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A lemma is obtained which guarantees that non-expansive mappings on contractive spaces have fixed points. An example shows that Schauder's fixed point theorem cannot be extended to contractive spaces, but a theorem for contractive spaces, analogous to a result of B. N. Sadovskii on convex spaces, is derived from the lemma. Finally, some local results for £-chainable contractive spaces are given.doi:10.1017/s1446788700021546 fatcat:ixl5cxqt4zgxtnj7lg5isvxytq