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Remarks on the fixed point problem of 2-metric spaces

  • Nguyen Van Dung
  • Nguyen Trung Hieu
  • Nguyen Thia Thanh Ly
  • Vo Duc Thinh
Open Access
Research

Abstract

In this paper, we prove a fixed point theorem on a 2-metric space and show that the main results in Lahiri et al. (Taiwan. J. Math. 15:337-352, 2011) and Singh et al. (J. Adv. Math. Stud. 5:71-76, 2012) may be obtained easily from the axioms of a 2-metric space. Examples are given to validate the results.

Keywords

Point Theorem Fixed Point Theorem Distinct Point Contraction Condition Unique Fixed Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction and preliminaries

There have been some generalizations of a metric space and its fixed point problem such as 2-metric spaces, D-metric spaces, G-metric spaces, cone metric spaces, complex-valued metric spaces. The notion of a 2-metric space was introduced by Gähler in [1]. Notice that a 2-metric is not a continuous function of its variables, whereas an ordinary metric is. This led Dhage to introduce the notion of a D-metric space in [2]. After that, in [3], Mustafa and Sims showed that most of topological properties of D-metric spaces were not correct. Then, in [4], they introduced the notion of a G-metric space and many fixed point theorems on G-metric spaces have been obtained. Unfortunately, in [5], Jleli and Samet showed that most of the obtained fixed point theorems on G-metric spaces can be deduced immediately from fixed point theorems on metric spaces or quasi-metric spaces. In [6], Huang and Zhang defined the notion of a cone metric space, which generalized a metric and a metric space, and proved some fixed point theorems for contractive maps on this space. After that, many authors extended some fixed point theorems on metric spaces to cone metric spaces. In [7], Feng and Mao introduced a metric on a cone metric space and then proved that a complete cone metric space is always a complete metric space. They verified that a contractive map on a cone metric space is a contractive map on a metric space, then fixed point theorems on a cone metric space are, essentially, fixed point theorems on a metric space. In [8], Azam, Fisher and Khan introduced the notion of a complex-valued metric space and some fixed point theorems on this space were stated. But in [9], Sastry, Naidu and Bekeshie showed that some fixed point theorems recently generalized to complex-valued metric spaces are consequences of their counter parts in the setting of metric spaces and hence are redundant.

Notice that in the above generalizations, only a 2-metric space is not topologically equivalent to an ordinary metric. Then there was no easy relationship between results obtained in 2-metric spaces and metric spaces. In particular, the fixed point theorems on 2-metric spaces and metric spaces may be unrelated easily. For the fixed point theorems on 2-metric spaces, the readers may refer to [10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

In this paper, we prove a fixed point theorem on a 2-metric space and show that the main results in [17] and [20] may be obtained easily from the axioms of a 2-metric space. Examples are given to validate the results.

Now we recall some notions and lemmas which will be useful in what follows.

Definition 1.1 ([1])

Let X be a non-empty set and let d : X × X × X R Open image in new window be a map satisfying the following conditions:
  1. 1.

    For every pair of distinct points a , b X Open image in new window, there exists a point c X Open image in new window such that d ( a , b , c ) 0 Open image in new window.

     
  2. 2.

    d ( a , b , c ) = 0 Open image in new window only if at least two of three points are the same.

     
  3. 3.

    The symmetry: d ( a , b , c ) = d ( a , c , b ) = d ( b , c , a ) = d ( b , a , c ) = d ( c , a , b ) = d ( c , b , a ) Open image in new window for all a , b , c X Open image in new window.

     
  4. 4.

    The rectangle inequality: d ( a , b , c ) d ( a , b , d ) + d ( b , c , d ) + d ( c , a , d ) Open image in new window for all a , b , c , d X Open image in new window.

     

Then d is called a 2-metric on X and ( X , d ) Open image in new window is called a 2-metric space which will be sometimes denoted by X if there is no confusion. Every member x X Open image in new window is called a point in X.

Remark 1.2
  1. 1.

    Every 2-metric is non-negative.

     
  2. 2.

    We may assume that every 2-metric space contains at least three distinct points.

     

2 Main results

Theorem 2.1 Let ( X , d ) Open image in new window be a 2-metric space and let T , F : X X Open image in new window be two maps. If d ( T x , F y , x ) = d ( T x , F y , y ) = 0 Open image in new window for all x , y X Open image in new window, then Tx is a fixed point of T and Fy is a fixed point of F for all x , y X Open image in new window.

Proof For all x , y X Open image in new window, we have
d ( T x , x , y ) d ( T x , x , F y ) + d ( x , y , F y ) + d ( y , T x , F y ) = d ( F y , y , x ) . Open image in new window
By interchanging the roles of x and y, T and F, we get d ( F y , y , x ) d ( T x , x , y ) Open image in new window. So,
d ( T x , x , y ) = d ( F y , y , x ) Open image in new window
(2.1)

for all x , y X Open image in new window. Then if Tx plays the role of x in (2.1), we have d ( T 2 x , T x , y ) = d ( F y , y , T x ) = 0 Open image in new window for all x , y X Open image in new window. Hence T 2 x = T x Open image in new window for all x X Open image in new window. This proves that Tx is a fixed point of T for all x X Open image in new window. Similarly, Fy is a fixed point of F for all y X Open image in new window. □

Corollary 2.2 ([21], Lemma 4.1)

Let ( X , d ) Open image in new window be a 2-metric space and let T : X X Open image in new window be a map. If d ( T x , T y , x ) = 0 Open image in new window for all x , y X Open image in new window, then Tx is a fixed point of T for all x X Open image in new window.

The following examples show that Theorem 2.1 is a proper generalization of Corollary 2.2.

Example 2.3 Let X = { 1 , 2 , 3 } Open image in new window and d ( x , y , z ) = min { | x y | , | y z | , | z x | } Open image in new window for all x , y , z X Open image in new window. Then ( X , d ) Open image in new window is a 2-metric space. Let T , F : X X Open image in new window be two maps defined by T 1 = 1 Open image in new window, T 2 = T 3 = 3 Open image in new window and F 1 = F 3 = 3 Open image in new window, F 2 = 2 Open image in new window. We have d ( T 2 , T 1 , 2 ) = d ( 3 , 1 , 2 ) 0 Open image in new window and d ( F 1 , F 2 , 1 ) = d ( 3 , 2 , 1 ) 0 Open image in new window. This proves that Corollary 2.2 is neither applicable to T nor F. On the other hand, Theorem 2.1 is applicable to T and F since d ( T x , F y , x ) = d ( T x , F y , y ) = 0 Open image in new window for all x , y X Open image in new window as in the Table 1.
Table 1

Calculations for maps in Example 2.3

x

y

d ( T x , F y , x ) Open image in new window

d ( T x , F y , y ) Open image in new window

1

1

d(T 1,F 1,1)=d(1,3,1)=0

d(T 1,F 1,1)=d(1,3,1)=0

1

2

d(T 1,F 2,1)=d(1,2,1)=0

d(T 1,F 2,2)=d(1,2,2)=0

1

3

d(T 1,F 3,1)=d(1,3,1)=0

d(T 1,F 3,3)=d(1,3,3)=0

2

1

d(T 2,F 1,2)=d(3,3,2)=0

d(T 2,F 1,1)=d(3,3,1)=0

2

2

d(T 2,F 2,2)=d(3,2,2)=0

d(T 2,F 2,2)=d(3,2,2)=0

2

3

d(T 2,F 3,2)=d(3,3,2)=0

d(T 2,F 3,3)=d(3,3,3)=0

3

1

d(T 3,F 1,3)=d(3,3,3)=0

d(T 3,F 1,1)=d(3,3,1)=0

3

2

d(T 3,F 2,3)=d(3,2,3)=0

d(T 3,F 2,2)=d(3,2,2)=0

3

3

d(T 3,F 3,3)=d(3,3,3)=0

d(T 3,F 3,3)=d(3,3,3)=0

Definition 2.4 ([17], Definition 12)

Let ( X , d ) Open image in new window be a 2-metric space and let T : X X Open image in new window be a map. T is said to be contractive if d ( T x , T y , a ) < d ( x , y , a ) Open image in new window for all x y a X Open image in new window, and d ( T x , T y , a ) = 0 Open image in new window if any two of x , y , a Open image in new window are equal.

Corollary 2.5 Let ( X , d ) Open image in new window be a 2-metric space and let T : X X Open image in new window be a contractive map. Then T is a constant map, i.e., there exists x 0 X Open image in new window such that T x = x 0 Open image in new window for all x X Open image in new window. In particular, T has a unique fixed point x 0 Open image in new window and the sequence { T n x } Open image in new window converges to x 0 Open image in new window for all x X Open image in new window.

Proof Since d ( T x , T y , x ) = 0 Open image in new window for all x X Open image in new window, it follows from Corollary 2.2 that Tx is a fixed point of T for all x X Open image in new window.

If T x T y Open image in new window for some x , y X Open image in new window, then there exists a X Open image in new window such that d ( T x , T y , a ) 0 Open image in new window. Thus, T x T y a Open image in new window. Notice that T 2 x = T x Open image in new window and T is a contractive map, so we have
d ( T x , T y , a ) = d ( T 2 x , T 2 y , a ) < d ( T x , T y , a ) . Open image in new window

It is a contradiction. Therefore T x = T y Open image in new window for all x , y X Open image in new window, i.e., T is a constant map. Let T x = x 0 Open image in new window for all x X Open image in new window. Then x 0 Open image in new window is the unique fixed point of T and the sequence { T n x } Open image in new window converges to  x 0 Open image in new window for all x X Open image in new window. □

The following example shows that the contraction of T in Corollary 2.5 is essential.

Example 2.6 Let ( X , d ) Open image in new window be a 2-metric space and let T : X X Open image in new window be the identical map where X has at least three points. Then T is a non-contractive map with more than one fixed point.

In [17], Lahiri, Das and Dey established Cantor’s intersection theorem and Baire category theorem in 2-metric spaces, and some fixed point theorems in 2-metric spaces have been proved sophisticatedly. By using the assumption of a contractive map, we show that the main results in [17] are direct consequences of Corollary 2.5. Moreover, the assumption of a contractive map is essential by Example 2.6.

Corollary 2.7 ([17], Theorem 7)

Let ( X , d ) Open image in new window be a complete bounded 2-metric space and let T : X X Open image in new window be a map such that d ( T x , T y , a ) α d ( x , y , a ) Open image in new window for some 0 < α < 1 Open image in new window and all x y a X Open image in new window, and d ( T x , T y , a ) = 0 Open image in new window if any two of x , y , a X Open image in new window are equal. Then T has a unique fixed point in X.

Corollary 2.8 ([17], Theorem 8)

Let ( X , d ) Open image in new window be a bounded 2-metric space and let T : X X Open image in new window be a map such that d ( T x , T y , a ) α d ( x , y , a ) Open image in new window for some 0 < α < 1 Open image in new window and all x , y , a X Open image in new window. Let there be a point x X Open image in new window such that the sequence of iterates { T n x } Open image in new window contains a subsequence { T n i x } Open image in new window that converges to x 0 X Open image in new window. Then x 0 Open image in new window is a unique fixed point of T.

Corollary 2.9 ([17], Theorem 9)

Let ( X , d ) Open image in new window be an uncounTable  2-metric space and let T : X X Open image in new window be a contractive map. If there exists a point x X Open image in new window such that the sequence of iterates { T n x } Open image in new window contains a subsequence { T n i x } Open image in new window converging to x 0 X Open image in new window, then x 0 Open image in new window is the unique fixed point of T.

Recently, Singh, Mishra and Stofile have proved the following result.

Theorem 2.10 ([20], Theorem 2.1)

Let ( X , d ) Open image in new window be a complete 2-metric space and T : X X Open image in new window. Define a non-decreasing function θ from [ 0 , 1 ) Open image in new window onto ( 1 2 , 1 ] Open image in new window by
θ ( r ) = { 1 if 0 r 5 1 2 , 1 r r 2 if 5 1 2 r 1 2 , 1 1 + r if 1 2 r < 1 . Open image in new window
Assume that there exists r [ 0 , 1 ) Open image in new window such that
θ ( r ) d ( x , T x , a ) d ( x , y , a ) implies d ( T x , T y , a ) r d ( x , y , a ) Open image in new window
(2.2)

for all x , y , a X Open image in new window. Then there exists a unique fixed point z of T. Moreover, lim T n x = z Open image in new window for any x X Open image in new window.

In the proof of the above theorem, Singh, Mishra and Stofile claimed that
d ( x n , x n + 1 , a ) d ( x n , z , a ) + d ( x n + 1 , z , a ) + d ( x n , x n + 1 , x n ) Open image in new window
(2.3)
in lines +4 and +5, page 73 of [20]. In fact,
d ( x n , x n + 1 , a ) d ( x n , z , a ) + d ( x n + 1 , z , a ) + d ( x n , x n + 1 , z ) . Open image in new window

The error inequality (2.3) was pointed out in [22].

Now, by choosing a = x Open image in new window in (2.2), we have θ ( r ) d ( x , T x , x ) = 0 d ( x , y , x ) = 0 Open image in new window. It implies that d ( T x , T y , x ) r d ( x , y , x ) = 0 Open image in new window for all x , y X Open image in new window. Then, by Corollary 2.2, T has a fixed point. For the uniqueness, let T have fixed points x , y Open image in new window. We have
θ ( r ) d ( x , T x , a ) = θ ( r ) d ( x , x , a ) = 0 d ( x , y , a ) . Open image in new window

It implies that d ( T x , T y , a ) = d ( x , y , a ) r d ( x , y , a ) Open image in new window for all a X Open image in new window. Then d ( x , y , a ) = 0 Open image in new window for all a X Open image in new window, that is, x = y Open image in new window.

The following example shows that we cannot replace the assumption ‘for all x , y , a X Open image in new window’ in the contraction condition (2.2) by the assumption ‘for all x , y , a X Open image in new window and a x Open image in new window’.

Example 2.11 Let X = { 1 , 2 , 3 } Open image in new window and d ( x , y , z ) = min { | x y | , | y z | , | z x | } Open image in new window for all x , y , z X Open image in new window. Then ( X , d ) Open image in new window is a complete 2-metric space. Let T : X X Open image in new window be a map defined by T 1 = 2 Open image in new window, T 2 = 3 Open image in new window, T 3 = 1 Open image in new window. We see that T has no fixed point. But, for all x , y , a X Open image in new window and a x Open image in new window, the contraction condition (2.2) holds as in the Table 2.
Table 2

Calculations for maps in Example 2.11

x

a

θ ( r ) d ( x , T x , a ) d ( x , y , a ) Open image in new window

y

d ( T x , T y , a ) r d ( x , y , a ) Open image in new window

1

2

θ(r)⋅d(1,T 1,2)=0 ≤ d(1,y,2)

1

d(T 1,T 1,2)=0 ≤ rd(1,1,2)

2

d(T 1,T 2,2)=0 ≤ rd(1,2,2)

3

d(T 1,T 3,2)=0 ≤ rd(1,3,2)

1

3

θ(r)⋅d(1,T 1,3)=θ(r)⋅d(1,2,3)≤d(1,y,3)

2

d(T 1,T 2,3)=0 ≤ rd(1,2,3)

2

3

θ(r)⋅d(2,T 2,3)=0 ≤ d(2,y,3)

1

d(T 2,T 1,3)=0 ≤ rd(2,1,3)

2

d(T 2,T 2,3)=0 ≤ rd(2,2,3)

3

d(T 2,T 3,3)=0 ≤ rd(2,3,3)

2

1

θ(r)⋅d(2,T 2,1)=θ(r)⋅d(2,3,1)≤d(2,y,1)

3

d(T 2,T 3,1)=0 ≤ rd(2,3,1)

3

1

θ(r)⋅d(3,T 3,1)=0 ≤ d(1,y,3)

1

d(T 3,T 1,1)=0 ≤ rd(3,1,1)

2

d(T 3,T 2,1)=0 ≤ rd(3,2,1)

3

d(T 3,T 3,1)=0 ≤ rd(3,3,1)

3

2

θ(r)⋅d(3,T 3,2)=θ(r)⋅d(3,1,2)≤d(2,y,3)

1

d(T 3,T 1,2)=0 ≤ rd(3,1,2)

Notes

Acknowledgements

The authors would like to thank the referees for their valuable comments.

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Copyright information

© Dung 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.

Authors and Affiliations

  • Nguyen Van Dung
    • 1
  • Nguyen Trung Hieu
    • 1
  • Nguyen Thia Thanh Ly
    • 1
  • Vo Duc Thinh
    • 1
  1. 1.Department of MathematicsDong Thap UniversityDong ThapVietnam

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