# Best proximity point theorems for rational proximal contractions

- Hemant Kumar Nashine
^{1}, - Poom Kumam
^{2}Email author and - Calogero Vetro
^{3}

**2013**:95

https://doi.org/10.1186/1687-1812-2013-95

© Nashine et al.; licensee Springer 2013

**Received: **12 January 2013

**Accepted: **26 March 2013

**Published: **12 April 2013

## Abstract

We provide sufficient conditions which warrant the existence and uniqueness of the best proximity point for two new types of contractions in the setting of metric spaces. The presented results extend, generalize and improve some known results from best proximity point theory and fixed-point theory. We also give some examples to illustrate and validate our definitions and results.

**MSC:**41A65, 46B20, 47H10.

## Keywords

## 1 Introduction

Let $(\mathcal{X},d)$ be a metric space and $\mathcal{T}$ be a self-mapping defined on a subset of $\mathcal{X}$. In this setting, the fixed-point theory is an important tool for solving equations of the kind $\mathcal{T}x=x$, whose solutions are the fixed points of the mapping $\mathcal{T}$. On the other hand, if $\mathcal{T}$ is not a self-mapping, say $\mathcal{T}:\mathcal{A}\to \mathcal{B}$ where $\mathcal{A}$ and ℬ are nonempty subsets of $\mathcal{X}$, then $\mathcal{T}$ does not necessarily have a fixed point. Consequently, the equation $\mathcal{T}x=x$ could have no solutions, and in this case, it is of a certain interest to determine an element *x* that is in some sense closest to $\mathcal{T}x$. Thus, we can say that the aim of the best proximity point theorems is to provide sufficient conditions to solve a minimization problem. In view of the fact that $d(x,\mathcal{T}x)$ is at least $d(\mathcal{A},\mathcal{B}):=inf\{d(x,y):x\in \mathcal{A}\text{and}y\in \mathcal{B}\}$, a best proximity point theorem concerns the global minimum of the real valued function $x\to d(x,\mathcal{T}x)$, that is, an indicator of the error involved for an approximate solution of the equation $\mathcal{T}x=x$, by complying the condition $d(x,\mathcal{T}x)=d(\mathcal{A},\mathcal{B})$. The notation of best proximity point is introduced in [1] but one of the most interesting results in this direction is due to Fan [2] and can be stated as follows.

**Theorem 1.1** *Let* $\mathcal{K}$ *be a nonempty*, *compact and convex subset of a normed space* ℰ. *Then for any continuous mapping* $\mathcal{T}:\mathcal{K}\to \mathcal{E}$, *there exists* $x\in \mathcal{K}$ *with* $\parallel x-\mathcal{T}x\parallel ={inf}_{y\in \mathcal{K}}\parallel \mathcal{T}x-y\parallel $.

Some generalizations and extensions of this theorem appeared in the literature by Prolla [3], Reich [4], Sehgal and Singh [5, 6], Vetrivel *et al.* [7] and others. It turns out that many of the contractive conditions which are investigated for fixed points ensure the existence of best proximity points. Some results of this kind are obtained in [1, 5–40]. Note that the authors often, in proving these results, assume restrictive compactness hypotheses on the domain and codomain of the involved nonself-mapping. Inspired by [29], we consider these hypotheses too restrictive in dealing with proximal contractions and so we prove that the compactness hypotheses can be successfully replaced by standard completeness hypotheses. Following this idea, we propose a new type of condition to study the existence and uniqueness of the best proximity point of a nonself-mapping by assuming both compactness hypotheses and standard completeness hypotheses. Precisely, we introduce the notions of rational proximal contractions of the first and second kinds, then we establish some corresponding best proximity point theorems for such contractions. Our definitions include some earlier definitions as special cases. In particular, the presented theorems contain the results given in [29].

## 2 Preliminaries

In this section, we give some basic notations and definitions that will be used in the sequel.

Sufficient conditions to ensure that ${\mathcal{A}}_{0}$ and ${\mathcal{B}}_{0}$ are nonempty are given in [41]. Also, observe that if $\mathcal{A}$ and ℬ are closed subsets of a normed linear space such that $d(\mathcal{A},\mathcal{B})>0$, then ${\mathcal{A}}_{0}$ and ${\mathcal{B}}_{0}$ are contained in the boundaries of $\mathcal{A}$ and ℬ, respectively; see [27].

Now, we give sequentially two definitions that are essential to state and prove our main results.

**Definition 2.1**Let $(\mathcal{X},d)$ be a metric space and $\mathcal{A}$ and ℬ be two nonempty subsets of $\mathcal{X}$. Then $\mathcal{T}:\mathcal{A}\to \mathcal{B}$ is said to be a rational proximal contraction of the first kind if there exist nonnegative real numbers

*α*,

*β*,

*γ*,

*δ*with $\alpha +\beta +2\gamma +2\delta <1$, such that the conditions

for all ${u}_{1},{u}_{2},{x}_{1},{x}_{2}\in \mathcal{A}$.

Note that, if $\beta =0$, then from (1) we get the definition of the generalized proximal contraction of the first kind with $\alpha +2\gamma +2\delta <1$; see [29].

**Definition 2.2**Let $(\mathcal{X},d)$ be a metric space and $\mathcal{A}$ and ℬ be two nonempty subsets of $\mathcal{X}$. Then $\mathcal{T}:\mathcal{A}\to \mathcal{B}$ is said to be a rational proximal contraction of the second kind if there exist nonnegative real numbers

*α*,

*β*,

*γ*,

*δ*with $\alpha +\beta +2\gamma +2\delta <1$ such that the conditions

for all ${u}_{1},{u}_{2},{x}_{1},{x}_{2}\in \mathcal{A}$.

Note that, if $\beta =0$, then from (2) we get the definition of the generalized proximal contraction of the second kind with $\alpha +2\gamma +2\delta <1$, see [29].

The following example illustrates that a rational proximal contraction of the second kind is not necessarily a rational proximal contraction of the first kind. Therefore, both Definitions 2.1 and 2.2 are consistent.

**Example 2.1**Let $\mathcal{X}=\mathbb{R}\times \mathbb{R}$ endowed with the usual metric

Then $d(\mathcal{A},\mathcal{B})=2$ and $\mathcal{T}$ is a rational proximal contraction of the second kind but not a rational proximal contraction of the first kind. Indeed, using Definition 2.2 and after routine calculations, one can show that the left-hand side of inequality (2) is equal to 0. On the other hand, using Definition 2.1 and after routine calculations, one can show that the left-hand side of inequality (1) is equal to 2 and so inequality (1) is not satisfied for all nonnegative real numbers *α*, *β*, *γ*, *δ* with $\alpha +\beta +2\gamma +2\delta <1$.

It is well known that the notion of approximative compactness plays an important role in the theory of approximation [12]. In particular, the notion of an approximatively compact set was introduced by Efimov and Stechkin [16] and the properties of approximatively compact sets have been largely studied. The boundendly compact sets that are the sets whose intersection with any closed ball is compact are useful examples of approximatively compact sets. It is shown in [14] that in every infinite-dimensional separable Banach space there exists a bounded approximatively compact set, which is not compact.

**Remark 2.1** Since $(\mathcal{X},d)$ is a metric space, the bounded compactness of a set is equivalent to its closure and the possibility of selecting from any bounded sequence contained in it a converging subsequence.

Here, for our further use, we give the following definition.

**Definition 2.3** Let $(\mathcal{X},d)$ be a metric space and $\mathcal{A}$ and ℬ be two nonempty subsets of $\mathcal{X}$. Then ℬ is said to be approximatively compact with respect to $\mathcal{A}$ if every sequence $\{{y}_{n}\}$ of ℬ, satisfying the condition $d(x,{y}_{n})\to d(x,\mathcal{B})$ for some *x* in $\mathcal{A}$, has a convergent subsequence.

Obviously, any set is approximatively compact with respect to itself.

## 3 Rational proximal contractions

Our first main result is the following best proximity point theorem for a rational proximal contraction of the first kind.

**Theorem 3.1**

*Let*$(\mathcal{X},d)$

*be a complete metric space and*$\mathcal{A}$

*and*ℬ

*be two nonempty*,

*closed subsets of*$\mathcal{X}$

*such that*ℬ

*is approximatively compact with respect to*$\mathcal{A}$.

*Assume that*${\mathcal{A}}_{0}$

*and*${\mathcal{B}}_{0}$

*are nonempty and*$\mathcal{T}:\mathcal{A}\to \mathcal{B}$

*is a nonself*-

*mapping such that*:

- (a)
$\mathcal{T}$

*is a rational proximal contraction of the first kind*; - (b)
$\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}$.

*Then there exists* $x\in \mathcal{A}$ *such that* ${B}_{\mathrm{est}}(\mathcal{T})=\{x\}$. *Further*, *for any fixed* ${x}_{0}\in {\mathcal{A}}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B})$, *converges to* *x*.

*Proof*Let ${x}_{0}\in {\mathcal{A}}_{0}$ (such a point there exists since ${\mathcal{A}}_{0}\ne \mathrm{\varnothing}$). Since $\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}$, then by the definition of ${\mathcal{B}}_{0}$, there exists ${x}_{1}\in {\mathcal{A}}_{0}$ such that $d({x}_{1},\mathcal{T}{x}_{0})=d(\mathcal{A},\mathcal{B})$. Again, since $\mathcal{T}{x}_{1}\in {\mathcal{B}}_{0}$, it follows that there is ${x}_{2}\in {\mathcal{A}}_{0}$ such that $d({x}_{2},\mathcal{T}{x}_{1})=d(\mathcal{A},\mathcal{B})$. Continuing this process, we can construct a sequence $\{{x}_{n}\}$ in ${\mathcal{A}}_{0}$, such that

*n*. Using the fact that $\mathcal{T}$ is a rational proximal contraction of the first kind, we have

where $k=\frac{\alpha +\gamma +\delta}{1-\beta -\gamma -\delta}<1$. Therefore, $\{{x}_{n}\}$ is a Cauchy sequence and, since $(\mathcal{X},d)$ is complete and $\mathcal{A}$ is closed, the sequence $\{{x}_{n}\}$ converges to some $x\in \mathcal{A}$.

*x*must be in ${\mathcal{A}}_{0}$. Since $\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}$, then $d(u,\mathcal{T}x)=d(\mathcal{A},\mathcal{B})$ for some $u\in \mathcal{A}$. Again, using the fact that $\mathcal{T}$ is a rational proximal contraction of the first kind, we get

*i.e.*, ${B}_{\mathrm{est}}(\mathcal{T})$ is singleton), assume that

*z*is another best proximity point of $\mathcal{T}$ so that

It follows immediately that $x=z$, since $\alpha +2\delta <1$. Hence, $\mathcal{T}$ has a unique best proximity point. □

As consequences of the Theorem 3.1, we state the following corollaries.

**Corollary 3.1**

*Let*$(\mathcal{X},d)$

*be a complete metric space and*$\mathcal{A}$

*and*ℬ

*be two nonempty*,

*closed subsets of*$\mathcal{X}$

*such that*ℬ

*is approximatively compact with respect to*$\mathcal{A}$.

*Assume that*${\mathcal{A}}_{0}$

*and*${\mathcal{B}}_{0}$

*are nonempty and*$\mathcal{T}:\mathcal{A}\to \mathcal{B}$

*is a nonself*-

*mapping such that*:

- (a)
$\mathcal{T}$

*is a generalized proximal contraction of the first kind*,*with*$\alpha +2\gamma +2\delta <1$; - (b)
$\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}$.

*Then*, *there exists* $x\in \mathcal{A}$ *such that* ${B}_{\mathrm{est}}(\mathcal{T})=\{x\}$. *Further*, *for any fixed* ${x}_{0}\in {\mathcal{A}}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B})$, *converges to the best proximity point* *x*.

**Corollary 3.2**

*Let*$(\mathcal{X},d)$

*be a complete metric space and*$\mathcal{A}$

*and*ℬ

*be two nonempty*,

*closed subsets of*$\mathcal{X}$

*such that*ℬ

*is approximatively compact with respect to*$\mathcal{A}$.

*Assume that*${\mathcal{A}}_{0}$

*and*${\mathcal{B}}_{0}$

*are nonempty and*$\mathcal{T}:\mathcal{A}\to \mathcal{B}$

*is a nonself*-

*mapping such that*:

- (a)
*There exists a nonnegative real number*$\alpha <1$*such that*,*for all*${u}_{1}$, ${u}_{2}$, ${x}_{1}$, ${x}_{2}$*in*$\mathcal{A}$,*the conditions*$d({u}_{1},\mathcal{T}{x}_{1})=d(\mathcal{A},\mathcal{B})$*and*$d({u}_{2},\mathcal{T}{x}_{2})=d(\mathcal{A},\mathcal{B})$*imply that*$d({u}_{1},{u}_{2})\le \alpha d({x}_{1},{x}_{2})$; - (b)
$\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}$.

*Then there exists* $x\in \mathcal{A}$ *such that* ${B}_{\mathrm{est}}(\mathcal{T})=\{x\}$. *Further*, *for any fixed* ${x}_{0}\in {\mathcal{A}}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B})$, *converges to the best proximity point* *x*.

The following fixed-point result can be considered as a special case of the Theorem 3.1, when $\mathcal{T}$ is a self-mapping.

**Corollary 3.3**

*Let*$(\mathcal{X},d)$

*be a complete metric space and*$\mathcal{T}$

*be a self*-

*mapping on*$\mathcal{X}$.

*Assume that there exist nonnegative real numbers*

*α*,

*β*,

*γ*,

*δ*

*with*$\alpha +\beta +2\gamma +2\delta <1$

*such that*

*for all* ${x}_{1},{x}_{2}\in \mathcal{X}$. *Then the mapping* $\mathcal{T}$ *has a unique fixed point*.

**Remark 3.1** Note that the Corollary 3.3 is a proper extension of the contraction mapping principle [13] because the continuity of the mapping $\mathcal{T}$ is not required. It is well known that a contraction mapping must be continuous.

Now, we state and prove a best proximity point theorem for a rational proximal contraction of the second kind.

**Theorem 3.2**

*Let*$(\mathcal{X},d)$

*be a complete metric space and*$\mathcal{A}$

*and*ℬ

*be two nonempty*,

*closed subsets of*$\mathcal{X}$

*such that*$\mathcal{A}$

*is approximatively compact with respect to*ℬ.

*Assume that*${\mathcal{A}}_{0}$

*and*${\mathcal{B}}_{0}$

*are nonempty and*$\mathcal{T}:\mathcal{A}\to \mathcal{B}$

*is a nonself*-

*mapping such that*:

- (a)
$\mathcal{T}$

*is a continuous rational proximal contraction of the second kind*; - (b)
$\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}$.

*Then there exists* $x\in {B}_{\mathrm{est}}(\mathcal{T})$ *and for any fixed* ${x}_{0}\in {\mathcal{A}}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B})$, *converges to* *x*, *and* $\mathcal{T}x=\mathcal{T}z$ *for all* $x,z\in {B}_{\mathrm{est}}(\mathcal{T})$.

*Proof*Following the same lines of the proof of the Theorem 3.1, it is possible to construct a sequence $\{{x}_{n}\}$ in ${\mathcal{A}}_{0}$ such that

*n*. Using the fact that $\mathcal{T}$ is a rational proximal contraction of the second kind, we have

where $k=\frac{\alpha +\gamma +\delta}{1-\beta -\gamma -\delta}<1$. Therefore, $\{\mathcal{T}{x}_{n}\}$ is a Cauchy sequence and, since $(\mathcal{X},d)$ is complete, then the sequence $\{\mathcal{T}{x}_{n}\}$ converges to some $y\in \mathcal{B}$.

*z*is another best proximity point of $\mathcal{T}$ so that

It follows immediately that $\mathcal{T}x=\mathcal{T}z$, since $\alpha +2\delta <1$. □

As consequences of the Theorem 3.2, we state the following corollaries.

**Corollary 3.4**

*Let*$(\mathcal{X},d)$

*be a complete metric space and*$\mathcal{A}$

*and*ℬ

*be two nonempty*,

*closed subsets of*$\mathcal{X}$

*such that*$\mathcal{A}$

*is approximatively compact with respect to*ℬ.

*Assume that*${\mathcal{A}}_{0}$

*and*${\mathcal{B}}_{0}$

*are nonempty and*$\mathcal{T}:\mathcal{A}\to \mathcal{B}$

*is a nonself*-

*mapping such that*:

- (a)
$\mathcal{T}$

*is a continuous generalized proximal contraction of the second kind*,*with*$\alpha +2\gamma +2\delta <1$; - (b)
$\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}$.

*Then*, *there exists* $x\in {B}_{\mathrm{est}}(\mathcal{T})$ *and for any fixed* ${x}_{0}\in {\mathcal{A}}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B})$, *converges to* *x*. *Further*, $\mathcal{T}x=\mathcal{T}z$ *for all* $x,z\in {B}_{\mathrm{est}}(\mathcal{T})$.

**Corollary 3.5**

*Let*$(\mathcal{X},d)$

*be a complete metric space and*$\mathcal{A}$

*and*ℬ

*be two nonempty*,

*closed subsets of*$\mathcal{X}$

*such that*$\mathcal{A}$

*is approximatively compact with respect to*ℬ.

*Assume that*${\mathcal{A}}_{0}$

*and*${\mathcal{B}}_{0}$

*are nonempty and*$\mathcal{T}:\mathcal{A}\to \mathcal{B}$

*is a nonself*-

*mapping such that*:

- (a)
*There exists a nonnegative real number*$\alpha <1$*such that*,*for all*${u}_{1}$, ${u}_{2}$, ${x}_{1}$, ${x}_{2}$*in*$\mathcal{A}$,*the conditions*$d({u}_{1},\mathcal{T}{x}_{1})=d(\mathcal{A},\mathcal{B})$*and*$d({u}_{2},\mathcal{T}{x}_{2})=d(\mathcal{A},\mathcal{B})$*imply that*$d(\mathcal{T}{u}_{1},\mathcal{T}{u}_{2})\le \alpha d(\mathcal{T}{x}_{1},\mathcal{T}{x}_{2})$; - (b)
$\mathcal{T}$

*is continuous*; - (c)
$\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}$.

*Then there exists* $x\in {B}_{\mathrm{est}}(\mathcal{T})$ *and for any fixed* ${x}_{0}\in {\mathcal{A}}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B})$, *converges to* *x*. *Further*, $\mathcal{T}x=\mathcal{T}z$ *for all* $x,z\in {B}_{\mathrm{est}}(\mathcal{T})$.

**Remark 3.2** Note that in the Theorem 3.1 is not required the continuity of the mapping $\mathcal{T}$. On the contrary, the continuity of $\mathcal{T}$ is an hypothesis of the Theorem 3.2.

Our next theorem concerns a nonself-mapping that is a rational proximal contraction of the first kind as well as a rational proximal contraction of the second kind. In this theorem, we consider only a completeness hypothesis without assuming the continuity of the nonself-mapping.

**Theorem 3.3**

*Let*$(\mathcal{X},d)$

*be a complete metric space and*$\mathcal{A}$

*and*ℬ

*be two nonempty*,

*closed subsets of*$\mathcal{X}$.

*Assume that*${\mathcal{A}}_{0}$

*and*${\mathcal{B}}_{0}$

*are nonempty and*$\mathcal{T}:\mathcal{A}\to \mathcal{B}$

*is a nonself*-

*mapping such that*:

- (a)
$\mathcal{T}$

*is a rational proximal contraction of the first and second kinds*; - (b)
${\mathcal{T}(\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}$.

*Then there exists a unique* $x\in {B}_{\mathrm{est}}(\mathcal{T})$. *Further*, *for any fixed* ${x}_{0}\in {\mathcal{A}}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},\mathcal{T}{x}_{n})=d(\mathcal{A},\mathcal{B})$, *converges to* *x*.

*Proof*Following the same lines of the proof of the Theorem 3.1, it is possible to construct a sequence $\{{x}_{n}\}$ in ${\mathcal{A}}_{0}$ such that

*n*. Also, using the same arguments in the proof of the Theorem 3.1, we deduce that the sequence $\{{x}_{n}\}$ is a Cauchy sequence, and hence converges to some $x\in \mathcal{A}$. Moreover, on the lines of the proof of the Theorem 3.2, we obtain that the sequence $\{\mathcal{T}{x}_{n}\}$ is a Cauchy sequence and hence converges to some $y\in \mathcal{B}$. Therefore, we have

*x*must be in ${\mathcal{A}}_{0}$. Since $\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}$, then $d(u,\mathcal{T}x)=d(\mathcal{A},\mathcal{B})$ for some $u\in \mathcal{A}$. Using the fact that $\mathcal{T}$ is a rational proximal contraction of the first kind, we get

that is, $x\in {B}_{\mathrm{est}}(\mathcal{T})$. Again, following the same lines of the proof of the Theorem 3.1, we prove the uniqueness of the best proximity point of the mapping $\mathcal{T}$. To avoid repetitions, we omit the details. □

**Example 3.1**Let $\mathcal{X}=\mathbb{R}$ endowed with the usual metric $d(x,y)=|x-y|$, for all $x,y\in \mathcal{X}$. Define $\mathcal{A}=[-1,1]$ and $\mathcal{B}=[-3,-2]\cup [2,3]$. Then, $d(\mathcal{A},\mathcal{B})=1$, ${\mathcal{A}}_{0}=\{-1,1\}$ and ${\mathcal{B}}_{0}=\{-2,2\}$. Also define $\mathcal{T}:\mathcal{A}\to \mathcal{B}$ by

It is easy to show that $\mathcal{T}$ is a rational proximal contraction of the first and second kinds and $\mathcal{T}({\mathcal{A}}_{0})\subseteq {\mathcal{B}}_{0}$. Then all the hypotheses of the Theorem 3.3 are satisfied and $d(1,\mathcal{T}(1))=d(\mathcal{A},\mathcal{B})$. Clearly, the Theorem 3.2 is not applicable in this case.

## Declarations

### Acknowledgements

The third author is supported by Università degli Studi di Palermo, Local University Project R. S. ex 60%. The second author was supported by the Commission on Higher Education, the Thailand Research Fund, and the King Mongkuts University of Technology Thonburi (Grant No. MRG5580213).

## Authors’ Affiliations

## References

- Eldred A, Veeramani PL: Existence and convergence of best proximity points.
*J. Math. Anal. Appl.*2006, 323: 1001–1006. doi:10.1016/j.jmaa.2005.10.081MATHMathSciNetView ArticleGoogle Scholar - Fan K: Extensions of two fixed point theorems of F. E. Browder.
*Math. Z.*1969, 112: 234–240. doi:10.1007/BF01110225MATHMathSciNetView ArticleGoogle Scholar - Prolla JB: Fixed point theorems for set valued mappings and existence of best approximations.
*Numer. Funct. Anal. Optim.*1982, 5: 449–455.MathSciNetView ArticleGoogle Scholar - Reich S: Approximate selections, best approximations, fixed points and invariant sets.
*J. Math. Anal. Appl.*1978, 62: 104–113. doi:10.1016/0022–247X(78)90222–6MATHMathSciNetView ArticleGoogle Scholar - Sehgal VM, Singh SP: A generalization to multifunctions of Fan’s best approximation theorem.
*Proc. Am. Math. Soc.*1988, 102: 534–537.MATHMathSciNetGoogle Scholar - Sehgal VM, Singh SP: A theorem on best approximations.
*Numer. Funct. Anal. Optim.*1989, 10: 181–184. doi:10.1080/01630568908816298MATHMathSciNetView ArticleGoogle Scholar - Vetrivel V, Veeramani P, Bhattacharyya P: Some extensions of Fan’s best approximation theorem.
*Numer. Funct. Anal. Optim.*1992, 13: 397–402. doi:10.1080/01630569208816486 10.1080/01630569208816486MATHMathSciNetView ArticleGoogle Scholar - Al-Thagafi MA, Shahzad N: Best proximity sets and equilibrium pairs for a finite family of multimaps.
*Fixed Point Theory Appl.*2008., 2008: Article ID 457069Google Scholar - Al-Thagafi MA, Shahzad N: Best proximity pairs and equilibrium pairs for Kakutani multimaps.
*Nonlinear Anal.*2009, 70(3):1209–1216. doi:10.1016/j.na.2008.02.004 10.1016/j.na.2008.02.004MATHMathSciNetView ArticleGoogle Scholar - Al-Thagafi MA, Shahzad N: Convergence and existence results for best proximity points.
*Nonlinear Anal.*2009, 70(10):3665–3671. doi:10.1016/j.na.2008.07.022 10.1016/j.na.2008.07.022MATHMathSciNetView ArticleGoogle Scholar - Anuradha J, Veeramani P: Proximal pointwise contraction.
*Topol. Appl.*2009, 156(18):2942–2948. doi:10.1016/j.topol.2009.01.017 10.1016/j.topol.2009.01.017MATHMathSciNetView ArticleGoogle Scholar - Balaganskii VS, Vlasov LP: The problem of the convexity of Chebyshev sets.
*Usp. Mat. Nauk*1996, 51: 125–188.MathSciNetView ArticleGoogle Scholar - Banach S: Sur les opérations dans les ensembles absraites et leurs applications.
*Fundam. Math.*1922, 3: 133–181.MATHGoogle Scholar - Borodin PA: An example of a bounded approximately compact set that is not compact.
*Russ. Math. Surv.*1994, 49: 153–154.View ArticleGoogle Scholar - Di Bari C, Suzuki T, Vetro C: Best proximity points for cyclic Meir-Keeler contractions.
*Nonlinear Anal.*2008, 69(11):3790–3794. doi:10.1016/j.na.2007.10.014 10.1016/j.na.2007.10.014MATHMathSciNetView ArticleGoogle Scholar - Efimov NV, Stechkin SB: Approximative compactness and Chebyshev sets.
*Dokl. Akad. Nauk SSSR*1961, 140: 522–524. (in Russian)MathSciNetGoogle Scholar - Eldred A, Kirk WA, Veeramani P: Proximinal normal structure and relatively nonexpanisve mappings.
*Stud. Math.*2005, 171(3):283–293. doi:10.4064/sm171–3-5MATHMathSciNetView ArticleGoogle Scholar - Karpagam S, Agrawal S: Best proximity point theorems for
*p*-cyclic Meir-Keeler contractions.*Fixed Point Theory Appl.*2009., 2009(9): Article ID 197308Google Scholar - Kim WK, Kum S, Lee KH: On general best proximity pairs and equilibrium pairs in free abstract economies.
*Nonlinear Anal.*2008, 68(8):2216–2227. doi:10.1016/j.na.2007.01.057 10.1016/j.na.2007.01.057MATHMathSciNetView ArticleGoogle Scholar - Mongkolkeha C, Kumam P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces.
*J. Optim. Theory Appl.*2012, 155: 215–226. doi:10.1007/s10957–012–9991-y 10.1007/s10957-012-9991-yMATHMathSciNetView ArticleGoogle Scholar - Mongkolkeha C, Kumam P: Some common best proximity points for proximity commuting mappings.
*Optim. Lett.*2012. doi:10.1007/s11590–012–0525–1Google Scholar - Mongkolkeha C, Cho YJ, Kumam P: Best proximity points for generalized proximal
*C*-contraction mappings in metric spaces with partial orders.*J. Inequal. Appl.*2013., 2013: Article ID 94. doi:10.1186/1029–242X-2013–94Google Scholar - Sadiq Basha S: Extensions of Banach’s contraction principle.
*Numer. Funct. Anal. Optim.*2010, 31: 569–576. doi:10.1080/01630563.2010.485713MATHMathSciNetView ArticleGoogle Scholar - Sadiq Basha S: Best proximity points: global optimal approximate solution.
*J. Glob. Optim.*2010. doi:10.1007/s10898–009–9521–0Google Scholar - Sadiq Basha S, Shahzad N, Jeyaraj R: Common best proximity points: global optimization of multi-objective functions.
*Appl. Math. Lett.*2011, 24: 883–886. doi:10.1016/j.aml.2010.12.043MATHMathSciNetView ArticleGoogle Scholar - Sadiq Basha S, Veeramani P: Best approximations and best proximity pairs.
*Acta Sci. Math.*1997, 63: 289–300.MathSciNetGoogle Scholar - Sadiq Basha S, Veeramani P: Best proximity pair theorems for multifunctions with open fibres.
*J. Approx. Theory*2000, 103: 119–129. doi:10.1006/jath.1999.3415 10.1006/jath.1999.3415MATHMathSciNetView ArticleGoogle Scholar - Sadiq Basha S, Veeramani P, Pai DV: Best proximity pair theorems.
*Indian J. Pure Appl. Math.*2001, 32: 1237–1246.MATHMathSciNetGoogle Scholar - Sadiq Basha S, Shahzad N: Best proximity point theorems for generalized proximal contractions.
*Fixed Point Theory Appl.*2012., 2012: Article ID 42Google Scholar - Sankar Raj V, Veeramani P: Best proximity pair theorems for relatively nonexpansive mappings.
*Appl. Gen. Topol.*2009, 10(1):21–28.MATHMathSciNetView ArticleGoogle Scholar - Shahzad N, Sadiq Basha S, Jeyaraj R: Common best proximity points: global optimal solutions.
*J. Optim. Theory Appl.*2011, 148: 69–78. doi:10.1007/s10957–010–9745–7 10.1007/s10957-010-9745-7MATHMathSciNetView ArticleGoogle Scholar - Sanhan W, Mongkolkeha C, Kumam P: Generalized proximal
*ψ*-contraction mappings and best proximity points.*Abstr. Appl. Anal.*2012., 2012: Article ID 896912Google Scholar - Sintunavarat W, Kumam K: Coupled best proximity point theorem in metric spaces.
*Fixed Point Theory Appl.*2012., 2012: Article ID 93Google Scholar - Srinivasan PS: Best proximity pair theorems.
*Acta Sci. Math.*2001, 67: 421–429.MATHGoogle Scholar - Suzuki T, Kikkawa M, Vetro C: The existence of best proximity points in metric spaces with the property UC.
*Nonlinear Anal.*2009, 71: 2918–2926. doi:10.1016/j.na.2009.01.173 10.1016/j.na.2009.01.173MATHMathSciNetView ArticleGoogle Scholar - Suzuki T, Vetro C: Three existence theorems for weak contractions of Matkowski type.
*Int. J. Math. Stat.*2010, 6: 110–120. 10.3844/jmssp.2010.110.115MathSciNetView ArticleGoogle Scholar - Vetro C: Best proximity points: convergence and existence theorems for
*p*-cyclic mappings.*Nonlinear Anal.*2010, 73: 2283–2291. doi:10.1016/j.na.2010.06.008 10.1016/j.na.2010.06.008MATHMathSciNetView ArticleGoogle Scholar - Wlodarczyk K, Plebaniak R, Banach A: Best proximity points for cyclic and noncyclic set-valued relatively quasiasymptotic contractions in uniform spaces.
*Nonlinear Anal.*2009, 70(9):3332–3341. doi:10.1016/j.na.2008.04.037 10.1016/j.na.2008.04.037MATHMathSciNetView ArticleGoogle Scholar - Wlodarczyk K, Plebaniak R, Banach A: Erratum to: best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces.
*Nonlinear Anal.*2009, 71: 3585–3586. doi:10.1016/j.na.2008.11.020MATHMathSciNetView ArticleGoogle Scholar - Wlodarczyk K, Plebaniak R, Obczynski C: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces.
*Nonlinear Anal.*2010, 72: 794–805. doi:10.1016/j.na.2009.07.024MATHMathSciNetView ArticleGoogle Scholar - Kirk WA, Reich S, Veeramani P: Proximinal retracts and best proximity pair theorems.
*Numer. Funct. Anal. Optim.*2003, 24: 851–862. doi:10.1081/NFA-120026380MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.