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

Ricceri, Biagio

Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 81, No. 3, 283-286 (1987).

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.

Keywords:

absolute extensor; structure of the set of fixed points of a multi-valued contraction with convex values; lipschitzian operator; Lipschitz constant