Azerbaijan Journal of Mathematics V. 5, No 1, 2015, January ISSN 2218-6816 Existence of Best Proximity Points in Regular Cone Met- ric Spaces L. Kumar * , T. Som Abstract. In this paper we have established some conditions which guarantee the existence of the distance between two subsets of a regular cone metric space. Under these conditions we have given a main result which guarantee the existence of best proximity points for cyclic contraction mappings in regular cone metric space, which extends the earlier result of Haghi et al(2011). Key Words and Phrases: Cone L-function, Cyclic contraction map, Lower bound, Regular cone metric space. 2010 Mathematics Subject Classifications: 47H10, 47H04, 41A65 1. Introduction Consider a self map T defined on the union of two subsets A, B of a metric space X .A mapping T : A ∪ B −→ A ∪ B is said to be cyclic provided that T (A) ⊆ B and T (B) ⊆ A. Let T be a cyclic map. If there exists a point x ∈ A ∪ B such that d(x,Tx)= d(A,B), then x is a best proximity point with regard to T , where dist(A,B) := inf {d(x,y):(x,y) ∈ A × B}. In 2003, Kirk et al. [9] proved the following extension of the Banach contraction principle for cyclic mappings. Theorem 1. [9] Let A,B be two non empty closed subsets of a complete metric space (X,d). Suppose that T is a cyclic mapping such that d(Tx,Ty) ≤ kd(x,y), for some k ∈ (0, 1) and for all (x,y) ∈ (A × B). Then T has a unique fixed point in A ∩ B. Further, in 2006, Eldered et al.[4] introduced the class of cyclic contractions and ob- tained best proximity point results for cyclic contraction mappings. Definition 1. [4] Let A,B be two non empty subsets of a metric space (X,d). A mapping T : A ∪ B −→ A ∪ B is said to be cyclic contraction if T is cyclic and d(Tx,Ty) ≤ kd(x,y) + (1 − k)dist(A,B), for some k ∈ (0, 1) and for all (x,y) ∈ A × B. * Corresponding author. http://www.azjm.org 44 c  2010 AZJM All rights reserved.