<p>A new family of cyclic mappings, known as pseudo cyclic contractions, is introduced. It is remarked that pseudo cyclic contractions are much weaker than the famous, classical cyclic contractions which in turn generalize self-contraction mappings. Further, we introduce best vicinity point theorems that furnish the sufficient conditions for the existence of a global optimal approximate solution, known as a best vicinity point, to fixed point equations. The main intent of this article is to establish a best vicinity point theorem for pseudo cyclic contractions that ascertains the existence of a best vicinity point for pseudo cyclic contractions.</p>