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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Banach space bifurcation theory

Author: David Westreich
Journal: Trans. Amer. Math. Soc. 171 (1972), 135-156
MSC: Primary 47H15
MathSciNet review: 0328706
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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 \vert\epsilon\vert < {\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.

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PII: S 0002-9947(1972)0328706-0
Keywords: Banach space, bifurcation, homogeneous equation, nonhomogeneous equation, Fredholm operator, index zero, nonlinear operator, Fréchet differentiable, real analytic, symmetric $ n$-linear form, simple eigenvalue, eigenvalue of odd multiplicity, implicit function theorem
Article copyright: © Copyright 1972 American Mathematical Society

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