Fixed point and selection theorems in hyperconvex spaces

It is shown that a set-valued mapping T* 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)) less than or equal to d(H) (T*(x), T*(y)) for all x, y is an element of 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.