A note on a continuation principle for compact perturbations of the identity (Q1198704)

This article deals with the equation \(F(x,\lambda)=0\) where \(F(x,\lambda)=x-k(x,\lambda)\), \((x,\lambda)\in {\mathcal O}\subseteq X\times\mathbb{R}^ n\), \(X\) is a real Banach space, \(k(x,\lambda)\) is continuous on \({\mathcal O}\) and compact on the open sets \({\mathcal O}(\varepsilon)\), \(\bigcup_ \varepsilon {\mathcal O}(\varepsilon)={\mathcal O}\), \(k(0,0)=0\), \(k\) is differentiable at \((0,0)\), and \(\ker DF(0,0)\) has dimension \(n\). It is shown that the connected component of the set of zeros of \(F\) containing \((0,0)\) is either unbounded, or approaches \(\partial{\mathcal O}\) in a well-defined sense, or intersects all the subspaces \(Y\) of codimension \(n\) in \(X\times\mathbb{R}^ n\) such that \(\ker DF(0,0)\cap Y=\{(0,0)\}\) at a point distinct from \((0,0)\).