# Best proximity points for Geraghty’s proximal contraction mappings

- Chirasak Mongkolkeha
^{1, 3}, - Yeol Je Cho
^{2}Email author and - Poom Kumam
^{1}Email author

**2013**:180

https://doi.org/10.1186/1687-1812-2013-180

© Mongkolkeha et al.; licensee Springer 2013

**Received: **20 February 2013

**Accepted: **15 June 2013

**Published: **8 July 2013

## Abstract

In this paper, we generalized the notion of proximal contractions of the first and second kinds by using Geraghty’s theorem and establish best proximity point theorems for proximal contractions. Our results improve and extend the recent results of Sadiq Basha and some others.

**MSC:**47H09, 47H10.

## Keywords

## 1 Introduction

Several problems can be modeled as equations of the form $Tx=x$, where *T* is a given self-mapping defined on a subset of a metric space, a normed linear space, a topological vector space or some suitable space. However, if *T* is a nonself-mapping from *A* to *B*, then the aforementioned equation does not necessarily admit a solution. In this case, it is contemplated to find an approximate solution *x* in *A* such that the error $d(x,Tx)$ is minimum, where *d* is the distance function. In view of the fact that $d(x,Tx)$ is at least $d(A,B)$, a best proximity point theorem guarantees the global minimization of $d(x,Tx)$ by the requirement that an approximate solution *x* satisfies the condition $d(x,Tx)=d(A,B)$. Such optimal approximate solutions are called best proximity points of the mapping *T*. Interestingly, best proximity theorems also serve as a natural generalization of fixed point theorems, for a best proximity point becomes a fixed point if the mapping under consideration is a self-mapping.

A classical best approximation theorem was introduced by Fan [1], that is, if *A* is a non-empty 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 authors, including Prolla [2], Reich [3], Sehgal and Singh [4, 5], derived the extensions of Fan’s theorem in many directions. Other works on the existence of a best proximity point for contractions can be seen in [6–14].

In 1922, Banach proved that every contractive mapping in a complete metric spaces has a unique fixed point, which is called Banach’s fixed point theorem or Banach’s contraction principle. Since Banach’s fixed point theorem, many authors have extended, improved and generalized this theorem in several ways. Some applications of Banach’s fixed point theorem can be found in [15–18]. One of such generalizations is due to Geraghty [19] as follows.

**Theorem 1.1** [19]

*Let*$(X,d)$

*be a complete metric space and let*

*f*

*be a self*-

*mapping on*

*X*

*such that for each*$x,y\in X$

*satisfying*

*where*$\alpha \in \mathcal{S}$,

*is the family of functions from*$[0,\mathrm{\infty})$

*into*$[0,1)$

*which satisfies the condition*

*Then the sequence* $\{{f}_{n}\}$ *converges to the unique fixed point of* *f* *in* *X*.

In 2005, Eldred *et al.* [20] obtained best proximity point theorems for relatively nonexpansive mappings. Best proximity point theorems for several types of contractions were established in [21–25].

Recently, Sadiq Basha in [26] gave necessary and sufficient conditions to claim the existence of a best proximity point for proximal contractions of the first kind and the second kind, which are non-self mapping analogues of contraction self-mappings, and also established some best proximity and convergence theorems.

The aim of this paper is to introduce the new classes of proximal contractions, which are more general than a class of proximal contractions of the first and second kinds, by giving the necessary condition to have best proximity points, and we also give some illustrative example of our main results. The results of this paper are extension and generalizations of the main result of Sadiq Basha in [26] and some results in the literature.

## 2 Preliminaries

*A*and

*B*of a metric space $(X,d)$, we recall the following notations and notions that will be used in what follows.

If $A\cap B\ne \mathrm{\varnothing}$, then ${A}_{0}$ and ${B}_{0}$ are nonempty. Further, it is interesting to notice that ${A}_{0}$ and ${B}_{0}$ are contained in the boundaries of *A* and *B*, respectively, provided *A* and *B* are closed subsets of a normed linear space such that $d(A,B)>0$ (see [27]).

**Definition 2.1** [26]

*proximal contraction of the first kind*if there exists $k\in [0,1)$ such that

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

It is easy to see that a self-mapping that is a proximal contraction of the first kind is precisely a contraction. However, a nonself-proximal contraction is not necessarily a contraction.

**Definition 2.2** [26]

*proximal contraction of the second kind*if there exists $k\in [0,1)$ such that

for all $a,b,x,y\in A$.

**Definition 2.3**Let $S:A\to B$ and $T:B\to A$ be mappings. The pair $(S,T)$ is called a

*proximal cyclic contraction pair*if there exists $k\in [0,1)$ such that

for all $a,x\in A$ and $b,y\in B$.

**Definition 2.4**Let $S:A\to B$ and $g:A\to A$ be an isometry. The mapping

*S*is said to preserve the

*isometric distance*with respect to

*g*if

for all $x,y\in A$.

**Definition 2.5**A point $x\in A$ is called a

*best proximity point*of the mapping $S:A\to B$ if it satisfies the condition that

It can be observed that a best proximity reduces to a fixed point if the underlying mapping is a self-mapping.

## 3 Main results

In this section, we introduce a new class of proximal contractions, the so-called Geraghty’s proximal contraction mappings, and prove best proximity theorems for this class.

**Definition 3.1**A mapping $T:A\to B$ is called

*Geraghty’s proximal contraction of the first kind*if, there exists $\beta \in \mathcal{S}$ such that

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

**Definition 3.2**A mapping $T:A\to B$ is called

*Geraghty’s proximal contraction of the second kind*if, there exists $\beta \in \mathcal{S}$ such that

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

It is easy to see that if we take $\beta (t)=k$, where $k\in [0,1)$, then Geraghty’s proximal contraction of the first kind and Geraghty’s proximal contraction of the second kind reduce to a proximal contraction of the first kind (Definition 2.1) and a proximal contraction of the second kind (Definition 2.2), respectively.

Next, we extend the result of Sadiq Basha [26] and Banach’s fixed point theorem to the case of nonself-mappings satisfying Geraghty’s proximal contraction condition.

**Theorem 3.3**

*Let*$(X,d)$

*be a complete metric space and let*

*A*,

*B*

*be nonempty closed subsets of*

*X*

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

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

*are nonempty*.

*Let*$S:A\to B$, $T:B\to A$

*and*$g:A\cup B\to A\cup B$

*satisfy the following conditions*:

- (a)
*S**and**T**are Geraghty’s proximal contractions of the first kind*; - (b)
*g**is an isometry*; - (c)
*the pair*$(S,T)$*is a proximal cyclic contraction*; - (d)
$S({A}_{0})\subseteq {B}_{0}$, $T({B}_{0})\subseteq {A}_{0}$;

- (e)
${A}_{0}\subseteq g({A}_{0})$

*and*${B}_{0}\subseteq g({B}_{0})$.

*Then there exists a unique point*$x\in A$

*and there exists a unique point*$y\in B$

*such that*

*Moreover*,

*for any fixed*${x}_{0}\in {A}_{0}$,

*the sequence*$\{{x}_{n}\}$

*defined by*

*converges to the element*

*x*.

*For any fixed*${y}_{0}\in {B}_{0}$,

*the sequence*$\{{y}_{n}\}$

*defined by*

*converges to the element* *y*.

*On the other hand*,

*a sequence*$\{{u}_{n}\}$

*in*

*A*

*converges to*

*x*

*if there exists a sequence of positive numbers*$\{{\u03f5}_{n}\}$

*such that*

*where* ${z}_{n+1}\in A$ *satisfies the condition that* $d(g{z}_{n+1},S{u}_{n})=d(A,B)$.

*Proof*Let ${x}_{0}$ be a fixed element in ${A}_{0}$. In view of the fact that $S({A}_{0})\subseteq {B}_{0}$ and ${A}_{0}\subseteq g({A}_{0})$, it follows that there exists an element ${x}_{1}\in {A}_{0}$ such that

*S*is Geraghty’s proximal contraction of the first kind,

*g*is an isometry and the property of

*β*, it follows that for each $n\ge 1$

Taking $k\to \mathrm{\infty}$ in the above inequality, by (3.2), (3.4) and $\beta \in \mathcal{S}$, we get $\epsilon =0$, which is a contradiction. So we know that the sequence $\{{x}_{n}\}$ is a Cauchy sequence. Hence $\{{x}_{n}\}$ converges to some element $x\in A$.

*g*is an isometry, we have

*S*is Geraghty’s proximal contraction of the first kind of

*S*, we get

*g*is an isometry and

*S*is Geraghty’s proximal contraction of the first kind, it follows that

which is a contradiction. Thus we have $x={x}^{\ast}$. Similarly, we can prove that $y={y}^{\ast}$.

*A*and $\{{\u03f5}_{n}\}$ be a sequence of positive real numbers such that

*S*is Geraghty’s proximal contraction of the first kind and

*g*is an isometry, we have

*N*such that ${\u03f5}_{n}\le \u03f5$ for all $n\ge N$. Observe that

*x*, we get

which is a contradiction. Thus ${r}^{\mathrm{\prime}}=0$ and hence $\{{u}_{n}\}$ is convergent to the point *x*. This completes the proof. □

If *g* is the identity mapping in Theorem 3.3, then we obtain the following.

**Corollary 3.4**

*Let*$(X,d)$

*be a complete metric space and let*

*A*,

*B*

*be nonempty closed subsets of*

*X*.

*Further*,

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

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

*are nonempty*.

*Let*$S:A\to B$, $T:B\to A$

*and*$g:A\cup B\to A\cup B$

*be the mappings satisfying the following conditions*:

- (a)
*S**and**T**are Geraghty’s proximal contractions of the first kind*; - (b)
$S({A}_{0})\subseteq {B}_{0}$, $T({B}_{0})\subseteq {A}_{0}$;

- (c)
*the pair*$(S,T)$*is a proximal cyclic contraction*.

*Then there exists a unique point*$x\in A$

*and there exists a unique point*$y\in B$

*such that*

If we take $\beta (t)=k$, where $0\le k<1$, we obtain the following corollary.

**Corollary 3.5** [26]

*Let*$(X,d)$

*be a complete metric space and let*

*A*,

*B*

*be nonempty closed subsets of*

*X*.

*Further*,

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

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

*are nonempty*.

*Let*$S:A\to B$, $T:B\to A$

*and*$g:A\cup B\to A\cup B$

*be the mappings satisfying the following conditions*:

- (a)
*S**and**T**are proximal contractions of the first kind*; - (b)
*g**is an isometry*; - (c)
*the pair*$(S,T)$*is a proximal cyclic contraction*; - (d)
$S({A}_{0})\subseteq {B}_{0}$, $T({B}_{0})\subseteq {A}_{0}$;

- (e)
${A}_{0}\subseteq g({A}_{0})$

*and*${B}_{0}\subseteq g({B}_{0})$.

*Then there exists a unique point*$x\in A$

*and there exists a unique point*$y\in B$

*such that*

*Moreover*,

*for any fixed*${x}_{0}\in {A}_{0}$,

*the sequence*$\{{x}_{n}\}$

*defined by*

*converges to the element*

*x*.

*For any fixed*${y}_{0}\in {B}_{0}$,

*the sequence*$\{{y}_{n}\}$

*defined by*

*converges to the element* *y*.

If *g* is the identity mapping in Corollary 3.5, we obtain the following corollary.

**Corollary 3.6**

*Let*$(X,d)$

*be a complete metric space and let*

*A*,

*B*

*be nonempty closed subsets of*

*X*.

*Further*,

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

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

*are nonempty*.

*Let*$S:A\to B$, $T:B\to A$

*and*$g:A\cup B\to A\cup B$

*be the mappings satisfying the following conditions*:

- (a)
*S**and**T**are proximal contractions of the first kind*; - (b)
$S({A}_{0})\subseteq {B}_{0}$, $T({B}_{0})\subseteq {A}_{0}$;

- (c)
*the pair*$(S,T)$*is a proximal cyclic contraction*.

*Then there exists a unique point*$x\in A$

*and there exists a unique point*$y\in B$

*such that*

Next, we establish a best proximity point theorem for nonself-mappings which are Geraghty’s proximal contractions of the first kind and the second kind.

**Theorem 3.7**

*Let*$(X,d)$

*be a complete metric space and let*

*A*,

*B*

*be nonempty closed subsets of*

*X*.

*Further*,

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

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

*are nonempty*.

*Let*$S:A\to B$

*and*$g:A\to A$

*be the mappings satisfying the following conditions*:

- (a)
*S**is Geraghty’s proximal contraction of the first and second kinds*; - (b)
*g**is an isometry*; - (c)
*S**preserves isometric distance with respect to**g*; - (d)
$S({A}_{0})\subseteq {B}_{0}$;

- (e)
${A}_{0}\subseteq g({A}_{0})$.

*Then there exists a unique point*$x\in A$

*such that*

*Moreover*,

*for any fixed*${x}_{0}\in {A}_{0}$,

*the sequence*$\{{x}_{n}\}$

*defined by*

*converges to the element* *x*.

*On the other hand*,

*a sequence*$\{{u}_{n}\}$

*in*

*A*

*converges to*

*x*

*if there exists a sequence*$\{{\u03f5}_{n}\}$

*of positive numbers such that*

*where* ${z}_{n+1}\in A$ *satisfies the condition that* $d(g{z}_{n+1},S{u}_{n})=d(A,B)$.

*Proof*Since $S({A}_{0})\subseteq {B}_{0}$ and ${A}_{0}\subseteq g({A}_{0})$, as in the proof of Theorem 3.3, we can construct the sequence $\{{x}_{n}\}$ in ${A}_{0}$ such that

*g*is an isometry and

*S*is Geraghty’s proximal contraction of the first kind, we see that

*S*is Geraghty’s proximal contraction of the second kind and preserves the isometric distance with respect to

*g*, we have

*g*is an isometry, we have $d(x,z)=d(gx,gz)=0$. Hence $x=z$ and so $x\in {A}_{0}$. Since $S({A}_{0})\subseteq {B}_{0}$, there exists $u\in A$ such that

*S*is Geraghty’s proximal contraction of the first kind, it follows from (3.16) and (3.19) that

for all $n\ge 1$. Taking $n\to \mathrm{\infty}$ in (3.20), it follows that the sequence $\{g{x}_{n}\}$ converges to a point *u*. Since *g* is continuous and ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$, we have $g{x}_{n}\to gx$ as $n\to \mathrm{\infty}$. By the uniqueness of the limit, we conclude that $u=gx$. Therefore, it follows that $d(gx,Sx)=d(u,Sx)=d(A,B)$.

The uniqueness and the remaining part of the proof follow from the proof of Theorem 3.3. This completes the proof. □

If *g* is the identity mapping in Theorem 3.7, then we obtain the following.

**Corollary 3.8**

*Let*$(X,d)$

*be a complete metric space and let*

*A*,

*B*

*be nonempty closed subsets of*

*X*.

*Further*,

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

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

*are nonempty*.

*Let*$S:A\to B$

*be the mappings satisfying the following conditions*:

- (a)
*S**is Geraghty’s proximal contraction of the first and second kinds*; - (b)
$S({A}_{0})\subseteq {B}_{0}$.

*Then there exists a unique point*$x\in A$

*such that*

*Moreover*,

*for any fixed*${x}_{0}\in {A}_{0}$,

*the sequence*$\{{x}_{n}\}$

*defined by*

*converges to the best proximity point* *x* *of* *S*.

If we take $\beta (t)=k$ in Theorem 3.7, where $0\le k<1$, we obtain the following.

**Corollary 3.9** [26]

*Let*$(X,d)$

*be a complete metric space and let*

*A*,

*B*

*be nonempty closed subsets of*

*X*.

*Further*,

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

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

*are nonempty*.

*Let*$S:A\to B$

*and*$g:A\to A$

*be the mappings satisfying the following conditions*:

- (a)
*S**is a proximal contraction of the first and second kinds*; - (b)
*g**is an isometry*; - (c)
*S**preserves isometric distance with respect to**g*; - (d)
$S({A}_{0})\subseteq {B}_{0}$;

- (e)
${A}_{0}\subseteq g({A}_{0})$.

*Then there exists a unique point*$x\in A$

*such that*

*Moreover*,

*for any fixed*${x}_{0}\in {A}_{0}$,

*the sequence*$\{{x}_{n}\}$

*defined by*

*converges to the element* *x*.

If *g* is the identity mapping in Corollary 3.9, then we obtain the following.

**Corollary 3.10**

*Let*$(X,d)$

*be a complete metric space and let*

*A*,

*B*

*be nonempty closed subsets of*

*X*.

*Further*,

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

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

*are nonempty*.

*Let*$S:A\to B$

*be a mapping satisfying the following conditions*:

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

*Then there exists a unique point*$x\in A$

*such that*

*Moreover*,

*for any fixed*${x}_{0}\in {A}_{0}$,

*the sequence*$\{{x}_{n}\}$

*defined by*

*converges to the best proximity point* *x* *of* *S*.

## 4 Examples

Next, we give an example to show that Definition 3.1 is different from Definition 2.1; moreover, we give an example which supports Theorem 3.3. First, we give some proposition for our example as follows.

**Proposition 4.1**

*Let*$f:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$

*be a function defined by*$f(t)=ln(1+t)$.

*Then we have the following inequality*:

*for all* $a,b\in [0,\mathrm{\infty})$.

*Proof*If $x=y$, we have done. Suppose that $x>y$. Then since we have

it follows that $ln(1+x)-ln(1+y)<ln(1+|x-y|)$. In the case $x<y$, by a similar argument, we can prove that inequality (4.1) holds. □

**Proposition 4.2**

*For each*$x,y\in \mathbb{R}$,

*we have that the following inequality holds*:

*Proof*Since

□

**Example 4.3**Consider the complete metric space ${\mathbb{R}}^{2}$ with Euclidean metric. Let

*S*is Geraghty’s proximal contractions of the first kind with $\beta \in \mathcal{S}$ defined by

*A*satisfying

Thus *S* is Geraghty’s proximal contraction of the first kind.

*S*is not a proximal contraction of the first kind. Suppose

*S*is a proximal contraction of the first kind, then for each $(0,{x}^{\ast}),(0,{y}^{\ast}),(0,{a}^{\ast}),(0,{b}^{\ast})\in A$ satisfying

which is a contradiction. Thus *S* is not a proximal contraction of the first kind.

**Example 4.4**Consider the complete metric space ${\mathbb{R}}^{2}$ with metric defined by

Then $d(A,B)=2$, ${A}_{0}=A$, ${B}_{0}=B$ and the mapping *g* is an isometry.

*S*and

*T*are Geraghty’s proximal contractions of the first kind with $\beta \in \mathcal{S}$ defined by

*A*satisfying

*S*is Geraghty’s proximal contraction of the first kind. Similarly, we can see that

*T*is Geraghty’s proximal contraction of the first kind. Next, we show that the pair $(S,T)$ is a proximal cyclic contraction. Let $(0,u),(0,x)\in A$ and $(2,v),(2,y)\in B$ be such that

## 5 Conclusions

This article has investigated the existence of an optimal approximate solution, the so-called best proximity point, for the generalized notion of proximal contractions of the first and second kinds, which were defined by Sadiq Basha in [26]. Furthermore, an algorithm for computing such an optimal approximate solution and example which supports our main results have been presented.

## Declarations

### Acknowledgements

Mr. Chirasak Mongkolkeha was supported from the Thailand Research Fund through the Royal Golden Jubilee Program under Grant PHD/0029/2553 for the Ph.D. Program at KMUTT, Thailand. This research was partially finished at Department of Mathematics Education, Gyeongsang National University, Republic of Korea. Also, the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (Grant No. 2012-0008170). The third author was supported by the Commission on Higher Education, the Thailand Research Fund and the King Mongkut’s University of Technology Thonburi (Grant No. MRG5580213).

## Authors’ Affiliations

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