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


References [Enhancements On Off] (What's this?)

  • [1] Melvyn S. Berger, ``A bifurcation theory for nonlinear elliptic partial differential equations and related systems,'' in Bifurcation theory and nonlinear eigenvalue problems, Joseph B. Keller and Stuart Antman (Editors), Benjamin, New York, 1969. MR 0241213 (39:2555)
  • [2] Melvyn S. Berger and M. S. Berger, Perspectives in nonlinearity. An introduction to nonlinear analysis, Benjamin, New York, 1968. MR 40 #4971. MR 0251744 (40:4971)
  • [3] J. Dieudonné, Foundations of modern analysis, Pure and Appl. Math., vol. 10, Academic Press, New York, 1960. MR 22 #11074. MR 0120319 (22:11074)
  • [4] M. A. Krasnosel'skiĭ, Topological methods in the theory of nonlinear integral equations, GITTL, Moscow, 1956; English transl., Macmillan, New York, 1964. MR 20 #3464; MR 28 #2414. MR 0159197 (28:2414)
  • [5] -, Positive solutions of operator equations, Fizmatgiz, Moscow, 1962; English transl., Noordhoff, Groningen, 1964. MR 26 #2862; MR 31 #6107. MR 0181881 (31:6107)
  • [6] L. A. Ljusternik and S. L. Sobolev, Elements of functional analysis, GITTL, Moscow, 1951; English transl., Ungar, New York, 1961. MR 25 #5361. MR 0141967 (25:5362)
  • [7] M. M. Vainberg and V. A. Trenogin, The Lyapunov-Schmidt methods in the theory of non-linear equations and their subsequent development, Uspehi Mat. Nauk 17 (1962), no. 2 (104), 13-75 = Russian Math. Surveys 17 (1962), no. 2, 1-60. MR 27 #4071. MR 0154113 (27:4071)

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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 $ n$-linear form, simple eigenvalue, eigenvalue of odd multiplicity, implicit function theorem
Article copyright: © Copyright 1972 American Mathematical Society

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