Existence Of Multiple Solutions For Differential System In Banach Space

In last few years, all sorts of nolinear problems have resulted from mathematics, physics, chemistry, biology, medicine, economics, engineering, sybernetics and so on. With solving these problems, many important methods and theory such as partial ordering method, topological degree method, the theory of cone and the variational method have been developed gradually. They become a very efiective theoretical tool to solve many nonlinear problems in the fields of the science and technology.This paper mainly investigates the existence of positive solutions for boundary value problems of nolinear differential systems by using the theory of cone. The existence and uniqueness of positive solutions for differential equations have been considered extensively since twenty years ago([7]-[10]). This paper discusses the existence of solutions for differential systems.Chapter 1 investigates the existence of multiple solutions for boundary value problem of a coupled system on the half-line where [0, +∞). Agarwal and O'Regen considered the existence of a coupled system on [0, 1] in the paper([7]-[8]). Xiaohong Ni gained three solutions using the fixed point theorem of Leggett-Williams in the paper[17]. But as far as we know, there is no paper to investigate the existence of solutions for coupled system on the half-line as this paper does. The main tools used here are the fixed point theorem of cone expansion and compression and the theorem of Leggett-Williams.Chapter 2 investigates the existence of multiple solutions for boundary value problem of a singular differential system on the half-linewhere There are some discussion about existence of solutions for boundary value problems of differential systems on finite interval in papers[7-14]while there are some discussion about the ones of differential equations on half-line such as the papers([16]-[17]). But to our knowlege, there is no paper to consider the same problems as we do. we mainly use the fixed point theorem of cone expansion and compression.Chapter 3 investigates the existence of multiple positive solutions for the higher order boundary value systems with p-Laplacian. As we know, there are no papers to consider this problem up to now. In this paper, we construct a special cone and get existence of multiple solutions using theory similar to chapter 2.Chapter 4 investigates the existence of positive solutions of the nonlinear singular impulsive boundary value problems in Banach space. As far as we know, no papers investigate the problems. We introduce a special operator and a cone to overcome the singular case. Finally, we gain the existence of solutions, using the method like chapter 2.