Open Access

On best proximity point of ψ-Geraghty contractions

Fixed Point Theory and Applications20132013:200

DOI: 10.1186/1687-1812-2013-200

Received: 12 March 2013

Accepted: 17 May 2013

Published: 24 July 2013

Abstract

Very recently, Caballero, Harjani and Sadarangani (Fixed Point Theory Appl. 2012:231, 2012) observed some best proximity point results for Geraghty contractions by using the P-property. In this paper, we introduce the notion of ψ-Geraghty contractions and show the existence and uniqueness of the best proximity point of such contractions in the setting of a metric space. We state examples to illustrate our result.

MSC: 41A65, 90C30, 47H10.

Keywords

best proximity point non-self mapping partial order metric space fixed point

1 Introduction and preliminaries

In nonlinear functional analysis, fixed point theory and best proximity point theory play a crucial role in the establishment of the existence of certain differential and integral equations. As a consequence, fixed point theory is very useful for various quantitative sciences that involve such equations. To list a few, certain branches of computer sciences, engineering and economics are well-known examples in which fixed point theory is used.

The most remarkable paper in this field was reported by Banach [1] in 1922. In this paper, Banach proved that every contraction in a complete metric space has a unique fixed point. Following this outstanding paper, many authors have extended, generalized and improved this remarkable fixed point theorem of Banach by changing either the conditions of the mappings or the construction of the space. In particular, one of the notable generalizations of Banach fixed point theorem was reported by Geraghty [2].

Theorem 1.1 (Geraghty [2])

Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq1_HTML.gif be a complete metric space and T : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq2_HTML.gif be an operator. Suppose that there exists β : ( 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq3_HTML.gif satisfying the condition
β ( t n ) 1 implies t n 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equa_HTML.gif
If T satisfies the following inequality:
d ( T x , T y ) β ( d ( x , y ) ) d ( x , y ) for any x , y X , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ1_HTML.gif
(1)

then T has a unique fixed point.

It is very natural that some mappings, especially non-self-mappings defined on a complete metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq4_HTML.gif, do not necessarily possess a fixed point, that is, d ( x , T x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq5_HTML.gif for all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq6_HTML.gif. In such situations, it is reasonable to search for the existence (and uniqueness) of a point x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq7_HTML.gif such that d ( x , T x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq8_HTML.gif is an approximation of an x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq6_HTML.gif such that d ( x , T x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq9_HTML.gif. In other words, one speculates to determine an approximate solution x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq10_HTML.gif that is optimal in the sense that the distance between x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq10_HTML.gif and T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq11_HTML.gif is minimum. Here, the point x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq12_HTML.gif is called a best proximity point.

This research subject has attracted attention of a number of authors; for example, see [223]. In this paper we generalize and improve certain results of Caballero et al. in [6]. Notice also that in the best proximity point theory, we usually consider a non-self-mapping. In fixed point theory, almost all maps are self-mappings. For the sake of completeness, we recall some basic definitions and fundamental results on the best proximity theory.

Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq1_HTML.gif be a metric space and A and B be nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq1_HTML.gif. A mapping T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq13_HTML.gif is called a k-contraction if there exists k ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq14_HTML.gif such that d ( T x , T y ) k d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq15_HTML.gif for any x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq16_HTML.gif. It is clear that a k-contraction coincides with the celebrated Banach fixed point theorem if one takes A = B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq17_HTML.gif, where A is a complete subset of X.

Let A and B be two nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq4_HTML.gif. We denote by A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq18_HTML.gif and B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq19_HTML.gif the following sets:
A 0 = { x A : d ( x , y ) = d ( A , B )  for some  y B } , B 0 = { y B : d ( x , y ) = d ( A , B )  for some  x A } , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ2_HTML.gif
(2)

where d ( A , B ) = inf { d ( x , y ) : x A , y B } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq20_HTML.gif.

In [13], the authors presented sufficient conditions which determine when the sets A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq18_HTML.gif and B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq19_HTML.gif are nonempty. In [19], the author introduced the following definition.

Definition 1.1 Let ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq21_HTML.gif be a pair of nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq4_HTML.gif with A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq22_HTML.gif. Then the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq21_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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq23_HTML.gif and y 1 , y 2 B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq24_HTML.gif,
d ( x 1 , y 1 ) = d ( A , B ) and 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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ3_HTML.gif
(3)
It can be easily seen that for any nonempty subset A of ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq4_HTML.gif, the pair ( A , A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq25_HTML.gif has the P-property. In [19], the author proved that any pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq21_HTML.gif of nonempty closed convex subsets of a real Hilbert space H satisfies the P-property. Now, we introduce the class F of those functions β : ( 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq3_HTML.gif satisfying the following condition:
β ( t n ) 1 implies t n 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ4_HTML.gif
(4)

Definition 1.2 (See [6])

Let A, B be two nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq4_HTML.gif. A mapping T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq26_HTML.gif is said to be a Geraghty-contraction if there exists β F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq27_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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ5_HTML.gif
(5)

Theorem 1.2 (See [6])

Let ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq21_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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq4_HTML.gif such that A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq18_HTML.gif is nonempty. Let T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq26_HTML.gif be a continuous Geraghty-contraction satisfying T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq28_HTML.gif. Suppose that the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq21_HTML.gif has the P-property. Then there exists a unique x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq29_HTML.gif in A such that d ( x , T x ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq30_HTML.gif.

In the following section, we improve the theorem above by using a distance function ψ in Definition 1.2. In particular, we introduce Definition 2.1 and broaden the scope of Theorem 1.2 to ψ-Geraghty-contractions.

2 Main results

We start this section with the definition of the following auxiliary function. Let Ψ denote the class of functions ψ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq31_HTML.gif which satisfy the following conditions:
  1. (a)

    ψ is nondecreasing;

     
  2. (b)

    ψ is subadditive, that is, ψ ( s + t ) ψ ( s ) + ψ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq32_HTML.gif;

     
  3. (c)

    ψ is continuous;

     
  4. (d)

    ψ ( t ) = 0 t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq33_HTML.gif.

     

We introduce the following contraction.

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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq4_HTML.gif. A mapping T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq26_HTML.gif is said to be a ψ-Geraghty contraction if there exists β F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq27_HTML.gif such that
ψ ( d ( T x , T y ) ) β ( ψ ( d ( x , y ) ) ) ψ ( d ( x , y ) ) for any  x , y A . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ6_HTML.gif
(6)
Remark 2.1 Notice that since β : ( 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq3_HTML.gif, we have
ψ ( d ( T x , T y ) ) β ( ψ ( d ( x , y ) ) ) ψ ( d ( x , y ) ) < ψ ( d ( x , y ) ) for any  x , y A  with  x y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ7_HTML.gif
(7)

We are now ready to state and prove our main theorem.

Theorem 2.1 Let ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq21_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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq4_HTML.gif such that A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq18_HTML.gif is nonempty. Let T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq26_HTML.gif be a ψ-Geraghty contraction satisfying T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq28_HTML.gif. Suppose that the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq21_HTML.gif has the P-property. Then there exists a unique x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq29_HTML.gif in A such that d ( x , T x ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq30_HTML.gif.

Proof Regarding that A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq18_HTML.gif is nonempty, we take x 0 A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq34_HTML.gif. Since T x 0 T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq35_HTML.gif, we can find x 1 A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq36_HTML.gif such that d ( x 1 , T x 0 ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq37_HTML.gif. Analogously, regarding the assumption T x 1 T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq38_HTML.gif, we determine x 2 A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq39_HTML.gif such that d ( x 2 , T x 1 ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq40_HTML.gif. Recursively, we obtain a sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq41_HTML.gif in A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq18_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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ8_HTML.gif
(8)
Since ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq21_HTML.gif has the P-property, we derive 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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ9_HTML.gif
(9)
If there exists n 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq42_HTML.gif such that d ( x n 0 , x n 0 + 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq43_HTML.gif, then the proof is completed. Indeed,
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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ10_HTML.gif
(10)
and consequently, T x n 0 1 = T x n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq44_HTML.gif. On the other hand, due to (8) we have
d ( x n 0 , T x n 0 1 ) = d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equb_HTML.gif
Therefore, we conclude that
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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ11_HTML.gif
(11)
For the rest of the proof, we suppose that d ( x n , x n + 1 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq45_HTML.gif for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq46_HTML.gif. Since T is a ψ-Geraghty contraction, for any , we have that
ψ ( d ( x n , x n + 1 ) ) = ψ ( d ( T x n 1 , T x n ) ) β ( ψ ( d ( x n 1 , x n ) ) ) ψ ( d ( x n 1 , x n ) ) < ψ ( d ( x n 1 , x n ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ12_HTML.gif
(12)
Consequently, { ψ ( d ( x n , x n + 1 ) ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq47_HTML.gif is a nonincreasing sequence and bounded below, and so lim n ψ ( d ( x n , x n + 1 ) ) = L https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq48_HTML.gif exists. Let lim n ψ ( d ( x n , x n + 1 ) ) = L 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq49_HTML.gif. Assume that L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq50_HTML.gif. Then, from (6), we have
ψ ( d ( x n + 1 , x n + 2 ) ) ψ ( d ( x n , x n + 1 ) ) β ( ψ ( d ( x n , x n + 1 ) ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equc_HTML.gif
for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq51_HTML.gif, which implies that
lim n β ( ψ ( d ( x n , x n + 1 ) ) ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equd_HTML.gif
On the other hand, since β F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq27_HTML.gif, we conclude L = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq52_HTML.gif, that is,
lim n ψ ( d ( x n , x n + 1 ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ13_HTML.gif
(13)
Notice that since d ( x n + 1 , T x n ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq53_HTML.gif for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq54_HTML.gif, for fixed p , q N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq55_HTML.gif, 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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq56_HTML.gif, and since ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq21_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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq57_HTML.gif. In what follows, we prove that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq41_HTML.gif is a Cauchy sequence. On the contrary, assume that we have
ε = lim sup m , n d ( x n , x m ) > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ14_HTML.gif
(14)
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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ15_HTML.gif
(15)
By (12) 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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq58_HTML.gif, by the comment mentioned above, regarding the discussion on the P-property above together with (12), (15) and the property of the function ψ, we derive that
ψ ( d ( x n , x m ) ) ψ ( d ( x n , x n + 1 ) + d ( T x n , T x m ) + d ( x m + 1 , x m ) ) ψ ( d ( x n , x n + 1 ) ) + ψ ( d ( T x n , T x m ) ) + ψ ( d ( x m + 1 , x m ) ) ψ ( d ( x n , x n + 1 ) ) + β ( ψ ( d ( x n , x m ) ) ) ψ ( d ( x n , x m ) ) + ψ ( d ( x m + 1 , x m ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ16_HTML.gif
(16)
By a simple manipulation, (16) yields that
ψ ( d ( x n , x m ) ) ( 1 β ( ψ ( d ( x n , x m ) ) ) ) 1 [ ψ ( d ( x n , x n + 1 ) ) + ψ ( d ( x m + 1 , x m ) ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ17_HTML.gif
(17)
By taking the properties of the function ψ into account, together with (13) and lim sup m , n d ( x n , x m ) = ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq59_HTML.gif and lim n d ( x n , x n + 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq60_HTML.gif, the last inequality yields
lim sup m , n ( 1 β ( ψ ( d ( x n , x m ) ) ) ) 1 = . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ18_HTML.gif
(18)
Therefore lim sup m , n β ( ψ ( d ( x n , x m ) ) ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq61_HTML.gif. By taking the fact β F https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq27_HTML.gif into account, we get
lim sup m , n ψ ( d ( x n , x m ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Eque_HTML.gif

Regarding the properties of the function ψ, the limit above contradicts the assumption (14). Therefore, { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq41_HTML.gif is a Cauchy sequence.

Since ( x n ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq62_HTML.gif and A is a closed subset of the complete metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq4_HTML.gif, we can find x A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq63_HTML.gif such that x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq64_HTML.gif.

We claim that T x n T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq65_HTML.gif. Suppose, on the contrary, that T x n T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq66_HTML.gif. This means that we can find ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq67_HTML.gif such that for each n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq68_HTML.gif, there exists p n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq69_HTML.gif with
ε < d ( T x p n , T x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ19_HTML.gif
(19)
Due to the properties of ψ, we get
0 < ψ ( ε ) ψ ( d ( T x p n , T x ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ20_HTML.gif
(20)
Using the fact that T is a ψ-Geraghty contraction, we have
ψ ( ε ) ψ ( d ( T x p n , T x ) ) β ( ψ ( d ( x p n , x ) ) ) ψ ( d ( x p n , x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ21_HTML.gif
(21)
for any n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq68_HTML.gif. Since x p n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq70_HTML.gif and ψ ( d ( x p n , x p n + 1 ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq71_HTML.gif, we can find n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq72_HTML.gif such that for n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq73_HTML.gif
ψ ( d ( x p n , x ) ) < ψ ( ε ) and ψ ( d ( x p n , x p n + 1 ) ) < ψ ( ε ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ22_HTML.gif
(22)
Consequently, for n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq73_HTML.gif we have
0 < ψ ( ε ) ψ ( d ( T x p n , T x ) ) β ( ψ ( d ( x p n , x ) ) ) ψ ( d ( x p n , x ) ) ψ ( d ( x p n , x ) ) < ψ ( ε ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ23_HTML.gif
(23)

a contradiction. Therefore, T x n T x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq65_HTML.gif.

Regarding the fact that the sequence { d ( x n + 1 , T x n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq74_HTML.gif is a constant sequence with value d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq75_HTML.gif, we derive
d ( x , T x ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ24_HTML.gif
(24)

which is equivalent to saying that x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq29_HTML.gif is the best proximity point of T. This completes the proof of the existence of a best proximity point.

We shall show the uniqueness of the best proximity point of T. Suppose that x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq29_HTML.gif and y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq76_HTML.gif are two distinct best proximity points of T, that is, x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq77_HTML.gif. This implies that
d ( x , T x ) = d ( A , B ) = d ( y , T y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ25_HTML.gif
(25)
Using the P-property, we have
d ( x , x 2 ) = d ( T x , T y ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ26_HTML.gif
(26)
Using the fact that T is a ψ-Geraghty contraction, we have
ψ ( d ( x , y ) ) = ψ ( d ( T x , T y ) ) β ( ψ ( d ( x , y ) ) ) ψ ( d ( x , y ) ) < ψ ( d ( x , y ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equ27_HTML.gif
(27)

a contradiction. This completes the proof. □

Notice that the pair ( A , A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq25_HTML.gif satisfies the P-property for any nonempty subset A of X. Consequently, we have the following corollary.

Corollary 2.1 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq1_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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq78_HTML.gif be a ψ-Geraghty-contraction. Then T has a unique fixed point.

Proof Apply Theorem 2.1 with A = B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq79_HTML.gif. □

If we take ψ ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq80_HTML.gif we obtain Theorem 1.2 as a corollary of Theorem 2.1.

Corollary 2.2 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq1_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%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq78_HTML.gif be a Geraghty-contraction. Then T has a unique fixed point.

Proof Apply Theorem 2.1 with A = B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq79_HTML.gif and ψ ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq80_HTML.gif. □

In order to illustrate our results, we present the following example.

Example 2.1 Suppose that X = R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq81_HTML.gif with the metric
d ( ( x , y ) , ( x , y ) ) = max { | x x | , | y y | } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equf_HTML.gif
and consider the closed subsets
A = { ( 0 , x ) : 0 x < } , B = { ( 1 , x ) : 0 x < } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equg_HTML.gif

and ψ ( t ) = t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq82_HTML.gif and β ( t ) = ln ( 1 + t ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq83_HTML.gif.

Set T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq84_HTML.gif to be the mapping defined by
T ( ( 0 , x ) ) = ( 1 , ln ( 1 + x ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equh_HTML.gif

Since d ( A , B ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq85_HTML.gif, the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq21_HTML.gif has the P-property.

Notice that A 0 = A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq86_HTML.gif and B 0 = B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq87_HTML.gif and T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq28_HTML.gif.

Without loss of generality, we assume that x > x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq88_HTML.gif. Moreover,
d ( T ( 0 , x ) , T ( 0 , x ) ) = | ln ( 1 + x ) ln ( 1 + x ) | = | ln ( 1 + x 1 + x ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equi_HTML.gif
and
d ( ( 0 , x ) , ( 0 , x ) ) = | x x | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equj_HTML.gif
We have
ψ ( d ( T ( 0 , x ) , T ( 0 , x ) ) ) = 1 2 | ln ( 1 + x 1 + x ) | = 1 2 | ln ( 1 + x + x x 1 + x ) | = 1 2 | ln ( 1 + x x 1 + x ) | 1 2 ln ( 1 + | x x | ) ( since  ln ( 1 + t )  is strictly increasing ) ln ( 1 + 1 2 | x x | ) ln ( 1 + 1 2 | x x | ) 1 2 | x x | 1 2 | x x | = β ( ψ ( d ( ( 0 , x ) , ( 0 , x ) ) ) ) ψ ( d ( ( 0 , x ) , ( 0 , x ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equk_HTML.gif

where β : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq89_HTML.gif is defined as β ( t ) = ln ( 1 + t ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq90_HTML.gif.

Therefore, T is a ψ-Geraghty-contraction. Notice that the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq91_HTML.gif satisfies the P-property. Indeed, if
d ( ( 0 , x ) , ( 1 , x ) ) = max { 1 , | x x | } = d ( A , B ) = 1 , d ( ( 0 , y ) , ( 1 , y ) ) = max { 1 , | y y | } = d ( A , B ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equl_HTML.gif
then x = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq92_HTML.gif and y = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq93_HTML.gif and hence
| x y | = d ( ( 0 , x ) , ( 0 , y ) ) = d ( ( 1 , x ) , ( 1 , y ) ) = | x y | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equm_HTML.gif
Therefore, since the assumptions of Theorem 2.1 are satisfied, by Theorem 2.1 there exists a unique ( 0 , x ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq94_HTML.gif such that
d ( ( 0 , x ) , T ( 0 , x ) ) = 1 = d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equn_HTML.gif

More precisely, the point ( 0 , 0 ) A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq95_HTML.gif is the best proximity point of T.

Example 2.2 Let X = R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq96_HTML.gif and d ( x , y ) = | x y | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq97_HTML.gif be a metric on X. Suppose A = [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq98_HTML.gif and B = [ 15 / 8 , 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq99_HTML.gif are two closed subsets of . Define T : A B https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq84_HTML.gif by T x = 1 8 x + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq100_HTML.gif. Define β : ( 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq101_HTML.gif by β ( t ) = 1 1 + t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq102_HTML.gif and ψ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq103_HTML.gif by ψ ( t ) = 1 2 t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq104_HTML.gif. Clearly, d ( A , B ) = 7 / 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq105_HTML.gif. Now we have
A 0 = { x A : d ( x , y ) = d ( A , B ) = 7 / 8  for some  y B } = { 1 } , B 0 = { y B : d ( x , y ) = d ( A , B ) = 7 / 8  for some  x A } = { 15 / 8 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equo_HTML.gif
Also, T ( A 0 ) = B 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq106_HTML.gif. Further, clearly, the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq21_HTML.gif has the P-property. Let x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq107_HTML.gif. Note that, if x = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq108_HTML.gif, then condition (6) holds. Hence, we assume that x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq109_HTML.gif. We shall show that (6) holds. Suppose, on the contrary, there exist x 0 , y 0 A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq110_HTML.gif such that
ψ ( d ( T x 0 , T y 0 ) ) > β ( ψ ( d ( x 0 , y 0 ) ) ) ψ ( d ( x 0 , y 0 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equp_HTML.gif
and so
1 2 × 1 8 | x 0 y 0 | > 1 2 | x 0 y 0 | 1 + 1 2 | x 0 y 0 | , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_Equq_HTML.gif

which yields that 14 < | x 0 y 0 | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq111_HTML.gif, a contradiction. Therefore condition (6) holds for all x , y A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq107_HTML.gif. Hence, the conditions of Theorem 2.1 hold and T has a unique best proximity point. Here, x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-200/MediaObjects/13663_2013_Article_533_IEq112_HTML.gif is the best proximity point of T.

Declarations

Acknowledgements

The author expresses his gratitude to the anonymous referees for constructive and useful remarks, comments and suggestions.

Authors’ Affiliations

(1)
Department of Mathematics, Atilim University

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