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.

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.

Let, and be nonempty subsets of a matric space. Then the mapping is called cyclic if, and. Consider the following optimization problem Let, certainly if the condition be true for some then it is BEST answer for optimization problem, that we called it triple BEST PROXIMITY POINT of. In this paper, first we introduce the notion of 3-cyclic summing Meir-Keeler contractions as a generalization of 3-cyclic summing contractions, then we obtain the conditions for the existence of a triple BEST PROXIMITY POINT for these class of mappings in the metric spaces with property UC. Our results in this paper are true for a n-cyclic summing Meir-Keeler contraction just we work with order 3 for the simplicity of proofs. Note that, our results are generalizations of some existing theorems with shorter and simpler proofs. Note that, our results are generalizations of some existing theorems with shorter and simpler proofs.

This paper is concerned with the BEST PROXIMITY pair problem in Hilbert spaces. Given two subsets A and B of a Hilbert space H and the set-valued maps F: A® 2B and G: A0®2A0, where A0= {xÎ A: ||x - y||=d (A, B) for some yÎ B}, BEST PROXIMITY pair theorems provide sufficient conditions that ensure the existence of anx0 2 A such thatd(G (x0), F (x0)) =d (A, B).

In this paper, we aim to introduce Ciric type G-contractions using directed graphs in metric spaces and then to investigate the existence and uniqueness of BEST PROXIMITY POINTs for them. We also discuss the main theorem and list some consequences of it.

In this paper, we introduce a new type of graph contraction using a special class of functions and give a BEST PROXIMITY POINT theorem for this contraction in complete metric spaces endowed with a graph under two different conditions. We then support our main theorem by a non-trivial example and give some consequences of BEST PROXIMITY POINT of it for usual graphs.

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.