In [J. Math. Anal. Appl. 323, No. 2, 1001–1006 (2006; Zbl 1105.54021)], A. A. Eldered and P. Veeramani introduced cyclic contraction maps and gave sufficient conditions for the existence and convergence of a unique best proximity for such map on a uniformly convex space. Further, this result was extended by C. Di Bari, T. Suzuki and C. Vetro [Nonlinear Anal., Theory Methods Appl. 69, No. 11 (A), 3790–3794 (2008; Zbl 1169.54021)].
In this paper, the authors give sufficient conditions for the existence and convergence of the best proximity point for contraction maps of the Meir–Keeler type for which is an extension of the results given in [Di Bari et al., loc. cit.]