Abstract
We provide a positive answer to a question raised by Al-Thagafi and Shahzad [Nonlinear Anal. 70 (2009), 3665-3671] about the existence of a best proximity point for a cyclic phi-contraction map in a reflexive Banach space by using appreciable generalized notions. In turn, this also gives a positive answer to a question raised by Eldred and Veeramani [J. Math. Anal. Appl. 323 (2006), 1001-1006]. To this end, we introduce a new concept of C-proximity point and a new class of maps, called cyclic C-phi-contractions, which contains cyclic contraction maps and cyclic phi-contraction maps as subclasses. Convergence and existence results of best C-proximity points for cyclic C-phi-contraction maps are also obtained.