Global Bifurcation in 2m-Order Generic Systems of Nonlinear Boundary Value Problems

Abstract

We consider the systems of $(-1{)}^m{u}^{(2m)}=\lambda u+\lambda v+uf(t,u,v),t\in (\mathrm{0,1}),u^{(2i)}(0)=u^{(2i)}(1)=0$, and $0\le i\le m-1,(-1{)}^m{v}^{(2m)}=\mu u+\mu v+vg(t,u,v),t\in (\mathrm{0,1}),v^{(2i)}(0)=v^{(2i)}(1)=0,0\le i\le m-1$, where $\lambda ,\mu \in R$ are real parameters. $f,g:[\mathrm{0,1}]\times R^2\to R$ are $C^k,k\ge 3$ functions and $f(t,\mathrm{0,0})=g(t,\mathrm{0,0})=0,t\in [\mathrm{0,1}]$. It will be shown that if the functions, $f$ and $g$ are “generic” then the solution set of the systems consists of a countable collection of 2-dimensional, $C^k$ manifolds.