in this paper, we prove some approximate fixed point results for proximinal valued $beta$-contractive multifunctions on metric spaces. we show that our results generalize some old fixed point results in the literature.

The main purpose of this paper is to find t-best approximations in fuzzy normed spaces. We introduce the notions of t-proximinal sets and F-approximations and prove some interesting theorems. In particular, we investigate the set of all t-best approximations to an element from a set.

In this paper, we consider the concepts farthest points and nearest points in normed linear spaces, We obtain a necessary and coecient conditions for proximinal, Chebyshev, remotal and uniquely remotal subsets in normed linear spaces. Also, we consider -remotality, -proximinality, coproximinality and co-remotality.

the main purpose of this paper is to find t-best approximations in fuzzy normed spaces. we introduce the notions of t-proximinal sets and f-approximations and prove some interesting theorems. in particular, we investigate the set of all t-best approximations to an element from a set.

For arbitrary semimetric space $$(X, d)$$ and disjoint proximinal subsets $$A$$ , $$B$$ of $$X$$ we define the graph as a bipartite with parts whose edges $$\{a, b\}$$ satisfy equality $$d(a, b) = {\text {dist}}(A, B)$$ . We characterize spaces graphs have at most one edge vertices degree $$1$$ only. This allows us to describe necessary sufficient conditions for uniqueness best proximity pairs ...