In this paper, we discuss the existence of best proximity points of new generalized proximal contractions of metric spaces. Moreover, we obtain a completeness characterization of underlying metric space via the best proximity points. Some new best proximity point theorems have been derived as consequences of main results in (partially ordered) metric spaces.

This paper deals with the existence of non-empty fixed point sets of newly introduced generalized set-valued $F$-contractions of $b$-metric spaces. Some illustrative examples show that the new results in this paper generalize properly, unify and extend some related results in the existing literature. Moreover, we extract some important consequences of the results in $b$-metric spaces. Particularly, by setting $b$-metric constant equal to one, we obtain some specific cases showing notable enhancement of existing results yet in metric spaces.