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A note on ‘A best proximity point theorem for Geraghty-contractions’

Fixed Point Theory and Applications20132013:99

https://doi.org/10.1186/1687-1812-2013-99

  • Received: 3 January 2013
  • Accepted: 26 March 2013
  • Published:

Abstract

In Caballero et al. (Fixed Point Theory Appl. (2012). doi:10.1186/1687-1812-2012-231), the authors prove a best proximity point theorem for Geraghty nonself contraction. In this note, not only P-property has been weakened, but also an improved best proximity point theorem will be presented by a short and simple proof. An example which satisfies weak P-property but not P-property has been presented to demonstrate our results.

MSC:47H05, 47H09, 47H10.

Keywords

  • Geraghty-contractions
  • fixed point
  • best proximity point
  • weak P-property

1 Introduction and preliminaries

Let A and B be nonempty subsets of a metric space ( X , d ) . An operator T : A B is said to be contractive if there exists k [ 0 , 1 ) such that d ( T x , T y ) k d ( x , y ) for any x , y A . The well-known Banach contraction principle says: Let ( X , d ) be a complete metric space, and T : X X be a contraction of X into itself. Then T has a unique fixed point in X.

In 1973, Geraghty introduced the Geraghty-contraction and obtained Theorem 1.2.

Definition 1.1 ([1])

Let ( X , d ) be a metric space. A mapping T : X X is said to be a Geraghty-contraction if there exists such that for any x , y X
d ( T x , T y ) β ( d ( x , y ) ) d ( x , y ) ,
where the class Γ denotes those functions β : [ 0 , ) [ 0 , 1 ) satisfying the following condition:
β ( t n ) 1 t n 0 .

Theorem 1.2 ([1])

Let ( X , d ) be a complete metric space and T : X X be an operator. Suppose that there exists such that for any x , y X ,
d ( T x , T y ) β ( d ( x , y ) ) d ( x , y ) .

Then T has a unique fixed point.

Obviously, Theorem 1.2 is an extensive version of Banach contraction principle. In 2012, Caballero et al. introduced generalized Geraghty-contraction as follows.

Definition 1.3 ([2])

Let A, B be two nonempty subsets of a metric space ( X , d ) . A mapping T : A B is said to be a Geraghty-contraction if there exists such that for any x , y A
d ( T x , T y ) β ( d ( x , y ) ) d ( x , y ) ,
where the class denotes those functions β : [ 0 , ) [ 0 , 1 ) satisfying the following condition:
β ( t n ) 1 t n 0 .

Now we need the following notations and basic facts.

Let A and B be two nonempty subsets of a metric space ( X , d ) . We denote by A 0 and B 0 the following sets:

where d ( A , B ) = inf { d ( x , y ) : x A  and  y B } .

In [3], the authors give sufficient conditions for when the sets A 0 and B 0 are nonempty. In [4], the author presents the following definition and proves that any pair ( A , B ) of nonempty, closed and convex subsets of a real Hilbert space H satisfies the P-property.

Definition 1.4 ([2])

Let ( A , B ) be a pair of nonempty subsets of a metric space ( X , d ) with A 0 . Then the pair ( A , B ) is said to have the P-property if and only if for any x 1 , x 2 A 0 and y 1 , y 2 B 0 ,
{ 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 ) .

Let A, B be two nonempty subsets of a complete metric space and consider a mapping T : A B . The best proximity point problem is whether we can find an element x 0 A such that d ( x 0 , T x 0 ) = min { d ( x , T x ) : x A } . Since d ( x , T x ) d ( A , B ) for any x A , in fact, the optimal solution to this problem is the one for which the value d ( A , B ) is attained.

In [2], the authors give a generalization of Theorem 1.2 by considering a nonself map and they get the following theorem.

Theorem 1.5 ([2])

Let ( A , B ) be a pair of nonempty closed subsets of a complete metric space ( X , d ) such that A 0 is nonempty. Let T : A B be a Geraghty-contraction satisfying T ( A 0 ) B 0 . Suppose that the pair ( A , B ) has the P-property. Then there exists a unique x in A such that d ( x , T x ) = d ( A , B ) .

Remark In [2], the proof of Theorem 1.5 is unnecessarily complex. In this note, not only P-property has been weakened, but also an improved best proximity point theorem will be presented by a short and simple proof. An example which satisfies weak P-property but not P-property has been presented to demonstrate our results.

2 Main results

Before giving our main results, we first introduce the notion of weak P-property.

Weak P-property Let ( A , B ) be a pair of nonempty subsets of a metric space ( X , d ) with A 0 . Then the pair ( A , B ) is said to have the weak P-property if and only if for any x 1 , x 2 A 0 and y 1 , y 2 B 0 ,
{ 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 ) .

Now we are in a position to give our main results.

Theorem 2.1 Let ( A , B ) be a pair of nonempty closed subsets of a complete metric space ( X , d ) such that A 0 . Let T : A B be a Geraghty-contraction satisfying T ( A 0 ) B 0 . Suppose that the pair ( A , B ) has the weak P-property. Then there exists a unique x in A such that d ( x , T x ) = d ( A , B ) .

Proof We first prove that B 0 is closed. Let { y n } B 0 be a sequence such that y n q B . It follows from the weak P-property that
d ( y n , y m ) 0 d ( x n , x m ) 0 ,

as n , m , where x n , x m A 0 and d ( x n , y n ) = d ( A , B ) , d ( x m , y m ) = d ( A , B ) . Then { x n } is a Cauchy sequence so that { x n } converges strongly to a point p A . By the continuity of metric d we have d ( p , q ) = d ( A , B ) , that is, q B 0 , and hence B 0 is closed.

Let A ¯ 0 be the closure of A 0 , we claim that T ( A ¯ 0 ) B 0 . In fact, if x A ¯ 0 A 0 , then there exists a sequence { x n } A 0 such that x n x . By the continuity of T and the closeness of B 0 , we have T x = lim n T x n B 0 . That is T ( A ¯ 0 ) B 0 .

Define an operator P A 0 : T ( A ¯ 0 ) A 0 , by P A 0 y = { x A 0 : d ( x , y ) = d ( A , B ) } . Since the pair ( A , B ) has weak P-property and T is a Geraghty-contraction, we have
d ( P A 0 T x 1 , P A 0 T x 2 ) d ( T x 1 , T x 2 ) β ( d ( x 1 , x 2 ) ) d ( x 1 , x 2 )
for any x 1 , x 2 A ¯ 0 . This shows that P A 0 T : A ¯ 0 A ¯ 0 is a Geraghty-contraction from complete metric subspace A ¯ 0 into itself. Using Theorem 1.2, we can get P A 0 T has a unique fixed point x . That is P A 0 T x = x A 0 . It implies that
d ( x , T x ) = d ( A , B ) .

Therefore, x is the unique one in A 0 such that d ( x , T x ) = d ( A , B ) . It is easy to see that x is also the unique one in A such that d ( x , T x ) = d ( A , B ) . □

Remark In Theorem 2.1, P-property is weakened to weak P-property. Therefore, Theorem 2.1 is an improved result of Theorem 1.5. In addition, our proof is shorter and simpler than that in [2]. In fact, our proof process is less than one page. However, the proof process in [2] is three pages.

3 Example

Now we present an example which satisfies weak P-property but not P-property.

Consider ( R 2 , d ) , where d is the Euclidean distance and the subsets A = { ( 0 , 0 ) } and B = { y = 1 + 1 x 2 } .

Obviously, A 0 = { ( 0 , 0 ) } , B 0 = { ( 1 , 1 ) , ( 1 , 1 ) } and d ( A , B ) = 2 . Furthermore,
d ( ( 0 , 0 ) , ( 1 , 1 ) ) = d ( ( 0 , 0 ) , ( 1 , 1 ) ) = 2 ,
however,
0 = d ( ( 0 , 0 ) , ( 0 , 0 ) ) < d ( ( 1 , 1 ) , ( 1 , 1 ) ) = 2 .

We can see that the pair ( A , B ) satisfies the weak P-property but not the P-property.

Declarations

Acknowledgements

This project is supported by the National Natural Science Foundation of China under grant (11071279).

Authors’ Affiliations

(1)
Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, P.R. China

References

  1. Geraghty M: On contractive mappings. Proc. Am. Math. Soc. 1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5MATHMathSciNetView ArticleGoogle Scholar
  2. Caballero J, et al.: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. 2012. doi:10.1186/1687–1812–2012–231Google Scholar
  3. Kirk WA, Reich S, Veeramani P: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 2003, 24: 851–862. 10.1081/NFA-120026380MATHMathSciNetView ArticleGoogle Scholar
  4. Sankar Raj, V: Banach contraction principle for non-self mappings. PreprintGoogle Scholar

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