Open Access

A best proximity point theorem for Geraghty-contractions

Fixed Point Theory and Applications20122012:231

DOI: 10.1186/1687-1812-2012-231

Received: 16 May 2012

Accepted: 10 December 2012

Published: 27 December 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).

Keywords

fixed point Geraghty-contraction P-property best proximity point

1 Introduction

Let A and B be nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq1_HTML.gif.

An operator T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq2_HTML.gif is said to be a k-contraction if there exists k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq3_HTML.gif such that d ( T x , T y ) k d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq4_HTML.gif for any x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq5_HTML.gif. 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 of those functions β : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq6_HTML.gif satisfying the following condition:
β ( t n ) 1 implies t n 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equa_HTML.gif

Theorem 1.1 ([1])

Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq7_HTML.gif be a complete metric space and T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq8_HTML.gif be an operator. Suppose that there exists β F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq9_HTML.gif such that for any x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq10_HTML.gif,
d ( T x , T y ) β ( d ( x , y ) ) d ( x , y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equ1_HTML.gif
(1)

Then T has a unique fixed point.

Since the constant functions f ( t ) = k https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq11_HTML.gif, where k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq3_HTML.gif, belong to , Theorem 1.1 extends Banach’s contraction principle.

Remark 1.1 Since the functions belonging to are strictly smaller than one, condition (1) implies that
d ( T x , T y ) < d ( x , y ) for any  x , y X  with  x y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equb_HTML.gif

Therefore, any operator T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq8_HTML.gif 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq7_HTML.gif and T : A X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq12_HTML.gif be a mapping. The solutions of the equation T x = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq13_HTML.gif are fixed points of T. Consequently, T ( A ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq14_HTML.gif 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 , T x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq15_HTML.gif for any x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq16_HTML.gif and the mapping T : A X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq17_HTML.gif does not have any fixed point. In this setting, our aim is to find an element x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq16_HTML.gif such that d ( x , T x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq18_HTML.gif 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 B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq2_HTML.gif.

A natural question is whether one can find an element x 0 A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq19_HTML.gif such that d ( x 0 , T x 0 ) = min { d ( x , T x ) : x A } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq20_HTML.gif. Since d ( x , T x ) d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq21_HTML.gif for any x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq16_HTML.gif, the optimal solution to this problem will be the one for which the value d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq22_HTML.gif is attained by the real valued function φ : A R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq23_HTML.gif given by φ ( x ) = d ( x , T x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq24_HTML.gif.

Some results about best proximity points can be found in [29].

2 Notations and basic facts

Let A and B be two nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq7_HTML.gif.

We denote by A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq25_HTML.gif and B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq26_HTML.gif the following sets:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equc_HTML.gif

where d ( A , B ) = inf { d ( x , y ) : x A  and  y B } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq27_HTML.gif.

In [8], the authors present sufficient conditions which determine when the sets A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq25_HTML.gif and B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq26_HTML.gif are nonempty.

Now, we present the following definition.

Definition 2.1 Let A, B be two nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq7_HTML.gif. A mapping T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq2_HTML.gif is said to be a Geraghty-contraction if there exists β F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq9_HTML.gif such that
d ( T x , T y ) β ( d ( x , y ) ) d ( x , y ) for any  x , y A . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equd_HTML.gif
Notice that since β : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq28_HTML.gif, we have
d ( T x , T y ) β ( d ( x , y ) ) d ( x , y ) < d ( x , y ) for any  x , y A  with  x y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Eque_HTML.gif

Therefore, every Geraghty-contraction is a contractive mapping.

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

Definition 2.2 ([10])

Let ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq29_HTML.gif be a pair of nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq7_HTML.gif with A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq30_HTML.gif. Then the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq29_HTML.gif is said to have the P-property if and only if for any x 1 , x 2 A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq31_HTML.gif and y 1 , y 2 B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq32_HTML.gif,
d ( x 1 , y 1 ) = d ( A , B ) d ( x 2 , y 2 ) = d ( A , B ) } d ( x 1 , x 2 ) = d ( y 1 , y 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equf_HTML.gif

It is easily seen that for any nonempty subset A of ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq7_HTML.gif, the pair ( A , A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq33_HTML.gif has the P-property.

In [10], the author proves that any pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq29_HTML.gif 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq29_HTML.gif be a pair of nonempty closed subsets of a complete metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq7_HTML.gif such that A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq25_HTML.gif is nonempty. Let T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq2_HTML.gif be a Geraghty-contraction satisfying T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq34_HTML.gif. Suppose that the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq29_HTML.gif has the P-property. Then there exists a unique x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq35_HTML.gif in A such that d ( x , T x ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq36_HTML.gif.

Proof Since A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq25_HTML.gif is nonempty, we take x 0 A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq19_HTML.gif.

As T x 0 T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq37_HTML.gif, we can find x 1 A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq38_HTML.gif such that d ( x 1 , T x 0 ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq39_HTML.gif. Similarly, since T x 1 T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq40_HTML.gif, there exists x 2 A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq41_HTML.gif such that d ( x 2 , T x 1 ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq42_HTML.gif. Repeating this process, we can get a sequence ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq43_HTML.gif in A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq25_HTML.gif satisfying
d ( x n + 1 , T x n ) = d ( A , B ) for any  n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equg_HTML.gif
Since ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq29_HTML.gif has the P-property, we have that
d ( x n , x n + 1 ) = d ( T x n 1 , T x n ) for any  n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equh_HTML.gif
Taking into account that T is a Geraghty-contraction, for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq44_HTML.gif, we have that
d ( x n , x n + 1 ) = d ( T x n 1 , T x n ) β ( d ( x n 1 , x n ) ) d ( x n 1 , x n ) < d ( x n 1 , x n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equ2_HTML.gif
(2)

Suppose that there exists n 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq45_HTML.gif such that d ( x n 0 , x n 0 + 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq46_HTML.gif.

In this case,
0 = d ( x n 0 , x n 0 + 1 ) = d ( T x n 0 1 , T x n 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equi_HTML.gif

and consequently, T x n 0 1 = T x n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq47_HTML.gif.

Therefore,
d ( A , B ) = d ( x n 0 , T x n 0 1 ) = d ( x n 0 , T x n 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equj_HTML.gif

and this is the desired result.

In the contrary case, suppose that d ( x n , x n + 1 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq48_HTML.gif for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq44_HTML.gif.

By (2), ( d ( x n , x n + 1 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq49_HTML.gif is a decreasing sequence of nonnegative real numbers, and hence there exists r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq50_HTML.gif such that
lim n d ( x n , x n + 1 ) = r . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equk_HTML.gif

In the sequel, we prove that r = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq51_HTML.gif.

Assume r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq52_HTML.gif, then from (2) we have
0 < d ( x n , x n + 1 ) d ( x n 1 , x n ) β ( d ( x n 1 , x n ) ) < 1 for any  n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equl_HTML.gif

The last inequality implies that lim n β ( d ( x n 1 , x n ) ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq53_HTML.gif and since β F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq9_HTML.gif, we obtain r = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq51_HTML.gif and this contradicts our assumption.

Therefore,
lim n d ( x n , x n + 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equ3_HTML.gif
(3)

Notice that since d ( x n + 1 , T x n ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq54_HTML.gif for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq55_HTML.gif, for p , q N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq56_HTML.gif fixed, we have d ( x p , T x p 1 ) = d ( x q , T x q 1 ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq57_HTML.gif, and since ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq29_HTML.gif satisfies the P-property, d ( x p , x q ) = d ( T x p 1 , T x q 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq58_HTML.gif.

In what follows, we prove that ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq43_HTML.gif is a Cauchy sequence.

In the contrary case, we have that
lim sup m , n d ( x n , x m ) > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equm_HTML.gif
By using the triangular inequality,
d ( x n , x m ) d ( x n , x n + 1 ) + d ( x n + 1 , x m + 1 ) + d ( x m + 1 , x m ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equn_HTML.gif
By (2) and since d ( x n + 1 , x m + 1 ) = d ( T x n , T x m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq59_HTML.gif, by the above mentioned comment, we have
d ( x n , x m ) d ( x n , x n + 1 ) + d ( T x n , T x m ) + d ( x m + 1 , x m ) d ( x n , x n + 1 ) + β ( d ( x n , x m ) ) d ( x n , x m ) + d ( x m + 1 , x m ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equo_HTML.gif
which gives us
d ( x n , x m ) ( 1 β ( d ( x n , x m ) ) ) 1 [ d ( x n , x n + 1 ) + d ( x m + 1 , x m ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equp_HTML.gif
Since lim sup m , n d ( x n , x m ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq60_HTML.gif and by (3), lim sup n d ( x n , x n + 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq61_HTML.gif, from the last inequality it follows that
lim sup m , n ( 1 β ( d ( x n , x m ) ) ) 1 = . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equq_HTML.gif

Therefore, lim sup m , n β ( d ( x n , x m ) ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq62_HTML.gif.

Taking into account that β F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq9_HTML.gif, we get lim sup m , n d ( x n , x m ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq63_HTML.gif and this contradicts our assumption.

Therefore, ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq43_HTML.gif is a Cauchy sequence.

Since ( x n ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq64_HTML.gif and A is a closed subset of the complete metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq7_HTML.gif, we can find x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq65_HTML.gif such that x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq66_HTML.gif.

Since any Geraghty-contraction is a contractive mapping and hence continuous, we have T x n T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq67_HTML.gif.

This implies that d ( x n + 1 , T x n ) d ( x , T x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq68_HTML.gif.

Taking into account that the sequence ( d ( x n + 1 , T x n ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq69_HTML.gif is a constant sequence with value d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq22_HTML.gif, we deduce
d ( x , T x ) = d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equr_HTML.gif

This means that x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq35_HTML.gif is a best proximity point of T.

This proves the part of existence of our theorem.

For the uniqueness, suppose that x 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq70_HTML.gif and x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq71_HTML.gif are two best proximity points of T with x 1 x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq72_HTML.gif.

This means that
d ( x i , T x i ) = d ( A , B ) for  i = 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equs_HTML.gif
Using the P-property, we have
d ( x 1 , x 2 ) = d ( T x 1 , T x 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equt_HTML.gif
Using the fact that T is a Geraghty-contraction, we have
d ( x 1 , x 2 ) = d ( T x 1 , T x 2 ) β ( d ( x 1 , x 2 ) ) d ( x 1 , x 2 ) < d ( x 1 , x 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equu_HTML.gif

which is a contradiction.

Therefore, x 1 = x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq73_HTML.gif.

This finishes the proof. □

4 Examples

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

Example 4.1 Consider X = R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq74_HTML.gif with the usual metric.

Let A and B be the subsets of X defined by
A = { 0 } × [ 0 , ) and B = { 1 } × [ 0 , ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equv_HTML.gif

Obviously, d ( A , B ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq75_HTML.gif and A, B are nonempty closed subsets of X.

Moreover, it is easily seen that A 0 = A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq76_HTML.gif and B 0 = B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq77_HTML.gif.

Let T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq2_HTML.gif be the mapping defined as
T ( 0 , x ) = ( 1 , ln ( 1 + x ) ) for any  ( 0 , x ) A . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equw_HTML.gif

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

In fact, for ( 0 , x ) , ( 0 , y ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq78_HTML.gif with x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq79_HTML.gif, we have
d ( T ( 0 , x ) , T ( 0 , y ) ) = d ( ( 1 , ln ( 1 + x ) ) , ( 1 , ln ( 1 + y ) ) ) = | ln ( 1 + x ) ln ( 1 + y ) | = | ln ( 1 + x 1 + y ) | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equ4_HTML.gif
(4)
Now, we prove that
| ln ( 1 + x 1 + y ) | ln ( 1 + | x y | ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equ5_HTML.gif
(5)

Suppose that x > y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq80_HTML.gif (the same reasoning works for y > x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq81_HTML.gif).

Then, since ϕ ( t ) = ln ( 1 + t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq82_HTML.gif is strictly increasing in [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq83_HTML.gif, we have
ln ( 1 + x 1 + y ) = ln ( 1 + y + x y 1 + y ) = ln ( 1 + x y 1 + y ) ln ( 1 + x y ) = ln ( 1 + | x y | ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equx_HTML.gif

This proves (5).

Taking into account (4) and (5), we have
d ( T ( 0 , x ) , T ( 0 , y ) ) = | ln ( 1 + x 1 + y ) | ln ( 1 + | x y | ) = ln ( 1 + | x y | ) | x y | | x y | = ϕ ( d ( ( 0 , x ) , ( 0 , y ) ) ) d ( ( 0 , x ) , ( 0 , y ) ) d ( ( 0 , x ) , ( 0 , y ) ) = β ( d ( ( 0 , x ) , ( 0 , y ) ) ) d ( ( 0 , x ) , ( 0 , y ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equ6_HTML.gif
(6)

where ϕ ( t ) = ln ( 1 + t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq82_HTML.gif for t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq84_HTML.gif, and β ( t ) = ϕ ( t ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq85_HTML.gif for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq86_HTML.gif and β ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq87_HTML.gif.

Obviously, when x = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq88_HTML.gif, the inequality (6) is satisfied.

It is easily seen that β ( t ) = ln ( 1 + t ) t F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq89_HTML.gif by using elemental calculus.

Therefore, T is a Geraghty-contraction.

Notice that the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq29_HTML.gif satisfies the P-property.

Indeed, if
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equy_HTML.gif
then x 1 = y 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq90_HTML.gif and x 2 = y 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq91_HTML.gif and consequently,
d ( ( 0 , x 1 ) , ( 0 , x 2 ) ) = | x 1 x 2 | = | y 1 y 2 | = d ( ( 1 , y 1 ) , ( 1 , y 2 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equz_HTML.gif

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

Obviously, this point is ( 0 , 0 ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq92_HTML.gif.

The condition A and B are nonempty closed subsets of the metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq7_HTML.gif is not a necessary condition for the existence of a unique best proximity point for a Geraghty-contraction T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq2_HTML.gif as it is proved with the following example.

Example 4.2 Consider X = R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq74_HTML.gif with the usual metric and the subsets of X given by
A = { 0 } × [ 0 , ) and B = { 1 } × [ 0 , π 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equaa_HTML.gif

Obviously, d ( A , B ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq75_HTML.gif and B is not a closed subset of X.

Note that A 0 = 0 × [ 0 , π 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq93_HTML.gif and B 0 = B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq77_HTML.gif.

We consider the mapping T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq2_HTML.gif defined as
T ( 0 , x ) = ( 1 , arctan x ) for any  ( 0 , x ) A . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equab_HTML.gif

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

In fact, for ( 0 , x ) , ( 0 , y ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq78_HTML.gif with x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq79_HTML.gif, we have
d ( T ( 0 , x ) , T ( 0 , y ) ) = d ( ( 1 , arctan x ) , ( 1 , arctan y ) ) = | arctan x arctan y | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equ7_HTML.gif
(7)
In what follows, we need to prove that
| arctan x arctan y | arctan | x y | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equ8_HTML.gif
(8)

In fact, suppose that x > y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq80_HTML.gif (the same argument works for y > x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq81_HTML.gif).

Put arctan x = α https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq94_HTML.gif and arctan y = β https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq95_HTML.gif (notice that α > β https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq96_HTML.gif since the function ϕ ( t ) = arctan t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq97_HTML.gif for t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq84_HTML.gif is strictly increasing).

Taking into account that
tan ( α β ) = tan α tan β 1 + tan α tan β https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equac_HTML.gif
and since α , β [ 0 , π 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq98_HTML.gif, we have that tan α , tan β [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq99_HTML.gif, and consequently, from the last inequality it follows that
tan ( α β ) tan α tan β . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equad_HTML.gif
Applying ϕ (notice that ϕ ( t ) = arctan t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq97_HTML.gif) to the last inequality and taking into account the increasing character of ϕ, we have
α β arctan ( tan α tan β ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equae_HTML.gif
or equivalently,
arctan x arctan y = α β arctan ( x y ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equaf_HTML.gif

and this proves (8).

By (7) and (8), we get
d ( T ( 0 , x ) , T ( 0 , y ) ) = | arctan x arctan y | arctan | x y | = arctan | x y | | x y | | x y | = β ( d ( 0 , x ) , d ( 0 , y ) ) d ( ( 0 , x ) , ( 0 , y ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equ9_HTML.gif
(9)

where β ( t ) = arctan t t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq100_HTML.gif for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq86_HTML.gif and β ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq87_HTML.gif. Obviously, the inequality (9) is satisfied for ( 0 , x ) , ( 0 , y ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq101_HTML.gif with x = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq88_HTML.gif.

Now, we prove that β F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq9_HTML.gif.

In fact, if β ( t n ) = arctan t n t n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq102_HTML.gif, then the sequence ( t n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq103_HTML.gif is a bounded sequence since in the contrary case, t n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq104_HTML.gif and thus β ( t n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq105_HTML.gif. Suppose that t n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq106_HTML.gif. This means that there exists ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq107_HTML.gif such that, for each n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq55_HTML.gif, there exists p n n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq108_HTML.gif with t p n ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq109_HTML.gif. The bounded character of ( t n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq103_HTML.gif gives us the existence of a subsequence ( t k n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq110_HTML.gif of ( t p n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq111_HTML.gif with ( t k n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq110_HTML.gif convergent. Suppose that t k n a https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq112_HTML.gif. From β ( t n ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq113_HTML.gif, we obtain arctan t k n t k n arctan a a = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq114_HTML.gif and, as the unique solution of arctan x = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq115_HTML.gif is x 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq116_HTML.gif, we obtain a = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq117_HTML.gif.

Thus, t k n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq118_HTML.gif and this contradicts the fact that t k n ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq119_HTML.gif for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq55_HTML.gif.

Therefore, t n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq120_HTML.gif and this proves that β F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq9_HTML.gif.

A similar argument to the one used in Example 4.1 proves that the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq29_HTML.gif has the P-property.

On the other hand, the point ( 0 , 0 ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq92_HTML.gif is a best proximity point for T since
d ( ( 0 , 0 ) , T ( 0 , 0 ) ) = d ( ( 0 , 0 ) , ( 1 , arctan 0 ) ) = d ( ( 0 , 0 ) , ( 1 , 0 ) ) = 1 = d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equag_HTML.gif

Moreover, ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq121_HTML.gif is the unique best proximity point for T.

Indeed, if ( 0 , x ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq122_HTML.gif is a best proximity point for T, then
1 = d ( A , B ) = d ( ( 0 , x ) , T ( 0 , x ) ) = d ( ( 0 , x ) , ( 1 , arctan x ) ) = 1 + ( x arctan x ) 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equah_HTML.gif
and this gives us
x = arctan x . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_Equai_HTML.gif

Taking into account that the unique solution of this equation is x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq123_HTML.gif, we have proved that T has a unique best proximity point which is ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq121_HTML.gif.

Notice that in this case B is not closed.

Since for any nonempty subset A of X, the pair ( A , A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq33_HTML.gif satisfies the P-property, we have the following corollary.

Corollary 4.1 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq7_HTML.gif be a complete metric space and A be a nonempty closed subset of X. Let T : A A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq124_HTML.gif be a Geraghty-contraction. Then T has a unique fixed point.

Proof Using Theorem 3.1 when A = B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq125_HTML.gif, the desired result follows. □

Notice that when A = X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-231/MediaObjects/13663_2012_Article_332_IEq126_HTML.gif, Corollary 4.1 is Theorem 1.1 due to Gerahty [1].

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria

References

  1. Geraghty M: On contractive mappings. Proc. Am. Math. Soc. 1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5MathSciNetView Article
  2. Eldred AA, Veeramani P: Existence and convergence of best proximity points. J. Math. Anal. Appl. 2006, 323: 1001–1006. 10.1016/j.jmaa.2005.10.081MathSciNetView Article
  3. Anuradha J, Veeramani P: Proximal pointwise contraction. Topol. Appl. 2009, 156: 2942–2948. 10.1016/j.topol.2009.01.017MathSciNetView Article
  4. Markin J, Shahzad N: Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces. Nonlinear Anal. 2009, 70: 2435–2441. 10.1016/j.na.2008.03.045MathSciNetView Article
  5. Sadiq Basha S, Veeramani P: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 2000, 103: 119–129. 10.1006/jath.1999.3415MathSciNetView Article
  6. Sankar Raj V, Veeramani P: Best proximity pair theorems for relatively nonexpansive mappings. Appl. Gen. Topol. 2009, 10: 21–28.MathSciNetView Article
  7. Al-Thagafi MA, Shahzad N: Convergence and existence results for best proximity points. Nonlinear Anal. 2009, 70: 3665–3671. 10.1016/j.na.2008.07.022MathSciNetView Article
  8. Kirk WA, Reich S, Veeramani P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 2003, 24: 851–862. 10.1081/NFA-120026380MathSciNetView Article
  9. Sankar Raj V: A best proximity theorem for weakly contractive non-self mappings. Nonlinear Anal. 2011, 74: 4804–4808. 10.1016/j.na.2011.04.052MathSciNetView Article
  10. Sankar Raj, V: Banach’s contraction principle for non-self mappings. Preprint

Copyright

© Caballero et al.; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.