On the error estimates for the sequence of successive approximations for cyclic φ--contractions in metric spaces.

In this paper, in the setting of metric spaces, we introduce the notion of strongly cyclic Sehgal type φ-contraction of type one as generalization of the notions of cyclic φ-contraction map in the sense of Păcurar-Rus and cyclic contraction map in the sense of Suzuki-Kikawa-Vetro. Then we study the existence and uniqueness of the best proximity points for such mappings by using the WUC property. In the following, while presenting an algorithm to determine the best proximity points, we also find a priori and a posteriori error estimates of the best proximity point for this algorithm associated with a strongly cyclic Sehgal type φ-contraction of type one, which is defined on a uniformly convex Banach space with a modulus of convexity of power type. Also, we give a positive answer to Zlatanov's question ['Error estimates for approximating best proximity points for cyclic contractive maps', Carpathian J. Math. 32(2) (2016), 265-270] on error estimates for the sequence of successive approximations for cyclic φ-contraction maps in the sense of Păcurar-Rus. As an important result, we obtain a generalization of Ćirić's Theorem, which itself is a generalization of the Banach contraction principle in a particular case. [ABSTRACT FROM AUTHOR]