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

*Fixed Point Theory and Applications*
**volume 2013**, Article number: 99 (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.

## 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])

Let (X,d) 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

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

**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])

Let *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

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

Now we need the following notations and basic facts.

Let *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])

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 *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}.

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

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

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}}\}.

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

however,

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

## References

Geraghty M: On contractive mappings.

*Proc. Am. Math. Soc.*1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5Caballero J,

*et al*.: A best proximity point theorem for Geraghty-contractions.*Fixed Point Theory Appl.*2012. doi:10.1186/1687–1812–2012–231Kirk WA, Reich S, Veeramani P: Proximinal retracts and best proximity pair theorems.

*Numer. Funct. Anal. Optim.*2003, 24: 851–862. 10.1081/NFA-120026380Sankar 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).

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

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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

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