A NOTE ON BEST PROXIMITY POINTS FOR $P$--SUMMING CYCLIC ORBITAL MEIR-KEELER CONTRACTIONS

Abstract

The classical Banach contraction theorem has a lot of applications [1]. One of the interesting generalizations of Banach contraction is the well known Meir Keeler contraction theorem [12]. In [3], the following notions are introduced.If A and B are non empty subsets of a metric space (X, d), and if T : A∪B → A∪B is such that T (A) ⊆ T (B) and T (B) ⊆ T (A), then T is called a cyclic map. A point x ∈ A∪B is called a best proximity point if d(x, Tx) = dist(A,B), where dist(A,B) = inf{d(x, y) : x ∈ A and y ∈ B}. In this paper a best proximity point is obtained for a map called cyclic contraction. It is further generalized in [2] by introducing a map called cyclic Meir Keeler contraction.