Some coupled common fixed points for a pair of mappings in partially ordered G-metric spaces

The purpose of this paper is to establish some coupled coincidence point theorems for a pair of mappings having a mixed g-monotone property in partially ordered G-metric spaces. Also, we present a result on the existence and uniqueness of coupled common fixed points. The results presented in the paper generalize and extend several well-known results in the literature. To illustrate our results, we give some examples.


Introduction
Fixed point theory is one of the famous and traditional theories in mathematics and has a large number of applications. The Banach contraction mapping is one of the pivotal results of analysis. It is very popular tool for solving existence problems in many different fields of mathematics. There are a lot of generalizations of the Banach contraction principle in the literature. Ran and Reurings [1] extended the Banach contraction principle in partially ordered sets with some applications to linear and nonlinear matrix equations. While Nieto and Rodŕiguez-López [2] extended the result of Ran and Reurings and applied their main theorems to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions. Bhaskar and Lakshmikantham [3] introduced the concept of mixed monotone mappings and obtained some coupled fixed point results. Also, they applied their results on a first-order differential equation with periodic boundary conditions. On the other hand, Mustafa and Sims [4] introduced G-metric space which is a generalization of metric spaces in which every triplet of the elements is assigned to a nonnegative real number. Recently, many researchers have *Correspondence: poom teun@hotmail.com; poom.kum@kmutt.ac.th 2 Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand Full list of author information is available at the end of the article obtained fixed point, common fixed point, coupled fixed point and coupled common fixed point results on metric spaces, G-metric spaces, partially ordered metric spaces, and partially ordered G-metric spaces (see e.g. [1][2][3][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and references cited therein). The purpose of this paper is to establish some coupled coincidence point results in partially ordered G-metric spaces for a pair of mappings having mixed g-monotone property. Also, we present a result on the existence and uniqueness of coupled common fixed points. We supply appropriate examples to make obvious the validity of the propositions of our results.

Preliminaries
In the sequel, R, R + , and N denote the set of real numbers, the set of nonnegative real numbers, and the set of positive integers, respectively. Definition 2.1. (See [21]). Let X be a non-empty set, and G : X ×X ×X → R + be a function satisfying the following properties: . . (symmetry in all three variables); http://www.iaumath.com/content/7/1/24 (G5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a ∈ X (rectangle inequality).
Then, the function G is called a generalized metric, or more specially, a G-metric on X and the pair (X, G) is called a G-metric space. It can be easily verified that every G-metric on X induces a metric d G on X given by Trivial examples of G-metric are as follows: [14]). Let (X, G) be a G-metric space. A mapping F : X × X → X is said to be continuous if for any two G-convergent sequences, {x n } and {y n } converging to x and y, respectively, {F(x n , y n )} is G-convergent to F(x, y).
Definition 2.11. Let X be a non-empty set and F : X × X → X and g : X → X. The mappings F and g are said to commute if F(gx, gy) = g(F(x, y)) for all x, y ∈ X.
Definition 2.12. Let (X, ) be a partially ordered set, and F : X → X. The mapping F is said to be non-decreasing if for x, y ∈ X, x y implies F(x) F(y); non-increasing if for x, y ∈ X, x y implies F(x) F(y).
Definition 2.13. Let (X, ) be a partially ordered set, and F : X × X → X and g : X → X. The mapping F is said to have the mixed g-monotone property if F(x, y) is monotone g-non-decreasing in x and monotone g-non-increasing in y, that is, for any x, y ∈ X, and y 1 , y 2 ∈ X, gy 1 gy 2 ⇒ F(x, y 1 ) F(x, y 2 ).
If g is identity mapping in Definition 2.13, then the mapping F is said to have the mixed monotone property. Definition 2.14. Let X be a non-empty set. An element (x, y) ∈ X × X is called a coupled coincidence point of the mappings F : X × X → X and g : X → X if F(x, y) = gx and F(y, x) = gy. If gx = x and gy = y, then (x, y) ∈ X × X is called a coupled common fixed point.
If g is identity mapping in Definition 2.14, then (x, y) ∈ X × X is called a coupled fixed point.

Main results
In this section, we prove some coupled common fixed point theorems in the context of ordered G-metric spaces. Theorem 3.1. Let (X, ) be a partially ordered set, and G be a G-metric on X such that (X, G) is a G-metric space. Suppose that F : X × X → X and g : X → X are continuous such that F has the mixed g-monotone property http://www.iaumath.com/content/7/1/24 on X such that there exist two elements x 0 , y 0 ∈ X with g(x 0 ) F(x 0 , y 0 ) and g(y 0 ) F(y 0 , x 0 ). Suppose that there exist non-negative real numbers α, β and L with α + β < 1 such that for all x, y, u, v, w, z ∈ X with gx gu gw and gy gv gz, either gu = gw or gv = gz. Furthermore, suppose that F(X × X) ⊆ g(X), g(X) is a G-complete subspace of X, and g commutes with F, then there exist x, y ∈ X such that F(x, y) = gx and gy = F(y, x), that is, F and g have a coupled coincidence If there exists n ∈ N such that gx n −1 = gx n and gy n −1 = gy n , then gx n −1 = F(x n −1 , y n −1 ) and gy n −1 = F(y n −1 , x n −1 ) that is a point (x n −1 , y n −1 ) ∈ X × X is a coupled coincidence point of F and g. Thus, we may assume that gx n−1 = gx n or gy n−1 = gy n for all n ∈ N.
Next, we claim that for all n ≥ 0, and gy n gy n+1 .
There fore, gy m , gy m )] = 0. Therefore, {gx n } and {gy n } are G-Cauchy sequences in g(X). Since g(X) is a G-complete subspace of X, there is (x, y) ∈ X × X such that {gx n } and {gy n } are respectively G-convergent to x and y.
Using continuity of g, we get Since gx n+1 = F(x n , y n ) and gy n+1 = F(y n , x n ), hence the commutativity of F and g yields that F(gx n , gy n ) = gF(x n , y n ) = g(gx n+1 ) and F(gy n , gx n ) = gF(y n , x n ) = g(gy n+1 ). Now, we show that F(x, y) = gx and F(y, x) = gy. The mapping F is continuous, so since the sequences {gx n } and {gy n } are respectively G-convergent to x and y; hence, using Definition 2.10, the sequence {F(gx n , gy n )} is G- convergent to F(x, y). Therefore, {g(gx n+1 )} is G-convergent to F(x, y). By uniqueness of the limit, we have F(x, y) = gx. Similarly, we can show that F(y, x) = gy. Hence, (x, y) is a coupled coincidence point of F and g.
In the next theorem, we replace the continuity of F with the following definition: Let (X, ) be a partially ordered set, and G be a G-metric on X. We say that (X, G, ) is regular if the following conditions hold: (1) if a non-decreasing sequence {x n } is such that x n → x, then x n x for all n, (2) if a non-increasing sequence {y n } is such that y n → y, then y y n for all n.

Theorem 3.3.
Let (X, ) be a partially ordered set, and G be a G-metric on X such that (X, G, ) is regular. Suppose that F : X × X → X and g : X → X are self mappings on X, such that F has the mixed g-monotone property on X such that there exist two elements x 0 , y 0 ∈ X with g(x 0 ) F(x 0 , y 0 ) and g(y 0 ) F(y 0 , x 0 ). Suppose that there exist non-negative real numbers α, β and L with α + β < 1 such that (3.1) satisfies for all x, y, u, v, w, z ∈ X with gx gu gw and gy gv gz, where either gu = gw or gv = gz. Further, suppose that F(X × X) ⊆ g(X), and (g(X), G) is a complete G-metric. Then, there exist x, y ∈ X such that F(x, y) = g(x) and gy = F(y, x), that is, F and g have a coupled coincidence point (x, y) ∈ X × X.
Proof. Following the proof of Theorem 3.1, we will get two G-Cauchy sequences {gx n } and {gy n } in the complete G-metric space (g(X), G). Then, there exist x, y ∈ X such that gx n → gx and gy n → gy as n → ∞. Since {gx n } is non-decreasing and {gy n } is non-increasing, using the regularity of (X, G, ), we have gx n gx and gy gy n for all n ≥ 0. If gx n = gx and gy n = gy for some n ≥ 0, then gx = gx n gx n +1 gx = gx n and gy gy n +1 gy n = gy, which implies that gx n = gx n +1 = F(x n , y n ) and gy n = gy n +1 = F(y n , x n ), that is, (x n , y n ) is a coupled coincidence point of F and g. Therefore, we suppose that gx n = gx or gy n = gy for all n ≥ 0. Using rectangle inequality, commutativity, and (3.1), we have G(gx n+1 , gx n+1 , F(x, y)) = G(F(x n , y n ), F(x n , y n ), F(x, y)) ≤ αG(gx n , gx n , gx) + βG(gy n , gy n , gy) + L min{G(gx n+1 , gx n , gx), G(gx n+1 , gx n , gx),G (F(x, y), gx n , gx n ), G(gx n+1 , gx n , gx), G(gx n+1 , gx n , gx), G (F(x, y), gx, gx)}. (3.10) Taking n → ∞, we get G(gx, gx, F(x, y)) = 0 and hence gx = F(x, y). Similarly, one can show that gy = F(y, x). Thus F and g have a coupled coincidence point. [4]. From the condition that either gu = gw or gv = gz, the inequality (3.1) does not reduce to any metric inequality with the metric d G . Therefore, the corresponding metric space (X, d G ) results are not applicable to Theorems 3.1 and 3.3.

Remark 3.1. A G-metric naturally induces a metric d G given by d G (x, y) = G(x, y, y)+G(x, x, y)
Taking L = 0 in Theorems 3.1 and 3.3, we have the following result:

Corollary 3.4. Let (X, ) be a partially ordered set, and G be a G-metric on X such that (X, G) is a G-metric space.
Suppose that F : X×X → X and g : X → X are continuous self mappings on X such that F has the mixed g-monotone property on X such that there exist two elements x 0 , y 0 ∈ X with g(x 0 ) F(x 0 , y 0 ) and g(y 0 ) F(y 0 , x 0 ). Suppose that there exist non-negative real numbers α, β with α + β < 1 such that G(F(x, y), F(u, v), F(w, z)) ≤ αG(gx, gu, gw)+βG(gy, gv, gz), (3.11) for all x, y, u, v, w, z ∈ X with gx gu gw and gy gv gz, where either gu = gw or gv = gz. Further, suppose F(X × X) ⊆ g(X), g(X) is a G-complete subspace of X and g commutes with F. Then, there exist x, y ∈ X such that F(x, y) = gx and gy = F(y, x), that is, F and g have a coupled coincidence point (x, y) ∈ X × X. X such that (X, G, ) is regular. Suppose that F : X × X → X and g : X → X are self mappings on X such that F has the mixed g-monotone property on X such that there exist two elements x 0 , y 0 ∈ X with g(x 0 ) F(x 0 , y 0 ) and g(y 0 ) F(y 0 , x 0 ). Suppose that there exist non-negative real numbers α, β with α + β < 1 such that (3.11) satisfies for all x, y, u, v, w, z ∈ X with gx gu gw and gy gv gz, where either gu = gw or gv = gz. Further, suppose that F(X × X) ⊆ g(X), and (g(X), G) is a complete G-metric, then there exist x, y ∈ X such that F(x, y) = g(x) and gy = F(y, x), that is, F and g have a coupled coincidence point (x, y) ∈ X × X. http://www.iaumath.com/content/7/1/24

Corollary 3.5. Let (X, ) be a partially ordered set, and G be a G-metric on
Taking α = β = k 2 , where k ∈[ 0, 1) and L = 0 in Theorems 3.1 and 3.3, we have the following result: Corollary 3.6. Let (X, ) be a partially ordered set, and G be a G-metric on X such that (X, G) is a G-metric space. Suppose that F : X × X → X and g : X → X are continuous such that F has the mixed g-monotone property on X such that there exist two elements x 0 , y 0 ∈ X with g(x 0 ) F(x 0 , y 0 ) and g(y 0 ) F(y 0 , x 0 ). Suppose that there exists k ∈[ 0, 1) such that G (F(x, y), F(u, v), F(z, w)) ≤ k 2 (G(gx, gu, gw)+G(gy, gv, gz)), (3.12) for all x, y, u, v, w, z ∈ X with gx gu gw and gy gv gz, where either gu = gw or gv = gz. Further, suppose F(X × X) ⊆ g(X), g(X) is a G-complete subspace of X and g commutes with F, then there exist x, y ∈ X such that F(x, y) = gx and gy = F(y, x), that is, F and g have a coupled coincidence point (x, y) ∈ X × X. Corollary 3.7. Let (X, ) be a partially ordered set, and G be a G-metric on X such that (X, G, ) is regular. Suppose that F : X × X → X and g : X → X are self mappings on X such that F has the mixed g-monotone property on X such that there exist two elements x 0 , y 0 ∈ X with g(x 0 ) F(x 0 , y 0 ) and g(y 0 ) F(y 0 , x 0 ). Suppose that there exists k ∈[ 0, 1) such that (3.12) satisfies for all x, y, u, v, w, z ∈ X with gx gu gw and gy gv gz, where either gu = gw or gv = gz. Further, suppose that F(X × X) ⊆ g(X), and (g(X), G) is a complete G-metric. Then, there exist x, y ∈ X such that F(x, y) = g(x) and gy = F(y, x), that is, F and g have a coupled coincidence point (x, y) ∈ X × X. Remark 3.2. Corollaries 3.6 and 3.7 are generalization of the results of Choudhury and Maity [14]. Now, we shall prove the existence and uniqueness of a coupled common fixed point. Note that if (X, ) is a partially ordered set, then we endow the product space X × X with the following partial order relation: (F(x, y), F(y, x)) and (F(z, t), F(t, z)). Then, F and g have a unique coupled common fixed point, that is, there exists a unique (x, y) ∈ X × X such that x = gx = F(x, y) and y = gy = F(y, x).

Theorem 3.8. In addition to the hypotheses of Theorem 3.1, suppose that for every
Proof. From Theorem 3.1, the set of coupled coincidence points of F and g is non-empty. Suppose that (x, y) and (z, t) are coupled coincidence points of F and g, that is, gx = F(x, y), gy = F(y, x), gz = F(z, t), and gt = F(t, z). We shall show that gx = gz and gy = gt. By the assumption, there exists (u, v) ∈ X × X such that (F(u, v), F(v, u)) is comparable with (F(x, y), F(y, x)) and (F(z, t), F(t, z)). Put u 0 = u and v 0 = v, and choose u 1 , v 1 ∈ X so that gu 1 = F(u 0 , v 0 ) and gv 1 = F(v 0 , u 0 ). Then, similarly as in the proof of Theorem 3.1, we can inductively define sequences {gu n } and {gv n } as gu n+1 = F(u n , v n ) and gv n+1 = F(v n , u n ) for all n. Further, set x 0 = x, y 0 = y, z 0 = z, and t 0 = t and on the same way define the sequences {gx n } and {gy n }, and {gz n } and {gt n }. Since (F(x, y), F(y, x)) = (gx 1 , gy 1 ) = (gx, gy) and (F(u, v), F(v, u)) = (gu 1 , gv 1 ) are comparable, then gx gu 1 and gy gv 1 . Now, we shall show that (gx, gy) and (gu n , gv n ) are comparable, that is, gx gu n and gy gv n for all n. Suppose that it holds for some n ≥ 0, then by the mixed g-monotone property of F, we have gu n+1 = F(u n , v n ) F(x, y) = gx and gv n+1 = F(v n , u n ) F(y, x) = gy. Hence, gx gu n and gy gv n hold for all n. Thus, from (3.1), we have ≤ αG(gx, gx, gu n ) + βG(gy, gy, gv n ) , y), gx, gu n ), Thus, lim n→∞ G(gx, gx, gu n+1 ) = 0 and lim n→∞ G(gy, gy, gv n+1 ) = 0. Similarly, we can prove that lim n→∞ G(gz, gz, gu n ) = 0 = lim n→∞ G(gt, gt, gv n ). Hence, gx = gz and gy = gt.
Finally, we provide some examples to illustrate our obtained Theorem 3.1.
Example 3.9. Let X = R be a set endowed with order x y ⇔ x ≤ y. Let the mapping G : X × X × X → R + be defined by G(x, y, z) = |x − y| + |y − z| + |z − x|, for all x, y, z ∈ X. Then, G is a G-metric on X. Define the mapping F : X × X → X and g : X → X by F(x, y) = x − 2y 8 for all (x, y) ∈ X × X and gx = x 2 for all x ∈ X.
For all x, y, u, v, w, z ∈ X, we have G (F(x, y), F(u, v), F(w, z)) for all x, y ∈ X.
Therefore, all hypotheses of Theorem 3.1 hold, and so F and g have a coupled coincidence point that is a point (0, 0) ∈ X × X. Moreover, this point is also coupled common fixed point of F and g.