Banach space bifurcation theory

Author:
David Westreich

Journal:
Trans. Amer. Math. Soc. **171** (1972), 135-156

MSC:
Primary 47H15

DOI:
https://doi.org/10.1090/S0002-9947-1972-0328706-0

MathSciNet review:
0328706

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]***Bifurcation theory and nonlinear eigenvalue problems*, Edited by Joseph B. Keller and Stuart Antman, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0241213****[2]**Melvyn Berger and Marion Berger,*Perspectives in nonlinearity. An introduction to nonlinear analysis.*, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR**0251744****[3]**J. Dieudonné,*Foundations of modern analysis*, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR**0120319****[4]**M. A. Krasnosel’skii,*Topological methods in the theory of nonlinear integral equations*, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR**0159197****[5]**M. A. Krasnosel′skiĭ,*Positive solutions of operator equations*, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964. MR**0181881****[6]**L. A. Liusternik and V. J. Sobolev,*Elements of functional analysis*, Russian Monographs and Texts on Advanced Mathematics and Physics, Vol. 5, Hindustan Publishing Corp., Delhi; Gordon and Breach Publishers, Inc., New York, 1961. MR**0141967****[7]**M. M. Vaĭnberg and V. A. Trenogin,*The Ljapunov and Schmidt methods in the theory of non-linear equations and their subsequent development*, Uspehi Mat. Nauk**17**(1962), no. 2 (104), 13–75 (Russian). MR**0154113**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
47H15

Retrieve articles in all journals with MSC: 47H15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0328706-0

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