Positive Solutions For Nonlinear Higher Order(k,n-k) Conjugate Boundary Problem

This paper researches the boundary problem of two kinds of high order nonlinear differential equations, i.e. the one is conjugate boundary value problem of nonlinear high order (k,n-k) differential equation, and the other is conjugate eigenvalue problem of nonlinear singular high order (k,n-k) differential equation. Moreover, the existence result of positive solution is established under proper assumed situation. This paper is divided into three parts, which mainly including the contents as following:First, it clarifies the research background and status of high order nonlinear differential equation boundary value problems, and gives a brief introduction of the researching problem and relative lemmas of this dissertation. Next, it researches the conjugate boundary value problem of nonlinear high order (k, n-k). (-1)n-ky(n)(x)=h(x)f(y),0<x<1,n≥1,0<k<n y(i)(0)=y(j)(1)=0,0≤i≤k-1,k≤j≤n-1Green’s function is structured by using 8 function, which further derive the integral representation of Green’s function and show it by series. Next, upper and lower bound estimation as well as derivative estimation are established, finally, the existence result of positive solution for this problem is set up by utilizing the " Krasnoselskii " fixed point theorem in cones, what’s more, the uniqueness of positive solution is proved when n= 2m,k= m. In the end, the singular higher order (k,n-k) conjugate eigenvalue problem (-1)n-ky(n)(x)=λh(x)f(y), 0<x<1, y(i)(0)=0,y(j)(1)=0,0≤i≤k-1,0≤j≤n-k-1Is researched, where λ>0 is a parameter. Based on priori estimation, the Krasnoselskii fixed point theorem in cones and the fixed point index theorem, it discusses whether positive solution is existed when λ is changed, which also proves that 0<λ*<+∞. When λ∈(0,λ*), there are two positive solutions. When λ∈(λ*,+∞), there is no positive solution. When λ=λ*, there is only one positive solution.