On some multi-valued mappings without contractivity conditions and common fixed points (Q1283619)

For \((X,d)\) a metric space, let \(\text{Cl}(X)\) be the collection of all nonempty closed subsets of \(X\) and \(H\) be the (generalized) Hausdorff metric generated by \(d\). Let \(\Phi\) be the family of those selfmaps \(\phi\) of \([0,\infty)\) which satisfy: \(\phi(t)< t\) \((t>0)\) and \(\limsup_{r\to t+}\phi(r)< t\) \((t>0)\). Let \(D(x,A)= \inf\{d(x, a):a\in A\}\) for \(x\in X\) and \(A\subset X\). Main result: If \(F\) and \(G\) are maps from a complete \((X,d)\) into \((\text{Cl}(X),H)\) and there is a \(\phi\in\Phi\) such that \[ H(Fx, Gy)\leq \phi(\max(d(x, y), D(x,Fx), D(y, Gy), \textstyle{{1\over 2}}(D(x, Gy)+ D(y, Fx))) \] for all \(x,y\in X\), then \(F\) and \(G\) have a common fixed point, i.e., there exists an \(x\in X\) with \(x\in Fx\cap Gx\). As always, a number of known results have been obtained as corollaries. Reviewer's remarks: (1) the author of Ref. [17], is I. Kubiaczyk (and not the reviewer); (2) the most notable misprint is in the title of the paper: for ``contractivity'' one should have ``continuity''; (3) the reviewer whishes to thank the author for correcting a statement the author made in [\textit{T. Kubiak}, Math. Japon. 30, 89--101 (1985; Zbl 0567.54030)].