Differential Boundary Value Problems

Along with the development of scientific technology, modern physics and ap-plied mathmatics, many kind of non-linear problems have emerged.Those prob-lems have increasingly aroused people's widespread attentions, which greatly urge the non-linear functional analysis to be better improved. The semi-positone prob-lems and singular boundary value problems are the topic points. Many authors have studied in every asperts. In this paper using cone theorem, fixed theorem as well as the knowledge of translation transformation, we discuss differential equation boundary value problems and talk about their postive solutions.The thesis is divided into three chapters according to contents:In Chapter 1, we use the cone theory and cone expansion and compression theorem to study the existence of postive solutions for fourth-order four-point boundary value problem wheref∈C([0,1]×(-∞,+∞)×(-∞,0], (-∞,+∞)), a,b,c,d, is parameter, 0≤ξ1≤ξ2≤1.The previous literature (see [2,7], etc.) of nonlinear term f nonnegative or sign-changing and having lower bound is removed. The operator generated by f is not necessary to be a cone mapping. Thus fixed point theorem and the fixed point index theorem on cone become invalid.In Chapter 2, we use fixed point theorem as well as the knowledge of trans-lation transformation to study the existence of postive solutions for fourth-order semi-positone boundary value problem whereλ>0,0<β<Ï€2is parameter, p, q∈L[0,1]f∈C((0,1)x[0,+∞), [0,+∞)), g∈C((0,1)×[0,+∞), (-∞,+∞)), the singularity of the nonlinear term f, g at t=0,1,u=0.Most previous literature around g(t,u)≡0 situation to discuss the problem(see[11-14]),and in this paper g(t,u)is sign-changing.In Chapter 3,we use the fixed point index theorem to investigate the ex-istence of multiple postive solutions of the following boundary value problems based on second-order functional differential equations where 0<Ï„<1,η(t)∈C([1,1+Ï„]),η(t)>0fort∈[1,1+Ï„],f:(0,1)×Dâ†'R is continuous,may be singular att=0,1.where D=C([1,1+Ï„],R0+),R0+= (0,+∞),R+=[0,+∞),R=(-∞,+∞).The previous literature(see[26,27,29], etc.)of nonlinear term f have lower bound andη(t)≡0 requirements is removed.