Banach space bifurcation theory

Abstract:

We consider the bifurcation problem for the nonlinear operator equation $x = \lambda Lx + T(\lambda ,x,y)$ in a real Banach space $X$. Here ${\lambda _0}$ is an eigenvalue of the bounded linear operator $L,X = N(I - {\lambda _0}L) \oplus R(I - {\lambda _0}L),T \in {C^1}$ and $T$ is of higher order in $x$. New techniques are developed to simplify the solution of the bifurcation problem. When ${\lambda _0}$ is a simple eigenvalue, ${\lambda _0}$ is shown to be a bifurcation point of the homogeneous equation (i.e. $y \equiv 0$) with respect to 0. All solutions near $({\lambda _0},0)$ are shown to be of the form $(\lambda (\epsilon ),x(\epsilon )),0 \leqslant |\epsilon | < {\epsilon _0},\lambda (\epsilon )$ and $x(\epsilon )$ are continuous and $\lambda (\epsilon )$ and $x(\epsilon )$ are in ${C^n}$ or real analytic as $T$ is in ${C^{n + 1}}$ or is real analytic. When $T$ is real analytic and $\lambda (\epsilon ){\lambda _0}$ then there are at most two solution branches, and each branch is an analytic function of $\lambda$ for $\lambda \ne {\lambda _0}$. If $T$ is odd and analytic, for each $\lambda \in ({\lambda _0} - \delta ,{\lambda _0})$ (or $\lambda \in ({\lambda _0},{\lambda _0} + \delta )$) there exist two nontrivial solutions near 0 and there are no solutions near 0 for $\lambda \in ({\lambda _0},{\lambda _0} + \delta )$ (or $\lambda \in ({\lambda _0} - \delta ,{\lambda _0})$). We then demonstrate that in each sufficiently small neighborhood of a solution of the homogeneous bifurcation problem there are solutions of the nonhomogeneous equation (i.e. $y \not \equiv 0$) depending continuously on a real parameter and on $y$. If ${\lambda _0}$ is an eigenvalue of odd multiplicity we prove it is a point of bifurcation of the homogeneous equation. With a strong restriction on the projection of $T$ onto the null space of $I - {\lambda _0}L$ we show ${\lambda _0}$ is a bifurcation point of the homogeneous equation when ${\lambda _0}$ is a double eigenvalue. Counterexamples to some of our results are given when the hypotheses are weakened.