Set-valued contractions and fixed points

Abstract

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 α is a family of set-valued contractions indexed over a directed set Λ and taking values in the space of all nonempty admissible subsets of M endowed with the Hausdorff metric, then the condition f β ( x)⊆ f α ( x) for all x∈ M and β⩾ α implies that the set of points x∈ M for which x∈⋂ α∈ Λ f β ( x) is nonempty and hyperconvex.