Bifurcation at multiple eigenvalues and stability of bifurcating solutions

If K is a bounded linear operator from the real Banach space U into the real Banach space V and $\text{?:U×}$$\text{R}$→V has the value zero at (0, 0), the existence and linear stability of the equilibrium solutions of the dynamical system

$\text{K}\text{du}\text{dt}\text{= ?(u, α)}$

which are close to the origin in U×$\text{R}$ are studied. It is assumed that $\text{?}{}_{\mathrm{u}}\text{(0, 0): U → V}$ is a Freholm operator of index zero. The only restriction on the dimension of the null space of $\text{?}{}_{\mathrm{u}}\text{(0, 0)}$ and the order of vanishing, at (0, 0), of ? restricted to the null space of $\text{D?(0,0)}$:U×$\text{R}$→V, is that they both be finite positive integers. The main result gives conditions under which the equation, which determines the equilibrium solutions in a neighborhood of the origin, also determines the stability of these equilibrium solutions.