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Best proximity point theorems for weakly contractive mapping and weakly Kannan mapping in partial metric spaces

Fixed Point Theory and Applications20142014:50

https://doi.org/10.1186/1687-1812-2014-50

Received: 24 October 2013

Accepted: 10 February 2014

Published: 25 February 2014

Abstract

The purpose of this paper is to obtain best proximity point theorems for a weakly contractive mapping and a weakly Kannan mapping in partial metric spaces. In this paper, the P-operator technique, which changes a non-self mapping to a self mapping, provides a key method. Many recent results in this area have been improved.

MSC:47H05, 47H09, 47H10.

Keywords

  • fixed point
  • best proximity point
  • weakly contractive mapping
  • P-property
  • partial metric

1 Introduction and preliminaries

Let us recall some basic definitions of a partial metric space and its properties which can be found in [1].

Definition 1.1 A partial metric on a nonempty set X is a function p : X × X R + such that for all x , y , z X :

(p1) x = y p ( x , x ) = p ( x , y ) = p ( y , y ) ,

(p2) p ( x , x ) p ( x , y ) ,

(p3) p ( x , y ) = p ( y , x ) ,

(p4) p ( x , y ) p ( x , z ) + p ( z , y ) p ( z , z ) .

A partial metric space is a pair ( X , p ) such that X is a nonempty set and p is a partial metric on X.

We can see from (p1) and (p2) that p ( x , y ) = 0 implies x = y . However, the converse is not necessarily true. A typical example of this situation is provided by the partial metric space ( R + , p max ) , where the function p max : R + × R + R + is defined by p max ( x , y ) = max { x , y } for all x , y R + . Other examples of partial metric spaces which are interesting from a computational point of view may be found in [1] and [2].

Following [1], each partial metric p on X generates a T 0 topology τ ( p ) on X, whose base is a family of open p-balls:
{ B p ( x , ε ) : x X , ε > 0 } ,

where B p ( x , ε ) = { y X : p ( x , y ) p ( x , x ) + ε } for all x X and ε > 0 . Definitions of convergence, Cauchy sequence, completeness and continuity on partial metric spaces are as follows:

(d1) A sequence { x n } in a partial metric space ( X , p ) converges to x if and only if p ( x , x ) = lim n p ( x , x n ) .

(d2) A sequence { x n } in a partial metric space ( X , p ) is called a Cauchy sequence if lim n , m p ( x n , x m ) exists and is finite.

(d3) A partial metric space ( X , p ) is called complete if every Cauchy sequence { x n } in X converges, with respect to τ ( p ) , to a point x X such that p ( x , x ) = lim n , m p ( x n , x m ) .

(d4) A mapping f : X X is said to be continuous at x 0 X if for every ε > 0 , there exists δ > 0 such that f ( B p ( x 0 , δ ) ) B p ( f ( x 0 ) , ε ) .

It can be easily verified that the function d p : X × X R + defined by
d p ( x , y ) = 2 p ( x , y ) p ( x , x ) p ( y , y )

is a metric on X. The following useful remarks were introduced in [1]:

(r1) If a sequence converges in a partial metric space ( X , p ) with respect to τ ( d p ) , then it converges with respect to τ ( p ) . Of course, the converse is not true.

(r2) A sequence { x n } n N in a partial metric space ( X , p ) is a Cauchy sequence if and only if it is a Cauchy sequence in the metric space ( X , d p ) .

(r3) A partial metric space ( X , p ) is complete if and only if the metric space ( X , d p ) is complete.

(r4) Given a sequence { x n } n N in a partial metric space ( X , p ) and x X , we have that
lim n d p ( x , x n ) = 0 p ( x , x ) = lim n p ( x , x n ) = lim n , m p ( x n , x m ) .

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 let T : X X be a contraction of X into itself; then T has a unique fixed point in X.

In the last fifty years, the Banach contraction principle has been extensively studied and generalized on many settings. One of the generalizations is a weakly contractive mapping.

Definition 1.2 ([3])

Let ( X , d ) be a metric space. A mapping f : X X is said to be weakly contractive provided that
d ( f ( x ) , f ( y ) ) α ¯ ( x , y ) d ( x , y )
for all x , y X , where the function α ¯ : X × X [ 0 , 1 ) , holds for every 0 < a < b that
θ ( a , b ) = sup { α ¯ ( x , y ) : a d ( x , y ) b } < 1 .

The fixed point theorem for a weakly contractive mapping was presented in [3].

Theorem 1.3 Let ( X , d ) be a complete metric space. If f : X X is a weakly contractive mapping, then f has a unique fixed point x and the Picard sequence of iterates { f n ( x ) } n N converges, for every x X , to x .

One type of contraction which is different from the Banach contraction is Kannan mappings. In [4], Kannan obtained the following fixed point theorem.

Theorem 1.4 ([4])

Let ( X , d ) be a complete metric space, and let f : X X be a mapping such that
d ( f ( x ) , f ( y ) ) α 2 [ d ( x , f ( x ) ) + d ( y , f ( y ) ) ]

for all x , y X and some α [ 0 , 1 ] , then f has a unique fixed point x X . Moreover, the Picard sequence of iterates { f n ( x ) } n N converges, for every x X , to x .

In [5], the authors introduced a more general weakly Kannan mapping and obtained its fixed point theorem.

Definition 1.5 ([5])

Let ( X , d ) be a metric space. A mapping f : X X is said to be weakly Kannan if there exists α ¯ : X × X [ 0 , 1 ) which satisfies, for every 0 < a b and for all x , y X , that
θ ( a , b ) = sup { α ¯ ( x , y ) : a d ( x , y ) b } < 1
and
d ( f ( x ) , f ( y ) ) α ¯ ( x , y ) 2 [ d ( x , f ( x ) ) + d ( y , f ( y ) ) ] .

Theorem 1.6 ([5])

Let ( X , d ) be a complete metric space. If f : X X is a weakly Kannan mapping, then f has a unique fixed point x and the Picard sequence of iterates { f n ( x ) } n N converges, for every x X , to x .

Recently, Alghamdi et al. [6] generalized the weakly contractive and weakly Kannan mappings to partial metric spaces and obtained the following fixed point theorems.

Definition 1.7 ([6])

Let ( X , p ) be a partial metric space. A mapping f : X X is said to be weakly contractive provided that there exists α ¯ : X × X [ 0 , 1 ) such that for every 0 a b ,
θ ( a , b ) = sup { α ¯ ( x , y ) : a p ( x , y ) b } < 1 ,
and for every x , y X ,
p ( f ( x ) , f ( y ) ) α ¯ ( x , y ) p ( x , y ) .

Definition 1.8 ([6])

Let ( X , p ) be a partial metric space. A mapping f : X X is said to be weakly Kannan if there exists α ¯ : X × X [ 0 , 1 ) which satisfies for every 0 < a b and for all x , y X that
θ ( a , b ) = sup { α ¯ ( x , y ) : a p ( x , y ) b } < 1
and
p ( f ( x ) , f ( y ) ) α ¯ ( x , y ) 2 [ p ( x , f ( x ) ) + p ( y , f ( y ) ) ] .

Theorem 1.9 ([6])

Let ( X , p ) be a complete partial metric space, and let f : X X be a weakly contractive mapping. Then f has a unique fixed point x X and the Picard sequence of iterates { f n ( x ) } n N converges, with respect to τ ( d p ) , for every x X , to x . Moreover, p ( x , x ) = 0 .

Theorem 1.10 ([6])

Let ( X , p ) be a complete partial metric space, and let f : X X be a weakly Kannan mapping. Then f has a unique fixed point x X and the Picard sequence of iterates { f n ( x ) } n N converges, with respect to τ ( d p ) , for every x X , to x . Moreover, p ( x , x ) = 0 .

In this paper, we first obtain best proximity point theorems for a weakly contractive mapping and a weakly Kannan mapping in partial metric spaces. The P-operator technique, which changes a non-self mapping to a self mapping, provides a key method. Many recent results in this area have been improved.

Before giving the main results, we need the following notations and basic facts.

Let A, B be two nonempty subsets of a complete partial metric space ( X , p ) and consider a mapping T : A B . The best proximity point problem is whether we can find an element x 0 A such that p ( x 0 , T x 0 ) = p ( A , B ) , where p ( A , B ) = inf { p ( x , y ) : x A  and  y B } . Since p ( x , T x ) p ( A , B ) for any x A , in fact, the optimal solution to this problem is the one for which the value p ( A , B ) is attained. Some works on the best proximity point problem can be found in [711].

Let A and B be two nonempty subsets of a partial metric space ( X , p ) . We denote by A 0 and B 0 the following sets:
A 0 = { x A : p ( x , y ) = p ( A , B )  for some  y B } , B 0 = { y B : p ( x , y ) = p ( A , B )  for some  x A } .

2 Best proximity point theorems in partial metric spaces

Definition 2.1 Let ( A , B ) be a pair of nonempty subsets of a partial metric space ( X , p ) . A mapping f : A B is said to be weakly contractive provided that
p ( f ( x ) , f ( y ) ) α ¯ ( x , y ) p ( x , y )
for all x , y A , where the function α ¯ : A × A [ 0 , 1 ) , holds for every 0 < a < b that
θ ( a , b ) = sup { α ¯ ( x , y ) : a p ( x , y ) b } < 1 .
Definition 2.2 Let ( A , B ) be a pair of nonempty subsets of a partial metric space ( X , p ) . A mapping f : A B is said to be weakly Kannan provided that
p ( f ( x ) , f ( y ) ) α ¯ ( x , y ) 2 [ p ( x , f ( x ) ) + p ( y , f ( y ) ) 2 p ( A , B ) ]
for all x , y A , where the function α ¯ : A × A [ 0 , 1 ) , holds for every 0 < a < b that
θ ( a , b ) = sup { α ¯ ( x , y ) : a p ( x , y ) b } < 1 .

We rewrite the P-property in the setting of partial metric spaces as follows.

Definition 2.3 Let ( A , B ) be a pair of nonempty subsets of a partial metric space ( X , p ) 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 ,
{ p ( x 1 , y 1 ) = p ( A , B ) , p ( x 2 , y 2 ) = p ( A , B ) p ( x 1 , x 2 ) = p ( y 1 , y 2 ) .

Lemma 2.4 Let ( X , p ) be a partial metric space, then p is a continuous function, that is, for any x n , y n , x , y X , if x n x , y n y , then p ( x n , y n ) p ( x , y ) as n .

Proof Since
p ( x n , y n ) p ( x n , x ) + p ( x , y n ) p ( x , x ) p ( x n , x ) + p ( x , y ) + p ( y , y n ) p ( x , x ) p ( y , y ) .
From the above inequality, we can get that
p ( x n , y n ) p ( x , y ) [ p ( x n , x ) p ( x , x ) ] + [ p ( y , y n ) p ( y , y ) ] 0 as  n .
On the other hand, we have
p ( x , y ) p ( x , x n ) + p ( x n , y ) p ( x n , x n ) p ( x , x n ) + p ( x n , y n ) + p ( y n , y ) p ( x n , x n ) p ( y n , y n ) .
Then we can obtain
p ( x , y ) p ( x n , y n ) [ p ( x , x n ) p ( x n , x n ) ] + [ p ( y n , y ) p ( y n , y n ) ] 0 as  n .
Above all, we can get that
| p ( x n , y n ) p ( x , y ) | 0 as  n .

This completes the proof. □

Remark For (r4) we know that, for any x n , y n , x , y X , if d p ( x n , x ) 0 , d p ( y n , y ) 0 , then p ( x n , y n ) p ( x , y ) as n .

Theorem 2.5 Let ( A , B ) be a pair of nonempty closed subsets of a complete partial metric space ( X , p ) such that A 0 . Let T : A B be a continuous weakly contractive mapping. Suppose that T ( A 0 ) B 0 and the pair ( A , B ) has the P-property. Then T has a unique best proximity point x A 0 and the iteration sequence { x 2 k } n = 0 defined by
x 2 k + 1 = T x 2 k , p ( x 2 k + 2 , x 2 k + 1 ) = p ( A , B ) , k = 0 , 1 , 2 ,

converges, with respect to τ ( d p ) , for every x 0 A 0 , to x .

Proof We first prove that B 0 is closed with respect to ( X , d p ) . Let { y n } B 0 be a sequence such that y n q B . It follows from the P-property that
d p ( y n , y m ) = 2 p ( y n , y m ) p ( y n , y n ) p ( y m , y m ) = 2 p ( x n , x m ) p ( x n , x n ) p ( x m , x m ) = d p ( x n , x m ) .
Hence
d p ( y n , y m ) 0 d p ( x n , x m ) 0

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

Let A ¯ 0 be the closure of A 0 in a metric space ( X , d p ) , 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 closedness 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 : p ( x , y ) = p ( A , B ) } . Since the pair ( A , B ) has the P-property, we have
p ( P A 0 T x 1 , P A 0 T x 2 ) = p ( T x 1 , T x 2 ) α ¯ ( x 1 , x 2 ) p ( 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 weak contraction from a complete partial metric subspace A ¯ 0 into itself. Using Theorem 1.9, we can get that P A 0 T has a unique fixed point x ; that is, P A 0 T x = x A 0 , which implies that
p ( x , T x ) = p ( A , B ) .
Therefore, x is the unique one in A 0 such that p ( x , T x ) = p ( A , B ) . And the Picard iteration sequence { ( P A 0 T ) n x 0 } n N converges, with respect to τ ( d p ) , for every x 0 A 0 , to x . Since the iteration sequence { x 2 k } n = 0 defined by
x 2 k + 1 = T x 2 k , p ( x 2 k + 2 , x 2 k + 1 ) = p ( A , B ) , k = 0 , 1 , 2 ,

is exactly the subsequence of { x n } , so that it converges, for every x 0 A 0 , to x . This completes the proof. □

Theorem 2.6 Let ( A , B ) be a pair of nonempty closed subsets of a complete partial metric space ( X , p ) such that A 0 . Let T : A B be a continuous weakly Kannan mapping. Suppose that T ( A 0 ) B 0 and the pair ( A , B ) has the P-property. Then T has a unique best proximity point x A 0 and the iteration sequence { x 2 k } n = 0 defined by
x 2 k + 1 = T x 2 k , p ( x 2 k + 2 , x 2 k + 1 ) = p ( A , B ) , k = 0 , 1 , 2 ,

converges, with respect to τ ( d p ) , for every x 0 A 0 , to x .

Proof We can prove that B 0 is closed and T ( A ¯ 0 ) B 0 in the same way as in Theorem 2.5. Now define an operator P A 0 : T ( A ¯ 0 ) A 0 by P A 0 y = { x A 0 : p ( x , y ) = p ( A , B ) } . Since the pair ( A , B ) has the P-property, we have
p ( P A 0 T x 1 , P A 0 T x 2 ) = p ( T x 1 , T x 2 ) α ¯ ( x , y ) 2 [ p ( x 1 , T x 1 ) + p ( x 2 , T x 2 ) 2 p ( A , B ) ] α ¯ ( x , y ) 2 [ p ( x 1 , P A 0 T x 1 ) + p ( P A 0 T x 1 , T x 1 ) + p ( x 2 , P A 0 T x 2 ) + p ( P A 0 T x 2 , T x 2 ) 2 p ( A , B ) ] = α ¯ ( x , y ) 2 [ p ( x 1 , P A 0 T x 1 ) + p ( x 2 , P A 0 T x 2 ) ]
for any x 1 , x 2 A ¯ 0 . This shows that P A 0 T : A ¯ 0 A ¯ 0 is a weakly Kannan mapping from a complete partial metric subspace A ¯ 0 into itself. Using Theorem 1.10, we can get that P A 0 T has a unique fixed point x ; that is, P A 0 T x = x A 0 , which implies that
p ( x , T x ) = p ( A , B ) .
Therefore, x is the unique one in A 0 such that p ( x , T x ) = p ( A , B ) . And the Picard iteration sequence { ( P A 0 T ) n x 0 } n N converges, with respect to τ ( d p ) , for every x 0 A 0 , to x . Since the iteration sequence { x 2 k } n = 0 defined by
x 2 k + 1 = T x 2 k , p ( x 2 k + 2 , x 2 k + 1 ) = p ( A , B ) , k = 0 , 1 , 2 ,

is exactly the subsequence of { x n } , so that it converges, for every x 0 A 0 , to x . This completes the proof. □

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, China

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Copyright

© Zhang and Su; licensee Springer. 2014

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 credited.

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