In the present paper, by introducing a new notion named as nonunique cyclic contractions, we give some best proximity point results for such mappings. Then, we indicate the shortcoming of the concept of best periodic proximity point which is defined for cyclic mapping by giving a simple example. To overcome this deficiency, we give a more suitable definition named as best cyclic periodic point. Finally, we obtain some best cyclic periodic point theorems, including the famous periodic point result of Ćirić   [Ćiri, L. B. On some maps with a nonunique fixed point. Institut Mathèmatique 17 (1974), 52–58.], for nonunique cyclic contractions. We also provide some illustrative and comparative examples to support our results.

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 Aslantas, Mustafa, Altun, Ishak , Sahin, Hakan