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

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.

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 Ï„(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)=2p(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 Ï„( d p ), then it converges with respect to Ï„(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(Tx,Ty)≤kd(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,Tx)≥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 [7–11].

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)|→0as 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 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 :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. □

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Acknowledgements

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

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Zhang, J., Su, Y. Best proximity point theorems for weakly contractive mapping and weakly Kannan mapping in partial metric spaces. Fixed Point Theory Appl 2014, 50 (2014). https://doi.org/10.1186/1687-1812-2014-50

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