Eigenvalue Problem Of Nonlinear Differential And Difference Equations

In this paper,we study the eigenvalue problem of two kinds of differential and difference systems.By using the method of upper and lower solutions and the fixed point theorem on cone,we give the existence of positive solutions for these two kinds of eigenvalue problems and summarized the results of multiple solutions.In chapter one,we first introduce the background of this paper.Then,divided into two sections,we introduce the development of the eigenvalue problem for differ-ential equations and difference equations respectively.Finally,we describe the research content of this paper and list the basic definitions and theorems used in the paper.In the second chapter,we consider a class of mixed boundary value problems for nonlinear differential systems with ?-Laplace operator:where,? is monotone increasing odd function,and ?(u'),?(v')? C1((0,1),R).Eigen-value ? and ? are nonnegative real numbers and not not all equal to 0.hi(i = 1,2)?C([0,1],R+),and in any nonzero measure on the[0,1]interval is not identical to 0.f,g is nonnegative,and satisfies superlinearity at infinity.In this paper,we first establish the theorem of upper and lower solutions for such boundary value problems,and give the existence result of positive solutions.Then the existence of two positive solutions is given by using the Guo-Krasnosel' skii fixed point theorem,and the relation between the number of positive solutions and the eigenvalues is obtained.In the third chapter,we study the Dirichlet boundary value problem of two order difference systems:Among them,T>1 is a positive integer,? is the forward difference operator,?u(k)=?u(k + 1)-?u(k).Eigenvalue ? and ? are nonnegative real numbers and not not all equal to 0,hi:[1,T]z?(0,+?),f,g are nonnegative and continuous functions.In this paper,we prove the existence of positive solutions for a class of boundary value problems when ? and ? are small,the existence of multiple solutions is also proved by using the fixed point index theory.