Set-valued contractions and fixed points

✍ Scribed by R Espı́nola; W.A Kirk

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 120 KB
• Volume: 54
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(03)00107-x

✦ Synopsis

A common ÿxed 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 ÿxed 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 is a family of set-valued contractions indexed over a directed set and taking values in the space of all nonempty admissible subsets of M endowed with the Hausdor metric, then the condition f ÿ (x) ⊆ f (x) for all x ∈ M and ÿ ¿ implies that the set of points x ∈ M for which x ∈ ∈ f ÿ (x) is nonempty and hyperconvex.