Une propriété topologique de l'ensemble des points fixes d'une contraction multivoque à valeurs convexes

Ricceri, Biagio

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 81 (1987) no. 3, pp. 283-286

Résumé

In this Note 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: $\alpha = \sup \, \{ \, \inf \,\{ \,\|u\|_{U} : u \in \Phi^{-1}(v) \} : v \in V, \, \|v\|_{V} \le 1 \}$. Then, for every $v \in V$ and every lipschitzian operator $\Psi : U \rightarrow V$, with Lipschitz constant $L$ such that $\alpha L 1$, the set $\{u \in U : \Phi (u) + \Psi (u) = v\}$ is non-empty and arc wise connected.