We assume that X is a normed linear space, W and M are subspaces of X. We develop a theory of best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W + M and the quotient space W /M.

In this paper, we extend the concept of best proximity point to 2-metric spaces and prove the existence of such points for contraction type non-self mappings in the setting of complete 2-metric spaces. Also, we presented an example to support our results.

Abstract. In the present research, an interesting common best proximity point theorem for pairs of non-self-mappings is presented. It satisfies a weakly contraction-like condition, thereby producing common optimal approximate solutions of certain simultaneous fixed point equations.

The main purpose of this article is to introduce particular subsets of R I , which are not necessarily convex, and we call them I m -quasi upward, or I m -quasi downward. We show that these sets can be translated to downward or upward sets. We introduce the connection of these sets with downward and upward subsets of R I , and discuss the best approximation of these sets. Also we introduce embedded I m -quasi upward and embedded I m -quasi downward subsets of a normed space X .Keywords Banach Lattice Space; Best approximation; Downward set; I m -quasi downward hull; I m -quasi downward set; I m -quasi upward hull; I m -quasi upward set; Proximinal set; Upward set.

In this paper, we find a way to give best simultaneous approximation of n arbitrary points in convex sets. First , we introduce a special hyperplane which is based on those n points. Then by using this hyperplane, we define best approximation of each point and achieve our purpose .