- Research
- Open Access
- Published:

# A best proximity point theorem for Geraghty-contractions

*Fixed Point Theory and Applications*
**volume 2012**, Article number: 231 (2012)

## Abstract

The purpose of this paper is to provide sufficient conditions for the existence of a unique best proximity point for Geraghty-contractions.

Our paper provides an extension of a result due to Geraghty (Proc. Am. Math. Soc. 40:604-608, 1973).

## 1 Introduction

Let *A* and *B* be nonempty subsets of a metric space (X,d).

An operator T:A\to B is said to be a k-contraction if there exists k\in [0,1) such that d(Tx,Ty)\le kd(x,y) for any x,y\in A. Banach’s contraction principle states that when *A* is a complete subset of *X* and *T* is a k-contraction which maps *A* into itself, then *T* has a unique fixed point in *A*.

A huge number of generalizations of this principle appear in the literature. Particularly, the following generalization of Banach’s contraction principle is due to Geraghty [1].

First, we introduce the class ℱ of those functions \beta :[0,\mathrm{\infty})\to [0,1) satisfying the following condition:

**Theorem 1.1** ([1])

*Let* (X,d) *be a complete metric space and* T:X\to X *be an operator*. *Suppose that there exists* \beta \in \mathcal{F} *such that for any* x,y\in X,

*Then* *T* *has a unique fixed point*.

Since the constant functions f(t)=k, where k\in [0,1), belong to ℱ, Theorem 1.1 extends Banach’s contraction principle.

**Remark 1.1** Since the functions belonging to ℱ are strictly smaller than one, condition (1) implies that

Therefore, any operator T:X\to X satisfying (1) is a continuous operator.

The aim of this paper is to give a generalization of Theorem 1.1 by considering a non-self map *T*.

First, we present a brief discussion about a best proximity point.

Let *A* be a nonempty subset of a metric space (X,d) and T:A\to X be a mapping. The solutions of the equation Tx=x are fixed points of *T*. Consequently, T(A)\cap A\ne \mathrm{\varnothing} is a necessary condition for the existence of a fixed point for the operator *T*. If this necessary condition does not hold, then d(x,Tx)>0 for any x\in A and the mapping T:A\to X does not have any fixed point. In this setting, our aim is to find an element x\in A such that d(x,Tx) is minimum in some sense. The best approximation theory and best proximity point analysis have been developed in this direction.

In our context, we consider two nonempty subsets *A* and *B* of a complete metric space and a mapping T:A\to B.

A natural question is whether one 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, the optimal solution to this problem will be the one for which the value d(A,B) is attained by the real valued function \phi :A\to \mathbb{R} given by \phi (x)=d(x,Tx).

Some results about best proximity points can be found in [2–9].

## 2 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 [8], the authors present sufficient conditions which determine when the sets {A}_{0} and {B}_{0} are nonempty.

Now, we present the following definition.

**Definition 2.1** 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 \beta \in \mathcal{F} such that

Notice that since \beta :[0,\mathrm{\infty})\to [0,1), we have

Therefore, every Geraghty-contraction is a contractive mapping.

In [10], the author introduces the following definition.

**Definition 2.2** ([10])

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},

It is easily seen that for any nonempty subset *A* of (X,d), the pair (A,A) has the *P*-property.

In [10], the author proves that any pair (A,B) of nonempty closed convex subsets of a real Hilbert space *H* satisfies the *P*-property.

## 3 Main results

We start this section presenting our main result.

**Theorem 3.1** *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).

*Proof* Since {A}_{0} is nonempty, we take {x}_{0}\in A.

As T{x}_{0}\in T({A}_{0})\subseteq {B}_{0}, we can find {x}_{1}\in {A}_{0} such that d({x}_{1},T{x}_{0})=d(A,B). Similarly, since T{x}_{1}\in T({A}_{0})\subseteq {B}_{0}, there exists {x}_{2}\in {A}_{0} such that d({x}_{2},T{x}_{1})=d(A,B). Repeating this process, we can get a sequence ({x}_{n}) in {A}_{0} satisfying

Since (A,B) has the *P*-property, we have that

Taking into account that *T* is a Geraghty-contraction, for any n\in \mathbb{N}, we have that

Suppose that there exists {n}_{0}\in \mathbb{N} such that d({x}_{{n}_{0}},{x}_{{n}_{0}+1})=0.

In this case,

and consequently, T{x}_{{n}_{0}-1}=T{x}_{{n}_{0}}.

Therefore,

and this is the desired result.

In the contrary case, suppose that d({x}_{n},{x}_{n+1})>0 for any n\in \mathbb{N}.

By (2), (d({x}_{n},{x}_{n+1})) is a decreasing sequence of nonnegative real numbers, and hence there exists r\ge 0 such that

In the sequel, we prove that r=0.

Assume r>0, then from (2) we have

The last inequality implies that {lim}_{n\to \mathrm{\infty}}\beta (d({x}_{n-1},{x}_{n}))=1 and since \beta \in \mathcal{F}, we obtain r=0 and this contradicts our assumption.

Therefore,

Notice that since d({x}_{n+1},T{x}_{n})=d(A,B) for any n\in \mathbb{N}, for p,q\in \mathbb{N} fixed, we have d({x}_{p},T{x}_{p-1})=d({x}_{q},T{x}_{q-1})=d(A,B), and since (A,B) satisfies the *P*-property, d({x}_{p},{x}_{q})=d(T{x}_{p-1},T{x}_{q-1}).

In what follows, we prove that ({x}_{n}) is a Cauchy sequence.

In the contrary case, we have that

By using the triangular inequality,

By (2) and since d({x}_{n+1},{x}_{m+1})=d(T{x}_{n},T{x}_{m}), by the above mentioned comment, we have

which gives us

Since {lim\hspace{0.17em}sup}_{m,n\to \mathrm{\infty}}d({x}_{n},{x}_{m})>0 and by (3), {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=0, from the last inequality it follows that

Therefore, {lim\hspace{0.17em}sup}_{m,n\to \mathrm{\infty}}\beta (d({x}_{n},{x}_{m}))=1.

Taking into account that \beta \in \mathcal{F}, we get {lim\hspace{0.17em}sup}_{m,n\to \mathrm{\infty}}d({x}_{n},{x}_{m})=0 and this contradicts our assumption.

Therefore, ({x}_{n}) is a Cauchy sequence.

Since ({x}_{n})\subset A and *A* is a closed subset of the complete metric space (X,d), we can find {x}^{\ast}\in A such that {x}_{n}\to {x}^{\ast}.

Since any Geraghty-contraction is a contractive mapping and hence continuous, we have T{x}_{n}\to T{x}^{\ast}.

This implies that d({x}_{n+1},T{x}_{n})\to d({x}^{\ast},T{x}^{\ast}).

Taking into account that the sequence (d({x}_{n+1},T{x}_{n})) is a constant sequence with value d(A,B), we deduce

This means that {x}^{\ast} is a best proximity point of *T*.

This proves the part of existence of our theorem.

For the uniqueness, suppose that {x}_{1} and {x}_{2} are two best proximity points of *T* with {x}_{1}\ne {x}_{2}.

This means that

Using the *P*-property, we have

Using the fact that *T* is a Geraghty-contraction, we have

which is a contradiction.

Therefore, {x}_{1}={x}_{2}.

This finishes the proof. □

## 4 Examples

In order to illustrate our results, we present some examples.

**Example 4.1** Consider X={\mathbb{R}}^{2} with the usual metric.

Let *A* and *B* be the subsets of *X* defined by

Obviously, d(A,B)=1 and *A*, *B* are nonempty closed subsets of *X*.

Moreover, it is easily seen that {A}_{0}=A and {B}_{0}=B.

Let T:A\to B be the mapping defined as

In the sequel, we check that *T* is a Geraghty-contraction.

In fact, for (0,x),(0,y)\in A with x\ne y, we have

Now, we prove that

Suppose that x>y (the same reasoning works for y>x).

Then, since \varphi (t)=ln(1+t) is strictly increasing in [0,\mathrm{\infty}), we have

This proves (5).

Taking into account (4) and (5), we have

where \varphi (t)=ln(1+t) for t\ge 0, and \beta (t)=\frac{\varphi (t)}{t} for t>0 and \beta (0)=0.

Obviously, when x=y, the inequality (6) is satisfied.

It is easily seen that \beta (t)=\frac{ln(1+t)}{t}\in \mathcal{F} by using elemental calculus.

Therefore, *T* is a Geraghty-contraction.

Notice that the pair (A,B) satisfies the *P*-property.

Indeed, if

then {x}_{1}={y}_{1} and {x}_{2}={y}_{2} and consequently,

By Theorem 3.1, *T* has a unique best proximity point.

Obviously, this point is (0,0)\in A.

The condition *A* and *B* are nonempty closed subsets of the metric space (X,d) is not a necessary condition for the existence of a unique best proximity point for a Geraghty-contraction T:A\to B as it is proved with the following example.

**Example 4.2** Consider X={\mathbb{R}}^{2} with the usual metric and the subsets of *X* given by

Obviously, d(A,B)=1 and *B* is not a closed subset of *X*.

Note that {A}_{0}=0\times [0,\frac{\pi}{2}) and {B}_{0}=B.

We consider the mapping T:A\to B defined as

Now, we check that *T* is a Geraghty-contraction.

In fact, for (0,x),(0,y)\in A with x\ne y, we have

In what follows, we need to prove that

In fact, suppose that x>y (the same argument works for y>x).

Put arctanx=\alpha and arctany=\beta (notice that \alpha >\beta since the function \varphi (t)=arctant for t\ge 0 is strictly increasing).

Taking into account that

and since \alpha ,\beta \in [0,\frac{\pi}{2}), we have that tan\alpha ,tan\beta \in [0,\mathrm{\infty}), and consequently, from the last inequality it follows that

Applying *ϕ* (notice that \varphi (t)=arctant) to the last inequality and taking into account the increasing character of *ϕ*, we have

or equivalently,

and this proves (8).

By (7) and (8), we get

where \beta (t)=\frac{arctant}{t} for t>0 and \beta (0)=0. Obviously, the inequality (9) is satisfied for (0,x),(0,y)\in A with x=y.

Now, we prove that \beta \in \mathcal{F}.

In fact, if \beta ({t}_{n})=\frac{arctan{t}_{n}}{{t}_{n}}\to 1, then the sequence ({t}_{n}) is a bounded sequence since in the contrary case, {t}_{n}\to \mathrm{\infty} and thus \beta ({t}_{n})\to 0. Suppose that {t}_{n}\nrightarrow 0. This means that there exists \u03f5>0 such that, for each n\in \mathbb{N}, there exists {p}_{n}\ge n with {t}_{{p}_{n}}\ge \u03f5. The bounded character of ({t}_{n}) gives us the existence of a subsequence ({t}_{{k}_{n}}) of ({t}_{{p}_{n}}) with ({t}_{{k}_{n}}) convergent. Suppose that {t}_{{k}_{n}}\to a. From \beta ({t}_{n})\to 1, we obtain \frac{arctan{t}_{{k}_{n}}}{{t}_{{k}_{n}}}\to \frac{arctana}{a}=1 and, as the unique solution of arctanx=x is {x}_{0}=0, we obtain a=0.

Thus, {t}_{{k}_{n}}\to 0 and this contradicts the fact that {t}_{{k}_{n}}\ge \u03f5 for any n\in \mathbb{N}.

Therefore, {t}_{n}\to 0 and this proves that \beta \in \mathcal{F}.

A similar argument to the one used in Example 4.1 proves that the pair (A,B) has the *P*-property.

On the other hand, the point (0,0)\in A is a best proximity point for *T* since

Moreover, (0,0) is the unique best proximity point for *T*.

Indeed, if (0,x)\in A is a best proximity point for *T*, then

and this gives us

Taking into account that the unique solution of this equation is x=0, we have proved that *T* has a unique best proximity point which is (0,0).

Notice that in this case *B* is not closed.

Since for any nonempty subset *A* of *X*, the pair (A,A) satisfies the *P*-property, we have the following corollary.

**Corollary 4.1** *Let* (X,d) *be a complete metric space and* *A* *be a nonempty closed subset of X*. *Let* T:A\to A *be a Geraghty*-*contraction*. *Then* *T* *has a unique fixed point*.

*Proof* Using Theorem 3.1 when A=B, the desired result follows. □

Notice that when A=X, Corollary 4.1 is Theorem 1.1 due to Gerahty [1].

## References

Geraghty M: On contractive mappings.

*Proc. Am. Math. Soc.*1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5Eldred AA, Veeramani P: Existence and convergence of best proximity points.

*J. Math. Anal. Appl.*2006, 323: 1001–1006. 10.1016/j.jmaa.2005.10.081Anuradha J, Veeramani P: Proximal pointwise contraction.

*Topol. Appl.*2009, 156: 2942–2948. 10.1016/j.topol.2009.01.017Markin J, Shahzad N: Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces.

*Nonlinear Anal.*2009, 70: 2435–2441. 10.1016/j.na.2008.03.045Sadiq Basha S, Veeramani P: Best proximity pair theorems for multifunctions with open fibres.

*J. Approx. Theory*2000, 103: 119–129. 10.1006/jath.1999.3415Sankar Raj V, Veeramani P: Best proximity pair theorems for relatively nonexpansive mappings.

*Appl. Gen. Topol.*2009, 10: 21–28.Al-Thagafi MA, Shahzad N: Convergence and existence results for best proximity points.

*Nonlinear Anal.*2009, 70: 3665–3671. 10.1016/j.na.2008.07.022Kirk 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: A best proximity theorem for weakly contractive non-self mappings.

*Nonlinear Anal.*2011, 74: 4804–4808. 10.1016/j.na.2011.04.052Sankar Raj, V: Banach’s contraction principle for non-self mappings. Preprint

## Acknowledgements

This research was partially supported by ‘Universidad de Las Palmas de Gran Canaria’, Project ULPGC 2010-006.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The three authors have contributed equally in this paper. They read and approval the final manuscript.

## Rights and permissions

**Open Access**
This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (
https://creativecommons.org/licenses/by/2.0
), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Caballero, J., Harjani, J. & Sadarangani, K. A best proximity point theorem for Geraghty-contractions.
*Fixed Point Theory Appl* **2012, **231 (2012). https://doi.org/10.1186/1687-1812-2012-231

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/1687-1812-2012-231

### Keywords

- fixed point
- Geraghty-contraction
*P*-property- best proximity point