Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, Las Palmas de Gran Canaria, 35017, Spain

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

An operator

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

First, we introduce the class ℱ of those functions

**Theorem 1.1** (

Since the constant functions

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

Therefore, any operator

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

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

Let

In our context, we consider two nonempty subsets

A natural question is whether one can find an element

Some results about best proximity points can be found in

2 Notations and basic facts

Let

We denote by

where

In

Now, we present the following definition.

**Definition 2.1** Let

Notice that since

Therefore, every Geraghty-contraction is a contractive mapping.

In

**Definition 2.2** (

Let

It is easily seen that for any nonempty subset

In

3 Main results

We start this section presenting our main result.

**Theorem 3.1**

As

Since

Taking into account that

Suppose that there exists

In this case,

and consequently,

Therefore,

and this is the desired result.

In the contrary case, suppose that

By (2),

In the sequel, we prove that

Assume

The last inequality implies that

Therefore,

Notice that since

In what follows, we prove that

In the contrary case, we have that

By using the triangular inequality,

By (2) and since

which gives us

Since

Therefore,

Taking into account that

Therefore,

Since

Since any Geraghty-contraction is a contractive mapping and hence continuous, we have

This implies that

Taking into account that the sequence

This means that

This proves the part of existence of our theorem.

For the uniqueness, suppose that

This means that

Using the

Using the fact that

which is a contradiction.

Therefore,

This finishes the proof. □

4 Examples

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

**Example 4.1** Consider

Let

Obviously,

Moreover, it is easily seen that

Let

In the sequel, we check that

In fact, for

Now, we prove that

Suppose that

Then, since

This proves (5).

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

where

Obviously, when

It is easily seen that

Therefore,

Notice that the pair

Indeed, if

then

By Theorem 3.1,

Obviously, this point is

The condition

**Example 4.2** Consider

Obviously,

Note that

We consider the mapping

Now, we check that

In fact, for

In what follows, we need to prove that

In fact, suppose that

Put

Taking into account that

and since

Applying

or equivalently,

and this proves (8).

By (7) and (8), we get

where

Now, we prove that

In fact, if

Thus,

Therefore,

A similar argument to the one used in Example 4.1 proves that the pair

On the other hand, the point

Moreover,

Indeed, if

and this gives us

Taking into account that the unique solution of this equation is

Notice that in this case

Since for any nonempty subset

**Corollary 4.1**

Notice that when

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.

Acknowledgements

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