Banach space bifurcation theory

Abstract

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