Abstract. In this paper, we show that the xed points set of self-mappings satisfying condition (C) on a nonempty closed convex subset of a convex metric space having property (D) is always closed and convex. Moreover, we prove that the xed points set of such mappings on a nonempty bounded closed convex subset of a uniformly convex complete metric space is always nonempty, closed and convex. Our results improve and extend some results in the literature.
