The Positive Solutions For High-order Ordinary Differential Equations Arise From EFK And SH Equations

In this paper, we study the existence and multiplicity of the positive solutions for nonlinear high-order ordinary differential equation arise from the problem of the periodic solutions for EFK and SH equations.By applying the fixed point theorem, we study the existence of positive solutions of sixth-order differential equation with two-points boundary value, we show that the equation has one , two or three positive solutions whenα,β,γsatisfy some suitable assumptions.By using the Krein-Rutman theorem and Bifurcation methods, we discuss the existence of positive solutions of sixth-order differential equation with more general nonlinear term. We obtain the existence results of the equation has at least one positive solution.By using fixed point theorem and Maximum principles, we consider the existence of positive solutions for sixth-order differential equation with periodic boundary value, by drawing into control functions, we show the equation has at least one positive solution provided the growth rates of nonlinear term are appropriate on some bounded subsets of its domain. Then we obtain the existence results of the equation has n and infinitely many positive solutions.