Abstract
By relating the set of branch points \begin{document}$ \mathcal{B} (f) $\end{document} of a Fredholm mapping \begin{document}$ f $\end{document} to linearized bifurcation, we show, among other things, that under mild local assumptions at a single point, the set \begin{document}$ \mathcal B(f) $\end{document} is sufficiently large to separate the domain of the mapping. In the variational case, we will also provide estimates from below for the number of connected components of the complement of \begin{document}$ \mathcal B(f). $\end{document}
Highlights
Let X and Y be real Banach spaces and f : U → Y be a CkFredholm mapping of index 0 defined on an open subset U of X
A point x∗ ∈ U is called a branch point for f provided that there is no neighborhood of x∗ on which f is one-to-one, that is to say, there are sequences {xn} and {zn} in U that converge to x∗ and for all n, f = f while xn = zn
Applying multi-parameter bifurcation results from [2, 16] to families associated to restrictions of the map f to k-dimensional planes through the point x∗ with k > 1, appropriate modifications of i) and ii) of Theorem 1.7 provide more general sufficient conditions for branching
Summary
Let X and Y be real Banach spaces and f : U → Y be a CkFredholm mapping of index 0 defined on an open subset U of X. Let U be an open connected subset of X and the mapping f : U → Y be both C2-Fredholm and proper on closed bounded sets.
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