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Une propriété topologique de l’ensemble des points fixes d’une contraction multivoque à valeurs convexes. (A topological property of the set of fixed points of a multivalued contraction with convex values). (French) Zbl 0666.47030
We first establish a result on the structure of the set of fixed points of a multi-valued contraction with convex values. As a consequence of this result, we then obtain the following theorem:
Let \((U,\| \cdot \|_ U)\), \((V,\| \cdot \|_ V)\) be two real Banach spaces and let \(\Phi\) be a continuous linear operator from U onto V. Put: \(\alpha =\sup \{\inf \{\| u\|_ U:\) \(u\in \Phi^{- 1}(v)\}:\) \(v\in V\), \(\| v\|_ V\leq 1\}\). Then, for every \(v\in V\) and every lipschitzian operator \(\Psi\) : \(U\to V\), with Lipschitz constant L such that \(\alpha L<1\), the set \(\{\) \(u\in U:\) \(\Phi (u)+\Psi (u)=v\}\) is non-empty and arcwise connected.

47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.