From the authors’ abstract: “We prove several fixed point theorems for multivalued and pairs of multivalued maps from an orbitally complete metric space X to B(X), the collection of nonempty bounded subsets of X.” Theorem 1 is typical for the results on single mappings: Suppose that \(F:X\to B(X)\) is continuous and that FA\(\in B(X)\) for every \(A\in B(X)\). Let X be F-orbitally complete and, for some \(p\in {\mathbb{N}}\), \(\delta (F^ px,F^ py)\leq Q(M(x,y))\) for all x,y\(\in X\), where \(M(x,y)=\max \{\delta (F^ rx,F^ sx),\quad \delta (F^ rx,F^ sy),\quad \delta (F^ ry,F^ sy):\;0\leq r,\quad s\leq p\text{ with } F^ 0x=\{x\}\},\) Q nondecreasing real-valued such that \(0<Q(s)<s\) for \(s>0\) and s/(s-Q(s)) nonincreasing, \(Q(0)=0\). Then F has a unique fixed point x and \(Fx=\{x\}\).