We consider a contraction map T of the Meir-Keeler type on the union of p subsets A 1 , . . . , A p , p ≥ 2 , of a metric space X, d to itself. We give sufficient conditions for the existence and convergence of a best proximity point for such a map.

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Let (X, d) be a metric space, and A1, A2,. .. , Ap be nonempty subsets of X. We introduce a self map T on X, called p-cyclic orbital contraction map on the union of A1, A2,. .. , Ap, and obtain a unique best proximity point of T , that is, a point x ∈ ∪ p i=1 Ai such that d(x, T x) = dist(Ai, Ai+1), 1 i p, where dist(Ai, Ai+1) = inf{d(x, y): x ∈ Ai, y ∈ Ai+1}.

In this manuscript, we introduce the concept of Ω -class of self mappings on a metric space and a notion of p-cyclic complete metric space for a natural number ( p ≥ 2 ) . We not only give sufficient conditions for the existence of best proximity points for the Ω -class self-mappings that are defined on p-cyclic complete metric space, but also discuss the convergence of best proximity points for those mappings.

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