Positive Solutions For The Nth-order Delay Differential Equations

In this paper, we are concerned with the existence of positive solutions for the nonlineax eigenvalue problem and the superlinear semipositone problem of the nth-order delay differential systems. The main results in this paper generalize some of the existing results in the literature. In section 2, we give some preliminary knowledge adn theorems firstly. In particular, we carefully do some research on the Green's function for the n-th order boundary value probelm and present two inequalities which are very important for the proof of the existence results later. In the first part of section 3, ordinary nonlinear eigenvalue problem is discussed. Both the existence and the multiplicity of the positive solutions are presented. In the second part of this section, the nonlinear term hasn't to be non-negetive. That is, we consider the existence of positive solutions with g regular, which is named as Semipositone problem. Our proofs are based on the well-known Guo-Krasnoselskii fixed-point theorem.