Global bifurcation in \(2m\)-order generic systems of nonlinear boundary value problems
Summary: We consider the system \[ \begin{aligned} (-1)^m u^{(2m)} &= \lambda u + \lambda v + uf(t, u, v),\quad t \in (0, 1),\\ u^{(2i)}(0)& = u^{(2i)}(1) = 0,\quad 0 \leq i \leq m - 1,\\ (-1)^m v^{(2m)}& = \mu u + \mu v + vg(t, u, v),\quad t \in (0, 1),\\ v^{(2i)}(1)& = 0,\quad 0 \leq i \leq m - 1,\end{aligned} \] where \(\lambda, \mu \in \mathbb R\) are real parameters. \(f, g : [0, 1] \times \mathbb R^2 \rightarrow \mathbb R\) are \(C^k,\;k \geq 3\) functions and \(f(t, 0, 0) \equiv g(t, 0, 0) \equiv 0,\;t \in [0, 1]\). It is shown that, if the functions, \(f\) and \(g\) are ``generic'', then the solution set of the system consists of a countable collection of two-dimensional \(C^k\) manifolds.
scientific article; zbMATH DE number 6134166
scientific article