Solutions For Several Kinds Of Boundary Value Problems Of Differential Equations

In last decades of years, all sorts of boundary value problems of differential equa-tions have resulted from mathematics, physics, engineering, sybernetics, biology, eco-nomics and so on. With solving these problems, many important methods and theory such as partial ordering method, upper and lower solutions method, fixed point the-ory, topological degree method, the theory of cone and the bifurcation theory have been developed gradually. They become very effective theoretical tools to solve many nonlinear problems in the fields of the science and technology.This paper mainly investigates the existence of solutions for several kinds of boundary value problems of nolinear differential equations by using fixed point the-ory, topological degree method, the theory of cone and the bifurcation theory. The existence and uniqueness of positive solutions for differential equations have been con-sidered extensively since twenty years ago([1]-[56]). This paper discusses the existence of solutions for several kinds of boundary value problems of nolinear differential equa-tions.Chapter 1 investigates the existence of multiple solutions for boundary value prob-lem of second order differential equation with p-Laplace operator and parameters whereφp(s)=|s|p-2 s,p>1,φp-1=φq,1/p+1/q=1, and 0<ξ1<ξ2<…<ξn<1. The authors considered the existence of one positive solution for this problem by using Krasnosel'skii fixed point theorem in the paper [19]. But as far as we know, there are few papers to investigate the existence of multiple solutions for this problem as this paper does. This chapter firstly obtains two positive solutions by using the fixed point theorem of cone expansion and compression, and secondly gets three solutions by the theorem of A very-Peterson.Chapter 2 studies the existence of multiple solutions for m-point boundary value problem of fourth order differential equations and the boundary value condition is where m≥3,ηi∈(0,1) andαi＞0, i= 1,…, m- 2 satisfying In papers ([26], [44], [45]), Professor Sun Jingxian obtained some fixed point theorems of operaters on lattice by using the theory of lattice. The authors in [53] investigated the existence of multiple solutions by using the fixed point index. But to our knowlege, there are few papers to consider the same problems as we do. We mainly use the fixed point index and a fixed point theorem of quasi-additive operator on lattice due to Professor Sun Jingxian.Chapter 3 considers the global behavior of the components of nodal solutions for Lidstone boundary value problem where n≥1, r> 0 is a given parameter. In papers ([32]-[37], [27]), Professor Ma Ruyun and Professor Liu Yansheng studied the global behavior of the components of nodal solutions for boundary value problems of second order and fourth order differ-ential equations by applying Rabinowitz's global bifurcation theorem respectively. As we know, there is no paper to consider this problem up to now. In this paper, we obtain the global behavior of the components of nodal solutions for this problem by applying Rabinowitz's global bifurcation theorem and fill this gap.