In the present paper, we study existence of best proximity pair in ultrametric spaces. We show, under suitable assumptions, that proximinal $$(A,B)$$ has a pair. As consequence generalize well known approximation result and derive some fixed point theorems. Moreover, provide examples to illustrate obtained results.

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.

Let X be a Banach space, C a closed subset of X , and T : C -+ C a nonexpansive mapping. Conditions are given which assure that if the fixed point set F ( T ) of T has nonempty interior then the Picard iterates of the mapping T always converge to a point of F ( T ) . If T is asymptotically regular, it suffices to assume that the closed subsets of X are densely proximinal and that nested spheres...

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.

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.