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

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 $\eta(d)<d$ ($h(\cdot, \cdot)$ 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.