Abstract and Applied Analysis
Volume 2011 (2011), Article ID 596971, 9 pages
http://dx.doi.org/10.1155/2011/596971
Research Article

Approximate Best Proximity Pairs in Metric Space

S. A. M. Mohsenalhosseini,1 H. Mazaheri,2 and M. A. Dehghan1

1Faculty of Mathematics, Valiasr Rafsanjan University, Rafsanjan, Iran
2Faculty of Mathematics, Yazd University, Yazd, Iran

Received 8 January 2011; Accepted 12 February 2011

Academic Editor: Norimichi Hirano

Copyright © 2011 S. A. M. Mohsenalhosseini et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝐴 and 𝐵 be nonempty subsets of a metric space 𝑋 and also 𝑇 ∶ 𝐴 ∪ 𝐵 → 𝐴 ∪ 𝐵 and 𝑇 ( 𝐴 ) ⊆ 𝐵 , 𝑇 ( 𝐵 ) ⊆ 𝐴 . We are going to consider element 𝑥 ∈ 𝐴 such that 𝑑 ( 𝑥 , 𝑇 𝑥 ) ≤ 𝑑 ( 𝐴 , 𝐵 ) + 𝜖 for some 𝜖 > 0 . We call pair ( 𝐴 , 𝐵 ) an approximate best proximity pair. In this paper, definitions of approximate best proximity pair for a map and two maps, their diameters, 𝑇 -minimizing a sequence are given in a metric space.