Positive Solutions Of Second Order Boundary Value Problems For Singular Differential Equations And Differential Systems

Recently, the existence of positive solutions for differential equations and differential systems boundary value problems has been studied by many authors. In chapter 1 of this thesis, the author explains the present condition of this question.In chapter 2, we consider the existence and multiplicity of positive solutions for the following singular second order m-point boundary value problems where (Lφ)(x)=(p(x)φ'(x))'+q(x)φ(x) andξi∈(0,1) with 0<ξ1,<ξ2,<…<ξm-2<1, ai∈[0,∞). h(x) is allowed to be singular at x=0 and x=1. In this chapter, we apply the Avery Five Functional Fixed Point Theorem to obtain the existence of multiple positive solutions. The conclusions extend and improve the main results of Zhang Guowei and Sun Jingxian. Similar conclusions hold for some other m-point boundary value conditions.In chapter 3, we consider the existence of positive solutions for the following boundary value problems of second order two-point differential systems whereαi,αi,γi,δi≥0 andρi=αiγi+αiδi+γiβi>0 (i=1,2). ai(t)(i=1,2) is allowed to be singular at t=0 and t=1. By using different convex functionals to compute fixed point index, the existence of positive solutions for second order two-point boundary value problems is obtained.