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# Optimal solutions for nonlinear proximal ${C}_{N}$-contraction mapping in metric space

- Marwan Amin Kutbi
^{1}, - Sumit Chandok
^{2}and - Wutiphol Sintunavarat
^{3}Email author

**2014**:193

https://doi.org/10.1186/1029-242X-2014-193

© Kutbi et al.; licensee Springer. 2014

**Received:**30 September 2013**Accepted:**28 April 2014**Published:**14 May 2014

## Abstract

The objective of this paper is to solve an optimal solutions for new nonlinear mapping in the setting of metric spaces. Our results extend, generalize, and improve some known results from best proximity point theory and fixed point theory. Examples are given to support our main results.

**MSC:**41A50, 47H10, 54H25.

## Keywords

- approximatively compact
- best proximity point
- proximal ${C}_{N}$-contraction mapping

## 1 Introduction

Fixed point theory is one of the famous and traditional theories in mathematics and has a large number of applications in various fields of pure and applied mathematics, as well as in physical, chemical, life, and social sciences. It is an important tool for solving equations of the form $Tx=x$, where *T* is a self-mapping on a subset of a metric space. On the other hand, if *T* is not a self-mapping, say $T:A\to B$ where *A* and *B* are nonempty subsets of a metric space, then *T* does not necessarily have a fixed point. Consequently, the equation $Tx=x$ 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 *Tx*. Here best approximation theorems explore the existence of an approximate solution whereas best proximity point theorems analyze the existence of an approximate solution that is optimal. 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 $d(x,Tx)$ is at least $d(A,B):=inf\{d(x,y):x\in A\text{and}y\in B\}$, a best proximity point theorem concerns the global minimum of the real valued function $x\to d(x,Tx)$, that is, an indicator of the error involved for an approximate solution of the equation $Tx=x$, by complying with the condition $d(x,Tx)=d(A,B)$. A classical best approximation theorem was introduced by Fan [1], that is, if *A* is a nonempty compact convex subset of a Hausdorff locally convex topological vector space *B* and $T:A\to B$ is a continuous mapping, then there exists an element $x\in A$ such that $d(x,Tx)=d(Tx,A)$. Afterward, several generalizations and extensions of this theorem appeared in the literature (see *e.g.* [2–4], and references cited therein). It turns out that many of the contractive conditions which are investigated for fixed points ensure the existence of best proximity points, and many results of this kind are obtained in [5–25].

where $T:A\to B$ belongs a to new class of non-self-mappings, and *A* and *B* are nonempty subsets of a metric space $(X,d)$. Also, we give some illustrative examples to support our results.

## 2 Preliminaries

*A*and

*B*be two nonempty subsets of

*X*and $T:A\to B$ be a non-self-mapping. The following notations will be used in the sequel:

Kirk *et al.* [9] gave sufficient conditions to ensure that ${A}_{0}$ and ${B}_{0}$ are nonempty. Also, we find that if *A* and *B* are closed subsets of a normed linear space such that $d(A,B)>0$, then ${A}_{0}$ and ${B}_{0}$ are contained in the boundaries of *A* and *B*, respectively (see [12]).

In 1961, Efimov and Stechkin [26] introduced the concept of an approximatively compact set. The properties of approximatively compact sets have been largely studied. It is well known that the concept of approximative compactness plays an important role in the theory of approximation [27]. Borodin [28] showed that in every infinite-dimensional separable Banach space there exists a bounded approximatively compact set which is not compact.

**Remark 2.1** For a metric space $(X,d)$, 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.

**Definition 2.1** Let *A* and *B* be two nonempty subsets of metric space $(X,d)$. Then *B* is said to be approximatively compact with respect to *A* if every sequence $\{{y}_{n}\}$ of *B*, satisfying the condition $d(x,{y}_{n})\to d(x,B)$ as $n\to \mathrm{\infty}$ for some $x\in A$, has a convergent subsequence.

We see that any set is approximatively compact with respect to itself.

## 3 Main results

In this section, we give sequentially two new classes of non-self-mappings that are essential to state and prove the existence of best proximity point theorems.

**Definition 3.1**Let $(X,d)$ be a metric space and

*A*and

*B*are two nonempty subsets of

*X*. A mapping $T:A\to B$ is said to be a

*proximal*${C}_{N}$-

*contraction of the first kind*if there exist nonnegative real numbers

*α*,

*β*, and

*L*with $\alpha +\beta <1$, such that the conditions

for all $u,v,x,y\in A$.

**Remark 3.1**If

*T*is a self-mapping on

*A*, then the requirement in the above definition reduces to the following generalized contractive condition which is useful in establishing a fixed point theorem:

for all $x,y\in X$.

**Definition 3.2**Let $(X,d)$ be a metric space and

*A*and

*B*be two nonempty subsets of

*X*. A mapping $T:A\to B$ is said to be a

*proximal*${C}_{N}$-

*contraction of the second kind*if there exist nonnegative real numbers

*α*,

*β*, and

*L*with $\alpha +\beta <1$, such that the conditions

for all $u,v,x,y\in A$.

Here, we give our first main result which is the best proximity point theorem for a proximal ${C}_{N}$-contraction of the first kind.

**Theorem 3.1**

*Let*$(X,d)$

*be a complete metric space and*

*A*

*and*

*B*

*be two nonempty*,

*closed subsets of*

*X*

*such that*

*B*

*is approximatively compact with respect to*

*A*.

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

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

*are nonempty and*$T:A\to B$

*is a non*-

*self*-

*mapping such that*:

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

*Then there exists a unique element* $x\in A$ *such that* $d(x,Tx)=d(A,B)$. *Moreover*, *for any fixed* ${x}_{0}\in {A}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},T{x}_{n})=d(A,B)$, *converges to* *x*.

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

*n*. Since

*T*is a proximal ${C}_{N}$-contraction, we have

*A*is closed, the sequence $\{{x}_{n}\}$ converges to some $x\in A$. Further, we have

*B*is approximatively compact with respect to

*A*, the sequence $\{T{x}_{n}\}$ has a subsequence $\{T{x}_{{n}_{k}}\}$ converging to some element $y\in B$. Therefore,

*x*must be a member of ${A}_{0}$. Because of the fact that $T({A}_{0})$ is contained in ${B}_{0}$, $d(u,Tx)=d(A,B)$ for some element

*u*in

*A*. Since

*T*is a proximal ${C}_{N}$-contraction of the first kind, we get

Taking the limit $n\to \mathrm{\infty}$, we have $x=u$. Thus, it follows that $d(x,Tx)=d(u,Tx)=d(A,B)$.

*z*is another best proximity point of

*T*so that $d(z,Tz)=d(A,B)$. Since

*T*is a proximal ${C}_{N}$-contraction of the first kind, we have

which implies that $x=z$. Hence *T* has a unique best proximity point. □

It is easy to see that the preceding result yields the following corollaries.

**Corollary 3.2**

*Let*$(X,d)$

*be a complete metric space and*

*A*

*and*

*B*

*be two nonempty*,

*closed subsets of*

*X*

*such that*

*B*

*is approximatively compact with respect to*

*A*.

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

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

*are nonempty and*$T:A\to B$

*is a non*-

*self*-

*mapping such that*:

- (a)
*there exist nonnegative real numbers**α**and**L**with*$\alpha <1$,*such that*,*for all*$u,v,x,y\in A$,*the conditions*$d(u,Tx)=d(A,B)$*and*$d(v,Ty)=d(A,B)$*imply that*$d(u,v)\le \alpha d(x,y)+Lmin\{d(y,u),d(x,v),d(x,u),d(y,v)\};$ - (b)
$T({A}_{0})\subseteq {B}_{0}$.

*Then there exists a unique element* $x\in A$ *such that* $d(x,Tx)=d(A,B)$. *Moreover*, *for any fixed* ${x}_{0}\in {A}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},T{x}_{n})=d(A,B)$, *converges to* *x*.

**Corollary 3.3**

*Let*$(X,d)$

*be a complete metric space and*

*A*

*and*

*B*

*be two nonempty*,

*closed subsets of*

*X*

*such that*

*B*

*is approximatively compact with respect to*

*A*.

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

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

*are nonempty and*$T:A\to B$

*is a non*-

*self*-

*mapping such that*:

- (a)
*there exist nonnegative real numbers*$\alpha <1$*such that*,*for all*$u,v,x,y\in A$,*the conditions*$d(u,Tx)=d(A,B)$*and*$d(v,Ty)=d(A,B)$*imply that*$d(u,v)\le \alpha d(x,y)$; - (b)
$T({A}_{0})\subseteq {B}_{0}$.

*Then there exists a unique element* $x\in A$ *such that* $d(x,Tx)=d(A,B)$. *Moreover*, *for any fixed* ${x}_{0}\in {A}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},T{x}_{n})=d(A,B)$, *converges to* *x*.

In Theorem 3.1, if *T* is a self-mapping, then we get the following fixed point theorem.

**Corollary 3.4**

*Let*

*T*

*be a self*-

*mapping of a complete metric space*$(X,d)$.

*Assume that there exist nonnegative real numbers*

*α*,

*β*,

*and*

*L*

*with*$\alpha +\beta <1$,

*such that*

*for all* $x,y\in X$. *Then* *T* *has a unique fixed point*.

**Remark 3.2** It is well known that a contraction mapping must be continuous. Therefore, Corollary 3.4 is a real proper extension of the Banach contraction mapping principle of Banach [29] because the continuity of the mapping *T* is not required.

Next, we give the existence of best proximity point theorem for proximal ${C}_{N}$-contraction of the second kind.

**Theorem 3.5**

*Let*$(X,d)$

*be a complete metric space and*

*A*

*and*

*B*

*be two nonempty*,

*closed subsets of*

*X*

*such that*

*A*

*is approximatively compact with respect to*

*B*.

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

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

*are nonempty and*$T:A\to B$

*is a non*-

*self*-

*mapping such that*:

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

*Then there exists* $x\in A$ *such that* $d(x,Tx)=d(A,B)$. *Moreover*, *for any fixed* ${x}_{0}\in {A}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},T{x}_{n})=d(A,B)$, *converges to* *x*. *Further*, $Tx=Tz$ *for all* *x* *and* *z* *belong to best proximity of* *T*.

*Proof*Following the arguments in Theorem 3.1, we can construct a sequence $\{{x}_{n}\}$ in ${A}_{0}$, such that

*n*. Since

*T*is a proximal ${C}_{N}$-contraction of the second kind, we have

*B*is closed, the sequence $\{T{x}_{n}\}$ converges to some $y\in B$. Further, we have

Therefore, $d(y,{x}_{n})\to d(y,A)$. Since *A* is approximatively compact with respect to *B*, then the sequence $\{{x}_{n}\}$ has a subsequence $\{{x}_{{n}_{k}}\}$ converging to some element $x\in A$. Now, using the continuity of *T*, we obtain $d(x,Tx)={lim}_{k\to \mathrm{\infty}}d({x}_{{n}_{k}+1},T{x}_{{n}_{k}})=d(A,B)$.

*z*is another best proximity point of

*T*so that $d(z,Tz)=d(A,B)$. Since

*T*is a proximal ${C}_{N}$-contraction of the second kind, we get

which implies that $Tx=Tz$. □

As consequences of Theorem 3.5, we state the following corollaries.

**Corollary 3.6**

*Let*$(X,d)$

*be a complete metric space and*

*A*

*and*

*B*

*be two nonempty*,

*closed subsets of*

*X*

*such that*

*A*

*is approximatively compact with respect to*

*B*.

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

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

*are nonempty and*$T:A\to B$

*is a non*-

*self*-

*mapping such that*:

- (a)
*there exist nonnegative real numbers**α**and**L**with*$\alpha <1$*such that*,*for all*$u,v,x,y\in A$,*the conditions*$d(u,Tx)=d(A,B)$*and*$d(v,Ty)=d(A,B)$*imply that*$d(Tu,Tv)\le \alpha d(Tx,Ty)+Lmin\{d(Ty,Tu),d(Tx,Tv),d(Tx,Tu),d(Ty,Tv)\};$ - (b)
*T**is a continuous*; - (c)
$T({A}_{0})\subseteq {B}_{0}$.

*Then there exists* $x\in A$ *such that* $d(x,Tx)=d(A,B)$. *Moreover*, *for any fixed* ${x}_{0}\in {A}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},T{x}_{n})=d(A,B)$, *converges to* *x*. *Further*, $Tx=Tz$ *for all* *x* *and* *z* *belong to best proximity of* *T*.

**Corollary 3.7**

*Let*$(X,d)$

*be a complete metric space and*

*A*

*and*

*B*

*be two nonempty*,

*closed subsets of*

*X*

*such that*

*A*

*is approximatively compact with respect to*

*B*.

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

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

*are nonempty and*$T:A\to B$

*is a non*-

*self*-

*mapping such that*:

- (a)
*there exists a nonnegative real number*$\alpha <1$*such that*,*for all*$u,v,x,y\in A$,*the conditions*$d(u,Tx)=d(A,B)$*and*$d(v,Ty)=d(A,B)$*imply that*$d(Tu,Tv)\le \alpha d(Tx,Ty)$; - (b)
*T**is a continuous*; - (c)
$T({A}_{0})\subseteq {B}_{0}$.

*Then there exists* $x\in A$ *such that* $d(x,Tx)=d(A,B)$. *Moreover*, *for any fixed* ${x}_{0}\in {A}_{0}$, *the sequence* $\{{x}_{n}\}$, *defined by* $d({x}_{n+1},T{x}_{n})=d(A,B)$, *converges to* *x*. *Further*, $Tx=Tz$ *for all* *x* *and* *z* *belong to the best proximity of* *T*.

In Theorem 3.5, if *T* is a self-mapping, then we get the following fixed point theorem.

**Corollary 3.8**

*Let*

*T*

*be a continuous self*-

*mapping of a complete metric space*$(X,d)$.

*Assume that there exist nonnegative real numbers*

*α*,

*β*,

*and*

*L*

*with*$\alpha +\beta <1$,

*such that*

*for* $x,y\in X$. *Then* *T* *has a unique fixed point*.

**Remark 3.3** It is easy to see that Corollary 3.8 is a special case of Corollary 3.4.

The next theorem, we give conditions for the existence of best proximity point for a non-self-mapping that is a proximal ${C}_{N}$-contraction of the first and second kind. In this theorem, we consider only a completeness hypothesis without assuming the continuity of the non-self-mapping and the approximatively compactness of subspace.

**Theorem 3.9**

*Let*$(X,d)$

*be a complete metric space and*

*A*

*and*

*B*

*be two nonempty*,

*closed subsets of*

*X*.

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

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

*are nonempty and*$T:A\to B$

*is a non*-

*self*-

*mapping such that*:

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

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

*Proof*Following the arguments in Theorem 3.1, we can construct a sequence $\{{x}_{n}\}$ in ${A}_{0}$, such that

*n*. Also, using similar arguments as in the proof of Theorem 3.1, we deduce that the sequence $\{{x}_{n}\}$ is a Cauchy sequence, and hence converges to some $x\in A$. Moreover, on the lines of Theorem 3.5, we find that the sequence $\{T{x}_{n}\}$ is a Cauchy sequence and hence converges to some $y\in B$. Therefore, we have $d(x,y)={lim}_{n\to \mathrm{\infty}}d({x}_{n+1},T{x}_{n})=d(A,B)$, hence

*x*must be in ${A}_{0}$. Since $T({A}_{0})\subseteq {B}_{0}$, then $d(u,Tx)=d(A,B)$ for some $u\in A$. Using the fact that

*T*is proximal ${C}_{N}$-contraction of first kind, we have

Taking the limit $n\to \mathrm{\infty}$, we have $x=u$. Thus, it follows that $d(x,Tx)=d(u,Tx)=d(A,B)$.

*z*is another best proximity point of

*T*so that $d(z,Tz)=d(A,B)$. Since

*T*is a proximal ${C}_{N}$-contraction of the first kind, we have

which implies that $x=z$. Hence *T* has a unique best proximity point. □

Finally, we give some illustrative example which demonstrate the validity of the hypotheses and degree of utility of our results.

**Example 3.10**Let $X=\mathbb{R}$ endowed with the usual metric $d(x,y)=|x-y|$, for all $x,y\in X$. Define two closed subsets

*A*,

*B*of

*X*by $A=[0,1]$ and $B=(-\mathrm{\infty},-2]\cup [3,\mathrm{\infty})$. Then $d(A,B)=2$, ${A}_{0}=\{0,1\}$ and ${B}_{0}=\{-2,3\}$. Also define $T:A\to B$ by

It is easy to see that $T({A}_{0})\subseteq {B}_{0}$ and *T* is a proximal ${C}_{N}$-contraction of the first and second kind. Now all the hypotheses of Theorem 3.9 are satisfied and $d(1,T(1))=d(A,B)$. Obviously, Theorem 3.5 is not applicable in this case since *T* is not continuous.

## Declarations

### Acknowledgements

The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.

## Authors’ Affiliations

## References

- Fan K:
**Extensions of two fixed point theorems of F. E. Browder.***Math. Z.*1969,**112:**234–240. 10.1007/BF01110225MathSciNetView ArticleMATHGoogle Scholar - Prolla JB:
**Fixed point theorems for set valued mappings and existence of best approximations.***Numer. Funct. Anal. Optim.*1982,**5:**449–455.MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Approximate selections, best approximations, fixed points and invariant sets.***J. Math. Anal. Appl.*1978,**62:**2–6.View ArticleMathSciNetMATHGoogle Scholar - Sehgal VM, Singh SP:
**A generalization to multifunctions of Fan’s best approximation theorem.***Proc. Am. Math. Soc.*1988,**102:**534–537.MathSciNetMATHGoogle 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. 10.1016/j.na.2008.02.004MathSciNetView ArticleMATHGoogle Scholar - Al-Thagafi MA, Shahzad N:
**Convergence and existence results for best proximity points.***Nonlinear Anal.*2009,**70**(10):3665–3671. 10.1016/j.na.2008.07.022MathSciNetView ArticleMATHGoogle Scholar - Di Bari C, Suzuki T, Vetro C:
**Best proximity points for cyclic Meir-Keeler contractions.***Nonlinear Anal.*2008,**69**(11):3790–3794. 10.1016/j.na.2007.10.014MathSciNetView ArticleMATHGoogle Scholar - Kirk WA, Reich S, Veeramani P:
**Proximinal retracts and best proximity pair theorems.***Numer. Funct. Anal. Optim.*2003,**24:**851–862. 10.1081/NFA-120026380MathSciNetView ArticleMATHGoogle 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. 10.1016/j.aml.2010.12.043MathSciNetView ArticleMATHGoogle Scholar - Sadiq Basha S, Veeramani P:
**Best approximations and best proximity pairs.***Acta Sci. Math.*1997,**63:**289–300.MathSciNetMATHGoogle Scholar - Sadiq Basha S, Veeramani P:
**Best proximity pair theorems for multifunctions with open fibres.***J. Approx. Theory*2000,**103:**119–129. 10.1006/jath.1999.3415MathSciNetView ArticleMATHGoogle Scholar - Sadiq Basha S, Veeramani P, Pai DV:
**Best proximity pair theorems.***Indian J. Pure Appl. Math.*2001,**32:**1237–1246.MathSciNetMATHGoogle 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. 10.4995/agt.2009.1784MathSciNetView ArticleMATHGoogle Scholar - Srinivasan PS:
**Best proximity pair theorems.***Acta Sci. Math.*2001,**67:**421–429.MathSciNetMATHGoogle 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. 10.1016/j.na.2009.01.173MathSciNetView ArticleMATHGoogle Scholar - Sintunavarat W, Kumam P:
**Coupled best proximity point theorem in metric spaces.***Fixed Point Theory Appl.*2012.,**2012:**Article ID 93Google Scholar - Sintunavarat W, Kumam P:
**The existence theorems of an optimal approximate solution for generalized proximal contraction mappings.***Abstr. Appl. Anal.*2013.,**2013:**Article ID 375604Google Scholar - Kumam P, Aydi H, Karapinar E, Sintunavarat W:
**Best proximity points and extension of Mizoguchi-Takahashi’s fixed point theorems.***Fixed Point Theory Appl.*2013.,**2013:**Article ID 242Google Scholar - Mongkolkeha C, Kumam P:
**Best proximity point theorems for generalized cyclic contractions in ordered metric spaces.***J. Optim. Theory Appl.*2012,**155**(1):215–226. 10.1007/s10957-012-9991-yMathSciNetView ArticleMATHGoogle Scholar - Mongkolkeha C, Kumam P:
**Some common best proximity points for proximity commuting mappings.***Optim. Lett.*2013,**7**(8):1825–1836. 10.1007/s11590-012-0525-1MathSciNetView ArticleMATHGoogle Scholar - Sanhan W, Mongkolkeha C, Cho YJ, Kumam P:
**Generalized proximal**ψ**-contraction mappings and best proximity points.***Abstr. Appl. Anal.*2012.,**2012:**Article ID 896912Google 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 94Google Scholar - Mongkolkeha C, Cho YJ, Kumam P:
**Best proximity points for Geraghty’s proximal contraction mappings.***Fixed Point Theory Appl.*2013.,**2013:**Article ID 180Google Scholar - Efimov NV, Stechkin SB:
**Approximative compactness and Chebyshev sets.***Dokl. Akad. Nauk SSSR*1961,**140:**522–524. (in Russian)MathSciNetMATHGoogle Scholar - Balaganskii VS, Vlasov LP:
**The problem of the convexity of Chebyshev sets.***Usp. Mat. Nauk*1996,**51:**125–188.MathSciNetView ArticleGoogle Scholar - Borodin PA:
**An example of a bounded approximately compact set that is not compact.***Russ. Math. Surv.*1994,**49:**153–154.View ArticleMATHGoogle Scholar - Banach S:
**Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales.***Fundam. Math.*1922,**3:**133–181.MATHGoogle Scholar

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