A note on some coupled fixed-point theorems on G-metric spaces

Open Access
Research

Abstract

The purpose of this paper is to extend some recent coupled fixed-point theorems in the context of G-metric space by essentially different and more natural way. We state some examples to illustrate our results.

MSC:46N40, 47H10, 54H25, 46T99.

Keywords

coupled fixed point coincidence point mixed g-monotone property ordered set G-metric space 

1 Introduction

In nonlinear functional analysis, one of the most productive tools is the fixed-point theory, which has numerous applications in many quantitative disciplines such as biology, chemistry, computer science, and additionally in many branches of engineering. In this theory, the Banach contraction principle can be considered as a cornerstone pioneering result which in elementary terms states that each contraction has a unique fixed point in a complete metric space. Due to its potential of applications in the fields above mentioned and many more, the fixed-point theory, in particular, the Banach contraction principle, attracts considerable attention from many authors (see, e.g., [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]). Especially, it is considered very natural and curious to investigate the existence and uniqueness of a fixed point for several contraction type mappings in various abstract spaces. A major example in this direction is the work of Mustafa and Sims [19] in which they introduced the concept of G-metric spaces as a generalization of (usual) metric spaces in 2004. After this remarkable paper, a number of papers have appeared on this topic in the literature (see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]).

For the sake of completeness, we recall some basic definitions and elementary results from the literature. Throughout this paper, N Open image in new window is the set of nonnegative integers, and N Open image in new window is the set of positive integers.

Definition 1 (See [19])

Let X be a nonempty set, G : X × X × X R + Open image in new window be a function satisfying the following properties:

(G1) G ( x , y , z ) = 0 Open image in new window if x = y = z Open image in new window,

(G2) 0 < G ( x , x , y ) Open image in new window for all x , y X Open image in new window with x y Open image in new window,

(G3) G ( x , x , y ) G ( x , y , z ) Open image in new window for all x , y , z X Open image in new window with y z Open image in new window,

(G4) G ( x , y , z ) = G ( x , z , y ) = G ( y , z , x ) = Open image in new window (symmetry in all three variables),

(G5) G ( x , y , z ) G ( x , a , a ) + G ( a , y , z ) Open image in new window for all x , y , z , a X Open image in new window (rectangle inequality).

Then the function G is called a generalized metric, or more specially, a G-metric on X, and the pair ( X , G ) Open image in new window is called a G-metric space.

Every G-metric on X defines a metric d G Open image in new window on X by
d G ( x , y ) = G ( x , y , y ) + G ( y , x , x ) , for all  x , y X . Open image in new window
(1.1)
Example 2 Let ( X , d ) Open image in new window be a metric space. The function G : X × X × X [ 0 , + ) Open image in new window, defined by
G ( x , y , z ) = max { d ( x , y ) , d ( y , z ) , d ( z , x ) } , Open image in new window
or
G ( x , y , z ) = d ( x , y ) + d ( y , z ) + d ( z , x ) , Open image in new window

for all x , y , z X Open image in new window, is a G-metric on X.

Definition 3 (See [19])

Let ( X , G ) Open image in new window be a G-metric space, and let { x n } Open image in new window be a sequence of points of X, therefore, we say that ( x n ) Open image in new window is G-convergent to x X Open image in new window if lim n , m + G ( x , x n , x m ) = 0 Open image in new window, that is, for any ε > 0 Open image in new window, there exists N N Open image in new window such that G ( x , x n , x m ) < ε Open image in new window, for all n , m N Open image in new window. We call x the limit of the sequence and write x n x Open image in new window or lim n + x n = x Open image in new window.

Proposition 4 (See [19])

Let ( X , G ) Open image in new windowbe a G-metric space. The following are equivalent:
  1. (1)

    { x n } Open image in new window is G-convergent to x,

     
  2. (2)

    G ( x n , x n , x ) 0 Open image in new window as n + Open image in new window,

     
  3. (3)

    G ( x n , x , x ) 0 Open image in new window as n + Open image in new window,

     
  4. (4)

    G ( x n , x m , x ) 0 Open image in new window as n , m + Open image in new window.

     

Definition 5 (See [19])

Let ( X , G ) Open image in new window be a G-metric space. A sequence { x n } Open image in new window is called a G-Cauchy sequence if, for any ε > 0 Open image in new window, there is N N Open image in new window such that G ( x n , x m , x l ) < ε Open image in new window for all m , n , l N Open image in new window, that is, G ( x n , x m , x l ) 0 Open image in new window as n , m , l + Open image in new window.

Proposition 6 (See [19])

Let ( X , G ) Open image in new windowbe a G-metric space. Then the following are equivalent:
  1. (1)

    the sequence { x n } Open image in new window is G-Cauchy,

     
  2. (2)

    for any ε > 0 Open image in new window, there exists N N Open image in new window such that G ( x n , x m , x m ) < ε Open image in new window, for all m , n N Open image in new window.

     

Definition 7 (See [19])

A G-metric space ( X , G ) Open image in new window is called G-complete if every G-Cauchy sequence is G-convergent in ( X , G ) Open image in new window.

Definition 8 Let ( X , G ) Open image in new window be a G-metric space. A mapping F : X × X × X X Open image in new window is said to be continuous if for any three G-convergent sequences { x n } Open image in new window, { y n } Open image in new window and { z n } Open image in new window converging to x, y, and z, respectively, { F ( x n , y n , z n ) } Open image in new window is G-convergent to F ( x , y , z ) Open image in new window.

Definition 9 Let F : X × X X Open image in new window and g : X X Open image in new window be mappings. The mappings F and g are said to commute if
g ( F ( x , y ) ) = F ( g ( x ) , g ( y ) ) , for all  x , y X . Open image in new window

In [27], Shatanawi proved the following theorems.

Theorem 10 Let ( X , G ) Open image in new windowbe a G-metric space. Let F : X × X X Open image in new windowand g : X X Open image in new windowbe two mappings such that
G ( F ( x , y ) , F ( u , v ) , F ( z , w ) ) k ( G ( g x , g u , g z ) + G ( g y , g v , g w ) ) for all  x , y , u , v , z , w . Open image in new window
(1.2)
Assume that F and g satisfy the following conditions:
  1. (1)

    F ( X × X ) g ( X ) Open image in new window,

     
  2. (2)

    g ( X ) Open image in new window is G-complete,

     
  3. (3)

    g is G-continuous and commutes with F.

     

If k [ 0 , 1 2 ) Open image in new window, then there is a unique x X Open image in new windowsuch that g x = F ( x , x ) = x Open image in new window.

Corollary 11 Let ( X , G ) Open image in new windowbe a complete G-metric space. Let F : X × X X Open image in new windowbe a mapping such that
G ( F ( x , y ) , F ( u , v ) , F ( u , v ) ) k ( G ( x , u , u ) + G ( y , v , v ) ) for all  x , y , u , v X . Open image in new window
(1.3)

If k [ 0 , 1 2 ) Open image in new window, then there is a unique x X Open image in new windowsuch that F ( x , x ) = x Open image in new window.

In this paper, we aim to extend the above coupled fixed-point results.

2 Main results

We start with an example to show the weakness of Theorem 10.

Example 12 Let X = [ 0 , 1 ] Open image in new window. Define G : X × X × X [ 0 , + ) Open image in new window by
G ( x , y , z ) = | x y | + | x z | + | y z | Open image in new window
for all x , y , z X Open image in new window. Then ( X , G ) Open image in new window is a G-metric space. Define a map F : X × X X Open image in new window by F ( x , y ) = 1 3 x + 1 8 y Open image in new window and g : X X Open image in new window by g ( x ) = x 2 Open image in new window for all x , y X Open image in new window. Then, for all x , y , u , v , z , w X Open image in new window with y = v = w Open image in new window, we have
G ( F ( x , y ) , F ( u , v ) , F ( z , w ) ) = G ( 1 3 x + 1 8 y , 1 3 u + 1 8 v , 1 3 z + 1 8 w ) = | x u | + | x z | + | u z | 3 Open image in new window
and
G ( g x , g u , g z ) + G ( g y , g v , g w ) = G ( x 2 , u 2 , z 2 ) + G ( y 2 , v 2 , w 2 ) = | x u | + | x z | + | u z | 2 . Open image in new window
Then it is easy to that there is no k [ 0 , 1 2 ) Open image in new window such that
G ( F ( x , y ) , F ( u , v ) , F ( z , w ) ) k [ G ( g x , g u , g z ) + G ( g y , g v , g w ) ] Open image in new window

for all x , y , u , v , z , w X Open image in new window. Thus, Theorem 10 cannot be applied to this example. However, it is easy to see that 0 is the unique point x X Open image in new window such that x = g x = F ( x , x ) Open image in new window.

We now state our first result which successively guarantee a coupled fixed point.

Theorem 13 Let ( X , G ) Open image in new windowbe a G-metric space. Let F : X × X X Open image in new windowand g : X X Open image in new windowbe two mappings such that
for all x , y , u , v X Open image in new window. Assume that F and g satisfy the following conditions:
  1. (1)

    F ( X × X ) g ( X ) Open image in new window,

     
  2. (2)

    g ( X ) Open image in new window is G-complete,

     
  3. (3)

    g is G-continuous and commutes with F.

     

If k [ 0 , 1 ) Open image in new window, then there is a unique x X Open image in new windowsuch that g x = F ( x , x ) = x Open image in new window.

Proof Take x 0 , y 0 X Open image in new window. Noting that F ( X × X ) g ( X ) Open image in new window, we can construct two sequences { x n } Open image in new window and { y n } Open image in new window in X such that
g x n + 1 = F ( x n , y n ) , g y n + 1 = F ( y n , x n ) , n N . Open image in new window
Let
M n = G ( g x n , g x n + 1 , g x n + 1 ) + G ( g y n , g y n + 1 , g y n + 1 ) , n N . Open image in new window
Then, by using (2.1), for each n N Open image in new window, we have
M n = G ( g x n , g x n + 1 , g x n + 1 ) + G ( g y n , g y n + 1 , g y n + 1 ) = G ( F ( x n 1 , y n 1 ) , F ( x n , y n ) , F ( x n , y n ) ) + G ( F ( y n 1 , x n 1 ) , F ( y n , x n ) , F ( y n , x n ) ) k [ G ( g x n 1 , g x n , g x n ) + G ( g y n 1 , g y n , g y n ) ] = k M n 1 , Open image in new window
which yields that
M n k n M 0 , n N . Open image in new window
(2.2)
Now, for all m , n N Open image in new window with m > n Open image in new window, by using rectangle inequality of G-metric and (2.2), we get
G ( g x n , g x m , g x m ) + G ( g y n , g y m , g y m ) G ( g x n , g x n + 1 , g x n + 1 ) + G ( g x n + 1 , g x m , g x m ) + G ( g y n , g y n + 1 , g y n + 1 ) + G ( g y n + 1 , g y m , g y m ) G ( g x n , g x n + 1 , g x n + 1 ) + G ( g x n + 1 , g x n + 2 , g x n + 2 ) + G ( g x n + 2 , g x m , g x m ) + G ( g y n , g x n + 1 , g y n + 1 ) + G ( g y n + 1 , g y n + 2 , g y n + 2 ) + G ( g y n + 2 , g y m , g y m ) G ( g x n , g x n + 1 , g x n + 1 ) + G ( g x n + 1 , g x n + 2 , g x n + 2 ) + + G ( g x m 1 , g x m , g x m ) + G ( g y n , g y n + 1 , g y n + 1 ) + G ( g y n + 1 , g y n + 2 , g y n + 2 ) + + G ( g y m 1 , g y m , g y m ) M n + M n + 1 + + M m 1 ( k n + k n + 1 + + k m 1 ) M 0 k n 1 k M 0 , Open image in new window
which yields that
lim n , m + G ( g x n , g x m , g x m ) + G ( g y n , g y m , g y m ) = 0 . Open image in new window

Then, by Proposition 6, we conclude that the sequences { g x n } Open image in new window and { g y n } Open image in new window are G-Cauchy.

Noting that g ( X ) Open image in new window is G-complete, there exist x , y g ( X ) Open image in new window such that { g x n } Open image in new window and { g y n } Open image in new window are G-convergent to x and y, respectively, i.e.,
lim n + G ( g x n , x , x ) = 0 , lim n + G ( g y n , y , y ) = 0 . Open image in new window
Also, since g is G-continuous, we get
lim n + G ( g g x n , g x , g x ) = 0 , lim n + G ( g g y n , g y , g y ) = 0 . Open image in new window
(2.3)
In addition, by (2.1) and the fact g commutes with F, we get
G ( g g x n + 1 , F ( x , y ) , F ( x , y ) ) + G ( g g y n + 1 , F ( y , x ) , F ( y , x ) ) = G ( g ( F ( x n , y n ) ) , F ( x , y ) , F ( x , y ) ) + G ( g ( F ( y n , x n ) ) , F ( y , x ) , F ( y , x ) ) = G ( F ( g x n , g y n ) , F ( x , y ) , F ( x , y ) ) + G ( F ( g y n , g x n ) , F ( y , x ) , F ( y , x ) ) k [ G ( g g x n , g x , g x ) + G ( g g y n , g y , g y ) ] . Open image in new window
Combining this with (2.3), we get
G ( g g x n + 1 , F ( x , y ) , F ( x , y ) ) + G ( g g y n + 1 , F ( y , x ) , F ( y , x ) ) 0 , n + . Open image in new window
On the other hand, by the fact that G is continuous on its variables (cf. [19]), we have
G ( g g x n + 1 , F ( x , y ) , F ( x , y ) ) + G ( g g y n + 1 , F ( y , x ) , F ( y , x ) ) G ( g x , F ( x , y ) , F ( x , y ) ) + G ( g y , F ( y , x ) , F ( y , x ) ) , n + . Open image in new window
Thus, we conclude that
G ( g x , F ( x , y ) , F ( x , y ) ) + G ( g y , F ( y , x ) , F ( y , x ) ) = 0 , Open image in new window
i.e.,
G ( g x , F ( x , y ) , F ( x , y ) ) = G ( g y , F ( y , x ) , F ( y , x ) ) = 0 , Open image in new window
which yields that
g x = F ( x , y ) , g y = F ( y , x ) . Open image in new window
Moreover, it follows from
G ( g x , g y , g y ) + G ( g y , g x , g x ) = G ( F ( x , y ) , F ( y , x ) , F ( y , x ) ) + G ( F ( y , x ) , F ( x , y ) , F ( x , y ) ) k [ G ( g x , g y , g y ) + G ( g y , g x , g x ) ] Open image in new window

that G ( g x , g y , g y ) + G ( g y , g x , g x ) = 0 Open image in new window. Thus, G ( g x , g y , g y ) = 0 Open image in new window, i.e., g x = g y Open image in new window.

Next, let us show that g x = F ( x , x ) = x Open image in new window. By using rectangle inequality of G-metric and (2.1), we have
G ( x , g x , g x ) + G ( y , g y , g y ) G ( x , g x n + 1 , g x n + 1 ) + G ( g x n + 1 , g x , g x ) + G ( y , g y n + 1 , g y n + 1 ) + G ( g y n + 1 , g y , g y ) [ G ( x , g x n + 1 , g x n + 1 ) + G ( y , g y n + 1 , g y n + 1 ) ] + [ G ( F ( x n , y n ) , F ( x , y ) , F ( x , y ) ) + G ( F ( y n , x n ) , F ( y , x ) , F ( y , x ) ) ] [ G ( x , g x n + 1 , g x n + 1 ) + G ( y , g y n + 1 , g y n + 1 ) ] + k [ G ( g x n , g x , g x ) + G ( g y n , g y , g y ) ] [ G ( x , g x n + 1 , g x n + 1 ) + G ( y , g y n + 1 , g y n + 1 ) ] + k [ G ( x , g x , g x ) + G ( y , g y , g y ) ] + k [ G ( g x n , x , x ) + G ( g y n , y , y ) ] , Open image in new window
which gives that
G ( x , g x , g x ) + G ( y , g y , g y ) G ( x , g x n + 1 , g x n + 1 ) + G ( y , g y n + 1 , g y n + 1 ) + k [ G ( g x n , x , x ) + G ( g y n , y , y ) ] 1 k . Open image in new window
Combing this with the fact that { g x n } Open image in new window and { g y n } Open image in new window are G-convergent to x and y, respectively, we conclude that
G ( x , g x , g x ) + G ( y , g y , g y ) = 0 , Open image in new window
which yields that
x = g x , y = g y . Open image in new window

Recalling that g x = g y Open image in new window and g x = F ( x , y ) Open image in new window, we get x = y Open image in new window and x = g x = F ( x , x ) Open image in new window.

It remains to show the uniqueness. Let u X Open image in new window be such that u = g u = F ( u , u ) Open image in new window. Then we have
2 G ( u , x , x ) = G ( F ( u , u ) , F ( x , x ) , F ( x , x ) ) + G ( F ( u , u ) , F ( x , x ) , F ( x , x ) ) k [ G ( g u , g x , g x ) + G ( g u , g x , g x ) ] 2 k G ( u , x , x ) , Open image in new window

which yields that ( 2 2 k ) G ( u , x , x ) 0 Open image in new window. Thus, G ( u , x , x ) = 0 Open image in new window, which means u = x Open image in new window. This completes the proof. □

Remark 14 It is easy to see that Theorem 10, appearing in [27], is a direct corollary of Theorem 13. On the other hand, Theorem 13 can deal with some cases, which Theorem 10 cannot be applied. For this, let us reconsider Example 12. In fact, for all x , y , u , v X Open image in new window, we have
G ( F ( x , y ) , F ( u , v ) , F ( u , v ) ) + G ( F ( y , x ) , F ( v , u ) , F ( v , u ) ) = G ( 1 3 x + 1 8 y , 1 3 u + 1 8 v , 1 3 u + 1 8 v ) + G ( 1 3 y + 1 8 x , 1 3 v + 1 8 u , 1 3 v + 1 8 u ) 11 ( | x u | + | y v | ) 12 = 11 12 [ G ( g x , g u , g u ) + G ( g y , g v , g v ) ] , Open image in new window

i.e., (2.1) holds. Other assumptions of Theorem 13 are easy to verify. So, by Theorem 13, there exists a unique x X Open image in new window such that g x = F ( x , x ) = x Open image in new window.

Letting g = I Open image in new window, we can get the following result.

Corollary 15 Let ( X , G ) Open image in new windowbe a complete G-metric space. Let F : X × X X Open image in new windowbe a mapping such that

for all x , y , u , v X Open image in new window. If k [ 0 , 1 ) Open image in new window, then there is a unique x X Open image in new windowsuch that F ( x , x ) = x Open image in new window.

Example 16 Let ( X , G ) Open image in new window be the same as in Example 12. Then ( X , G ) Open image in new window is a G-metric space. Also, it is not difficult to verify that ( X , G ) Open image in new window is G-complete. Define a map F : X × X X Open image in new window by F ( x , y ) = 1 1 16 x 2 5 16 y 2 Open image in new window for all x , y X Open image in new window. Then, for all x , y , u , v X Open image in new window, we have
G ( F ( x , y ) , F ( u , v ) , F ( u , v ) ) + G ( F ( y , x ) , F ( v , u ) , F ( v , u ) ) = G ( 1 1 16 x 2 5 16 y 2 , 1 1 16 u 2 5 16 v 2 , 1 1 16 u 2 5 16 v 2 ) + G ( 1 1 16 y 2 5 16 x 2 , 1 1 16 v 2 5 16 u 2 , 1 1 16 v 2 5 16 u 2 ) 1 8 | u 2 x 2 | + 5 8 | v 2 y 2 | + 1 8 | v 2 y 2 | + 5 8 | u 2 x 2 | = 3 4 | u 2 x 2 | + 3 4 | v 2 y 2 | 3 2 | u x | + 3 2 | v y | Open image in new window
and
G ( x , u , u ) + G ( y , v , v ) = 2 ( | x u | + | y v | ) . Open image in new window

Thus, the statement (2.4) of Corollary 15 is satisfied for any k [ 3 4 , 1 ) Open image in new window. Thus, there is a unique x X Open image in new window such that F ( x , x ) = x Open image in new window.

Remark 17 Corollary 11 cannot be applied to Example 16 since (1.3) does not hold. In fact, if (1.3) holds for some k [ 0 , 1 2 ) Open image in new window, then
9 40 = G ( 11 16 , 4 5 , 4 5 ) = G ( F ( 0 , 1 ) , F ( 0 , 4 5 ) , F ( 0 , 4 5 ) ) k [ G ( 0 , 0 , 0 ) + G ( 1 , 4 5 , 4 5 ) ] = 2 k 5 1 5 , Open image in new window

which is a contradiction.

Notes

Acknowledgements

The authors are indebted to the referees for their careful reading of the manuscript and valuable suggestions. Hui-Sheng Ding acknowledges support from the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province (20114BAB211002), the Jiangxi Provincial Education Department (GJJ12173), and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University.

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Copyright information

© Ding and Karapınar; licensee Springer 2012

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

  1. 1.College of Mathematics and Information Science, Jiangxi Normal UniversityNanchang, JiangxiPeople’s Republic of China
  2. 2.Department of MathematicsAtilim Universityİncek, AnkaraTurkey

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