`Abstract and Applied AnalysisVolumeΒ 2011, Article IDΒ 596971, 9 pageshttp://dx.doi.org/10.1155/2011/596971`
Research Article

## Approximate Best Proximity Pairs in Metric Space

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

Received 8 January 2011; Accepted 12 February 2011

Copyright Β© 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 and be nonempty subsets of a metric space and also and , . We are going to consider element such that for some . We call pair an approximate best proximity pair. In this paper, definitions of approximate best proximity pair for a map and two maps, their diameters, -minimizing a sequence are given in a metric space.

#### 1. Introduction

Let be a metric space and and nonempty subsets of , and is distance of and . If , then the pair is called a best proximity pair for and and put as the set of all best proximity pair . Best proximity pair evolves as a generalization of the concept of best approximation. That reader can find some important result of it in [1β4].

Now, as in [5] (see also [4, 6β11]), we can find the best proximity points of the sets and , by considering a map such that and . Best proximity pair also evolves as a generalization of the concept of fixed point of mappings. Because if , every best proximity point is a fixed point of .

We say that the point is an approximate best proximity point of the pair , if , for some .

In the following, we introduce a concept of approximate proximity pair that is stronger than proximity pair.

Definition 1.1. Let and be nonempty subsets of a metric space and a map such that , . put We say that the pair is an approximate best proximity pair if .

Example 1.2. Suppose that , , and with for . Then for some . Hence .

#### 2. Approximate Best Proximity

In this section, we will consider the existence of approximate best proximity points for the map , such that , , and its diameter.

Theorem 2.1. Let and be nonempty subsets of a metric space . Suppose that the mapping is satisfying , , and Then the pair is an approximate best proximity pair.

Proof. Let be given and such that ; then there exists such that If , then , and and .

Theorem 2.2. Let and be nonempty subsets of a metric space . Suppose that the mapping is satisfying , and for all , where and . Then the pair is an approximate best proximity pair.

Proof. If , then Therefore, Now if , then also Therefore, and so Therefore, by Theorem 2.1, ; then pair is an approximate best proximity pair.

Definition 2.3. Let and be nonempty subsets of a metric space . Suppose that the mapping is satisfying , . We say that the sequence is T-minimizing if

Theorem 2.4. Let and be nonempty subsets of a metric space , suppose that the mapping is satisfying , . If is a T-minimizing for some , then is an approximate best pair proximity.

Proof. Since therefore, by Theorem 2.1, ; then pair is an approximate best proximity pair.

Theorem 2.5. Let and be nonempty subsets of a normed space such that is compact. Suppose that the mapping is satisfying , , is continuous and where . Then is nonempty and compact.

Proof. Since compact, there exists a such that If , then which contradict to the definition of , ( and by (*) ). Therefore, for some and . Therefore, is nonempty.
Also, if , then , for some , and by compactness of , there exists a subsequence and a such that and so for some , hence is compact.

Example 2.6. If , and such that then is compact, and we have That is compact.

In the following, by for a set , we will understand the diameter of the set .

Definition 2.7. Let be a continuous map such that , and . We define diameter by

Theorem 2.8. Let , such that , and . If there exists an such that for all then

Proof. If , then Put , therefore, . Hence .

#### 3. Approximate Best Proximity for Two Maps

In this section, we will consider the existence of approximate best proximity points for two maps and , and its diameter.

Definition 3.1. Let and be nonempty subsets of a metric space and let two maps such that , . A point in is said to be an approximate-pair fixed point for in if there exists We say that the pair has the approximate-pair fixed property in if , where

Theorem 3.2. Let and be nonempty subsets of a metric space and let and be two maps such that , . If, for every , then has the approximate-pair fixed property.

Proof. For , Suppose . Since then for every . Put and . Hence and .

Theorem 3.3. Let and be nonempty subsets of a metric space and let and be two maps such that , and, for every , where and . Then if is an approximate fixed point for , or is an approximate fixed point for , then .

Proof. If , then Therefore, Now if , then also If is an approximate fixed point for , then there exists a and by (*) And ; also if is an approximate fixed point for , then there exists a and by (**) And . Therefore, .

Theorem 3.4. Let and be nonempty subsets of a metric space and let and be two continuous maps such that , . If, for every , where and , also let and be as follows: If has a convergent subsequence in , then there exists a such that .

Proof. We have If converges to , that is, , then Since is continuous, then Therefore, .

Definition 3.5. Let and be continues maps such that and . We define diameter by

Example 3.6. Suppose , , , and . Then and .

Theorem 3.7. Let and be continuous maps such that , . If there exists ββ, then

Proof. If , then Therefore, . Then .

#### References

1. K. Fan, βExtensions of two fixed point theorems of F. E. Browder,β Mathematische Zeitschrift, vol. 112, pp. 234β240, 1969.
2. W. K. Kim and K. H. Lee, βCorrigendum to "Existence of best proximity pairs and equilibrium pairs" [J. Math. Anal. Appl. 316 (2006) 433-446] (DOI:10.1016/j.jmaa.2005.04.053),β Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 1482β1483, 2007.
3. W. A. Kirk, S. Reich, and P. Veeramani, βProximinal retracts and best proximity pair theorems,β Numerical Functional Analysis and Optimization, vol. 24, no. 7-8, pp. 851β862, 2003.
4. V. Vetrivel, P. Veeramani, and P. Bhattacharyya, βSome extensions of Fan's best approximation theorem,β Numerical Functional Analysis and Optimization, vol. 13, no. 3-4, pp. 397β402, 1992.
5. I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Die Grundlehren der mathematischen Wissenschaften, Band 171, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest, Romania, 1970.
6. A. A. Eldred and P. Veeramani, βExistence and convergence of best proximity points,β Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1001β1006, 2006.
7. G. Beer and D. Pai, βProximal maps, prox maps and coincidence points,β Numerical Functional Analysis and Optimization, vol. 11, no. 5-6, pp. 429β448, 1990.
8. K. Włodarczyk, R. Plebaniak, and A. Banach, βBest proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces,β Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3332β3341, 2009.
9. K. Włodarczyk, R. Plebaniak, and A. Banach, βErratum to: "Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces" [Nonlinear Anal. (2008), doi: 10.1016/j.na.2008.04.037],β Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3585β3586, 2009.
10. K. Włodarczyk, R. Plebaniak, and C. Obczyński, βConvergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces,β Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 794β805, 2010.
11. X. B. Xu, βA result on best proximity pair of two sets,β Journal of Approximation Theory, vol. 54, no. 3, pp. 322β325, 1988.