In this paper we introduce geometric contraction map and give a new condition for the existence and uniqueness of BEST PROXIMITY point of geometric contractions. We also consider the convergence of iterates to PROXIMITY points in metric spaces.

This study has considered the problem of finding BEST PROXIMITY pair in fuzzy metric spaces and uniformly convex fuzzy Banach spaces for fuzzy cyclic contraction map. We prove the uniqueness of this point in uniformly fuzzy Banach spaces. We also give an algorithm to find a BEST PROXIMITY point for the map S in setting of a uniformly convex fuzzy Banach spaces.

BEST PROXIMITY point theorems for self-mappings were investigated with different conditions on spaces for contraction mappings. In this paper, we prove BEST PROXIMITY point theorems for proximal F∗,-weak contraction mappings.

In this paper, we establish and prove the existence of BEST PROXIMITY points for multivalued cyclic FF-contraction mappings in complete metric spaces. Our results improve and extend various results in literature.

In this article, we introduce a new notion of proximal contraction, named as generalized S-proximal contraction and derive a common BEST PROXIMITY point theorem for proximally commuting non-self mappings, thereby yielding the common optimal approximate solution of some  xed point equations when there is no common solution. We furnish illustrative examples to highlight our results. We extend some results existing in the literature.

In this article we survey the existence of BEST PROXIMITY points for a class of non-self mappings which satisfy a particular nonexpansiveness condition. In this way, we improve and extend a main result of Abkar and Gabeleh [A. Abkar, M. Gabeleh, BEST PROXIMITY points of non-self mappings, Top, 21, (2013), 287-295] which guarantees the existence of BEST PROXIMITY points for nonexpansive non-self mappings in the setting of uniformly convex Banach spaces.We also introduce a new notion, called relatively asymptotic center, on a nonempty, bounded, closed and convex pair of subsets of a Hadamard metric space and as a result of our main conclusions, we will show that the asymptotic center of any sequence in a nonempty, bounded, closed and convex subset of a Hadamard space is singleton. Moreover, we obtain the other existence results of BEST PROXIMITY points for generalized nonexpansive mappings using the appropriate geometric properties of Hadamard spaces. Finally, we provide some examples to illustrate our main results.

A new geometric notion on a nonempty and convex pair of subsets of a convex metric space X, called semi-normal structure, is introduced and used to investigate the existence of BEST PROXIMITY pairs for a new class of mappings, called strongly noncyclic relatively C-nonexpansive. We also study the structure of minimal sets of strongly noncyclic relatively C-nonexpansive mappings in the setting of convex metric spaces.

We generalise the notion of cyclic-noncyclic pairs by introducing the concept of $m$-contraction pairs in convex metric spaces. We shall then investigate the exixtence and convergence of coincidence-BEST PROXIMITY points, and finally, we will provide examples to illustrate the new findings.