# A note on ‘A best proximity point theorem for Geraghty-contractions’

## Abstract

In Caballero et al. (Fixed Point Theory Appl. (2012). doi:10.1186/1687-1812-2012-231), the authors prove a best proximity point theorem for Geraghty nonself contraction. In this note, not only P-property has been weakened, but also an improved best proximity point theorem will be presented by a short and simple proof. An example which satisfies weak P-property but not P-property has been presented to demonstrate our results.

MSC:47H05, 47H09, 47H10.

## 1 Introduction and preliminaries

Let A and B be nonempty subsets of a metric space $\left(X,d\right)$. An operator $T:A\to B$ is said to be contractive if there exists $k\in \left[0,1\right)$ such that $d\left(Tx,Ty\right)\le kd\left(x,y\right)$ for any $x,y\in A$. The well-known Banach contraction principle says: Let $\left(X,d\right)$ be a complete metric space, and $T:X\to X$ be a contraction of X into itself. Then T has a unique fixed point in X.

In 1973, Geraghty introduced the Geraghty-contraction and obtained Theorem 1.2.

Definition 1.1 ()

Let $\left(X,d\right)$ be a metric space. A mapping $T:X\to X$ is said to be a Geraghty-contraction if there exists such that for any $x,y\in X$

$d\left(Tx,Ty\right)\le \beta \left(d\left(x,y\right)\right)\cdot d\left(x,y\right),$

where the class Γ denotes those functions $\beta :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ satisfying the following condition:

$\beta \left({t}_{n}\right)\to 1\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}{t}_{n}\to 0.$

Theorem 1.2 ()

Let $\left(X,d\right)$ be a complete metric space and $T:X\to X$ be an operator. Suppose that there exists such that for any $x,y\in X$,

$d\left(Tx,Ty\right)\le \beta \left(d\left(x,y\right)\right)\cdot d\left(x,y\right).$

Then T has a unique fixed point.

Obviously, Theorem 1.2 is an extensive version of Banach contraction principle. In 2012, Caballero et al. introduced generalized Geraghty-contraction as follows.

Definition 1.3 ()

Let A, B be two nonempty subsets of a metric space $\left(X,d\right)$. A mapping $T:A\to B$ is said to be a Geraghty-contraction if there exists such that for any $x,y\in A$

$d\left(Tx,Ty\right)\le \beta \left(d\left(x,y\right)\right)\cdot d\left(x,y\right),$

where the class denotes those functions $\beta :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)$ satisfying the following condition:

$\beta \left({t}_{n}\right)\to 1\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}{t}_{n}\to 0.$

Now we need the following notations and basic facts.

Let A and B be two nonempty subsets of a metric space $\left(X,d\right)$. We denote by ${A}_{0}$ and ${B}_{0}$ the following sets: where .

In , the authors give sufficient conditions for when the sets ${A}_{0}$ and ${B}_{0}$ are nonempty. In , the author presents the following definition and proves that any pair $\left(A,B\right)$ of nonempty, closed and convex subsets of a real Hilbert space H satisfies the P-property.

Definition 1.4 ()

Let $\left(A,B\right)$ be a pair of nonempty subsets of a metric space $\left(X,d\right)$ with ${A}_{0}\ne \mathrm{\varnothing }$. Then the pair $\left(A,B\right)$ is said to have the P-property if and only if for any ${x}_{1},{x}_{2}\in {A}_{0}$ and ${y}_{1},{y}_{2}\in {B}_{0}$,

$\left\{\begin{array}{c}d\left({x}_{1},{y}_{1}\right)=d\left(A,B\right),\hfill \\ d\left({x}_{2},{y}_{2}\right)=d\left(A,B\right)\hfill \end{array}⇒\phantom{\rule{1em}{0ex}}d\left({x}_{1},{x}_{2}\right)=d\left({y}_{1},{y}_{2}\right).$

Let A, B be two nonempty subsets of a complete metric space and consider a mapping $T:A\to B$. The best proximity point problem is whether we can find an element ${x}_{0}\in A$ such that $d\left({x}_{0},T{x}_{0}\right)=min\left\{d\left(x,Tx\right):x\in A\right\}$. Since $d\left(x,Tx\right)\ge d\left(A,B\right)$ for any $x\in A$, in fact, the optimal solution to this problem is the one for which the value $d\left(A,B\right)$ is attained.

In , the authors give a generalization of Theorem 1.2 by considering a nonself map and they get the following theorem.

Theorem 1.5 ()

Let $\left(A,B\right)$ be a pair of nonempty closed subsets of a complete metric space $\left(X,d\right)$ such that ${A}_{0}$ is nonempty. Let $T:A\to B$ be a Geraghty-contraction satisfying $T\left({A}_{0}\right)\subseteq {B}_{0}$. Suppose that the pair $\left(A,B\right)$ has the P-property. Then there exists a unique ${x}^{\ast }$ in A such that $d\left({x}^{\ast },T{x}^{\ast }\right)=d\left(A,B\right)$.

Remark In , the proof of Theorem 1.5 is unnecessarily complex. In this note, not only P-property has been weakened, but also an improved best proximity point theorem will be presented by a short and simple proof. An example which satisfies weak P-property but not P-property has been presented to demonstrate our results.

## 2 Main results

Before giving our main results, we first introduce the notion of weak P-property.

Weak P-property Let $\left(A,B\right)$ be a pair of nonempty subsets of a metric space $\left(X,d\right)$ with ${A}_{0}\ne \mathrm{\varnothing }$. Then the pair $\left(A,B\right)$ is said to have the weak P-property if and only if for any ${x}_{1},{x}_{2}\in {A}_{0}$ and ${y}_{1},{y}_{2}\in {B}_{0}$,

$\left\{\begin{array}{c}d\left({x}_{1},{y}_{1}\right)=d\left(A,B\right),\hfill \\ d\left({x}_{2},{y}_{2}\right)=d\left(A,B\right)\hfill \end{array}⇒\phantom{\rule{1em}{0ex}}d\left({x}_{1},{x}_{2}\right)\le d\left({y}_{1},{y}_{2}\right).$

Now we are in a position to give our main results.

Theorem 2.1 Let $\left(A,B\right)$ be a pair of nonempty closed subsets of a complete metric space $\left(X,d\right)$ such that ${A}_{0}\ne \mathrm{\varnothing }$. Let $T:A\to B$ be a Geraghty-contraction satisfying $T\left({A}_{0}\right)\subseteq {B}_{0}$. Suppose that the pair $\left(A,B\right)$ has the weak P-property. Then there exists a unique ${x}^{\ast }$ in A such that $d\left({x}^{\ast },T{x}^{\ast }\right)=d\left(A,B\right)$.

Proof We first prove that ${B}_{0}$ is closed. Let $\left\{{y}_{n}\right\}\subseteq {B}_{0}$ be a sequence such that ${y}_{n}\to q\in B$. It follows from the weak P-property that

$d\left({y}_{n},{y}_{m}\right)\to 0\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}d\left({x}_{n},{x}_{m}\right)\to 0,$

as $n,m\to \mathrm{\infty }$, where ${x}_{n},{x}_{m}\in {A}_{0}$ and $d\left({x}_{n},{y}_{n}\right)=d\left(A,B\right)$, $d\left({x}_{m},{y}_{m}\right)=d\left(A,B\right)$. Then $\left\{{x}_{n}\right\}$ is a Cauchy sequence so that $\left\{{x}_{n}\right\}$ converges strongly to a point $p\in A$. By the continuity of metric d we have $d\left(p,q\right)=d\left(A,B\right)$, that is, $q\in {B}_{0}$, and hence ${B}_{0}$ is closed.

Let ${\overline{A}}_{0}$ be the closure of ${A}_{0}$, we claim that $T\left({\overline{A}}_{0}\right)\subseteq {B}_{0}$. In fact, if $x\in {\overline{A}}_{0}\setminus {A}_{0}$, then there exists a sequence $\left\{{x}_{n}\right\}\subseteq {A}_{0}$ such that ${x}_{n}\to x$. By the continuity of T and the closeness of ${B}_{0}$, we have $Tx={lim}_{n\to \mathrm{\infty }}T{x}_{n}\in {B}_{0}$. That is $T\left({\overline{A}}_{0}\right)\subseteq {B}_{0}$.

Define an operator ${P}_{{A}_{0}}:T\left({\overline{A}}_{0}\right)\to {A}_{0}$, by ${P}_{{A}_{0}}y=\left\{x\in {A}_{0}:d\left(x,y\right)=d\left(A,B\right)\right\}$. Since the pair $\left(A,B\right)$ has weak P-property and T is a Geraghty-contraction, we have

$d\left({P}_{{A}_{0}}T{x}_{1},{P}_{{A}_{0}}T{x}_{2}\right)\le d\left(T{x}_{1},T{x}_{2}\right)\le \beta \left(d\left({x}_{1},{x}_{2}\right)\right)\cdot d\left({x}_{1},{x}_{2}\right)$

for any ${x}_{1},{x}_{2}\in {\overline{A}}_{0}$. This shows that ${P}_{{A}_{0}}T:{\overline{A}}_{0}\to {\overline{A}}_{0}$ is a Geraghty-contraction from complete metric subspace ${\overline{A}}_{0}$ into itself. Using Theorem 1.2, we can get ${P}_{{A}_{0}}T$ has a unique fixed point ${x}^{\ast }$. That is ${P}_{{A}_{0}}T{x}^{\ast }={x}^{\ast }\in {A}_{0}$. It implies that

$d\left({x}^{\ast },T{x}^{\ast }\right)=d\left(A,B\right).$

Therefore, ${x}^{\ast }$ is the unique one in ${A}_{0}$ such that $d\left({x}^{\ast },T{x}^{\ast }\right)=d\left(A,B\right)$. It is easy to see that ${x}^{\ast }$ is also the unique one in A such that $d\left({x}^{\ast },T{x}^{\ast }\right)=d\left(A,B\right)$. □

Remark In Theorem 2.1, P-property is weakened to weak P-property. Therefore, Theorem 2.1 is an improved result of Theorem 1.5. In addition, our proof is shorter and simpler than that in . In fact, our proof process is less than one page. However, the proof process in  is three pages.

## 3 Example

Now we present an example which satisfies weak P-property but not P-property.

Consider $\left({R}^{2},d\right)$, where d is the Euclidean distance and the subsets $A=\left\{\left(0,0\right)\right\}$ and $B=\left\{y=1+\sqrt{1-{x}^{2}}\right\}$.

Obviously, ${A}_{0}=\left\{\left(0,0\right)\right\}$, ${B}_{0}=\left\{\left(-1,1\right),\left(1,1\right)\right\}$ and $d\left(A,B\right)=\sqrt{2}$. Furthermore,

$d\left(\left(0,0\right),\left(-1,1\right)\right)=d\left(\left(0,0\right),\left(1,1\right)\right)=\sqrt{2},$

however,

$0=d\left(\left(0,0\right),\left(0,0\right)\right)

We can see that the pair $\left(A,B\right)$ satisfies the weak P-property but not the P-property.

## References

1. Geraghty M: On contractive mappings. Proc. Am. Math. Soc. 1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5

2. Caballero J, et al.: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. 2012. doi:10.1186/1687–1812–2012–231

3. Kirk WA, Reich S, Veeramani P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 2003, 24: 851–862. 10.1081/NFA-120026380

4. Sankar Raj, V: Banach contraction principle for non-self mappings. Preprint

## Acknowledgements

This project is supported by the National Natural Science Foundation of China under grant (11071279).

## Author information

Authors

### Corresponding author

Correspondence to Yongfu Su.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All the authors contributed equally to the writing of the present article. All authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

Zhang, J., Su, Y. & Cheng, Q. A note on ‘A best proximity point theorem for Geraghty-contractions’. Fixed Point Theory Appl 2013, 99 (2013). https://doi.org/10.1186/1687-1812-2013-99

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1812-2013-99

### Keywords

• Geraghty-contractions
• fixed point
• best proximity point
• weak P-property 