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# A short note on the equivalence between ‘best proximity’ points and ‘fixed point’ results

- Mohamed Jleli
^{1}, - Erdal Karapınar
^{2}Email author and - Bessem Samet
^{1}

**2014**:246

https://doi.org/10.1186/1029-242X-2014-246

© Jleli et al.; licensee Springer 2014

**Received:**30 April 2014**Accepted:**4 June 2014**Published:**23 June 2014

## Abstract

In this short note, we notice that, unexpectedly, some existing fixed point results and recently announced best proximity point results are equivalent.

**MSC:**41A65, 90C30, 47H10.

## Keywords

- best proximity
*h*-orbitally complete- fixed point

## 1 Introduction and preliminaries

*S*of functions $\beta :[0,\mathrm{\infty})\to [0,1)$ satisfying the following condition:

The author defined contraction mappings via functions from this class and proved the following result.

**Theorem 1.1** (Geraghty [1])

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be an operator*.

*If*

*T*

*satisfies the following inequality*:

*where* $\beta \in S$, *then* *T* *has a unique fixed point*.

Theorem 1.1 was generalized in several ways, see *e.g.* [2–6]. Recently, Caballero *et al.* [2] introduced the following contraction.

**Definition 1.1** ([7])

*A*,

*B*be two nonempty subsets of a metric space $(X,d)$. A mapping $T:A\to B$ is said to be a Geraghty-contraction if there exists $\beta \in S$ such that

For the sake of completeness, we recall some basic definitions and fundamental results.

*X*. We consider the following notations:

Through this paper, ℕ denotes the set of natural numbers.

In [8], Sadiq Basha introduced the following concept.

**Definition 1.2** We say that *B* is approximatively compact with respect to *A* if and only if every sequence $\{{y}_{n}\}\subset B$ satisfying the condition that ${lim}_{n\to \mathrm{\infty}}d(x,{y}_{n})=d(x,B)$ for some *x* in *A*, has a convergent subsequence.

**Definition 1.3**A mapping $g:A\to A$ is called an isometry if

**Definition 1.4** (see *e.g.* [9])

*T*is said to preserve the isometric distance with respect to

*g*if and only if

- (I)
*φ*is continuous and nondecreasing; - (II)
$\phi (0)=0$;

- (III)
${lim}_{t\to \mathrm{\infty}}\phi (t)=\mathrm{\infty}$.

The following notions were introduced by Sadiq Basha [9].

**Definition 1.5**A mapping $T:A\to B$ is said to be a generalized proximal contraction of the first kind if and only if

where $x,y,u,v\in A$ and $\phi \in \mathrm{\Xi}$.

**Definition 1.6**A mapping $T:A\to B$ is said to be a generalized proximal contraction of the second kind if and only if

where $x,y,u,v\in A$ and $\phi \in \mathrm{\Xi}$.

Inspired by these definitions, Amini-Harandi [3] introduced the following definition.

- (I)
*ψ*is continuous and nondecreasing; - (II)
$\psi (0)=0$;

- (III)
$t\le \psi (t)$ for each $t\ge 0$.

**Definition 1.7**A mapping $T:A\to B$ is said to be a generalized Geraghty proximal contraction of the first kind if and only if

where $x,y,u,v\in A$ and $\psi \in \mathrm{\Psi}$, $\beta \in S$.

**Definition 1.8**A mapping $T:A\to B$ is said to be a generalized Geraghty proximal contraction of the second kind if and only if

where $x,y,u,v\in A$ and $\psi \in \mathrm{\Psi}$, $\beta \in S$.

The main result in [3] is the following.

**Theorem 1.2**

*Let*

*A*

*and*

*B*

*be two nonempty closed subsets of a complete metric space*$(X,d)$

*such that*${A}_{0}\ne \mathrm{\varnothing}$, ${B}_{0}\ne \mathrm{\varnothing}$

*and*

*B*

*is approximatively compact with respect to*

*A*.

*Suppose that the mappings*$g:A\to A$

*and*$T:A\to B$

*satisfy the following conditions*:

- (i)
*T**is a generalized Geraghty proximal contraction of the first kind*; - (ii)
$T({A}_{0})\subseteq {B}_{0}$;

- (iii)
*g**is an isometry*; - (iv)
${A}_{0}\subseteq g({A}_{0})$.

*Then there exists a unique element*${x}^{\ast}\in A$

*such that*

*Further*,

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

*the iterative sequence*$\{{x}_{n}\}\subset {A}_{0}$,

*defined by*

*converges to* ${x}^{\ast}$.

In this manuscript, we shall show that Theorem 1.2 is a particular case of existing fixed point theorems in the literature. Hence, the main result of [3] is not a real generalization.

## 2 Some useful results

- (I)
*ϕ*is continuous and nondecreasing; - (II)
$\varphi (t)=0$ if and only if $t=0$.

First we show that we get the more general form of the main result in [3] by replacing the class of distance functions Ψ by Φ in Definition 1.7.

**Theorem 2.1**

*Let*

*A*

*and*

*B*

*be two nonempty closed subsets of a complete metric space*$(X,d)$

*such that*${A}_{0}\ne \mathrm{\varnothing}$, ${B}_{0}\ne \mathrm{\varnothing}$,

*and*

*B*

*is approximatively compact with respect to*

*A*.

*Suppose that the mappings*$g:A\to A$

*and*$T:A\to B$

*satisfy the following conditions*:

- (i)$\begin{array}{r}d(u,Tx)=d(A,B)\\ d(v,Ty)=d(A,B)\end{array}\}\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\psi (d(u,v))\le \beta (d(x,y))\psi (d(x,y)),$

*where*$x,y,u,v\in A$

*and*$\psi \in \mathrm{\Phi}$, $\beta \in S$.

- (ii)
$T({A}_{0})\subseteq {B}_{0}$;

- (iii)
*g**is an isometry*; - (iv)
${A}_{0}\subseteq g({A}_{0})$.

*Then there exists a unique element*${x}^{\ast}\in A$

*such that*

*Further*,

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

*the iterative sequence*$\{{x}_{n}\}\subset {A}_{0}$,

*defined by*

*converges to* ${x}^{\ast}$.

*Proof*By following the lines in the proof of Theorem 3.1 in the paper of Amini-Harandi [3], we conclude that

It is sufficient to prove that $\{{x}_{n}\}$ is a Cauchy sequence.

*ψ*is continuous and (7) holds. Due to the property of

*ψ*, $\psi (\epsilon )>0$, we derive

Due to (7) we get $\epsilon =0$, a contradiction. Hence, $\{{x}_{n}\}$ is Cauchy.

The rest follows from the corresponding lines in the proof of Theorem 3.1 in the paper of Amini-Harandi [3]. □

The following theorem is due to [10].

**Theorem 2.2**

*Let*$(X,d)$

*be a complete metric space and*$T:X\to X$

*be an operator*.

*If*

*T*

*satisfies the following inequality*:

*where* $\beta \in S$ *and* $\psi \in \mathrm{\Phi}$, *then* *T* *has a unique fixed point*.

The following concept was introduced by Ćirić in [11].

**Definition 2.1** Let $(X,d)$ be a metric space and $f:X\to X$ be a self-mapping. We say that *X* is *f*-orbitally complete if and only if for any $x\in X$, if $\{{f}^{n}x\}$ is a Cauchy sequence, then it converges to some element in *X*.

It is evident that in Theorem 2.2, the notion of completeness of the metric space $(X,d)$ can be replaced by the notion of *f*-orbitally completeness. Consequently, we derive the following fixed point result.

**Lemma 2.1** (*cf.* [7])

*Let*$(X,d)$

*be a*

*f*-

*orbitally complete metric space*,

*where*$f:X\to X$

*is a self*-

*mapping satisfying the following condition*:

*where* $\psi \in \mathrm{\Psi}$, $\beta \in S$. *Then* *f* *has a unique fixed point* ${x}^{\ast}\in X$. *Moreover*, *for any* $x\in X$, *the sequence* $\{{f}^{n}x\}$ *converges to* ${x}^{\ast}$.

Regarding the analogy with the proof of Lemma 2.2 in [7].

**Lemma 2.2** (*cf.* [7])

*Let*$(A,B)$

*be a pair of closed subsets of a metric space*$(X,d)$.

*Suppose that the following conditions hold*:

- (i)
${A}_{0}\ne \mathrm{\varnothing}$;

- (ii)
*B**is approximatively compact with respect to**A*.

*Then the set* ${A}_{0}$ *is closed*.

**Lemma 2.3** (*cf.* [7])

*Let*

*A*

*and*

*B*

*be nonempty subsets of a metric space*$(X,d)$

*such that*${A}_{0}\ne \mathrm{\varnothing}$.

*Suppose that the mappings*$g:A\to A$

*and*$T:A\to B$

*satisfy the following conditions*:

- (i)
*T**is a generalized Geraghty*-*proximal contraction of the first kind*; - (ii)
$T({A}_{0})\subseteq {B}_{0}$;

- (iii)
*g**is an isometry*; - (iv)
${A}_{0}\subseteq g({A}_{0})$.

*Then there exists a self*-

*mapping*$f:{A}_{0}\to {A}_{0}$

*satisfying the condition*:

Regarding the analogy with the proof of Lemma 2.4 in [7].

**Lemma 2.4** ([12])

*Let*$(X,d)$

*be a metric space and let*$({x}_{n})$

*be a sequence in*

*X*

*such that*$(d({x}_{n+1},{x}_{n}))$

*is non*-

*increasing and*

*If*$({x}_{n})$

*is not a Cauchy sequence*,

*then there exist*$\epsilon >0$

*and two sequences*$({m}_{k})$

*and*$({n}_{k})$

*of positive integers such that*${m}_{k}>{n}_{k}\ge k$

*and the following four sequences tend to*

*ε*

*when*$k\to +\mathrm{\infty}$:

## 3 Main results

**Theorem 3.1** *Theorem * 2.1 *is a consequence of Theorem * 2.2.

*Proof*Suppose that all the assumptions of Theorem 1.2 are satisfied. From Lemma 2.3, there exists a self-mapping $f:{A}_{0}\to {A}_{0}$ satisfying (13). Then, for all $(x,y)\in {A}_{0}\times {A}_{0}$, we have

*T*is a generalized Geraghty-proximal contraction of the first kind and

*g*is an isometry, we obtain

for every pair $(x,y)\in {A}_{0}\times {A}_{0}$. Thus *f* satisfies inequality (11).

Since $(X,d)$ is complete, $({A}_{0},d)$ is also complete. From Theorem 2.2, the self-mapping $h:{A}_{0}\to {A}_{0}$ has a unique fixed point ${x}^{\ast}\in {A}_{0}$.

*T*is a generalized Geraghty-proximal contraction of the first kind and

*g*is an isometry, we have ${x}^{\ast}\in {A}_{0}$ is a fixed point of

*f*if and only if ${x}^{\ast}\in A$ and $d(g{x}^{\ast},T{x}^{\ast})=d(A,B)$. Then there exists a unique ${x}^{\ast}\in {A}_{0}$ such that $d(g{x}^{\ast},T{x}^{\ast})=d(A,B)$. Now, let $a\in {A}_{0}$ be an arbitrary point. Consider a sequence $\{{a}_{n}\}\subset {A}_{0}$ satisfying

*T*is a generalized Geraghty-proximal contraction of the first kind and

*g*is an isometry, it follows from (13) that

This ends the proof. □

## 4 Consequences

**Theorem 4.1**

*Let*

*A*

*and*

*B*

*be two nonempty closed subsets of a complete metric space*$(X,d)$

*such that*${A}_{0}\ne \mathrm{\varnothing}$

*and*${B}_{0}\ne \mathrm{\varnothing}$.

*Suppose that the mappings*$g:A\to A$

*and*$T:A\to B$

*satisfy the following conditions*:

- (i)
*T**is a generalized Geraghty proximal contraction of the first and second kind*,$\begin{array}{c}\begin{array}{r}d(u,Tx)=d(A,B)\\ d(v,Ty)=d(A,B)\end{array}\}\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\psi (d(u,v))\le \beta (d(x,y))\psi (d(x,y)),\hfill \\ \begin{array}{r}d(u,Tx)=d(A,B)\\ d(v,Ty)=d(A,B)\end{array}\}\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}d(Tu,Tv)\le \beta (d(Tx,Ty))\psi (d(Tx,Ty)),\hfill \end{array}$

*where*$x,y,u,v\in A$

*and*$\psi \in \mathrm{\Phi}$, $\beta \in S$;

- (ii)
$T({A}_{0})\subseteq {B}_{0}$;

- (iii)
*g**is an isometry*; - (iv)
${A}_{0}\subseteq g({A}_{0})$.

*Then there exists a unique element*${x}^{\ast}\in A$

*such that*

*Further*,

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

*the iterative sequence*$\{{x}_{n}\}\subset {A}_{0}$,

*defined by*

*converges to* ${x}^{\ast}$.

**Theorem 4.2** *Theorem * 4.1 *is a consequence of Lemma * 2.1.

*Proof*Suppose that all the assumptions of Theorem 4.1 are satisfied. From Lemma 2.3, there exists a self-mapping $f:{A}_{0}\to {A}_{0}$ satisfying (13). Then, for all $(x,y)\in {A}_{0}\times {A}_{0}$, we have

*T*is a generalized Geraghty-proximal contraction of the first kind and

*g*is an isometry, we obtain

for every pair $(x,y)\in {A}_{0}\times {A}_{0}$. Thus *f* satisfies inequality (11).

*f*-orbitally complete. Indeed, let ${x}_{0}\in {A}_{0}$ and consider the sequence $\{{x}_{n}\}\subset {A}_{0}$ defined by ${x}_{n}={f}^{n}{x}_{0}$ for all $n\in \mathbb{N}$. Suppose that $\{{x}_{n}\}$ is a Cauchy sequence, we have to prove that $\{{x}_{n}\}$ converges to some element in ${A}_{0}$. Since $(X,d)$ is complete and

*A*is closed, there exists some $z\in A$ such that

*f*, for all $n\in \mathbb{N}$, we have

*T*is a generalized Geraghty-proximal contraction of the second kind) that

*T*preserves the isometric distance with respect to

*g*, we obtain

*B*is closed, there exists some $b\in B$ such that

*f*, we get

*g*is an isometry, it is continuous. Now, we have

This implies that $gz\in {A}_{0}$. On the other hand, since ${A}_{0}\subseteq g({A}_{0})$ and *g* is an isometry, we obtain $z\in {A}_{0}$. Thus, we proved that ${A}_{0}$ is *f*-orbitally complete. Now, applying Lemma 2.1, we find that *f* has a unique fixed point ${x}^{\ast}\in {A}_{0}$.

The rest follows from the lines of the proof of Theorem 3.1. □

## Declarations

### Acknowledgements

The authors thank the Visiting Professor Programming at King Saud University for funding this work. The authors thank to anonymous referees for their remarkable comments, suggestion and ideas that helps to improve this paper.

## Authors’ Affiliations

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