Abstarct: In 1992, the Banach proved that in each mapping the contraction in complete metric spaces has a unique point.In this thesis, we generalize the idea of mapping C -weak non-self mappings and study the best approximation theorems for this class. These results are the generalizations of the work done by Harjani and colleagues. Our goal is to provide a constant point checking field for weak contractions mappings .Which is in perfect metric spaces that has a small sequence. We also introduce the concept of pairwise fixed points in metric spaces. In addition, we continue to examine the proximity of generalized C -contractible maps