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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.

MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.