Solutions For Several Kinds Of Boundary Value Problems Of Differential Equations With P-Laplace Operator

In last decades of years, all sorts of boundary value problems of differential equations have resulted from mathematics, physics, engineering, sybernetics, biology, economics and so on. With solving these problems, many important methods and theories such as fixed point theory, partial ordering method, upper and lower solutions method, 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 nonlinear differential equations with p-Laplace operator by using the theory of cone and the fixed point index theory. The existence and multiple of solutions for differential equations have been considered extensively since twenty years ago([l]-[35]). This paper discusses the existence of solutions for several kinds of boundary value problems of nonlinear differential equations.Chapter 1 investigates the existence of two positive solutions for three-point boundary value problems with p-Laplace operator and parameters on time scales(?) where (?) is p-Laplace operator: (?). Note(?) the inverse of (?), that is(?) - In the reference [1], Hong considered the existence of three solutions for the similar problem by using Avery-Peterson fixed point theorem. But the solution of the equation investigated in [1] does not depend on the parameter, while the existence of our positive solutions discussed in this chapter is restricted by the parameter, which is very significant in the engineering ; In addition, although the reference [1] obtained the existence of three solutions, there probably be a trivial solution, moreover, it called for very strong conditions to ensure the existence of the solution. As a contrary, the conditions needed in our paper are relative weaker, and we obtain two positive solutions by using the fixed point theorem of cone expansion and compression. Therefore, it is very convenient to use the results.Chapter 2 considers the existence of twin positive solutions for multi-point boundary value problems with p-Laplace operator and parameters(?)where(?) is p-Laplace operator: (?) Note (?) the inverse of (?), that is (?), and (?) Recently, the authors considered the existence of one positive solution for this problem by using Krasnoselskii fixed point theorem in the reference [2]. But as far as we know, there are few papers to investigate the existence of multiple positive solutions for this problem. This chapter obtains two positive solutions by using the fixed point theorem of cone expansion and compression, and the main results depend on a parameter.Chapter 3 studies the existence of positive solutions for third-order four-point boundary value problem with p-Laplace operator(? )where (?) is p-Laplace operator:(?) is the inverse of (?). The solution of the equation investigated in [3] does not depend on the parameter, while the existence of our positive solutions discussed in this chapter by using the fixed point index theorem is restricted by theparameter; In addition, although the reference [3] obtained the existence of multiple solutions, it called for very strong conditions to ensure the existence of the solutions, and there are few nonlinear terms to satisfy the conditions, which would lead to the inconvenience for the application. As a contrary, the conditions needed in this chapter are very relaxed, and there is a kind of nonlinear functions which are satisfied the conditions. Therefore, it is very convenient to use the results. parameter; In addition, although the reference [3] obtained the existence of multiplesolutions, it called for very strong conditions to ensure the existence of the solutions,and there are few nonlinear terms to satisfy the conditions, which would lead to theinconvenience for the application. As a contrary, the conditions needed in this chapterare very relaxed, and there is a kind of nonlinear functions which are satisfied theconditions. Therefore, it is very convenient to use the results.