Set-valued contractions and fixed points

A common fixed point theorem is proved for a family of set-valued contraction mappings in gauge spaces. This result is related to a recent result of Frigon for 'generalized contractions' and it includes a method for approximating the fixed point. The remainder of the paper is devoted to results for families of set-valued contraction mappings in hyperconvex spaces. It is proved, for example, that if M is a hyperconvex metric space and f(alpha) is a family of set-valued contractions indexed over a directed set A and taking values in the space of all nonempty admissible subsets of M endowed with the Hausdorff metric, then the condition f(beta)(x) subset of or equal to f(alpha)(x) for all x is an element of M and beta greater than or equal to alpha implies that the set of points x is an element of M for which x is an element ofboolean AND(alphais an element ofLambda) f(beta)(x) is nonempty and hyperconvex. (C) 2003 Elsevier Science Ltd. All rights reserved.