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Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one. (English) Zbl 1027.58009
The paper is devoted to the study of the spectral bifurcation of non-trivial solutions to the equation $x=\lambda Lx+H(x,\lambda)$, where: $L$ is a linear and compact operator on the Banach space $E$; $H:E\times {\Bbb R}\rightarrow E$ is completely continuous; $H(0,\lambda)=0$ on ${\Bbb R}$, and $\|x\|^{-1}H(x,\lambda)\rightarrow 0$ as $\|x\|\rightarrow 0$ locally uniformly in $\lambda$. One of the purposes of this paper is to construct a counterexample to show that a conjectured global solution structure for bifurcation of non-trivial solutions from a simple eigenvalue of the linearization at zero really can occur. The paper also contains additional results and counterexamples concerning the bifurcation from an eigenvalue of geometric multiplicity 1 and odd algebraic multiplicity.

58E07Abstract bifurcation theory
35B32Bifurcation (PDE)
47J15Abstract bifurcation theory
37G10Bifurcations of singular points
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