On the existence problem for a fixed point of a generalized contracting multivalued mapping

E. S. Zhukovskiy

Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 136, pp. 372-381

Résumé

We discuss the still unresolved question, posed in [S. Reich, Some Fixed Point Problems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 57:8 (1974), 194–198], of existence in a complete metric space $X$ of a fixed point for a generalized contracting multivalued map $\Phi: X \rightrightarrows X $ having closed values $ \Phi (x) \subset X$ for all $ x \in X. $ Generalized contraction is understood as a natural extension of the Browder–Krasnoselsky definition of this property to multivalued maps: \begin{equation*} \forall x, u \in X \ \ h \bigl(\varphi(x), \varphi(u) \bigr) \leq \eta \bigl(\rho(x, u) \bigr), \end{equation*} where the function $ \eta: \mathbb {R}_+\to\mathbb{R}_+$ is increasing, right continuous, and for all $d>0,$\linebreak $\eta(d)$ ($h(\cdot, \cdot)$ denotes the Hausdorff distance between sets in the space $X\!$). We give an outline of the statements obtained in the literature that solve the S. Reich problem with additional requirements on the generalized contraction $\Phi.$ In the simplest case, when the multivalued generalized contraction map $\Phi$ acts in $\mathbb{R},$ without any additional conditions, we prove the existence of a fixed point for this map.

Détail

Keywords: fixed point, generalized contraction, multivalued map in metric space, the Browder–Krasnoselsky fixed point theorem.