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

Abstract

The purpose of this paper is to establish some coupled coincidence point theorems for a pair of map**s 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 map** 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 map**s 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 non-negative real number. Recently, many researchers have 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. [1–3, 5–20] 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 map**s 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, $\mathbb{R}$, ${\mathbb{R}}_{+}$, and $\mathbb{N}$ 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 $G:X\times X\times X\to {\mathbb{R}}_{+}$ be a function satisfying the following properties:

(G1) G(x,y,z)=0 if x=y=z;

(G2) 0<G(x,x,y) for all x,y∈X with x≠y;

(G3) G(x,x,y)≤G(x,y,z) for all x,y,z∈X with y≠z;

(G4) G(x,y,z)=G(x,z,y)=G(y,z,x)=… (symmetry in all three variables);

(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

$$d_G(x,y)=G(x,y,y)+G(y,x,x),$$

for all x,y∈X.

Trivial examples of G-metric are as follows:

Example 2.2

Let (X,d) be a metric space. The function $G:X\times X\times X\to {\mathbb{R}}_{+}$ defined by

$$G(x,y,z)=max\left\{d\right(x,y),d(y,z),d(z,x\left)\right\},$$

or

$$G(x,y,z)=d(x,y)+d(y,z)+d(z,x),$$

for all x,y,z∈X, is a G-metric on X

The concepts of convergence, continuity, completeness, and Cauchy sequence have also been defined in [21].

Definition 2.3

(See [21]). Let (X,G) be a G-metric space, and let {x _n } be a sequence of points of X. We say that {x _n } is G-convergent to x∈X if l i m_(n,m)→∞G(x,x _n ,x _m )=0, that is, for any ε>0, there exists $N\in \mathbb{N}$ such that G(x,x _n ,x _m )<ε, for all n,m≥N. We call x the limit of the sequence and write x _n →x as n→∞ or limn→∞ x _n =x.

It has been shown in [21] that the G-metric induces a Hausdorff topology, and the convergence described in the above definition is relative to this topology. So, a sequence can converge at the most to one point.

Proposition 2.4

(See [[21]]). Let ( X,G ) be a G -metric space. The following are equivalent:

• (1) {x _n } is G -convergent to x.

• (2) G(x _n ,x _n ,x)→0 as n→+∞.

• (3) G(x _n ,x,x)→0 as n→+∞.

• (4) G(x _n ,x _m ,x)→0 as n,m→+∞.

Definition 2.5

(See [21]). Let (X,G) be a G-metric space. A sequence {x _n } is called a G-Cauchy sequence if, for any ε>0, there is $N\in \mathbb{N}$ such that G(x _n ,x _m ,x _l )<ε for all m,n,l≥N, that is, G(x _n ,x _m ,x _l )→0 as n,m,l→+∞.

Proposition 2.6

(See [[4]]). Let (X,G) be a G -metric space. Then, the following are equivalent:

• (1) the sequence {x _n } is G -Cauchy;

• (2) for any ε>0, there exists $N\in \mathbb{N}$ such that G(x _n ,x _m ,x _m )<ε, for all m,n≥N.

Proposition 2.7

(See [[21]]). Let ( X,G) be a G -metric space. A map** f: X→X is G -continuous at x∈X if and only if it is G -sequentially continuous at x, that is, whenever {x _n } is G -convergent to x, {f(x _n )} is G -convergent to f(x).

Proposition 2.8

(See [[21]]). Let (X,G)be a G -metric space. Then, the function G(x,y,z) is jointly continuous in all three of its variables.

Definition 2.9

(See [21]). A G-metric space (X,G) is called G-complete if every G-Cauchy sequence is G-convergent in (X,G).

Definition 2.10

(See [14]). Let (X,G) be a G-metric space. A map** 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 map**s F and g are said to commute if F(g x,g y)=g(F(x,y)) for all x,y∈X.

Definition 2.12

Let (X,≼) be a partially ordered set, and F:X→X. The map** 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 map** 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,

$$x_1,x_2\in X,g{x}_1\preccurlyeq g{x}_2\Rightarrow F(x_1,y)\preccurlyeq F(x_2,y),$$

and

$$y_1,y_2\in X,g{y}_1\preccurlyeq g{y}_2\Rightarrow F(x,y_1)\succcurlyeq F(x,y_2).$$

If g is identity map** in Definition 2.13, then the map** 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 map**s F:X×X→X and g:X→X if F(x,y)=g x and F(y,x)=g y. If g x=x and g y=y, then (x,y)∈X×X is called a coupled common fixed point.

If g is identity map** 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 on X such that there exist two elements x₀,y₀∈X with g(x₀)≼F(x₀,y₀) and g(y₀)≽F(y₀,x₀). Suppose that there exist non-negative real numbers α,β and L with α+β<1 such that

$$\begin{array}{l}G\left(F\right(x,y),F(u,v),F(w,z\left)\right)\\ \le & \mathrm{\alpha G}(\mathit{\text{gx}},\mathit{\text{gu}},\mathit{\text{gw}})+\mathrm{\beta G}(\mathit{\text{gy}},\mathit{\text{gv}},\mathit{\text{gz}})+Lmin\left\{G\right(F(x,y),\\ \mathit{\text{gu}},\mathit{\text{gw}}),G(F(u,v),\mathit{\text{gx}},\mathit{\text{gw}}),\\ G\left(F\right(w,z),\mathit{\text{gx}},\mathit{\text{gu}}),G\left(F\right(x,y),\mathit{\text{gx}},\mathit{\text{gx}}),G\left(F\right(u,v),\\ \mathit{\text{gu}},\mathit{\text{gu}}),G(F(w,z),\mathit{\text{gw}},\mathit{\text{gw}}\left)\right\}\end{array}$$

(3.1)

for all x,y,u,v,w,z∈X with g x≽g u≽g w and g y≼g v≼g z, either g u≠g w or g v≠g z. 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)=g x and g y=F(y,x), that is, F and g have a coupled coincidence point (x,y)∈X×X.

Proof

Let x₀,y₀∈X be such that g x₀≼F(x₀,y₀) and g y₀≽F(y₀,x₀). Since F(X×X)⊆g(X), we can construct sequences {x _n } and {y _n } in X such that

$$g{x}_{n+1}=F(x_n,y_n)\text{and}g{y}_{n+1}=F(y_n,x_n),\forall n\ge 0.$$

(3.2)

If there exists $n^{\star }\in \mathbb{N}$ such that $g{x}_{n^{\star }-1}=g{x}_{n^{\star }}$ and $g{y}_{n^{\star }-1}=g{y}_{n^{\star }}$, then $g{x}_{n^{\star }-1}=F(x_{n^{\star }-1},y_{n^{\star }-1})$ and $g{y}_{n^{\star }-1}=F(y_{n^{\star }-1},x_{n^{\star }-1})$ that is a point $(x_{n^{\star }-1},y_{n^{\star }-1})\in X\times X$ is a coupled coincidence point of F and g. Thus, we may assume that g x_n−1≠g x _n or g y_n−1≠g y _n for all $n\in \mathbb{N}$.

Next, we claim that for all n≥0,

$$g{x}_n\preccurlyeq g{x}_{n+1},$$

(3.3)

and

$$g{y}_n\succcurlyeq g{y}_{n+1}.$$

(3.4)

We shall use the mathematical induction. Let n=0. Since g x₀≼F(x₀,y₀) and g y₀≽F(y₀,x₀), in view of g x₁=F(x₀,y₀) and g y₁=F(y₀,x₀), we have g x₀≼g x₁ and g y₀≽g y₁, that is, (3.3) and (3.4) hold for n=0. Suppose (3.3) and (3.4) hold for some n≥0. As F has the mixed g-monotone property, and g x _n ≼g x_n+1 and g y _n ≽g y_n+1, from (3.2), we get

$$\begin{array}{ll}g{x}_{n+1}& =F(x_n,y_n)\preccurlyeq F(x_{n+1},y_n)\preccurlyeq F(x_{n+1},y_{n+1})\\ =g{x}_{n+2},\end{array}$$

(3.5)

and

$$\begin{array}{ll}g{y}_{n+1}& =F(y_n,x_n)\succcurlyeq F(y_{n+1},x_n)\succcurlyeq F(y_{n+1},x_{n+1})\\ =g{y}_{n+2}.\end{array}$$

(3.6)

Now, from (3.5) and (3.6), we obtain that g x_n+1≼g x_n+2 and g y_n+1≽g y_n+2. Thus, by the mathematical induction, we conclude that (3.3) and (3.4) hold for all n≥0. Therefore,

$$g{x}_0\preccurlyeq g{x}_1\preccurlyeq g{x}_2\preccurlyeq \dots \preccurlyeq g{x}_n\preccurlyeq g{x}_{n+1}\preccurlyeq \dots$$

(3.7)

and

$$g{y}_0\succcurlyeq g{y}_1\succcurlyeq g{y}_2\succcurlyeq \dots \succcurlyeq g{y}_n\succcurlyeq g{y}_{n+1}\succcurlyeq \dots .$$

(3.8)

Since g x _n ≽g x_n−1 and g y _n ≼g y_n−1, where g x _n ≠g x_n−1 or g y _n ≠g y_n−1 for all $n\in \mathbb{N}$, from (3.1) and (3.2), we have

$$\begin{array}{ll}G(g{x}_{n+1},g{x}_{n+1},g{x}_n)& =G\left(F\right(x_n,y_n),F(x_n,y_n),F(x_{n-1},y_{n-1}\left)\right)\\ \le \mathrm{\alpha G}(g{x}_n,g{x}_n,g{x}_{n-1})+\mathrm{\beta G}(g{y}_n,g{y}_n,g{y}_{n-1})\\ +Lmin\left\{G\right(F(x_n,y_n),g{x}_n,g{x}_{n-1}),\\ G\left(F\right(x_n,y_n),g{x}_n,g{x}_{n-1}),G\left(F\right(x_{n-1},y_{n-1}),\\ g{x}_n,g{x}_n),G(F(x_n,y_n),g{x}_n,g{x}_n),\\ G\left(F\right(x_n,y_n),g{x}_n,g{x}_n),G\left(F\right(x_{n-1},y_{n-1}),\\ g{x}_{n-1},g{x}_{n-1}\left)\right\}\end{array}$$

(3.9)

which implies that G(g x_n+1,g x_n+1,g x _n )≤α G(g x _n ,g x _n , g x_n−1)+β G(g y _n ,g y _n ,g y_n−1). Similarly, we have G(g y_n+1,g y_n+1,g y _n )≤α G(g y _n ,g y _n ,g y_n−1)+β G(g x _n , g x _n ,g x_n−1). Hence, G(g x_n+1,g x_n+1,g x _n )+G(g y_n+1, g y_n+1,g y _n )≤(α+β)(G(g x _n ,g x _n ,g x_n−1)+G(g y _n ,g y _n , g y_n−1)). Set {ϱ _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 )} and δ=α+β<1, we have

$$0\le {\varrho }_n\le \delta {\varrho }_{n-1}\le {\delta }^2{\varrho }_{n-2}\le \dots \le {\delta }^n{\varrho }_0.$$

Now, we shall prove that {g x _n } and {g y _n } are G-Cauchy sequences. For each m>n, we have

$$\begin{array}{l}G(g{x}_n,g{x}_m,g{x}_m)\le G(x_n,x_{n+1},x_{n+1})\\ +G(x_{n+1},x_{n+2},x_{n+2})+\dots +G(g{x}_{m-1},g{x}_m,g{x}_m)\end{array}$$

and

$$\begin{array}{l}G(g{y}_n,g{y}_m,g{y}_m)\le G(y_n,y_{n+1},y_{n+1})\\ +G(y_{n+1},y_{n+2},y_{n+2})+\dots +G(g{y}_{m-1},g{y}_m,g{y}_m).\end{array}$$

There fore,

$$\begin{array}{ll}G(g{x}_n,g{x}_m,g{x}_m)+G(g{y}_n,g{y}_m,g{y}_m)& \le {\varrho }_n+{\varrho }_{n+1}+\dots +\\ {\varrho }_{m-1}\\ \le ({\delta }^n+{\delta }^{n+1}+\dots +\\ {\delta }^{m-1}){\varrho }_0\\ \le \frac{{\delta }^n}{1-\delta }{\varrho }_0\end{array}$$

which implies that limm,n→∞[G(g x _n ,g x _m ,g x _m )+G(g y _n , g y _m ,g y _m )]=0. Therefore, {g x _n } and {g y _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 {g x _n } and {g y _n } are respectively G-convergent to x and y.

Using continuity of g, we get

$$\underset{n\to \infty }{lim}g\left(g{x}_n\right)=g\left(\underset{n\to \infty }{lim}g{x}_n\right)=\mathit{\text{gx}}$$

and

$$\underset{n\to \infty }{lim}g\left(g{y}_n\right)=g\left(\underset{n\to \infty }{lim}g{y}_n\right)=\mathrm{gy.}$$

Since g x_n+1=F(x _n ,y _n ) and g y_n+1=F(y _n ,x _n ), hence the commutativity of F and g yields that F(g x _n ,g y _n )=g F(x _n ,y _n )=g(g x_n+1) and F(g y _n ,g x _n )=g F(y _n ,x _n )=g(g y_n+1).

Now, we show that F(x,y)=g x and F(y,x)=g y.

The map** F is continuous, so since the sequences {g x _n } and {g y _n } are respectively G-convergent to x and y; hence, using Definition 2.10, the sequence {F(g x _n ,g y _n )} is G-convergent to F(x,y). Therefore, {g(g x_n+1)} is G-convergent to F(x,y). By uniqueness of the limit, we have F(x,y)=g x. Similarly, we can show that F(y,x)=g y. 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:

Definition 3.2

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 map**s on X, such that F has the mixed g -monotone property on X such that there exist two elements x₀,y₀∈X with g(x₀)≼F(x₀,y₀) and g(y₀)≽F(y₀,x₀). 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 g x≽g u≽g w and g y≼g v≼g z, where either g u≠g w or g v≠g z. 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 g y=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 {g x _n } and {g y _n } in the complete G-metric space (g(X),G). Then, there exist x,y∈X such that g x _n →g x and g y _n →g y as n→∞. Since {g x _n } is non-decreasing and {g y _n } is non-increasing, using the regularity of (X,G,≼), we have g x _n ≼g x and g y≼g y _n for all n≥0. If $g{x}_{n^{\star }}=\mathit{\text{gx}}$ and $g{y}_{n^{\star }}=\mathit{\text{gy}}$ for some n^⋆≥0, then $\mathit{\text{gx}}=g{x}_{n^{\star }}\preccurlyeq g{x}_{n^{\star }+1}\preccurlyeq \mathit{\text{gx}}=g{x}_{n^{\star }}$ and $\mathit{\text{gy}}\preccurlyeq g{y}_{n^{\star }+1}\preccurlyeq g{y}_{n^{\star }}=\mathit{\text{gy}}$, which implies that

$$g{x}_{n^{\star }}=g{x}_{n^{\star }+1}=F(x_{n^{\star }},y_{n^{\star }})$$

and

$$g{y}_{n^{\star }}=g{y}_{n^{\star }+1}=F(y_{n^{\star }},x_{n^{\star }}),$$

that is, $(x_{n^{\star }},y_{n^{\star }})$ is a coupled coincidence point of F and g. Therefore, we suppose that g x _n ≠g x or g y _n ≠g y for all n≥0. Using rectangle inequality, commutativity, and (3.1), we have

$$\begin{array}{ll}G(g{x}_{n+1},g{x}_{n+1},F(x,y\left)\right)& =G\left(F\right(x_n,y_n),F(x_n,y_n),F(x,y\left)\right)\\ \le \mathrm{\alpha G}(g{x}_n,g{x}_n,\mathit{\text{gx}})+\mathrm{\beta G}(g{y}_n,g{y}_n,\mathit{\text{gy}})\\ +Lmin\left\{G\right(g{x}_{n+1},g{x}_n,\mathit{\text{gx}}),\\ G(g{x}_{n+1},g{x}_n,\mathit{\text{gx}}),G\left(F\right(x,y),g{x}_n,g{x}_n),\\ G(g{x}_{n+1},g{x}_n,\mathit{\text{gx}}),\\ G(g{x}_{n+1},g{x}_n,\mathit{\text{gx}}),G\left(F\right(x,y),\mathit{\text{gx}},\mathit{\text{gx}})\}.\end{array}$$

(3.10)

Taking n→∞, we get G(g x,g x,F(x,y))=0 and hence g x=F(x,y). Similarly, one can show that g y=F(y,x). Thus F and g have a coupled coincidence point. □

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) [4]. From the condition that either g u≠g w or g v≠g z, 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.

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 map**s on X such that F has the mixed g -monotone property on X such that there exist two elements x₀,y₀∈X with g(x₀)≼F(x₀,y₀) and g(y₀)≽F(y₀,x₀). Suppose that there exist non-negative real numbers α,β with α+β<1 such that

$$G\left(F\right(x,y),F(u,v),F(w,z\left)\right)\le \mathrm{\alpha G}(\mathit{\text{gx}},\mathit{\text{gu}},\mathit{\text{gw}})+\mathrm{\beta G}(\mathit{\text{gy}},\mathit{\text{gv}},\mathit{\text{gz}}),$$

(3.11)

for all x,y,u,v,w,z∈X with g x≽g u≽g w and g y≼g v≼g z, where either g u≠g w or g v≠g z. 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)=g x and g y=F(y,x), that is, F and g have a coupled coincidence point (x,y)∈X×X.

Corollary 3.5

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 map**s on X such that F has the mixed g -monotone property on X such that there exist two elements x₀,y₀∈X with g(x₀)≼F(x₀,y₀) and g(y₀)≽F(y₀,x₀). 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 g x≽g u≽g w and g y≼g v≼g z, where either g u≠g w or g v≠g z. 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 g y=F(y,x), that is, F and g have a coupled coincidence point (x,y)∈X×X.

Taking $\alpha =\beta =\frac{k}{2}$, 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 x₀,y₀∈X with g(x₀)≼F(x₀,y₀) and g(y₀)≽F(y₀,x₀). Suppose that there exists k∈[0,1) such that

$$G\left(F\right(x,y),F(u,v),F(z,w\left)\right)\le \frac{k}{2}\left(G\right(\mathit{\text{gx}},\mathit{\text{gu}},\mathit{\text{gw}})+G(\mathit{\text{gy}},\mathit{\text{gv}},\mathit{\text{gz}}\left)\right),$$

(3.12)

for all x,y,u,v,w,z∈X with g x≽g u≽g w and g y≼g v≼g z, where either g u≠g w or g v≠g z. 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)=g x and g y=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 map**s on X such that F has the mixed g -monotone property on X such that there exist two elements x₀,y₀∈X with g(x₀)≼F(x₀,y₀) and g(y₀)≽F(y₀,x₀). Suppose that there exists k∈[0,1) such that (3.12) satisfies for all x,y,u,v,w,z∈X with g x≽g u≽g w and g y≼g v≼g z, where either g u≠g w or g v≠g z. 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 g y=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:

$$\text{for}(x,y),(u,v)\in X\times X,(u,v)\preccurlyeq (x,y)\iff x\preccurlyeq u,y\succcurlyeq \mathrm{v.}$$

Theorem 3.8

In addition to the hypotheses of Theorem 3.1, suppose that for every (x,y),(z,t)∈X×X, there exists (u,v)∈X×X such that (F(u,v),F(v,u)) is comparable to (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=g x=F(x,y) and y=g y=F(y,x).

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, g x=F(x,y), g y=F(y,x), g z=F(z,t), and g t=F(t,z). We shall show that g x=g z and g y=g t. 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 u₀=u and v₀=v, and choose u₁,v₁∈X so that g u₁=F(u₀,v₀) and g v₁=F(v₀,u₀). Then, similarly as in the proof of Theorem 3.1, we can inductively define sequences {g u _n } and {g v _n } as g u_n+1=F(u _n ,v _n ) and g v_n+1=F(v _n ,u _n ) for all n. Further, set x₀=x, y₀=y, z₀=z, and t₀=t and on the same way define the sequences {g x _n } and {g y _n }, and {g z _n } and {g t _n }. Since (F(x,y),F(y,x))=(g x₁,g y₁)=(g x,g y) and (F(u,v),F(v,u))=(g u₁,g v₁) are comparable, then g x≽g u₁ and g y≼g v₁. Now, we shall show that (g x,g y) and (g u _n ,g v _n ) are comparable, that is, g x≽g u _n and g y≼g v _n for all n. Suppose that it holds for some n≥0, then by the mixed g-monotone property of F, we have g u_n+1=F(u _n ,v _n )≼F(x,y)=g x and g v_n+1=F(v _n ,u _n )≽F(y,x)=g y. Hence, g x≽g u _n and g y≼g v _n hold for all n. Thus, from (3.1), we have

$$\begin{array}{ll}G(\mathit{\text{gx}},\mathit{\text{gx}},g{u}_{n+1})& =G\left(F\right(x,y),F(x,y),F(u_n,v_n\left)\right)\\ \le \mathrm{\alpha G}(\mathit{\text{gx}},\mathit{\text{gx}},g{u}_n)+\mathrm{\beta G}(\mathit{\text{gy}},\mathit{\text{gy}},g{v}_n)\\ +Lmin\left\{G\right(F(x,y),\mathit{\text{gx}},g{u}_n),\\ G\left(F\right(x,y),\mathit{\text{gx}},g{u}_n),\\ G\left(F\right(u_n,v_n),\mathit{\text{gx}},\mathit{\text{gx}}),\\ G\left(F\right(x,y),\mathit{\text{gx}},\mathit{\text{gx}}),\\ G\left(F\right(x,y),\mathit{\text{gx}},\mathit{\text{gx}}),\\ G\left(F\right(u_n,v_n),g{u}_n,g{u}_n)\}\end{array}$$

(3.13)

which implies that G(g x,g x,g u_n+1)≤α G(g x,g x,g u _n )+β G(g y,g y,g v _n ). Similarly, we can prove that G(g y,g y, g v_n+1)≤α G(g y,g y,g v _n )+β G(g x,g x,g u _n ). Hence,

$$\begin{array}{ll}G(\mathit{\text{gx}},\mathit{\text{gx}},g{u}_{n+1})+G(\mathit{\text{gy}},g{v}_{n+1})& \le (\alpha +\beta )\left[G\right(\mathit{\text{gx}},\mathit{\text{gx}},g{u}_n)\\ +G(\mathit{\text{gy}},\mathit{\text{gy}},g{v}_n)]\\ \le {(\alpha +\beta )}^2\left[G\right(\mathit{\text{gx}},\mathit{\text{gx}},g{u}_{n-1})\\ +G(\mathit{\text{gy}},\mathit{\text{gy}},g{v}_{n-1})]\\ \dots \\ \le {(\alpha +\beta )}^{n+1}\left[G\right(\mathit{\text{gx}},\mathit{\text{gx}},g{u}_0)\\ +G(\mathit{\text{gy}},\mathit{\text{gyg}}{v}_0)].\end{array}$$

On taking limit, n→∞, we get

$$\underset{n\to \infty }{lim}\left[G\right(\mathit{\text{gx}},\mathit{\text{gx}},g{u}_{n+1})+G(\mathit{\text{gy}},\mathit{\text{gy}},g{v}_{n+1}\left)\right]=0.$$

(3.14)

Thus, limn→∞ G(g x,g x,g u_n+1)=0 and limn→∞ G(g y,g y,g v_n+1)=0. Similarly, we can prove that limn→∞ G(g z,g z,g u _n )=0= limn→∞ G(g t,g t,g v _n ). Hence,

$$\mathit{\text{gx}}=\mathit{\text{gz}}\text{and}\mathit{\text{gy}}=\mathrm{gt.}$$

(3.15)

Since g x=F(x,y) and g y=F(y,x), by the commutativity of F and g, we have

$$\begin{array}{ll}\hfill g\left(g\right(x\left)\right)=g\left(F\right(x,y\left)\right)=F(\mathit{\text{gx}},\mathit{\text{gy}}),\text{and}g\left(\mathit{\text{gy}}\right)& =g\left(F\right(y,x\left)\right)\\ =F(\mathit{\text{gy}},\mathit{\text{gx}}).\end{array}$$

(3.16)

Denote g x=p and g y=q. Then, g p=F(p,q) and g q=F(q,p). Thus, (p,q) is a coupled coincidence point. Then, from (3.15), with z=p and t=q, it follows that g p=g x and g q=g y, that is, g p=p and g q=q. Hence, p=g p=F(p,q) and q=g q=F(q,p). Therefore, (p,q) is a coupled common fixed point of F and g. To prove the uniqueness, assume that (r,s) is another coupled common fixed point. Then, by (3.15), we have r=g r=g p=p and s=g s=g q=q. Hence, we get the result. □

Finally, we provide some examples to illustrate our obtained Theorem 3.1.

Example 3.9

Let $X=\mathbb{R}$ be a set endowed with order x≼y⇔x≤y. Let the map** $G:X\times X\times X\to {\mathbb{R}}_{+}$ 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 map** F:X×X→X and g:X→X by

$$F(x,y)=\frac{x-2y}{8}\text{for all}(x,y)\in X\times X$$

and

$$\mathit{\text{gx}}=\frac{x}{2}\text{for all}x\in \mathrm{X.}$$

Then, the following properties hold:

• (1) F and g are continuous;

• (2) F has a mixed g-monotone property;

• (3) there exist x ₀,y ₀∈X with g(x ₀)≼F(x ₀,y ₀) and g(y ₀)≽F(y ₀,x ₀);

• (4) F satisfies condition (3.1). Indeed, we show that F satisfies condition (3.1).

For all x,y,u,v,w,z∈X, we have

$$\begin{array}{l}G\left(F\right(x,y),F(u,v),F(w,z\left)\right)\\ =\left|\frac{x-2y}{8}-\frac{u-2v}{8}\right|+\left|\frac{u-2v}{8}-\frac{w-2z}{8}\right|\\ +\left|\frac{w-2z}{8}-\frac{x-2y}{8}\right|\\ \le \frac{1}{4}\left(\left|\frac{x}{2}-\frac{u}{2}\right|+\left|\frac{u}{2}-\frac{w}{2}\right|+\left|\frac{w}{2}-\frac{x}{2}\right|\right)\\ +\frac{2}{4}\left(\left|\frac{y}{2}-\frac{v}{2}\right|+\left|\frac{v}{2}-\frac{z}{2}\right|+\left|\frac{z}{2}-\frac{y}{2}\right|\right)\\ =\frac{1}{4}G(\mathit{\text{gx}},\mathit{\text{gu}},\mathit{\text{gw}})+\frac{1}{2}G(\mathit{\text{gy}},\mathit{\text{gv}},\mathit{\text{gz}})\\ \le \frac{1}{4}G(\mathit{\text{gx}},\mathit{\text{gu}},\mathit{\text{gw}})+\frac{1}{2}G(\mathit{\text{gy}},\mathit{\text{gv}},\mathit{\text{gz}})\\ +Lmin\left\{G\right(F(x,y),\mathit{\text{gu}},\mathit{\text{gw}}),G(F(u,v),\mathit{\text{gx}},\mathit{\text{gw}}),\\ G\left(F\right(w,z),\mathit{\text{gx}},\mathit{\text{gu}}),G\left(F\right(x,y),\mathit{\text{gx}},\mathit{\text{gx}}),\\ G\left(F\right(u,v),\mathit{\text{gu}},\mathit{\text{gu}}),G\left(F\right(w,z),\mathit{\text{gw}},\mathit{\text{gw}})\}\end{array}$$

for all L≥0. Hence, F satisfies condition (3.1) for $\alpha =\frac{1}{4},\beta =\frac{1}{2}$ and for each L≥0.

• (5) F(X×X)⊆g(X) and g(X) is a G-complete subspace of X;

• (6) F and g are commutes. Indeed, we have

  $$F(\mathit{\text{gx}},\mathit{\text{gy}})=F(\frac{x}{2},\frac{y}{2})=\frac{\frac{x}{2}-y}{8}=\frac{\frac{x-2y}{8}}{2}=g\left(F\right(x,y\left)\right)$$

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.