Let X denote an arbitrary nonempty set, (M,d) a metric space, and S,T two mappings defined on X and taking values in M. \textit{K. Goebel} [Bull. Acad. Pol. Sci., Sér. Math. Astron. Phys. 16, 733-735 (1968; Zbl 0165.498)] has shown that if T(X) is a complete subspace of M with S(X)\(\subset T(X)\), then the condition d(SxSy)\(\leq kd(TxTy)\) for fixed \(k\in (0,1)\) and all x,y\(\in X\) is sufficient to ensure that S and T have a coincidence point z (i.e., \(Sz=Tz)\). The present paper contains extensions of this result to the case where S takes values among the closed bounded subsets of M. Variants of this extension are also discussed.