y)≥β(t), (t, y)∈(0, T)×(0,δ] to overcome the difficulty from the singularity and sign changing.The thesis not only prove each existence theory, but give an cxamplc behind that.