Open Access

A generalization for the best proximity point of Geraghty-contractions

Journal of Inequalities and Applications20132013:286

DOI: 10.1186/1029-242X-2013-286

Received: 6 February 2013

Accepted: 22 May 2013

Published: 6 June 2013

Abstract

In this paper, we introduce the notion of Geraghty-contractions and consider the related best proximity point in the context of a metric space. We state an example to illustrate our result.

MSC: 47H10, 54H25, 46J10, 46J15.

1 Introduction and preliminaries

Fixed point theory and best proximity theory are very important tools in nonlinear functional analysis. These related research areas have wide application potential in various branches of mathematics and different disciplines such as economics, engineering. One of the most impressive results in this direction, known as the Banach contraction mapping principle, was given by Banach: Every contraction on a complete metric space has a unique fixed point. This celebrated result has been generalized in several ways in various abstract spaces. In particular, one of the interesting generalizations of the Banach contraction mapping principle was given by Geraghty [1].

Theorem 1 (Geraghty [1])

Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq1_HTML.gif be a complete metric space and let T : X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq2_HTML.gif be an operator. Suppose that there exists β : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq3_HTML.gif satisfying the condition
β ( t n ) 1 implies t n 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_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%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ1_HTML.gif
(1.1)

then T has a unique fixed point.

It is clear that some mapping on a complete metric space has no fixed point, that is, d ( x , T x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq4_HTML.gif for all x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq5_HTML.gif. In this case, it is natural to ask the existence and uniqueness of the smallest value of d ( x , T x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq6_HTML.gif. This is the main motivation of a best proximity point. This research subject has attracted attention of a number of authors; see, e.g., [119].

First we recall fundamental definitions and basic results in this direction.

Let A and B be nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq1_HTML.gif. A mapping T : A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq7_HTML.gif is called a k-contraction if there exists k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq8_HTML.gif such that d ( T x , T y ) k d ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq9_HTML.gif for any x , y A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq10_HTML.gif. Notice that the k-contraction coincides with the Banach contraction mapping principle if one takes A = B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq11_HTML.gif, where A is a complete subset of X. A point x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq12_HTML.gif is called the best proximity of T if d ( x , T x ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq13_HTML.gif, where d ( A , B ) = inf { d ( x , y ) : x A , y B } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq14_HTML.gif.

Let A and B be two nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq15_HTML.gif. We denote by A 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq16_HTML.gif and B 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq17_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%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ2_HTML.gif
(1.2)
We denote by F the set of all functions β : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq18_HTML.gif satisfying the following property:
β ( t n ) 1 implies t n 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ3_HTML.gif
(1.3)

Definition 2 (See [2])

Let A, B be two nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq15_HTML.gif. A mapping T : A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq19_HTML.gif is said to be a Geraghty-contraction if there exists β F https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq20_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%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ4_HTML.gif
(1.4)

Very recently Raj [10, 11] introduced the notion of P-property as follows.

Definition 3 Let ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq21_HTML.gif be a pair of nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq15_HTML.gif with A 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq22_HTML.gif. Then the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_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%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq23_HTML.gif and y 1 , y 2 B 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_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%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ5_HTML.gif
(1.5)

Example 4 (See, e.g., [11])

Let A be a nonempty subset of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq15_HTML.gif. It is evident that the pair ( A , A ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq25_HTML.gif has the P-property. Let ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq21_HTML.gif be any pair of nonempty, closed, convex subsets of a real Hilbert space H. Then ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq21_HTML.gif has the P-property.

Theorem 5 (See [2])

Let ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_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%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq15_HTML.gif such that A 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq16_HTML.gif is nonempty. Let T : A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq19_HTML.gif be a continuous, Geraghty-contraction satisfying T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq26_HTML.gif. Suppose that the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq21_HTML.gif has the P-property. Then there exists a unique x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq12_HTML.gif in A such that d ( x , T x ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq13_HTML.gif.

The subject of this paper is to generalize, improve and extend the results of Caballero, Harjani and Sadarangani [2]. For this purpose, we first define the notion of generalized Geraghty-contraction as follows.

Definition 6 Let A, B be two nonempty subsets of a metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq15_HTML.gif. A mapping T : A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq19_HTML.gif is said to be a generalized Geraghty-contraction if there exists β F https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq20_HTML.gif such that
d ( T x , T y ) β ( M ( x , y ) ) [ M ( x , y ) d ( A , B ) ] for any  x , y A , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ6_HTML.gif
(1.6)

where M ( x , y ) = max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq27_HTML.gif.

Remark 7 Notice that since β : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq3_HTML.gif, we have
d ( T x , T y ) β ( M ( x , y ) ) [ M ( x , y ) d ( A , B ) ] < M ( x , y ) for any  x , y A  with  x y , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ7_HTML.gif
(1.7)

where M ( x , y ) = max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq27_HTML.gif.

2 Main results

We start this section with our main result.

Theorem 8 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq15_HTML.gif be a complete metric space. Suppose that ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq21_HTML.gif is a pair of nonempty closed subsets of X and A 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq16_HTML.gif is nonempty. Suppose also that the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq21_HTML.gif has the P-property. If a non-self-mapping T : A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq19_HTML.gif is a generalized Geraghty-contraction satisfying T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq26_HTML.gif, then there exists a unique best proximity point, that is, there exists x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq12_HTML.gif in A such that d ( x , T x ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq13_HTML.gif.

Proof Let us fix an element x 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq28_HTML.gif in A 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq16_HTML.gif. Since T x 0 T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq29_HTML.gif, we can find x 1 A 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq30_HTML.gif such that d ( x 1 , T x 0 ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq31_HTML.gif. Further, as T x 1 T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq32_HTML.gif, there is an element x 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq33_HTML.gif in A 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq16_HTML.gif such that d ( x 2 , T x 1 ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq34_HTML.gif. Recursively, we obtain a sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq35_HTML.gif in A 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq16_HTML.gif with the following property:
d ( x n + 1 , T x n ) = d ( A , B ) for any  n N . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ8_HTML.gif
(2.1)
Due to the fact that the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_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%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ9_HTML.gif
(2.2)
From (2.1), we get
d ( x n 1 , T x n 1 ) d ( x n 1 , x n ) + d ( x n , T x n 1 ) = d ( x n 1 , x n ) + d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equb_HTML.gif
On the other hand, by (2.1) and (2.2) we obtain that
d ( x n , T x n ) d ( x n , T x n 1 ) + d ( T x n 1 , T x n ) = d ( x n , x n + 1 ) + d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equc_HTML.gif
Consequently, we have
M ( x n 1 , x n ) = max { d ( x n 1 , x n ) , d ( x n 1 , T x n 1 ) , d ( x n , T x n ) } max { d ( x n 1 , x n ) , d ( x n , x n + 1 ) } + d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ10_HTML.gif
(2.3)
If there exists n 0 N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq36_HTML.gif such that d ( x n 0 , x n 0 + 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq37_HTML.gif, then the proof is completed. In fact, due to (2.2), we have
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%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ11_HTML.gif
(2.4)
which yields that T x n 0 1 = T x n 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq38_HTML.gif. Hence, equation (2.1) implies 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%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ12_HTML.gif
(2.5)
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%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq39_HTML.gif for any n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq40_HTML.gif. Owing to the fact T is a generalized Geraghty-contraction, we derive that
d ( x n , x n + 1 ) = d ( T x n 1 , T x n ) β ( M ( x n 1 , x n ) ) ( M ( x n 1 , x n ) d ( A , B ) ) < M ( x n 1 , x n ) d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ13_HTML.gif
(2.6)
Then, by (2.3) and (2.6), we deduce that
d ( x n , x n + 1 ) < M ( x n 1 , x n ) d ( A , B ) max { d ( x n 1 , x n ) , d ( x n , x n + 1 ) } . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equd_HTML.gif
Suppose that max { d ( x n 1 , x n ) , d ( x n , x n + 1 ) } = d ( x n , x n + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq41_HTML.gif. Then we get that
d ( x n , x n + 1 ) < d ( x n , x n + 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Eque_HTML.gif
a contradiction. As a result, we conclude that max { d ( x n 1 , x n ) , d ( x n , x n + 1 ) } = d ( x n 1 , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq42_HTML.gif and hence
M ( x n 1 , x n ) max { d ( x n 1 , x n ) , d ( x n , x n + 1 ) } + d ( A , B ) = d ( x n 1 , x n ) + d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ14_HTML.gif
(2.7)
By (2.6), we get
d ( x n , x n + 1 ) = d ( T x n 1 , T x n ) β ( M ( 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%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ15_HTML.gif
(2.8)
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq40_HTML.gif. Consequently, { d ( x n , x n + 1 ) } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq43_HTML.gif is a nonincreasing sequence and bounded below. Thus, there exists L 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq44_HTML.gif such that lim n ( d ( x n , x n + 1 ) ) = L https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq45_HTML.gif. We shall show that L = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq46_HTML.gif. Suppose, on the contrary, L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq47_HTML.gif. Then, by (2.8), we have
d ( x n + 1 , x n + 2 ) d ( x n , x n + 1 ) β ( M ( x n , x n + 1 ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equf_HTML.gif
for each n 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq48_HTML.gif. In what follows,
lim n β ( M ( x n , x n + 1 ) ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equg_HTML.gif
On the other hand, since β F https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq49_HTML.gif, we conclude lim n M ( x n , x n + 1 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq50_HTML.gif, that is,
L = lim n d ( x n , x n + 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ16_HTML.gif
(2.9)
Since, d ( x n , T x n 1 ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq51_HTML.gif holds for all n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq40_HTML.gif and ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq21_HTML.gif satisfies the P-property, then, for all m , n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq52_HTML.gif, we can write, d ( x m , x n ) = d ( T x m 1 , T x n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq53_HTML.gif. We also have
d ( x l , T x l ) d ( x l , x l + 1 ) + d ( x l + 1 , T x l ) = d ( x l , x l + 1 ) + d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equh_HTML.gif
for all l N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq54_HTML.gif. It follows that
M ( x m , x n ) = max { d ( x m , x n ) , d ( x m , T x m ) , d ( x n , T x n ) } max { d ( x m , x n ) , d ( x m , x m + 1 ) , d ( x n , x n + 1 ) } + d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equi_HTML.gif
Taking (2.9) into consideration, we find
lim m , n M ( x m , x n ) lim m , n d ( x m , x n ) + d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ17_HTML.gif
(2.10)
We shall show that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq35_HTML.gif is a Cauchy sequence. Suppose, on the contrary, that we have
ε = lim sup m , n d ( x n , x m ) > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ18_HTML.gif
(2.11)
Due to the triangular inequality, we have
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%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ19_HTML.gif
(2.12)
Regarding (1.6) and (2.12), 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 ( M ( x n , x m ) ) ( M ( x n , x m ) d ( A , B ) ) + d ( x m + 1 , x m ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ20_HTML.gif
(2.13)
Taking (2.10), (2.13) and (2.9) into account, we derive that
lim m , n d ( x n , x m ) lim m , n β ( M ( x n , x m ) ) lim m , n ( M ( x m , x n ) d ( A , B ) ) lim m , n β ( M ( x n , x m ) ) lim m , n d ( x m , x n ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equj_HTML.gif
Owing to (2.11), we get
1 lim m , n β ( M ( x n , x m ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equk_HTML.gif
which implies lim m , n β ( M ( x n , x m ) ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq55_HTML.gif. By the property of β, we have lim m , n M ( x n , x m ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq56_HTML.gif. Consequently, we have lim m , n d ( x n , x m ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq57_HTML.gif, a contradiction. Hence, we conclude that the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq35_HTML.gif is Cauchy. Since A is a closed subset of the complete metric space ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq15_HTML.gif and { x n } A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq58_HTML.gif, and we can find x A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq59_HTML.gif such that x n x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq60_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq61_HTML.gif. We assert that d ( x , T x ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq13_HTML.gif. Suppose, on the contrary, that d ( x , T x ) > d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq62_HTML.gif. First, we obtain the following inequalities:
d ( x , T x ) d ( x , T x n ) + d ( T x n , T x ) d ( x , x n + 1 ) + d ( x n + 1 , T x n ) + d ( T x n , T x ) d ( x , x n + 1 ) + d ( A , B ) + d ( T x n , T x ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equl_HTML.gif
Letting n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq61_HTML.gif in the inequalities above, we conclude that
d ( x , T x ) d ( A , B ) lim n d ( T x n , T x ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equm_HTML.gif
On the other hand, we obtain
d ( x n , T x n ) d ( x n , x n + 1 ) + d ( x n + 1 , T x n ) = d ( x n , x n + 1 ) + d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equn_HTML.gif
Taking limit as n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq61_HTML.gif in the inequality above, we find
lim n d ( x n , T x n ) d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equo_HTML.gif
So, we deduce that lim n d ( x n , T x n ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq63_HTML.gif. As a consequence, we derive
lim n M ( x n , x ) = max { lim n d ( x , x n ) , lim n d ( x n , T x n ) , d ( x , T x ) } = d ( x , T x ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equp_HTML.gif
and hence
lim n M ( x n , x ) d ( A , B ) = d ( x , T x ) d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ21_HTML.gif
(2.14)
Combining (1.6) and (2.14), we find
d ( x , T x ) d ( A , B ) lim n d ( T x n , T x ) lim n [ β ( M ( x n , x ) ) ( M ( x n , x ) d ( A , B ) ) ] = lim n β ( M ( x n , x ) ) ( d ( x , T x ) d ( A , B ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ22_HTML.gif
(2.15)
Since d ( x , T x ) d ( A , B ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq64_HTML.gif together with (2.15), we get 1 lim n β ( M ( x n , x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq65_HTML.gif. Hence, we have
lim n β ( M ( x n , x ) ) = 1 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equq_HTML.gif
which yields
lim n M ( x n , x ) = d ( x , T x ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equr_HTML.gif

As a result, we deduce that d ( x , T x ) = 0 > d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq66_HTML.gif, a contradiction. So, d ( x , T x ) d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq67_HTML.gif and hence d ( x , T x ) = d ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq68_HTML.gif, x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq12_HTML.gif is a best proximity point of T. Hence, we conclude that T has a best proximity point.

We claim that the best proximity point of T is unique.

Suppose, on the contrary, that x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq12_HTML.gif and y https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq69_HTML.gif are two distinct best proximity points of T. Thus, we have
d ( x , T x ) = d ( A , B ) = d ( y , T y ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ23_HTML.gif
(2.16)
By using the P-property, we find
d ( x , y ) = d ( T x , T y ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equ24_HTML.gif
(2.17)
and
M ( x , y ) = max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } = max { d ( x , y ) , d ( A , B ) , d ( A , B ) } = d ( x , y ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equs_HTML.gif
Due to the fact that T is a generalized Geraghty-contraction, we have
d ( x , y ) = d ( T x , T y ) β ( M ( x , y ) ) ( M ( x , y ) d ( A , B ) ) = β ( d ( x , y ) ) ( d ( x , y ) d ( A , B ) ) β ( d ( x , y ) ) d ( x , y ) < d ( x , y ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equt_HTML.gif

a contradiction. This completes the proof. □

Remark 9 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq15_HTML.gif be a metric space and A be any nonempty subset of X. It is evident that a pair ( A , A ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq25_HTML.gif satisfies the P-property.

Corollary 10 Suppose that ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq1_HTML.gif is a complete metric space and A is a nonempty closed subset of X. If a self-mapping T : A A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq70_HTML.gif is a generalized Geraghty-contraction, then it has a unique fixed point.

Proof Taking Remark 9 into consideration, we conclude the desired result by applying Theorem 8 with A = B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq71_HTML.gif. □

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

Example 11 Suppose that X = R 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq72_HTML.gif with the metric
d ( ( x , y ) , ( x , y ) ) = max { | x x | , | y y | } , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equu_HTML.gif
and consider the closed subsets
A = { ( x , 0 ) : 0 x 1 } , B = { ( x , 0 ) : 1 x 0 } , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equv_HTML.gif
and let T : A B https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq73_HTML.gif be the mapping defined by
T ( ( x , 0 ) ) = ( x 1 + x , 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equw_HTML.gif

Since d ( A , B ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq74_HTML.gif, the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq21_HTML.gif has the P-property.

Notice that A 0 = ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq75_HTML.gif and B 0 = ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq76_HTML.gif and T ( A 0 ) B 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq26_HTML.gif.

Moreover,
d ( T ( x , 0 ) , T ( x , 0 ) ) = d ( ( x 1 + x , 0 ) , ( x 1 + x , 0 ) ) = | x 1 + x + x 1 + x | = | x x | ( 1 + x ) ( 1 + x ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equx_HTML.gif
and, as ( 1 + x ) ( 1 + x ) 1 + | x x | https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq77_HTML.gif, we have
d ( T ( x , 0 ) , T ( x , 0 ) ) = | x x | ( 1 + x ) ( 1 + x ) | x x | 1 + | x x | = β ( | x x | ) = β ( d ( ( x , 0 ) , ( x , 0 ) ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equy_HTML.gif

where β : [ 0 , ) [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq78_HTML.gif is defined as β ( t ) = t 1 + t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq79_HTML.gif.

Notice that β is nondecreasing since β ( t ) = 1 ( 1 + t ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq80_HTML.gif.

Therefore,
d ( T ( x , 0 ) , T ( x , 0 ) ) β ( d ( ( x , 0 ) , ( x , 0 ) ) ) β ( M ( ( x , 0 ) , ( x , 0 ) ) ) = β ( M ( ( x , 0 ) , ( x , 0 ) ) ) M ( ( x , 0 ) , ( x , 0 ) ) M ( ( x , 0 ) , ( x , 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equz_HTML.gif

and it is easily seen that the function γ ( t ) = β ( t ) t = 1 1 + t https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq81_HTML.gif belongs to F.

Therefore, since the assumptions of Theorem 8 are satisfied, by Theorem 8 there exists a unique ( x , 0 ) A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq82_HTML.gif such that
d ( ( x , 0 ) , T ( x , 0 ) ) = 0 = d ( A , B ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_Equaa_HTML.gif

The point ( x , 0 ) A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq82_HTML.gif is ( 0 , 0 ) A https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-286/MediaObjects/13660_2013_Article_724_IEq83_HTML.gif.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science and Arts, Amasya University
(2)
Department of Mathematics, Atilim University
(3)
Department of Mathematics, University of 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. Caballero J, Harjani J, Sadarangani K: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. 2012., 2012: Article ID 231
  3. 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
  4. Anuradha J, Veeramani P: Proximal pointwise contraction. Topol. Appl. 2009, 156: 2942–2948. 10.1016/j.topol.2009.01.017MathSciNetView Article
  5. 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
  6. Basha SS, Veeramani P: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 2000, 103: 119–129. 10.1006/jath.1999.3415MathSciNetView Article
  7. Raj VS, Veeramani P: Best proximity pair theorems for relatively nonexpansive mappings. Appl. Gen. Topol. 2009, 10: 21–28.MathSciNetView Article
  8. 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
  9. 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
  10. Raj VS: A best proximity theorem for weakly contractive non-self mappings. Nonlinear Anal. 2011, 74: 4804–4808. 10.1016/j.na.2011.04.052MathSciNetView Article
  11. Raj, VS: Banach’s contraction principle for non-self mappings. Preprint
  12. Karapınar E: Best proximity points of cyclic mappings. Appl. Math. Lett. 2012, 25(11):1761–1766. 10.1016/j.aml.2012.02.008MathSciNetView Article
  13. Karapınar E, Erhan ÝM: Best proximity point on different type contractions. Appl. Math. Inf. Sci. 2011, 3(3):342–353.
  14. Karapınar E: Best proximity points of Kannan type cyclic weak ϕ -contractions in ordered metric spaces. An. Stiint. Univ. Ovidius Constanta 2012, 20(3):51–64.MathSciNet
  15. Mongkolkeha C, Kumam P: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. 2012, 155: 215–226. 10.1007/s10957-012-9991-yMathSciNetView Article
  16. Mongkolkeha C, Kumam P: Some common best proximity points for proximity commuting mappings. Optim. Lett. 2012. doi:10.1007/s11590–012–0525–1 in press
  17. Jleli M, Samet B: Best proximity points for α - ψ -proximal contractive type mappings and applications. Bull. Sci. Math. 2013. doi:10.1016/j.bulsci.2013.02.003
  18. Mongkolkeha C, Cho YJ, Kumam P: Best proximity points for generalized proximal C -contraction mappings in metric spaces with partial orders. J. Inequal. Appl. 2013., 2013: Article ID 94
  19. De la Sen M: Fixed point and best proximity theorems under two classes of integral-type contractive conditions in uniform metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 510974

Copyright

© Bilgili et al.; licensee Springer 2013

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.