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In this paper we study the problem of existence of fixed points of k-Lipschitzian and uniformly fc-Lipschitzian mappings (k > 1) defined on nonempty closed convex subset of Banach space. Using very simple method we extend Kirk and Linhart's result [5, 8] in the case of involution of order n = 3. 1991 Mathematics Subject Classification: 47H09, 47H10.doi:10.1515/dema-2005-0217 fatcat:quydmsbtmbaqpinw6cx5lmcixi