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

- Jingling Zhang
^{1}, - Yongfu Su
^{1}Email author and - Qingqing Cheng
^{1}

**2013**:99

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

© Zhang et al.; licensee Springer 2013

**Received: **3 January 2013

**Accepted: **26 March 2013

**Published: **12 April 2013

## 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.

## Keywords

*P*-property

## 1 Introduction and preliminaries

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

*Geraghty-contraction*if there exists such that for any $x,y\in X$

**Theorem 1.2** ([1])

*Let*$(X,d)$

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

*be an operator*.

*Suppose that there exists*

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

*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** ([2])

*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 such that for any $x,y\in A$

Now we need the following notations and basic facts.

*A*and

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

where $d(A,B)=inf\{d(x,y):x\in A\text{and}y\in B\}$.

In [3], the authors give sufficient conditions for when the sets ${A}_{0}$ and ${B}_{0}$ are nonempty. In [4], the author presents the following definition and proves that any pair $(A,B)$ of nonempty, closed and convex subsets of a real Hilbert space *H* satisfies the *P*-property.

**Definition 1.4** ([2])

*P-property*if and only if for any ${x}_{1},{x}_{2}\in {A}_{0}$ and ${y}_{1},{y}_{2}\in {B}_{0}$,

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 $d({x}_{0},T{x}_{0})=min\{d(x,Tx):x\in A\}$. Since $d(x,Tx)\ge d(A,B)$ for any $x\in A$, in fact, the optimal solution to this problem is the one for which the value $d(A,B)$ is attained.

In [2], the authors give a generalization of Theorem 1.2 by considering a nonself map and they get the following theorem.

**Theorem 1.5** ([2])

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

**Remark** In [2], 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 $(A,B)$ be a pair of nonempty subsets of a metric space $(X,d)$ with ${A}_{0}\ne \mathrm{\varnothing}$. Then the pair $(A,B)$ is said to have the

*weak P-property*if and only if for any ${x}_{1},{x}_{2}\in {A}_{0}$ and ${y}_{1},{y}_{2}\in {B}_{0}$,

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

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

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

*P*-property that

as $n,m\to \mathrm{\infty}$, where ${x}_{n},{x}_{m}\in {A}_{0}$ and $d({x}_{n},{y}_{n})=d(A,B)$, $d({x}_{m},{y}_{m})=d(A,B)$. Then $\{{x}_{n}\}$ is a Cauchy sequence so that $\{{x}_{n}\}$ converges strongly to a point $p\in A$. By the continuity of metric *d* we have $d(p,q)=d(A,B)$, 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 $T({\overline{A}}_{0})\subseteq {B}_{0}$. In fact, if $x\in {\overline{A}}_{0}\setminus {A}_{0}$, then there exists a sequence $\{{x}_{n}\}\subseteq {A}_{0}$ 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 $T({\overline{A}}_{0})\subseteq {B}_{0}$.

*P*-property and

*T*is a Geraghty-contraction, we have

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

**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 [2]. In fact, our proof process is less than one page. However, the proof process in [2] is three pages.

## 3 Example

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

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

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

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

## References

- Geraghty M: On contractive mappings.
*Proc. Am. Math. Soc.*1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5MATHMathSciNetView ArticleGoogle Scholar - Caballero J,
*et al*.: A best proximity point theorem for Geraghty-contractions.*Fixed Point Theory Appl.*2012. doi:10.1186/1687–1812–2012–231Google 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-120026380MATHMathSciNetView ArticleGoogle Scholar - Sankar Raj, V: Banach contraction principle for non-self mappings. PreprintGoogle Scholar

## Copyright

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