Authors’ summary: We provide a positive answer to a question raised by A. A. Eldred
and P. Veeramani
[J. Math. Anal. Appl. 323, No. 2, 1001–1006 (2006; Zbl 1105.54021
)] about the existence of a best proximity point for a cyclic contraction map in a reflexive Banach space. Moreover, we introduce a new class of maps, called cyclic
-contractions, which contains the cyclic contraction maps as a subclass. Convergence and existence results of best proximity points for cyclic
-contraction maps are also obtained.