A Local Bifurcation Theorem for C1-Fredholm Maps

The Krasnosel'skii Bifurcation Theorem is generalized to C1-Fredholm maps. Let X and Y be Banach spaces, F: R × X → Y be C1-Fredholm of index 1 and$F(\lambda, 0) \equiv 0$. If$I \subseteq R$is a closed, bounded interval at whose endpoints ∂ F/∂ x ∂ F/∂ x (λ, 0) is invertible, and the parity of ∂ F/∂ x (λ, 0) on I is -1, then I contains a bifurcation point of the equation F(λ, x) = 0. At isolated potential bifurcation points, this sufficient condition for bifurcation is also necessary.