Open Access

Coupled fixed point theorems on partially ordered G-metric spaces

Fixed Point Theory and Applications20122012:174

DOI: 10.1186/1687-1812-2012-174

Received: 7 July 2012

Accepted: 26 September 2012

Published: 11 October 2012

Abstract

The purpose of this paper is to extend some recent coupled fixed point theorems in the context of partially ordered G-metric spaces in a virtually different and more natural way.

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

Keywords

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

1 Introduction and preliminaries

The notion of metric space was introduced by Fréchet [1] in 1906. In almost all fields of quantitative sciences which require the use of analysis, metric spaces play a major role. Internet search engines, image classification, protein classification (see, e.g., [2]) can be listed as examples in which metric spaces have been extensively used to solve major problems. It is conceivable that metric spaces will be needed to explore new problems that will arise in quantitative sciences in the future. Therefore, it is necessary to consider various generalizations of metrics and metric spaces to broaden the scope of applied sciences. In this respect, cone metric spaces, fuzzy metric spaces, partial metric spaces, quasi-metric spaces and b-metric spaces can be given as the main examples. Applications of these different approaches to metrics and metric spaces make it evident that fixed point theorems are important not only for the branches of mainstream mathematics, but also for many divisions of applied sciences.

Inspired by this motivation Mustafa and Sims [3] introduced the notion of a G-metric space in 2004 (see also [47]). In their introductory paper, the authors investigated versions of the celebrated theorems of the fixed point theory such as the Banach contraction mapping principle [8] from the point of view of G-metrics. Another fundamental aspect in the theory of existence and uniqueness of fixed points was considered by Ran and Reurings [9] in partially ordered metric spaces. After Ran and Reurings’ pioneering work, several authors have focused on the fixed points in ordered metric spaces and have used the obtained results to discuss the existence and uniqueness of solutions of differential equations, more precisely, of boundary value problems (see, e.g., [1020]). Upon the introduction of the notion of coupled fixed points by Guo and Laksmikantham [14], Gnana-Bhaskar and Lakshmikantham [15] obtained interesting results related to differential equations with periodic boundary conditions by developing the mixed monotone property in the context of partially ordered metric spaces. As a continuation of this trend, many authors conducted research on the coupled fixed point theory and many results in this direction were published (see, for example, [2135]).

In this paper, we prove the theorem that amalgamates these three seminal approaches in the study of fixed point theory, the so called G-metrics, coupled fixed points and partially ordered spaces.

We shall start with some necessary definitions and a detailed overview of the fundamental results developed in the remarkable works mentioned above. Throughout this paper, and N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq1_HTML.gif denote the set of non-negative integers and the set of positive integers respectively.

Definition 1 (See [3])

Let X be a non-empty set, G : X × X × X R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq2_HTML.gif be a function satisfying the following properties:

(G1) G ( x , y , z ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq3_HTML.gif if x = y = z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq4_HTML.gif,

(G2) G ( x , x , y ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq5_HTML.gif for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq6_HTML.gif with x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq7_HTML.gif,

(G3) G ( x , x , y ) G ( x , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq8_HTML.gif for all x , y , z X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq9_HTML.gif with y z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq10_HTML.gif,

(G4) G ( x , y , z ) = G ( x , z , y ) = G ( y , z , x ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq11_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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq12_HTML.gif for all x , y , z , a X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq13_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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq14_HTML.gif is called a G-metric space.

It can be easily verified that every G-metric on X induces a metric d G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq15_HTML.gif on X given 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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ1_HTML.gif
(1.1)

Trivial examples of G-metric are as follows.

Example 2 Let ( X , d ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq16_HTML.gif be a metric space. The function G : X × X × X [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq17_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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equb_HTML.gif

for all x , y , z X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq18_HTML.gif, is a G-metric on X.

The concepts of convergence, continuity, completeness and Cauchy sequence have also been defined in [3].

Definition 3 (See [3])

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq19_HTML.gif be a G-metric space, and let { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq20_HTML.gif be a sequence of points of X. We say that { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq20_HTML.gif is G-convergent to x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq21_HTML.gif if lim n , m + G ( x , x n , x m ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq22_HTML.gif, that is, if for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq23_HTML.gif, there exists N N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq24_HTML.gif such that G ( x , x n , x m ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq25_HTML.gif for all n , m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq26_HTML.gif. We call x the limit of the sequence and write x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq27_HTML.gif or lim n + x n = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq28_HTML.gif.

Proposition 4 (See [3])

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq19_HTML.gif be a G-metric space. The following statements are equivalent:
  1. (1)

    { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq20_HTML.gif is G-convergent to x,

     
  2. (2)

    G ( x n , x n , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq29_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq30_HTML.gif,

     
  3. (3)

    G ( x n , x , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq31_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq30_HTML.gif,

     
  4. (4)

    G ( x n , x m , x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq32_HTML.gif as n , m + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq33_HTML.gif.

     

Definition 5 (See [3])

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq19_HTML.gif be a G-metric space. A sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq20_HTML.gif is called G-Cauchy sequence if for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq23_HTML.gif, there is N N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq34_HTML.gif such that G ( x n , x m , x l ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq35_HTML.gif for all m , n , l N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq36_HTML.gif, that is, G ( x n , x m , x l ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq37_HTML.gif as n , m , l + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq38_HTML.gif.

Proposition 6 (See [3])

Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq19_HTML.gif be a G-metric space. The following statements are equivalent:
  1. (1)

    The sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq20_HTML.gif is G-Cauchy.

     
  2. (2)

    For any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq23_HTML.gif, there exists N N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq34_HTML.gif such that G ( x n , x m , x m ) < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq39_HTML.gif, for all m , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq40_HTML.gif.

     

Definition 7 (See [3])

A G-metric space ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq14_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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq14_HTML.gif.

Definition 8 Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq14_HTML.gif be a G-metric space. A mapping F : X × X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq41_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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq20_HTML.gif, { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq42_HTML.gif and { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq43_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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq44_HTML.gif is G-convergent to F ( x , y , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq45_HTML.gif.

We define below g-ordered complete G-metric spaces.

Definition 9 Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq46_HTML.gif be a partially ordered set, ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq19_HTML.gif be a G-metric space and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq47_HTML.gif be a mapping. A partially ordered G-metric space, ( X , G , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq48_HTML.gif, is called g-ordered complete if for each G-convergent sequence { x n } n = 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq49_HTML.gif, the following conditions hold:

( O C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq50_HTML.gif) If { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq20_HTML.gif is a non-increasing sequence in X such that x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq51_HTML.gif, then g x g x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq52_HTML.gif n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq53_HTML.gif.

( O C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq54_HTML.gif) If { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq20_HTML.gif is a non-decreasing sequence in X such that x n x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq51_HTML.gif, then g x g x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq55_HTML.gif n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq53_HTML.gif.

In particular, if g is the identity mapping in ( O C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq50_HTML.gif) and ( O C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq54_HTML.gif), the partially ordered G-metric space, ( X , G , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq48_HTML.gif, is called ordered complete.

We next recall some basic notions from the coupled fixed point theory. In 1987 Guo and Lakshmikantham [14] defined the concept of a coupled fixed point. In 2006, in order to prove the existence and uniqueness of the coupled fixed point of an operator F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq56_HTML.gif on a partially ordered metric space, Gnana-Bhaskar and Lakshmikantham [15] reconsidered the notion of a coupled fixed point via the mixed monotone property.

Definition 10 ([15])

Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq57_HTML.gif be a partially ordered set and F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq58_HTML.gif. The mapping F is said to have the mixed monotone property if F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq59_HTML.gif is monotone non-decreasing in x and is monotone non-increasing in y, that is, for any x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq6_HTML.gif,
x 1 x 2 F ( x 1 , y ) F ( x 2 , y ) , for  x 1 , x 2 X , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equc_HTML.gif
and
y 1 y 2 F ( x , y 2 ) F ( x , y 1 ) , for  y 1 , y 2 X . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equd_HTML.gif

Definition 11 ([15])

An element ( x , y ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq60_HTML.gif is called a coupled fixed point of the mapping F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq61_HTML.gif if
x = F ( x , y ) and y = F ( y , x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Eque_HTML.gif

The results in [15] were extended by Lakshmikantham and Ćirić in [16] by defining the mixed g-monotone property.

Definition 12 Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq46_HTML.gif be a partially ordered set, F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq62_HTML.gif and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq63_HTML.gif. The function F is said to have mixed g-monotone property if F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq59_HTML.gif is monotone g-non-decreasing in x and is monotone g-non-increasing in y, that is, for any x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq64_HTML.gif,
g ( x 1 ) g ( x 2 ) F ( x 1 , y ) F ( x 2 , y ) , for  x 1 , x 2 X , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ2_HTML.gif
(1.2)
and
g ( y 1 ) g ( y 2 ) F ( x , y 2 ) F ( x , y 1 ) , for  y 1 , y 2 X . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ3_HTML.gif
(1.3)

It is clear that Definition 12 reduces to Definition 10 when g is the identity mapping.

Definition 13 An element ( x , y ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq65_HTML.gif is called a coupled coincidence point of the mappings F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq66_HTML.gif and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq63_HTML.gif if
F ( x , y ) = g ( x ) , F ( y , x ) = g ( y ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equf_HTML.gif
and a common coupled fixed point of F and g if
F ( x , y ) = g ( x ) = x , F ( y , x ) = g ( y ) = y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equg_HTML.gif
Definition 14 The mappings F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq66_HTML.gif and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq63_HTML.gif 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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equh_HTML.gif

Throughout the rest of the paper, we shall use the notation gx instead of g ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq67_HTML.gif, where g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq68_HTML.gif and x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq21_HTML.gif, for brevity. In [35], Nashine proved the following theorems.

Theorem 15 Let ( X , G , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq69_HTML.gif be a partially ordered G-metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq70_HTML.gif and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq47_HTML.gif be mappings such that F has the mixed g-monotone property, and let there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq71_HTML.gif such that g x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq72_HTML.gif and F ( y 0 , x 0 ) g y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq73_HTML.gif. Suppose that there exists k [ 0 , 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq74_HTML.gif such that for all x , y , u , v , w , z X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq75_HTML.gif the following holds:
G ( F ( x , y ) , F ( u , v ) , F ( w , z ) ) k [ G ( g x , g u , g w ) + G ( g y , g v , g z ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ4_HTML.gif
(1.4)
for all g w g u g x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq76_HTML.gif and g y g v g z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq77_HTML.gif, where either g u g z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq78_HTML.gif or g v g w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq79_HTML.gif. Assume the following hypotheses:
  1. (i)

    F ( X × X ) g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq80_HTML.gif,

     
  2. (ii)

    g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq81_HTML.gif is G-complete,

     
  3. (iii)

    g is G-continuous and commutes with F.

     

Then F and g have a coupled coincidence point, that is, there exists ( x , y ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq82_HTML.gif such that g x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq83_HTML.gif and g y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq84_HTML.gif. If g u = g z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq85_HTML.gif and g v = g w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq86_HTML.gif, then F and g have a common fixed point, that is, there exists x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq21_HTML.gif such that g x = F ( x , x ) = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq87_HTML.gif.

Theorem 16 If in the above theorem, we replace the condition (ii) by the assumption that X is g-ordered complete, then we have the conclusions of Theorem  15.

We next give the definition of G-compatible mappings inspired by the definition of compatible mappings in [13].

Definition 17 Let ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq14_HTML.gif be a G-metric space. The mappings F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq88_HTML.gif, g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq68_HTML.gif are said to be G-compatible if
lim n G ( g F ( x n , y n ) , g F ( x n , y n ) , F ( g x n , g y n ) ) = 0 = lim n G ( g F ( x n , y n ) , F ( g x n , g y n ) , F ( g x n , g y n ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equi_HTML.gif
and
lim n G ( g F ( y n , x n ) , g F ( y n , x n ) , F ( g y n , g x n ) ) = 0 = lim n G ( g F ( y n , x n ) , F ( g y n , g x n ) , F ( g y n , g x n ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equj_HTML.gif

where { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq20_HTML.gif and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq89_HTML.gif are sequences in X such that lim n F ( x n , y n ) = lim n g x n = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq90_HTML.gif and lim n F ( y n , x n ) = lim n g y n = y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq91_HTML.gif for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq6_HTML.gif are satisfied.

In this paper, we aim to extend the results on coupled fixed points mentioned above. Our results improve, enrich and extend some existing theorems in the literature. We also give examples to illustrate our results. This paper can also be considered as a continuation of the works of Berinde [11, 12].

2 Main results

We start with an example which shows the weakness of Theorem 15.

Example 18 Let X = R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq92_HTML.gif. Define G : X × X × X [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq93_HTML.gif by
G ( x , y , z ) = | x y | + | x z | + | y z | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equk_HTML.gif
for all x , y , z X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq94_HTML.gif. Let be usual order. Then ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq14_HTML.gif is a G-metric space. Define a map F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq61_HTML.gif by F ( x , y ) = 1 8 x + 5 8 y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq95_HTML.gif and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq47_HTML.gif by g ( x ) = 7 x 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq96_HTML.gif for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq64_HTML.gif. Let x = u = z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq97_HTML.gif. Then we have
G ( F ( x , y ) , F ( u , v ) , F ( z , w ) ) = G ( 1 8 x + 5 8 y , 1 8 u + 5 8 v , 1 8 z + 5 8 w ) = 5 8 | v y | + 5 8 | w y | + 5 8 | w v | , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ5_HTML.gif
(2.1)
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 ) = 7 8 [ | y v | + | y w | + | v w | ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ6_HTML.gif
(2.2)

It is clear that there is no k [ 0 , 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq74_HTML.gif for which the statement (1.4) of Theorem 15 holds. Notice, however, that ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq98_HTML.gif is the unique coupled coincidence point of F and g. In fact, it is a common fixed point of F and g, that is, F ( 0 , 0 ) = g 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq99_HTML.gif.

We now state our first result which successively guarantees the existence of a coupled coincidence point.

Theorem 19 Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq46_HTML.gif be a partially ordered set and ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq19_HTML.gif be a G-complete G-metric space. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq100_HTML.gif and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq101_HTML.gif be two mappings such that F has the mixed g-monotone property on X and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ7_HTML.gif
(2.3)
for all x , y , u , v , z , w X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq102_HTML.gif with g x g u g w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq103_HTML.gif, g y g v g z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq77_HTML.gif. Assume that F ( X × X ) g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq104_HTML.gif, g is G-continuous and that F and g are G-compatible mappings. Suppose further that either
  1. (a)

    F is continuous or

     
  2. (b)

    ( X , G , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq69_HTML.gif is g-ordered complete.

     

Suppose also that there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq105_HTML.gif such that g x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq106_HTML.gif and F ( y 0 , x 0 ) g y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq107_HTML.gif. If k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq108_HTML.gif, then F and g have a coupled coincidence point, that is, there exists ( x , y ) ( X × X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq109_HTML.gif such that g ( x ) = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq110_HTML.gif and g ( y ) = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq111_HTML.gif.

Proof Let x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq112_HTML.gif be such that g x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq72_HTML.gif and F ( y 0 , x 0 ) g y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq113_HTML.gif. Using the fact that F ( X × X ) g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq104_HTML.gif, we can construct two sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq20_HTML.gif and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq89_HTML.gif in X in the following way:
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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ8_HTML.gif
(2.4)
We shall prove that for all n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq114_HTML.gif,
g x n g x n + 1 and g y n g y n + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ9_HTML.gif
(2.5)
Since g x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq72_HTML.gif and F ( y 0 , x 0 ) g y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq113_HTML.gif and g x 1 = F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq115_HTML.gif and F ( y 0 , x 0 ) = g y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq116_HTML.gif, we have g x 0 g x 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq117_HTML.gif and g y 1 g y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq118_HTML.gif, that is, (2.5) holds for n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq119_HTML.gif. Assume that (2.5) holds for some n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq120_HTML.gif. Since F has the mixed g-monotone property, from (2.4), we have
g x n + 1 = F ( x n , y n ) F ( x n + 1 , y n ) F ( x n + 1 , y n + 1 ) = g x n + 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ10_HTML.gif
(2.6)
and
g y n + 1 = F ( y n , x n ) F ( y n + 1 , x n ) F ( y n + 1 , x n + 1 ) = g y n + 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ11_HTML.gif
(2.7)
By mathematical induction, it follows that (2.5) holds for all n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq121_HTML.gif, that is,
g x 0 g x 1 g x 2 g x n g x n + 1 g x n + 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ12_HTML.gif
(2.8)
and
g y 0 g y 1 g y 2 g y n g y n + 1 g y n + 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ13_HTML.gif
(2.9)
If there exists n 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq122_HTML.gif such that ( g x n 0 + 1 , g y n 0 + 1 ) = ( g x n 0 , g y n 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq123_HTML.gif, then F and g have a coupled coincidence point. Indeed, in that case we would have
( g x n 0 + 1 , g y n 0 + 1 ) = ( F ( x n 0 , y n 0 ) , F ( y n 0 , x n 0 ) ) = ( g x n 0 , g y n 0 ) F ( x n 0 , y n 0 ) = g x n 0 and F ( y n 0 , x n 0 ) = g y n 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equl_HTML.gif

We suppose that ( g x n + 1 , g y n + 1 ) ( g x n , g y n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq124_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq125_HTML.gif. More precisely, we assume that either g x n + 1 = F ( x n , y n ) g x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq126_HTML.gif or g y n + 1 = F ( y n , x n ) g y n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq127_HTML.gif.

For n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq128_HTML.gif, we set
t n = G ( g x n + 1 , g x n + 1 , g x n ) + G ( g y n + 1 , g y n + 1 , g y n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equm_HTML.gif
Then by using (2.3) and (2.6), for each n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq128_HTML.gif, we have
t n = G ( g x n + 1 , g x n + 1 , g x n ) + G ( g y n + 1 , g y n + 1 , g y n ) = G ( F ( x n , y n ) , F ( x n , y n ) , F ( x n 1 , y n 1 ) ) + G ( F ( y n , x n ) , F ( y n , x n ) , F ( y n 1 , x n 1 ) ) k [ G ( g x n , g x n , g x n 1 ) + G ( g y n , g y n , g y n 1 ) ] = k t n 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equn_HTML.gif
which yields that
t n k n t 0 , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ14_HTML.gif
(2.10)
Now, for all m , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq129_HTML.gif with m > n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq130_HTML.gif, by using rectangle inequality (G5) of G-metric and (2.10), we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equo_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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equp_HTML.gif

Then by Proposition 6, we conclude that the sequences { g x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq131_HTML.gif and { g y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq132_HTML.gif are G-Cauchy.

Noting that g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq81_HTML.gif is G-complete, there exist x , y g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq133_HTML.gif such that { g x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq134_HTML.gif and { g y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq132_HTML.gif are G-convergent to x and y respectively, i.e.,
lim n + F ( x n , y n ) = lim n + g x n + 1 = x , lim n + F ( y n , x n ) = lim n + g y n + 1 = y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ15_HTML.gif
(2.11)
Since F and g are G-compatible mappings, by (2.11), we have
lim n G ( g F ( x n , y n ) , F ( g x n , g y n ) , F ( g x n , g y n ) ) = 0 , lim n G ( g F ( y n , x n ) , F ( g y n , g x n ) , F ( g y n , g x n ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ16_HTML.gif
(2.12)
Suppose that the condition (a) holds. For all n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq120_HTML.gif, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ17_HTML.gif
(2.13)
Letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq135_HTML.gif in the above inequality, using (2.11), (2.12) and the continuities of F and g, we have
lim n 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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equq_HTML.gif
Hence, we derive that g x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq83_HTML.gif and g y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq84_HTML.gif, that is, ( x , y ) X 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq136_HTML.gif is a coupled coincidence point of F and g. Suppose that the condition (b) holds. By (2.8), (2.9) and (2.11), we have
g g x g x and g g y g y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ18_HTML.gif
(2.14)
Due to the fact that F and g are G-compatible mappings and g is continuous, by (2.11) and (2.12), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ19_HTML.gif
(2.15)
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ20_HTML.gif
(2.16)
Keeping (2.15) and (2.16) in mind, we consider now
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ21_HTML.gif
(2.17)
Letting n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq135_HTML.gif in the above inequality, by using (2.15), (2.16) and the continuity of g, we conclude that
0 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%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ22_HTML.gif
(2.18)

By (G1), we have g x = F ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq83_HTML.gif and g y = F ( y , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq84_HTML.gif. Consequently, the element ( x , y ) X × X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq137_HTML.gif is a coupled coincidence point of the mappings F and g. □

Corollary 20 Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq46_HTML.gif be a partially ordered set and ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq19_HTML.gif be a G-metric space such that ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq14_HTML.gif is G-complete. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq100_HTML.gif and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq101_HTML.gif be two mappings such that F has the mixed g-monotone property on X and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ23_HTML.gif
(2.19)
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq138_HTML.gif with g x g u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq139_HTML.gif, g y g v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq140_HTML.gif. Assume that F ( X × X ) g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq104_HTML.gif, the self-mapping g is G-continuous and F and g are G-compatible mappings. Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)

    ( X , G , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq69_HTML.gif is g-ordered complete.

     

Suppose also that there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq105_HTML.gif such that g x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq106_HTML.gif and g y 0 F ( y 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq141_HTML.gif. If k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq108_HTML.gif, then F and g have a coupled coincidence point.

Proof It is sufficient to take z = u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq142_HTML.gif and w = v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq143_HTML.gif in Theorem 19. □

Corollary 21 Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq46_HTML.gif be a partially ordered set and ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq19_HTML.gif be a G-metric space such that ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq14_HTML.gif is G-complete. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq100_HTML.gif and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq101_HTML.gif be two mappings such that F has the mixed g-monotone property on X and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ24_HTML.gif
(2.20)
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq138_HTML.gif with g x g u g w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq103_HTML.gif, g y g v g z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq144_HTML.gif. Assume that F ( X × X ) g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq104_HTML.gif and that the self-mapping g is G-continuous and commutes with F. Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)

    ( X , G , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq69_HTML.gif is g-ordered complete.

     

Suppose further that there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq105_HTML.gif such that g x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq106_HTML.gif and g y 0 F ( y 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq141_HTML.gif. If k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq108_HTML.gif, then F and g have a coupled coincidence point.

Proof Since g commutes with F, then F and g are G-compatible mappings. Thus, the result follows from Theorem 19. □

Corollary 22 Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq46_HTML.gif be a partially ordered set and ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq19_HTML.gif be a G-metric space such that ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq14_HTML.gif is G-complete. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq100_HTML.gif and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq101_HTML.gif be two mappings such that F has the mixed g-monotone property on X and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ25_HTML.gif
(2.21)
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq138_HTML.gif with g x g u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq139_HTML.gif, g y g v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq140_HTML.gif. Assume that F ( X × X ) g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq104_HTML.gif and that g is G-continuous and commutes with F. Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)

    ( X , G , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq69_HTML.gif is g-ordered complete.

     

Assume also that there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq105_HTML.gif such that g x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq106_HTML.gif and g y 0 F ( y 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq141_HTML.gif. If k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq108_HTML.gif, then F and g have a coupled coincidence point.

Proof Since g commutes with F, then F and g are G-compatible mappings. Thus, the result follows from Corollary 20. □

Letting g = I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq145_HTML.gif in Theorem 19 and in Corollary 20, we get the following results.

Corollary 23 Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq46_HTML.gif be a partially ordered set and ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq19_HTML.gif be a G-metric space such that ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq14_HTML.gif is G-complete. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq100_HTML.gif be a mapping having the mixed monotone property on X and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ26_HTML.gif
(2.22)
for all x , y , u , v , z , w X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq102_HTML.gif with x u w https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq146_HTML.gif, y v z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq147_HTML.gif. Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)

    ( X , G , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq69_HTML.gif is ordered complete.

     

Suppose also that there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq105_HTML.gif such that x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq148_HTML.gif and y 0 F ( y 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq149_HTML.gif. If k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq108_HTML.gif, then F has a coupled fixed point.

Corollary 24 Let ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq46_HTML.gif be a partially ordered set and ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq19_HTML.gif be a G-metric space such that ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq14_HTML.gif is G-complete. Let F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq100_HTML.gif be a mapping having the mixed monotone property on X and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ27_HTML.gif
(2.23)
for all x , y , u , v X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq138_HTML.gif with x u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq150_HTML.gif, y v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq151_HTML.gif. Suppose that either
  1. (a)

    F is continuous or

     
  2. (b)

    ( X , G , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq69_HTML.gif is ordered complete.

     

Suppose further that there exist x 0 , y 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq105_HTML.gif such that x 0 F ( x 0 , y 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq148_HTML.gif and y 0 F ( y 0 , x 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq149_HTML.gif. If k [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq108_HTML.gif, then F has a coupled fixed point.

Example 25 Let us recall Example 18. We have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ28_HTML.gif
(2.24)
and
G ( g x , g u , g z ) + G ( g y , g v , g w ) = G ( 7 x 8 , 7 u 8 , 7 z 8 ) + G ( 7 y 8 , 7 v 8 , w 8 ) = 7 8 [ ( | u x | + | z x | + | z u | ) + ( | v y | + | w y | + | w v | ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ29_HTML.gif
(2.25)

It is clear that there any k [ 6 7 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq152_HTML.gif provides the statement (2.3) of Theorem 19.

Notice that ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq98_HTML.gif is the unique coupled coincidence point of F and g which is also common coupled fixed point, that is, F ( 0 , 0 ) = g 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq99_HTML.gif.

Example 26 Let X = R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq92_HTML.gif. Define G : X × X × X [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq93_HTML.gif by
G ( x , y , z ) = | x y | + | x z | + | y z | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equr_HTML.gif

for all x , y , z X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq94_HTML.gif. Let be usual order. Then ( X , G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq14_HTML.gif is a G-metric space.

Define a map F : X × X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq61_HTML.gif by
F ( x , y ) = 1 8 x 3 + 5 8 y 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equs_HTML.gif
and g : X X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq47_HTML.gif by g ( x ) = x 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq153_HTML.gif for all x , y X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq64_HTML.gif. Then F ( X × X ) = X = g ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq154_HTML.gif. We observe that
G ( F ( x , y ) , F ( u , v ) , F ( z , w ) ) + G ( F ( y , x ) , F ( v , u ) , F ( v , u ) ) = G ( 1 8 x 3 + 5 8 y 3 , 1 8 u 3 + 5 8 v 3 , 1 8 z 3 + 5 8 w 3 ) + G ( 1 8 y 3 + 5 8 x 3 , 1 8 v 3 + 5 8 u 3 , 1 8 w 3 + 5 8 z 3 ) = 5 8 | v 3 y 3 | + 5 8 | w 3 y 3 | + 5 8 | w 3 v 3 | + 1 8 | u 3 x 3 | + 1 8 | z 3 x 3 | + 1 8 | z 3 u 3 | + 1 8 | v 3 y 3 | + 1 8 | w 3 y 3 | + 1 8 | w 3 v 3 | + 5 8 | u 3 x 3 | + 5 8 | z 3 x 3 | + 5 8 | z 3 u 3 | = 6 8 [ | v 3 y 3 | + | w 3 y 3 | + | w 3 v 3 | + | u 3 x 3 | + | z 3 x 3 | + | z 3 u 3 | ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equt_HTML.gif
and
G ( g x , g u , g z ) + G ( g y , g v , g w ) = G ( x 3 , u 3 , z 3 ) + G ( y 3 , v 3 , w 3 ) = ( | x 3 u 3 | + | x 3 z 3 | + | u 3 z 3 | ) + ( | y 3 v 3 | + | y 3 w 3 | + | v 3 w 3 | ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ30_HTML.gif
(2.26)

then the statement (2.3) of Theorem 19 is satisfied for any k ( 3 4 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq155_HTML.gif and ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq98_HTML.gif.

Notice that if we replace the condition (2.3) of Theorem 19 with the condition (1.4) of Theorem 15 [21], that is,
G ( F ( x , y ) , F ( u , v ) , F ( w , z ) ) k [ G ( g x , g u , g w ) + G ( g y , g v , g z ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ31_HTML.gif
(2.27)

where k [ 0 , 1 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq156_HTML.gif, then the coupled coincidence point exists even though the contractive condition is not satisfied.

More precisely, consider x = u = z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq97_HTML.gif. Then we have
G ( F ( x , y ) , F ( u , v ) , F ( z , w ) ) = G ( 1 8 x 3 + 5 8 y 3 , 1 8 u 3 + 5 8 v 3 , 1 8 z 3 + 5 8 w 3 ) = 5 8 | v 3 y 3 | + 5 8 | w 3 y 3 | + 5 8 | w 3 v 3 | https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ32_HTML.gif
(2.28)
and
G ( g x , g u , g z ) + G ( g y , g v , g w ) = G ( x 3 , u 3 , z 3 ) + G ( y 3 , v 3 , w 3 ) = | y 3 v 3 | + | y 3 w 3 | + | v 3 w 3 | . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_Equ33_HTML.gif
(2.29)

It is clear that the condition (2.27) holds for k > 5 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-174/MediaObjects/13663_2012_Article_282_IEq157_HTML.gif.

Declarations

Acknowledgements

The second author gratefully acknowledges the support provided by the Department of Mathematics and Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT) during his stay at the Department of Mathematical and Statistical Sciences, University of Alberta as a visitor for the short term research.

Authors’ Affiliations

(1)
Department of Mathematics, Atilim University
(2)
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)
(3)
Department of Mathematical and Statistical Sciences, University of Alberta

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