Positive Solutions Of Second-order Singular Boundary Value Problem With General Differential Operator

The existence of solutions for second-order singular boundary value problems has very rich research background in theoretical and practical applications.In this thesis,we are concerned with the following boundary value problems (?) where ?,?,?,??0,?2 +?2>0,?2+?2>0.The coefficient functions a(t),b(t)of the differential operator u" + a(t)u' + b(t)u are allowed to be singular at t = 0,1.The methods we used are mainly the fixed point theorem of cone expansion and compression and bifurcation theory.The main results of this thesis are as follows:1.Consider the case:?=1,? = ?=1,?=0,i.e.Dirichlet boundary condition.We obtained the existence result of at least one positive solution under the nonlinearity satisfying either superlinear or sublinear conditions by using the fixed point theorem of cone expansion and compression.2.Consider the case:the Sturm-Liouville boundary condition.We divided the nonlinear term f into nine cases.In different cases,we obtained the range of parameter ? for the existence positive solutions.The results generalized the partial result in nonsingular case seperately.The results systematically explained that the number of solutions varies with parameters A.3 When weight function h(t)is allowed to be sign-changing and f satisfies cer-tain growth conditions,we got the shape of the connected component at Dirichlet boundary condition by using bifurcation theory.It is S-shape.Then we got the range of parameter when the existence results at least three,two and one positive solution.