A local bifurcation theorem for 𝐶¹-Fredholm maps

Fitzpatrick, P. M.; Pejsachowicz, Jacobo

doi:10.1090/S0002-9939-1990-1009988-5

A local bifurcation theorem for $C^ 1$-Fredholm maps
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Proc. Amer. Math. Soc. 109 (1990), 995-1002 Request permission

Abstract:

The Krasnosel’skii Bifurcation Theorem is generalized to ${C^1}$-Fredholm maps. Let $X$ and $Y$ be Banach spaces, $F:R \times X \to Y$ be ${C^1}$-Fredholm of index 1 and $F(\lambda ,0) \equiv 0$ . If $I \subseteq R$ is a closed, bounded interval at whose endpoints $\frac {{\partial F}}{{\partial x}}\frac {{\partial F}}{{\partial x}}(\lambda ,0)$ is invertible, and the parity of $\frac {{\partial F}}{{\partial x}}(\lambda ,0)$ on $I$ is -1, then $I$ contains a bifurcation point of the equation $F(\lambda ,x) = 0$. At isolated potential bifurcation points, this sufficient condition for bifurcation is also necessary.

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