Fixed point and selection theorems in hyperconvex spaces

It is shown that a set-valued mapping $T^{\ast}$ of a hyperconvex metric space $M$ which takes values in the space of nonempty externally hyperconvex subsets of $M$ always has a lipschitzian single valued selection $T$ which satisfies $d(T(x),T(y))\leq d_{H}(T^{\ast}(x),T^{\ast}(y))$ for all $x,y\in M$. (Here $d_{H}$ denotes the usual Hausdorff distance.) This fact is used to show that the space of all bounded $\lambda$-lipschitzian self-mappings of $M$ is itself hyperconvex. Several related results are also obtained.