Existence Of Positive Solutions Of Nonlinear Boundary Value Problems

As we all know,mathematics is a tool for the nature of the subject,and it can provide services to physical, chemical, biological and other fields.Therefore,some very important branches of mathematics also come into being,such as applied mathematics, basic mathematics, probability theory, control theory and so on.Nonlinear analysis has become one important research direction in modern mathematics, especially in recent decades,along with science's and technology's development, it is more and more by some mathematicians and other people who love mathematics research and study, which has formed in such a variety today and eye-catching hot topics, such as the boundary value problem, the singular problem, multi-solution of the problem, etc.,so these non-linear problems have aroused extensive attention.The nonlinear functional analysis is an important branch in nonlinear analy-sis, and it has been domestic and international mathematics and science emphasis on community because it can explain well various the natural phenomenon.The boundary value problem of nonlinear differential equation stems from nonlin-ear problems of math and science, constructs general theories and methods, and plays an important role in dealing with all kinds of nonlinear integral equations, nonlinear differential equations and partial differential equations.It is without doubt that in examining these issues are related to equations, in particular non-linear differential equations, which stem from the applied math-ematics, the physics, the cybernetics and several kinds of application discipline. It is one of most active domains of functional analysis at present. The nonlin-ear differential equation boundary value problem in Banach space is also the hot spot which has been discussed in recent years. So it has become a very important domain of differential equation research at present.In this paper, we use the cone theory, the fixed point theory, to study the existence for some kinds of boundary value problems for nonlinear differential equations.The thesis is divided into three chapters.In Chapter 1, we investigate the positive solutions of third order boundary value problems with integral boundary conditions in Banach space. whereλis a positive parameter,f:[0,∞)â†'[0,∞),κ0,κ1:[0,1]â†'[0,∞) are positive and continuous, a> 0, b> 0. We use the cone theory and cone expansion,compression fixed point theorem to obtain the positive solutions for boundary value problem (1.1.1).In Chapter 2, we talk about the positive solutions of three-point boundary problems for third order differential equations where 1<α< 2,λ> 0are parameters.a(t)can be singular at t= 0,1. The fixed point theory in cone is used to obtain the existence of positive solutions for boundary value problem (2.1.1) and generalize and improve the results in [21].In Chapter 3, we use the cone theory and cone expansion and compression fixed point theorem to investigate the positive solutions of second-order four-point boundary value problem in Banach space: whereα>0,β>0,0≤ξ<η≤1,f∈C([0,∞), [0,∞)). In this paper we mainly discuss that the above problem at least exist one positive solution when fis super linear or sublinear.we generalize and improve the results of relavent results.