  In this paper, we introduce the notions of proximally completeness,
proximally closedness and proximally continuity and utilize the same to prove a
result on existence and uniqueness of best proximity points in the setting of
metric space (not necessarily complete). Our newly proved result enriches,
sharpens, improves and modifies the proximal contraction principle of Basha [J.
Optim. Theory Appl. 2011:151 (2011), 210-216]. In order to illustrate the
effectiveness of our finding, we discuss the sufficient conditions ensuring the
existence of a unique solution of certain variational inequality problem.
