Positive Solutions Of Initial And Boundary Value Problems For Singular Differential Equations With P-Laplacian Operator

Singular differential equation with operator has a wide range of mathematics and physics application background.In recent years,the existence of positive solu-tions for singular differential equations has been widely studied.Initial and bound-ary value problems with operator stem from models in Mathematics and Physics. So the study of this problems is useful both theoretically and practically.Now many scholars home and abroad have payed attention to it.The theory of singular dif-ferential equations with p-Laplacian operator is an important branch of differential equation in the recent years, especially,when p=2,namely,φp(x)=x,some ex-istence theorems have been obtained.For instance, Yang Guangchong and Ge Weigao at home, Donal O’Regan, Ravi P.Agarwal and Stanck abroad have obtained many results under different conditions. These results are mostly on the conditions of f>0, but the case that f can change sign is studied very few. Now in the thesis we will discuss initial and boundary value problems with p-Laplacian operator and sign-changing nonlin-earties.The thesis contains two chapters.We will study the positive solution for the initial and boundary value problems of the second order singular differential equa-tions with p-Laplacian operator and sign-changing nonlinearties by 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 second order singular initial differential equations with p-Laplacian operator where f(t,y,y’)can change sign and be singular at y=0.φp(s)is p-Laplacian operator,φp(s)=|s|p-2s,p>1,s∈R. Setφq(s) as the inverse ofφp(s),that isφq(s)=|s|q-2s,s∈R,and 1/p+1/q=1.In the second chapter,we consider the second order singular boundary differen-tial equations with p-Laplacion operator where f(t,y,y’)can change sign and be singular at y=0 or y’=0.φp(s) is p-Laplacian operator,φp(s)=|s|p-28,p>1,s∈R.Setφq(s)as the inverse ofφp(s),that isφq(s)=|s|q-2s,s∈R,and 1/p+1/q=1.Donal O’Regan and Ravi P.Agarwal discussed the existence of positive solution for(3)and(4) where,f(t,y,y’) can change sign and be singular at y=0 or(and) y’=0, where f(t,y,py’)can change sign and be singular at y=0 and py’=0.They used the conditions(a)f(t,u,p)≤h(u)[g(p)+r(p)];(b)f(t,u,p)≥ΨH,L(t)uγ,(t,u,p)∈[0,1]×[0,H]×(0,L] and then construct the approximate equation without singular-ity.In the thesis we improve their work and consider the existence of positive solu-tions for initial and boundary value problems with operator 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 non-linearties.