The Positive Solutions For High-Order Nonlinear Ordinary Differential Systems

In this paper, we study the existence of solutions for high-order ordinary differential equation systems and delay differential equations, subject to different conditions.By applying the fixed point theorem, we study the existence of positive solutions of 2p-2q order differential equation systems with two independent parameters. When the nonlinear terms satisfy some certain conditions and the parametersλandμbelong to appropriate intervals respectively, we show that the systems have at least one and two positive solutions. This result is the generalization of some existing results mention to high-order and different order of differential systems and the independence of two parameters.By using Mountain pass theorem and Fountain theorem, we consider the existence of solutions for nonlinear 2p-2q order differential systems with variational structure. Under the conditions of Ambrosetti-Rabinowitz type superlinear and odd duality, we show that the systems have at least one nontrivial solution and infinitely many solutions.By using the fixed point theorem in cone and Leggett-Williams point theorem, we discuss the existence of positive solutions for 2n-order delay differential equations. We obtain the existence results of the equation has one, two or three positive solutions.