Proximal pointwise contraction

In this paper, we introduce the notion of proximal pointwise contraction and obtain the existence of a best proximity point on a pair of weakly compact convex subset of a Banach space and generalize a result of [W.A. Kirk, Mappings of generalized contractive type, J. Math. Anal. Appl. 32 (1970) 567–570; W.A. Kirk, H.K. Xu, Asymptotic pointwise contractions, Nonlinear Anal. 69 (2008) 4706–4712].