Geodesic
Fixed point theorems for $n$-periodic mappings in Banach spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 1, pp. 33-42.

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Using modified Halpern iterations, by elementary method, we extend and improve results obtained by W.A. Kirk (Proc. Amer. Math. Soc. {\bf 29} (1971), 294) and others, which have recently been presented in Chapter 11 of {\it Handbook of Metric Fixed Point Theory\/} (2001).
Classification : 47H09, 47H10
Keywords: lipchitzian mapping; uniformly lipschitzian mapping; $n$-periodic mapping; fixed point 
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Górnicki, Jarosław; Pupka, Krzysztof. Fixed point theorems for $n$-periodic mappings in Banach spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 1, pp. 33-42. http://geodesic.mathdoc.fr/item/CMUC_2005__46_1_a3/