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 Φ: X⇉X having closed values Φ(x)⊂X for all x∈X. Generalized contraction is understood as a natural extension of the Browder–Krasnoselsky definition of this property to multivalued maps: ∀x,u∈X h(φ(x),φ(u))≤ η(ρ(x,u)), where the function η:R_+→R_+ is increasing, right continuous, and for all d>0, η(d)<d (h(•,•) 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 Φ. In the simplest case, when the multivalued generalized contraction map Φ acts in R, without any additional conditions, we prove the existence of a fixed point for this map.