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

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.

1 Introduction and preliminaries

Let A and B be nonempty subsets of a metric space (X,d). An operator T:AB is said to be contractive if there exists k[0,1) such that d(Tx,Ty)kd(x,y) for any x,yA. The well-known Banach contraction principle says: Let (X,d) be a complete metric space, and T:XX 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:XX is said to be a Geraghty-contraction if there exists such that for any x,yX

d(Tx,Ty)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:XX be an operator. Suppose that there exists such that for any x,yX,

d(Tx,Ty)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:AB is said to be a Geraghty-contraction if there exists such that for any x,yA

d(Tx,Ty)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):xA and yB}.

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:AB. 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,Tx):xA}. Since d(x,Tx)d(A,B) for any xA, 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:AB 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:AB 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 qB. It follows from the weak P-property that

d( y n , y m )0d( 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 pA. 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 Tx= 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.

References

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Acknowledgements

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

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Correspondence to Yongfu Su.

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All the authors contributed equally to the writing of the present article. All authors read and approved the final manuscript.

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Zhang, J., Su, Y. & Cheng, Q. A note on ‘A best proximity point theorem for Geraghty-contractions’. Fixed Point Theory Appl 2013, 99 (2013). https://doi.org/10.1186/1687-1812-2013-99

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Keywords

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