Banach space bifurcation theory
Abstract: We consider the bifurcation problem for the nonlinear operator equation in a real Banach space . Here is an eigenvalue of the bounded linear operator and is of higher order in .
New techniques are developed to simplify the solution of the bifurcation problem. When is a simple eigenvalue, is shown to be a bifurcation point of the homogeneous equation (i.e. ) with respect to 0. All solutions near are shown to be of the form and are continuous and and are in or real analytic as is in or is real analytic. When is real analytic and then there are at most two solution branches, and each branch is an analytic function of for . If is odd and analytic, for each (or ) there exist two nontrivial solutions near 0 and there are no solutions near 0 for (or ).
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. ) depending continuously on a real parameter and on .
If 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 onto the null space of we show is a bifurcation point of the homogeneous equation when is a double eigenvalue.
Counterexamples to some of our results are given when the hypotheses are weakened.
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Keywords: Banach space, bifurcation, homogeneous equation, nonhomogeneous equation, Fredholm operator, index zero, nonlinear operator, Fréchet differentiable, real analytic, symmetric -linear form, simple eigenvalue, eigenvalue of odd multiplicity, implicit function theorem
Article copyright: © Copyright 1972 American Mathematical Society