Journal of Inequalities and Applications

, 2012:170

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

Open AccessResearch

DOI: 10.1186/1029-242X-2012-170

Cite this article as:
Ding, H. & Karapınar, E. J Inequal Appl (2012) 2012: 170. doi:10.1186/1029-242X-2012-170
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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 pointcoincidence pointmixed g-monotone propertyordered setG-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., [430]). 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., [18, 10, 12, 1829]).

For the sake of completeness, we recall some basic definitions and elementary results from the literature. Throughout this paper, N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq1_HTML.gif is the set of nonnegative integers, and N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq2_HTML.gif is the set of positive integers.

Definition 1 (See [19])

Let X be a nonempty set, G : X × X × X R + https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq3_HTML.gif be a function satisfying the following properties:

(G1) G ( x , y , z ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq4_HTML.gif if x = y = z https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq5_HTML.gif,

(G2) 0 < G ( x , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq6_HTML.gif for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq7_HTML.gif with x y https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq8_HTML.gif,

(G3) G ( x , x , y ) G ( x , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq9_HTML.gif for all x , y , z X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq10_HTML.gif with y z https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq11_HTML.gif,

(G4) G ( x , y , z ) = G ( x , z , y ) = G ( y , z , x ) = https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq12_HTML.gif (symmetry in all three variables),

(G5) G ( x , y , z ) G ( x , a , a ) + G ( a , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq13_HTML.gif for all x , y , z , a X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq14_HTML.gif (rectangle inequality).

Then the function G is called a generalized metric, or more specially, a G-metric on X, and the pair ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq15_HTML.gif is called a G-metric space.

Every G-metric on X defines a metric d G https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq16_HTML.gif on X by
d G ( x , y ) = G ( x , y , y ) + G ( y , x , x ) , for all  x , y X . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equ1_HTML.gif
(1.1)
Example 2 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq17_HTML.gif be a metric space. The function G : X × X × X [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq18_HTML.gif, defined by
G ( x , y , z ) = max { d ( x , y ) , d ( y , z ) , d ( z , x ) } , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equa_HTML.gif
or
G ( x , y , z ) = d ( x , y ) + d ( y , z ) + d ( z , x ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equb_HTML.gif

for all x , y , z X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq19_HTML.gif, is a G-metric on X.

Definition 3 (See [19])

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq20_HTML.gif be a G-metric space, and let { x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq21_HTML.gif be a sequence of points of X, therefore, we say that ( x n ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq22_HTML.gif is G-convergent to x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq23_HTML.gif if lim n , m + G ( x , x n , x m ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq24_HTML.gif, that is, for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq25_HTML.gif, there exists N N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq26_HTML.gif such that G ( x , x n , x m ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq27_HTML.gif, for all n , m N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq28_HTML.gif. We call x the limit of the sequence and write x n x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq29_HTML.gif or lim n + x n = x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq30_HTML.gif.

Proposition 4 (See [19])

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq20_HTML.gifbe aG-metric space. The following are equivalent:
  1. (1)

    { x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq21_HTML.gifisG-convergent tox,

     
  2. (2)

    G ( x n , x n , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq31_HTML.gifas n + https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq32_HTML.gif,

     
  3. (3)

    G ( x n , x , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq33_HTML.gifas n + https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq32_HTML.gif,

     
  4. (4)

    G ( x n , x m , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq34_HTML.gifas n , m + https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq35_HTML.gif.

     

Definition 5 (See [19])

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq20_HTML.gif be a G-metric space. A sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq21_HTML.gif is called a G-Cauchy sequence if, for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq25_HTML.gif, there is N N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq26_HTML.gif such that G ( x n , x m , x l ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq36_HTML.gif for all m , n , l N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq37_HTML.gif, that is, G ( x n , x m , x l ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq38_HTML.gif as n , m , l + https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq39_HTML.gif.

Proposition 6 (See [19])

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq20_HTML.gifbe aG-metric space. Then the following are equivalent:
  1. (1)

    the sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq21_HTML.gifisG-Cauchy,

     
  2. (2)

    for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq25_HTML.gif, there exists N N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq26_HTML.gifsuch that G ( x n , x m , x m ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq40_HTML.gif, for all m , n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq41_HTML.gif.

     

Definition 7 (See [19])

A G-metric space ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq15_HTML.gif is called G-complete if every G-Cauchy sequence is G-convergent in ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq15_HTML.gif.

Definition 8 Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq15_HTML.gif be a G-metric space. A mapping F : X × X × X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq42_HTML.gif is said to be continuous if for any three G-convergent sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq21_HTML.gif, { y n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq43_HTML.gif and { z n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq44_HTML.gif converging to x, y, and z, respectively, { F ( x n , y n , z n ) } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq45_HTML.gif is G-convergent to F ( x , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq46_HTML.gif.

Definition 9 Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq47_HTML.gif and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq48_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equc_HTML.gif

In [27], Shatanawi proved the following theorems.

Theorem 10Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq15_HTML.gifbe aG-metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq49_HTML.gifand g : X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq50_HTML.gifbe 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equ2_HTML.gif
(1.2)
Assume that F and g satisfy the following conditions:
  1. (1)

    F ( X × X ) g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq51_HTML.gif,

     
  2. (2)

    g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq52_HTML.gifisG-complete,

     
  3. (3)

    gisG-continuous and commutes withF.

     

If k [ 0 , 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq53_HTML.gif, then there is a unique x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq54_HTML.gifsuch that g x = F ( x , x ) = x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq55_HTML.gif.

Corollary 11Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq15_HTML.gifbe a completeG-metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq56_HTML.gifbe 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equ3_HTML.gif
(1.3)

If k [ 0 , 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq53_HTML.gif, then there is a unique x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq54_HTML.gifsuch that F ( x , x ) = x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq57_HTML.gif.

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 ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq58_HTML.gif. Define G : X × X × X [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq59_HTML.gif by
G ( x , y , z ) = | x y | + | x z | + | y z | https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equd_HTML.gif
for all x , y , z X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq60_HTML.gif. Then ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq15_HTML.gif is a G-metric space. Define a map F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq61_HTML.gif by F ( x , y ) = 1 3 x + 1 8 y https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq62_HTML.gif and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq63_HTML.gif by g ( x ) = x 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq64_HTML.gif for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq65_HTML.gif. Then, for all x , y , u , v , z , w X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq66_HTML.gif with y = v = w https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq67_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Eque_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equf_HTML.gif
Then it is easy to that there is no k [ 0 , 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq53_HTML.gif 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 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equg_HTML.gif

for all x , y , u , v , z , w X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq66_HTML.gif. Thus, Theorem 10 cannot be applied to this example. However, it is easy to see that 0 is the unique point x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq54_HTML.gif such that x = g x = F ( x , x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq68_HTML.gif.

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

Theorem 13Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq15_HTML.gifbe aG-metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq49_HTML.gifand g : X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq50_HTML.gifbe two mappings such that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equ4_HTML.gif
(2.1)
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq69_HTML.gif. Assume thatFandgsatisfy the following conditions:
  1. (1)

    F ( X × X ) g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq51_HTML.gif,

     
  2. (2)

    g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq52_HTML.gifisG-complete,

     
  3. (3)

    gisG-continuous and commutes withF.

     

If k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq70_HTML.gif, then there is a unique x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq54_HTML.gifsuch that g x = F ( x , x ) = x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq71_HTML.gif.

Proof Take x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq72_HTML.gif. Noting that F ( X × X ) g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq51_HTML.gif, we can construct two sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq21_HTML.gif and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq73_HTML.gif in X such that
g x n + 1 = F ( x n , y n ) , g y n + 1 = F ( y n , x n ) , n N . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equh_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equi_HTML.gif
Then, by using (2.1), for each n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq74_HTML.gif, 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equj_HTML.gif
which yields that
M n k n M 0 , n N . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equ5_HTML.gif
(2.2)
Now, for all m , n N https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq75_HTML.gif with m > n https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq76_HTML.gif, 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equk_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equl_HTML.gif

Then, by Proposition 6, we conclude that the sequences { g x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq77_HTML.gif and { g y n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq78_HTML.gif are G-Cauchy.

Noting that g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq52_HTML.gif is G-complete, there exist x , y g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq79_HTML.gif such that { g x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq80_HTML.gif and { g y n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq78_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equm_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equ6_HTML.gif
(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 ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equn_HTML.gif
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 + . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equo_HTML.gif
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 + . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equp_HTML.gif
Thus, we conclude that
G ( g x , F ( x , y ) , F ( x , y ) ) + G ( g y , F ( y , x ) , F ( y , x ) ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equq_HTML.gif
i.e.,
G ( g x , F ( x , y ) , F ( x , y ) ) = G ( g y , F ( y , x ) , F ( y , x ) ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equr_HTML.gif
which yields that
g x = F ( x , y ) , g y = F ( y , x ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equs_HTML.gif
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 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equt_HTML.gif

that G ( g x , g y , g y ) + G ( g y , g x , g x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq81_HTML.gif. Thus, G ( g x , g y , g y ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq82_HTML.gif, i.e., g x = g y https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq83_HTML.gif.

Next, let us show that g x = F ( x , x ) = x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq55_HTML.gif. 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 ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equu_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equv_HTML.gif
Combing this with the fact that { g x n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq80_HTML.gif and { g y n } https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq78_HTML.gif are G-convergent to x and y, respectively, we conclude that
G ( x , g x , g x ) + G ( y , g y , g y ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equw_HTML.gif
which yields that
x = g x , y = g y . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equx_HTML.gif

Recalling that g x = g y https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq83_HTML.gif and g x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq84_HTML.gif, we get x = y https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq85_HTML.gif and x = g x = F ( x , x ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq86_HTML.gif.

It remains to show the uniqueness. Let u X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq87_HTML.gif be such that u = g u = F ( u , u ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq88_HTML.gif. 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equy_HTML.gif

which yields that ( 2 2 k ) G ( u , x , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq89_HTML.gif. Thus, G ( u , x , x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq90_HTML.gif, which means u = x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq91_HTML.gif. 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 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq69_HTML.gif, 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 ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equz_HTML.gif

i.e., (2.1) holds. Other assumptions of Theorem 13 are easy to verify. So, by Theorem 13, there exists a unique x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq54_HTML.gif such that g x = F ( x , x ) = x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq92_HTML.gif.

Letting g = I https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq93_HTML.gif, we can get the following result.

Corollary 15Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq15_HTML.gifbe a completeG-metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq56_HTML.gifbe a mapping such that
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equ7_HTML.gif
(2.4)

for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq69_HTML.gif. If k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq70_HTML.gif, then there is a unique x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq54_HTML.gifsuch that F ( x , x ) = x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq57_HTML.gif.

Example 16 Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq15_HTML.gif be the same as in Example 12. Then ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq15_HTML.gif is a G-metric space. Also, it is not difficult to verify that ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq15_HTML.gif is G-complete. Define a map F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq61_HTML.gif by F ( x , y ) = 1 1 16 x 2 5 16 y 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq94_HTML.gif for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq65_HTML.gif. Then, for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq69_HTML.gif, 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 | https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equaa_HTML.gif
and
G ( x , u , u ) + G ( y , v , v ) = 2 ( | x u | + | y v | ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equab_HTML.gif

Thus, the statement (2.4) of Corollary 15 is satisfied for any k [ 3 4 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq95_HTML.gif. Thus, there is a unique x X https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq54_HTML.gif such that F ( x , x ) = x https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq57_HTML.gif.

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 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_IEq53_HTML.gif, 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 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-170/MediaObjects/13660_2012_Article_308_Equac_HTML.gif

which is a contradiction.

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.

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