Coupled fixed-point theorems for contraction mapping induced by cone ball-metric in partially ordered spaces

Abstract

Recently, Chen et al. (Appl. Math. Lett. 25:692-697, 2012) introduced the concept of the cone ball-metric and studied the common fixed-point theorems for the stronger Meir-Keeler cone-type function in cone ball-metric spaces. The purpose of this paper is to establish the coupled fixed-point theorems for nonlinear contraction mappings, which have a mixed monotone property by using the cone ball-metric. Also, we give some examples to validate our main results. At the end of this paper, we give an open problem for further investigation.

MSC:47H10, 54H25.

1 Introduction

Fixed-point theory has been the most attractive topic to hundreds of researchers since 1922 with the celebrated Banach’s contraction principle [11]. This principle provides a technique for solving a variety of applied problems in various branches of mathematics. Moreover, it provides the applications in many fields such as chemistry, biology, statistics, economics, computer science, and engineering. The Banach’s contraction principle has been extended and improved by many mathematicians (see [7, 9, 13, 15, 24, 31–34] and others).

In 2004, the Banach’s contraction principle was extended to metric spaces endowed with a partial ordering by Ran and Reurings [26]. Afterward, many generalizations and applications of the work of Ran and Reurings exist in the literature (see in [6, 17, 25]). For example, Nieto and Rodríguez-López [25] extended results of Ran and Reurings for nondecreasing mappings and studied a unique solution for a first-order ordinary differential equation with periodic boundary conditions.

In 2006, Bhaskar and Lakshmikantham [12] first introduced the concept of the mixed monotone property. Furthermore, they proved some coupled fixed-point theorems for mapping that satisfy the mixed monotone property and give some applications in the existence and uniqueness of a solution for a periodic boundary value problem

$$u^{\prime }(t)=f(t,u(t)),t\in [0,T],u(0)=u(T)$$

(1.1)

where the function f satisfies certain conditions. Afterward, several authors studied and extended coupled fixed-point theorems of Bhaskar and Lakshmikantham [12] to different generalized condition (see, e.g., [4, 5, 8, 22, 23, 28, 30, 35, 36]).

On the other hand, the concept of cone metric spaces is a generalization of metric spaces, where each pair of points is assigned to a member of a real Banach space with a cone. This cone naturally induces a partial order in the Banach spaces. The concept of the cone metric space was reintroduced in the work of Huang and Zhang [18] where they also established the Banach’s contraction mapping principle in this space. Afterward, several authors have studied fixed point and coupled fixed-point problems in cone metric spaces. Some of these works are noted in [1–3, 10, 20, 21, 29, 38]. Recently, Chen et al.[14] introduced the concept of cone ball-metric spaces and proved some fixed- point theorems in these spaces for mappings satisfying a contraction involving a stronger Meir-Keeler cone-type function.

Motivated by the interesting concept of cone ball-metric spaces of Chen et al.[14], in this paper, we establish some coupled fixed-point theorems for a contraction mapping induced by the cone ball-metric in partially ordered spaces and also study the condition claim of the uniqueness of a coupled fixed point. An open problem is also given at the end for further investigation.

2 Preliminaries

In this section, we shall recall some definitions and mathematical preliminaries.

Definition 2.1 Recall that a binary relation ⪯ on a nonempty set X is said to be an order relation (and X equipped with ⪯ is called a partially ordered set) if it satisfies the following three properties:

• (i) reflexivity: $x⪯x$ for all $x\in X$,

• (ii) antisymmetry: $x⪯y$ and $y⪯x$ imply $x=y$,

• (iii) transitivity: $x⪯y$ and $y⪯z$ imply $x⪯z$.

Throughout this paper $(X,⪯)$ denotes a partially ordered set. By $x⪰y$ holds, we mean that $y⪯x$ holds and by $x\prec y$ holds we mean that $x⪯y$ holds, but $x\ne y$. If $(X,⪯)$ is a partially ordered set and $f:X\to X$ is such that, for all $x,y\in X$, $x⪯y$ implies $f(x)⪯f(y)$, then a mapping f is said to be nondecreasing. Similarly, a nonincreasing mapping is also defined.

Definition 2.2 ([12])

Let $(X,⪯)$ be a partial ordered set and $F:X\times X\to X$ be a mapping. The mapping F is said to has the mixed monotone property if F is monotone nondecreasing in its first argument and is monotone nonincreasing in its second argument, that is, for any $x,y\in X$,

$$x_1,x_2\in X,x_1⪯x_2⟹F(x_1,y)⪯F(x_2,y)$$

(2.1)

and

$$y_1,y_2\in X,y_1⪯y_2⟹F(x,y_1)⪰F(x,y_2).$$

(2.2)

Definition 2.3 ([12])

Let X be a nonempty set. An element $(x,y)\in X\times X$ is called a coupled fixed point of the mapping $F:X\times X\to X$ if $x=F(x,y)$ and $y=F(y,x)$.

Next, we give some notations and lemmas of cone metric spaces which are reintroduced by Huang and Zhang [18].

Let E be a real Banach space and $0_E$ denote the zero element in E. A cone P is a subset of E such that

($C_1$) P is nonempty closed and $P\ne \{0_E\}$;

($C_2$) if a, b are nonnegative real numbers and $x,y\in P$, then $ax+by\in P$;

($C_3$) $P\cap (-P)=\{0_E\}$.

For any cone $P\subset E$, the partial ordering ${\le }_E$ with respect to P defined by $x{\le }_{E}y$ if and only if $y-x\in P$. We write $x{<}_{E}y$ to indicate that $x{\le }_{E}y$, but $x\ne y$, while $x\ll y$ stands for $y-x\in int(P)$, where $int(P)$ denotes the interior of P.

A cone P is said to be normal if there is a number $K>0$ such that, for all $x,y\in E$,

$$0_E{\le }_{E}x{\le }_{E}y⟹\parallel x\parallel \le K\parallel y\parallel .$$

The least positive number satisfying above is called the normal constant of P.

The cone P is said to be regular if every increasing sequence which is bounded from above is convergent, that is, if $\{x_n\}$ is a sequence in E such that

$$x_1{\le }_E{x}_2{\le }_E\cdots {\le }_{E}y$$

for some $y\in E$, then there is $x\in E$ such that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x\parallel =0$. Equivalently, the cone P is said to be regular if every decreasing sequence which is bounded from below is convergent. It is well known that a regular cone is a normal cone (see also [27]).

Remark 2.4 ([18])

• (1) If E be a real Banach space with a cone P in E and $a{\le }_{E}ka$, where $a\in E$ and $0<k<1$, then $a=0_E$.

• (2) If $c\in int(P)$, $0_E{\le }_E{a}_n$ and $a_n\to 0_E$, then there exists a positive integer N such that $a_n\ll c$ for all $n\ge N$.

Lemma 2.5 ([21])

If E be a real Banach space with a cone P in E, then we have the following:

• (1) If $0_E{\le }_{E}x{\le }_{E}y$ and k is a nonnegative real number, then $0_E{\le }_{E}kx{\le }_{E}ky$.

• (2) If $0_E{\le }_E{x}_n{\le }_E{y}_n$ for all $n\in \mathbb{N}$ and $x_n\to x$, $y_n\to y$ as $n\to \mathrm{\infty }$, then $0_E{\le }_{E}x{\le }_{E}y$.

Lemma 2.6 ([19])

If E be a real Banach space with a cone P in E, then we have the following: for all$a,b,c\in E$,

• (1) If $a{\le }_{E}b$ and $b\ll c$, then $a\ll c$.

• (2) If $a\ll b$ and $b\ll c$, then $a\ll c$.

Using the notation of a cone, we have following definitions of cone metric space.

Definition 2.7 ([18])

Let X be a nonempty set and E be a real Banach space equipped with the partial ordering ${\le }_E$ with respect to the cone $P\subset E$. Suppose that the mapping $d:X\times X\to E$ satisfies the following conditions:

($C{M}_1$) $0_E{<}_{E}d(x,y)$ for all $x,y\in X$ with $x\ne y$ and $d(x,y)=0_E$ if and only if $x=y$;

($C{M}_2$) $d(x,y)=d(y,x)$ for all $x,y\in X$;

($C{M}_3$) $d(x,y){\le }_{E}d(x,z)+d(z,y)$ for all $x,y,z\in X$.

Then d is called a cone metric on X and $(X,d)$ is called a cone metric space.

For other basic properties on a cone metric space, the reader can refer to [18].

Next, we give the concept of a cone ball-metric space introduced by Chen et al.[14] and its properties.

In the following, we always suppose that E is a real Banach space endowed with a cone P with the apex at the origin $0_E$, $int(P)\ne \mathrm{\varnothing }$ and a linear ordering ${\le }_E$ with respect to P.

Definition 2.8 ([14])

Let $(X,d)$ be a cone metric space. A cone ball-metric with respect to the cone metric d is a function $\mathcal{B}:X\times X\times X\to E$ defined by

$$\mathcal{B}(x,y,z)=inf\{2\gamma :\{x,y,z\}\subset B_{\gamma }\},$$

where

$$B_{\gamma }(x):=\{y\in X:d(x,y)<\gamma \}$$

for $x\in X$ is a ball in X with the center x and radius $\gamma \gg 0_E$. The ordered pair $(X,\mathcal{B})$ is called a cone ball-metric space.

Proposition 2.9 ([14])

If$(X,\mathcal{B})$is a cone ball-metric space, then the following statements hold:

(${\mathcal{B}}_1$) $\mathcal{B}(x,y,z)=0_E$if$x=y=z$;

(${\mathcal{B}}_2$) $\mathcal{B}(x,x,y)>0_E$for all$x,y\in X$with$x\ne y$;

(${\mathcal{B}}_3$) $\mathcal{B}(x,x,y)\le \mathcal{B}(x,y,z)$for all$x,y,z\in X$;

(${\mathcal{B}}_4$) $\mathcal{B}(x,y,z)=\mathcal{B}(x,z,y)=\mathcal{B}(y,z,x)=\cdots$ (symmetry in all three variables);

(${\mathcal{B}}_5$) $\mathcal{B}(x,y,z)\le \mathcal{B}(x,w,w)+\mathcal{B}(w,y,z)$for all$x,y,z,w\in X$ (rectangle inequality);

(${\mathcal{B}}_6$) $\mathcal{B}(x,x,y)=\mathcal{B}(x,y,y)$for all$x,y\in X$.

Definition 2.10 ([14])

Let $(X,\mathcal{B})$ be a cone ball-metric space and $\{x_n\}$ be a sequence in X. We say that $\{x_n\}$ is called:

• (1) A Cauchy sequence if, for any $ϵ\in E$ with $0_E\ll ϵ$, there exists a positive integer N such that $\mathcal{B}(x_n,x_m,x_l)\ll ϵ$ for all $n,m,l>N$.

• (2) A convergent sequence if, for any $ϵ\in E$ with $0_E\ll ϵ$, there exists a positive integer N such that, for all $n,m>N$, $\mathcal{B}(x,x_n,x_m)\ll ϵ$ for some $x\in X$. Here, x is called the limit of the sequence $\{x_n\}$ and is denoted by $\underset{n\to \mathrm{\infty }}{lim}{x}_n$ or $x_n\to x$.

Remark 2.11 We can prove easily that $\{x_n\}$ is a Cauchy sequence if and only if, for any $ϵ\in E$ with $0_E\ll ϵ$, there exists a positive integer N such that $\mathcal{B}(x_n,x_m,x_m)\ll ϵ$ for all $n,m>N$.

Definition 2.12 ([14])

A cone ball-metric space $(X,\mathcal{B})$ is said to be complete if every Cauchy sequence is convergent in X.

Proposition 2.13 ([14])

Let$(X,\mathcal{B})$be a cone ball-metric space and$\{x_n\}$be a sequence of points of X. Then the following are equivalent:

• (1) $x_n\to x$ as $n\to \mathrm{\infty }$.

• (2) $\mathcal{B}(x_n,x_n,x)\to 0_E$ as $n\to \mathrm{\infty }$.

• (3) $\mathcal{B}(x_n,x,x)\to 0_E$ as $n\to \mathrm{\infty }$.

• (4) $\mathcal{B}(x_n,x_m,x)\to 0_E$ as $n,m\to \mathrm{\infty }$.

Proposition 2.14 ([14])

Let$(X,\mathcal{B})$be a cone ball-metric space, $\{x_n\}$be a sequence of points of X and$x,y\in X$. If$x_n\to x$and$x_n\to y$as$n\to \mathrm{\infty }$, then$x=y$.

Proposition 2.15 ([14])

Let$(X,\mathcal{B})$be a cone ball-metric space and$\{x_n\}$, $\{y_m\}$and$\{z_l\}$be tree sequences in X. If$x_n\to x$, $y_m\to y$, and$z_l\to z$as$n,m,l\to \mathrm{\infty }$, then$\mathcal{B}(x_n,y_m,z_l)\to \mathcal{B}(x,y,z)$as$n,m,l\to \mathrm{\infty }$.

Definition 2.16 Let $(X,\mathcal{B})$ be a cone ball-metric space. A mapping $F:X\times X\to X$ is said to be continuous if for any two convergent sequences $\{x_n\}$ and $\{y_n\}$ converging to x and y in X, respectively, then $\{F(x_n,y_n)\}$ is convergent to $F(x,y)$.

3 Existence of coupled fixed point in cone ball-metric spaces

Let Δ denote the class of all functions $\delta :(int(P)\cup \{0_E\})\times (int(P)\cup \{0_E\})\to [0,1)$ which satisfies the following condition.

For any sequences $\{a_n\}$ and $\{b_n\}$ in $int(P)\cup \{0_E\}$,

$$\delta (a_n,b_n)\to 1⟹a_n,b_n\to 0_E.$$

Let $E=\mathbb{R}$ be a usual norm space with a cone $P=\{z\in \mathbb{R}|z\ge 0\}$ and $X=\mathbb{R}$ be a cone metric space with a metric $d(x,y)=|x-y|$ for all $x,y\in X$. The following are examples of the functions in Δ under above setting:

• (1) ${\delta }_1(a,b)=k$, where $k\in [0,1)$.

• (2) ${\delta }_2(a,b)=\{\begin{array}{cc}\frac{ln(1+ka+lb)}{ka+lb};\hfill & a>0\text{ or }b>0,\hfill \\ r\in [0,1);\hfill & a=b=0,\hfill \end{array}\text{where }k,l\in [0,1)$.

• (3) ${\delta }_3(a,b)=\{\begin{array}{cc}\frac{ln(1+max\{a,b\})}{max\{a,b\}};\hfill & a>0\text{ or }b>0,\hfill \\ r\in [0,1);\hfill & a=b=0.\hfill \end{array}$

Next, we prove our main theorems.

Theorem 3.1 Let$(X,⪯)$be a partially ordered set and$\mathcal{B}$be a cone ball-metric induced by the cone metric d on X with a regular cone P such that$(X,\mathcal{B})$is a complete cone ball-metric space. Let$F:X\times X\to X$be a continuous mapping having the mixed monotone property on X. Suppose that there exists$\delta \in \mathrm{\Delta }$such that

(3.1)

for all$x,y,z,u,v,w\in X$for which$x⪰u⪰w$and$y⪯v⪯z$. If there exists$x_0,y_0\in X$such that

$$x_0⪯F(x_0,y_0)\mathit{\text{and}}{y}_0⪰F(y_0,x_0),$$

then F has a coupled fixed point, that is, there exist$x,y\in X$such that

$$x=F(x,y)\mathit{\text{and}}y=F(y,x).$$

Proof We construct two sequences $\{x_n\}$, $\{y_n\}$ in X such that, for all $n\ge 0$,

$$x_{n+1}=F(x_n,y_n)\text{and}{y}_{n+1}=F(y_n,x_n).$$

(3.2)

First, by induction, we show that, for all $n\ge 0$,

$$x_n⪯x_{n+1}\text{and}{y}_n⪰y_{n+1}.$$

(3.3)

From $x_0⪯F(x_0,y_0)=x_1$ and $y_0⪰F(y_0,x_0)⪰y_1$, in case $n=0$, (3.3) holds. Assume that (3.3) holds for some $n\ge 0$. Then we get

$$x_n⪯x_{n+1}\text{and}{y}_n⪰y_{n+1}.$$

(3.4)

Since F has the mixed monotone property, it follows from (3.4) and (2.1) that

$$F(x_n,y)⪯F(x_{n+1},y)\text{and}F(y_{n+1},x)⪯F(y_n,x)$$

(3.5)

for all $x,y\in X$. From (3.4) and (2.2), we have

$$F(y,x_n)⪰F(y,x_{n+1})\text{and}F(x,y_{n+1})⪰F(x,y_n),$$

(3.6)

for all $x,y\in X$.

If we take $y=y_n$ and $x=x_n$ in (3.5), we get

$$x_{n+1}=F(x_n,y_n)⪯F(x_{n+1},y_n)\text{and}F(y_{n+1},x_n)⪯F(y_n,x_n)=y_{n+1}.$$

(3.7)

If we take $y=y_{n+1}$ and $x=x_{n+1}$ in (3.6), we get

$$F(y_{n+1},x_n)⪰F(y_{n+1},x_{n+1})=y_{n+2}\text{and}{x}_{n+2}=F(x_{n+1},y_{n+1})⪰F(x_{n+1},y_n).$$

(3.8)

From (3.7) and (3.8), we also have

$$x_{n+1}⪯x_{n+2}\text{and}{y}_{n+1}⪰y_{n+2}.$$

(3.9)

Consequently, by induction, we have (3.3) holds for all $n\ge 0$. This implies

$$x_0⪯x_1⪯x_2⪯\cdots ⪯x_n⪯x_{n+1}⪯\cdots$$

(3.10)

and

$$y_0⪰y_1⪰y_2⪰\cdots ⪰y_n⪰y_{n+1}⪰\cdots .$$

(3.11)

If there exists an integer number $n^{\star }\ge 0$ such that

$$\mathcal{B}(x_{n^{\star }+1},x_{n^{\star }+1},x_{n^{\star }})+\mathcal{B}(y_{n^{\star }+1},y_{n^{\star }+1},y_{n^{\star }})=0_E,$$

then we have

$$\mathcal{B}(x_{n^{\star }+1},x_{n^{\star }+1},x_{n^{\star }})=\mathcal{B}(y_{n^{\star }+1},y_{n^{\star }+1},y_{n^{\star }})=0_E,$$

which implies $x_{n^{\star }}=x_{n^{\star }+1}$ and $y_{n^{\star }}=y_{n^{\star }+1}$. Therefore, $x_{n^{\star }}=F(x_{n^{\star }},y_{n^{\star }})$ and $y_{n^{\star }}=F(y_{n^{\star }},x_{n^{\star }})$ and so $(x_{n^{\star }},y_{n^{\star }})$ is a coupled fixed point of F.

Now, we assume that $\mathcal{B}(x_{n+1},x_{n+1},x_n)+\mathcal{B}(y_{n+1},y_{n+1},y_n){>}_E{0}_E$ for all $n\ge 0$. Since $x_{n-1}⪯x_n$ and $y_{n-1}⪰y_n$ for all $n\in \mathbb{N}$, by (3.1) and (3.2), we have

(3.12)

So, we have the sequence $\{{\mathcal{B}}_n\}$ defined by ${\mathcal{B}}_n:=\mathcal{B}(x_n,x_n,x_{n-1})+\mathcal{B}(y_n,y_n,y_{n-1})$ is a decreasing sequence. Since P is regular, there exists $\mathcal{B}{\ge }_E{0}_E$ such that ${\mathcal{B}}_n\to \mathcal{B}$ as $n\to \mathrm{\infty }$.

Next, we prove that $\mathcal{B}=0_E$. Suppose that $\mathcal{B}{>}_E{0}_E$. From (3.12), letting $n\to \mathrm{\infty }$, we have

$$\delta (\mathcal{B}(x_n,x_n,x_{n-1}),\mathcal{B}(y_n,y_n,y_{n-1}))\to 1.$$

Since $\delta \in \mathrm{\Delta }$, we have $\mathcal{B}(x_n,x_n,x_{n-1})\to 0_E$ and $\mathcal{B}(y_n,y_n,y_{n-1})\to 0_E$ and then

$$\mathcal{B}(x_n,x_n,x_{n-1})+\mathcal{B}(y_n,y_n,y_{n-1})\to 0_E,$$

(3.13)

which contradictions with $\mathcal{B}{>}_E{0}_E$. Consequently, we must get

$$\mathcal{B}(x_n,x_n,x_{n-1})+\mathcal{B}(y_n,y_n,y_{n-1})\to 0_E.$$

By the property (${\mathcal{B}}_6$), we have

$${\mathcal{B}}_n^{\prime }:=\mathcal{B}(x_n,x_{n-1},x_{n-1})+\mathcal{B}(y_n,y_{n-1},y_{n-1})\to 0_E.$$

(3.14)

Now, we show that $\{x_n\}$ and $\{y_n\}$ are Cauchy sequences in cone ball-metric space $(X,\mathcal{B})$. Suppose on the contrary that at least one of $\{x_n\}$ and $\{y_n\}$ are not a Cauchy sequence in $(X,\mathcal{B})$. Then there exists $ϵ\in E$ such that $0_E\ll ϵ$ and sequences of positive integers $\{n(k)\}$ and $\{m(k)\}$ such that for all positive integers k,

$$n(k)>m(k)\ge k$$

and

$$\mathcal{B}(x_{n(k)},x_{m(k)},x_{m(k)})+\mathcal{B}(y_{n(k)},y_{m(k)},y_{m(k)}){>}_Eϵ.$$

(3.15)

Further, for the integer $m(k)$, we can choose $n(k)$ is the smallest integer for which (3.15) holds. Then we have

$$\mathcal{B}(x_{n(k)-1},x_{m(k)},x_{m(k)})+\mathcal{B}(y_{n(k)-1},y_{m(k)},y_{m(k)}){\le }_Eϵ.$$

(3.16)

Using (3.15) and (3.16) and the rectangle inequality, we have

$$\begin{array}{rcl}ϵ& {<}_E& r_k:=\mathcal{B}(x_{n(k)},x_{m(k)},x_{m(k)})+\mathcal{B}(y_{n(k)},y_{m(k)},y_{m(k)})\\ {\le }_E& \mathcal{B}(x_{n(k)},x_{n(k)-1},x_{n(k)-1})+\mathcal{B}(x_{n(k)-1},x_{m(k)},x_{m(k)})\\ +\mathcal{B}(y_{n(k)},y_{n(k)-1},y_{n(k)-1})+\mathcal{B}(y_{n(k)-1},y_{m(k)},y_{m(k)})\\ {\le }_E& \mathcal{B}(x_{n(k)},x_{n(k)-1},x_{n(k)-1})+\mathcal{B}(y_{n(k)},y_{n(k)-1},y_{n(k)-1})+ϵ.\end{array}$$

Letting $k\to \mathrm{\infty }$, we have

$$r_k=\mathcal{B}(x_{n(k)},x_{m(k)},x_{m(k)})+\mathcal{B}(y_{n(k)},y_{m(k)},y_{m(k)})\to ϵ.$$

(3.17)

By the rectangle inequality, we get

$$\begin{array}{rcl}{r}_k& =& \mathcal{B}(x_{n(k)},x_{m(k)},x_{m(k)})+\mathcal{B}(y_{n(k)},y_{m(k)},y_{m(k)})\\ {\le }_E& \mathcal{B}(x_{n(k)},x_{n(k)+1},x_{n(k)+1})+\mathcal{B}(x_{n(k)+1},x_{m(k)+1},x_{m(k)+1})+\mathcal{B}(x_{m(k)+1},x_{m(k)},x_{m(k)})\\ +\mathcal{B}(y_{n(k)},y_{n(k)+1},y_{n(k)+1})+\mathcal{B}(y_{n(k)+1},y_{m(k)+1},y_{m(k)+1})+\mathcal{B}(y_{m(k)+1},y_{m(k)},y_{m(k)})\\ =& [\mathcal{B}(x_{n(k)+1},x_{m(k)+1},x_{m(k)+1})+\mathcal{B}(y_{n(k)+1},y_{m(k)+1},y_{m(k)+1})]\\ +[\mathcal{B}(x_{n(k)},x_{n(k)+1},x_{n(k)+1})+\mathcal{B}(y_{n(k)},y_{n(k)+1},y_{n(k)+1})]\\ +[\mathcal{B}(x_{m(k)+1},x_{m(k)},x_{m(k)})+\mathcal{B}(y_{m(k)+1},y_{m(k)},y_{m(k)})]\\ =& [\mathcal{B}(x_{n(k)+1},x_{m(k)+1},x_{m(k)+1})+\mathcal{B}(y_{n(k)+1},y_{m(k)+1},y_{m(k)+1})]+{\mathcal{B}}_{n(k)+1}+{\mathcal{B}}_{m(k)+1}^{\prime }\\ =& [\mathcal{B}(x_{m(k)+1},x_{m(k)+1},x_{n(k)+1})+\mathcal{B}(y_{m(k)+1},y_{m(k)+1},y_{n(k)+1})]+{\mathcal{B}}_{n(k)+1}+{\mathcal{B}}_{m(k)+1}^{\prime }\\ =& [\mathcal{B}(F(x_{m(k)},y_{m(k)}),F(x_{m(k)},y_{m(k)}),F(x_{n(k)},y_{n(k)}))\\ +\mathcal{B}(F(y_{m(k)},x_{m(k)}),F(y_{m(k)},x_{m(k)}),F(y_{n(k)},x_{n(k)}))]+{\mathcal{B}}_{n(k)+1}+{\mathcal{B}}_{m(k)+1}^{\prime }\\ {\le }_E& \delta (\mathcal{B}(x_{m(k)},x_{m(k)},x_{n(k)}),\mathcal{B}(y_{m(k)},y_{m(k)},y_{n(k)}))\\ \times (\mathcal{B}(x_{m(k)},x_{m(k)},x_{n(k)})+\mathcal{B}(y_{m(k)},y_{m(k)},y_{n(k)}))+{\mathcal{B}}_{n(k)+1}+{\mathcal{B}}_{m(k)+1}^{\prime }\\ =& \delta (\mathcal{B}(x_{n(k)},x_{m(k)},x_{m(k)}),\mathcal{B}(y_{n(k)},y_{m(k)},y_{m(k)}))r_k+{\mathcal{B}}_{n(k)+1}+{\mathcal{B}}_{m(k)+1}^{\prime }\\ {<}_E& r_k+{\mathcal{B}}_{n(k)+1}+{\mathcal{B}}_{m(k)+1}^{\prime }.\end{array}$$

Taking $k\to \mathrm{\infty }$ and using (3.13), (3.14), and (3.17), we get

$$\delta (\mathcal{B}(x_{n(k)},x_{m(k)},x_{m(k)}),\mathcal{B}(y_{n(k)},y_{m(k)},y_{m(k)}))\to 1.$$

Since $\delta \in \mathrm{\Delta }$, we have $\mathcal{B}(x_{n(k)},x_{m(k)},x_{m(k)})\to 0_E$ and $\mathcal{B}(y_{n(k)},y_{m(k)},y_{m(k)})\to 0_E$, that is,

$$\mathcal{B}(x_{n(k)},x_{m(k)},x_{m(k)})+\mathcal{B}(y_{n(k)},y_{m(k)},y_{m(k)})\to 0_E,$$

which contradictions with (3.17). Therefore, $\{x_n\}$ and $\{y_n\}$ are Cauchy sequences in $(X,\mathcal{B})$. Since X complete, we get $x_n\to x$ and $y_n\to y$ as $n\to \mathrm{\infty }$ for some $x,y\in X$.

Finally, we prove that $(x,y)$ is a coupled fixed point of F. Since F is a continuous, taking $n\to \mathrm{\infty }$ in (3.2), we get

$$x=\underset{n\to \mathrm{\infty }}{lim}{x}_{n+1}=\underset{n\to \mathrm{\infty }}{lim}F(x_n,y_n)=F(\underset{n\to \mathrm{\infty }}{lim}{x}_n,\underset{n\to \mathrm{\infty }}{lim}{y}_n)=F(x,y)$$

and

$$y=\underset{n\to \mathrm{\infty }}{lim}{y}_{n+1}=\underset{n\to \mathrm{\infty }}{lim}F(y_n,x_n)=F(\underset{n\to \mathrm{\infty }}{lim}{y}_n,\underset{n\to \mathrm{\infty }}{lim}{x}_n)=F(y,x).$$

Therefore, $x=F(x,y)$ and $y=F(y,x)$, that is, F has a coupled fixed point. This completes the proof. □

Corollary 3.2 Let$(X,⪯)$be a partially ordered set and$\mathcal{B}$be a cone ball-metric induced by the cone metric d on X with a regular cone P such that$(X,\mathcal{B})$is a complete cone ball-metric space. Let$F:X\times X\to X$be a continuous mapping having the mixed monotone property on X. Suppose that there exists$k\in [0,1)$such that

(3.18)

for all$x,y,z,u,v,w\in X$for which$x⪰u⪰w$and$y⪯v⪯z$. If there exists$x_0,y_0\in X$such that

$$x_0⪯F(x_0,y_0)\mathit{\text{and}}{y}_0⪰F(y_0,x_0),$$

then F has a coupled fixed point, that is, there exist$x,y\in X$such that

$$x=F(x,y)\mathit{\text{and}}y=F(y,x).$$

In the next theorem, we omit the continuity hypothesis of F.

Theorem 3.3 Let$(X,⪯)$be a partially ordered set and$\mathcal{B}$be a cone ball-metric induced by the cone metric d on X with a regular cone P such that$(X,\mathcal{B})$is a complete cone ball-metric space. Let$F:X\times X\to X$be a mapping having the mixed monotone property on X. Suppose that there exists$\delta \in \mathrm{\Delta }$such that

(3.19)

for all$x,y,z,u,v,w\in X$for which$x⪰u⪰w$and$y⪯v⪯z$. If there exists$x_0,y_0\in X$such that

$$x_0⪯F(x_0,y_0)\mathit{\text{and}}{y}_0⪰F(y_0,x_0)$$

and X has the following property:

• (i) if a nondecreasing sequence $\{x_n\}$ converges to x, then $x_n⪯x$ for all $n\in \mathbb{N}$,

• (ii) if a nonincreasing sequence $\{y_n\}$ converges to y, then $y⪰y_n$ for all $n\in \mathbb{N}$,

then F has a coupled fixed point, that is, there exist$x,y\in X$such that

$$x=F(x,y)\mathit{\text{and}}y=F(y,x).$$

Proof By the similar the proof as in Theorem 3.1, we have the nondecreasing sequence $\{x_n\}$ converges to x and the nonincreasing sequence $\{y_n\}$ converges to y for some $x,y\in X$. By (i), (ii), we get $x_n⪯x$ and $y_n⪰y$ for all $n\in \mathbb{N}$. Thus, by the rectangle inequality of $\mathcal{B}$, we get

Taking the limit as $n\to \mathrm{\infty }$, we have $\mathcal{B}(F(x,y),x,x)+\mathcal{B}(F(y,x),y,y)=0_E$, and thus $x=F(x,y)$ and $y=F(y,x)$. Therefore, F has a coupled fixed point $(x,y)$ in $X\times X$. This completes the proof. □

Corollary 3.4 Let$(X,⪯)$be a partially ordered set and$\mathcal{B}$be a cone ball-metric induced by the cone metric d on X with a regular cone P such that$(X,\mathcal{B})$is a complete cone ball-metric space. Let$F:X\times X\to X$be a mapping having the mixed monotone property on X. Suppose that there exists$k\in [0,1)$such that

(3.20)

for all$x,y,z,u,v,w\in X$for which$x⪰u⪰w$and$y⪯v⪯z$. If there exists$x_0,y_0\in X$such that

$$x_0⪯F(x_0,y_0)\mathit{\text{and}}{y}_0⪰F(y_0,x_0)$$

and X has the following property:

• (i) if a nondecreasing sequence $\{x_n\}$ converges to x, then $x_n⪯x$ for all $n\in \mathbb{N}$,

• (ii) if a nonincreasing sequence $\{y_n\}$ converges to y, then $y⪰y_n$ for all $n\in \mathbb{N}$,

then F has a coupled fixed point, that is, there exist$x,y\in X$such that

$$x=F(x,y)\mathit{\text{and}}y=F(y,x).$$

Theorem 3.5 In addition to the hypotheses in Theorem 3.1, suppose that$x_0$, $y_0$are comparable then$x=y$, that is, $x=F(x,x)$.

Proof From Theorem 3.1, we have two sequences $\{x_n\}$, $\{y_n\}$ in X such that, for all $n\ge 0$,

$$x_{n+1}=F(x_n,y_n)\text{and}{y}_{n+1}=F(y_n,x_n)$$

and also $x_n\to x$ and $y_n\to y$ as $n\to \mathrm{\infty }$. Now, we assume that $x_0⪯y_0$. Since F has the mixed monotone property, we have $x_n⪯y_n$ for all $n\in \mathbb{N}$. From (3.1) and property of cone-ball metric $\mathcal{B}$, we have

This implies

$$\begin{array}{rcl}\mathcal{B}(y_{n+1},x_{n+1},x_{n+1})& {\le }_E& \delta (\mathcal{B}(y_n,x_n,x_n),\mathcal{B}(y_n,x_n,x_n))[\mathcal{B}(y_n,x_n,x_n)]\\ {<}_E& \mathcal{B}(y_n,x_n,x_n).\end{array}$$

(3.21)

So we have $\{\mathcal{B}(y_n,x_n,x_n)\}$ is a decreasing sequence. Similar to the prove in Theorem 3.1, we get $\mathcal{B}(y_n,x_n,x_n)\to 0_E$ as $n\to \mathrm{\infty }$.

By the rectangular inequality and (3.21), we have

$$\begin{array}{rcl}\mathcal{B}(y,x,x)& {\le }_E& \mathcal{B}(y,y_{n+1},y_{n+1})+\mathcal{B}(y_{n+1},x_{n+1},x_{n+1})+\mathcal{B}(x_{n+1},x,x)\\ {<}_E& \mathcal{B}(y,y_{n+1},y_{n+1})+\mathcal{B}(y_n,x_n,x_n)+\mathcal{B}(x_{n+1},x,x).\end{array}$$

From above inequality, taking $n\to \mathrm{\infty }$, we obtain that $\mathcal{B}(y,x,x)=0_E$ and then $x=y$. This completes the proof. □

Theorem 3.6 In addition to the hypotheses in Theorem 3.3, suppose that$x_0$, $y_0$are comparable then$x=y$, that is, $x=F(x,x)$.

Proof By the similar method as in the prove of Theorem 3.5 and by applying Theorem 3.3, we can get the conclusion. □

Theorem 3.7 Let$(X,⪯)$be a partially ordered set and$\mathcal{B}$be a cone ball-metric induced by the cone metric d on X with a regular cone P such that$(X,\mathcal{B})$is a complete cone ball-metric space. Let$F:X\times X\to X$be a mapping having the mixed monotone property on X. Suppose that there exists$\vartheta \in \mathrm{\Delta }$such that

$$\mathcal{B}(F(x,y),F(u,v),F(w,z)){\le }_E\frac{1}{2}\vartheta (\mathcal{B}(x,u,w),\mathcal{B}(y,v,z))(\mathcal{B}(x,u,w)+\mathcal{B}(y,v,z))$$

(3.22)

for all$x,y,z,u,v,w\in X$for which$x⪰u⪰w$and$y⪯v⪯z$. If there exists$x_0,y_0\in X$such that

$$x_0⪯F(x_0,y_0)\mathit{\text{and}}{y}_0⪰F(y_0,x_0)$$

and either

• (a) F is continuous or

• (b) X has the following property:

• (i) if a nondecreasing sequence $\{x_n\}$ converges to x, then $x_n⪯x$ for all $n\in \mathbb{N}$,

• (ii) if a nonincreasing sequence $\{y_n\}$ converges to y, then $y⪰y_n$ for all $n\in \mathbb{N}$,

then F has a coupled fixed point.

Proof For any $x,y,z,u,v,w\in X$ with $x⪰u⪰w$ and $y⪯v⪯z$, it follows from (3.22) that

$$\mathcal{B}(F(x,y),F(u,v),F(w,z)){\le }_E\frac{1}{2}\vartheta (\mathcal{B}(x,u,w),\mathcal{B}(y,v,z))(\mathcal{B}(x,u,w)+\mathcal{B}(y,v,z))$$

(3.23)

and

$$\begin{array}{rcl}\mathcal{B}(F(y,x),F(v,u),F(z,w))& =& \mathcal{B}(F(z,w),F(v,u),F(y,x))\\ {\le }_E& \frac{1}{2}\vartheta (\mathcal{B}(z,v,y),\mathcal{B}(w,u,x))(\mathcal{B}(z,v,y)+\mathcal{B}(w,u,x))\\ =& \frac{1}{2}\vartheta (\mathcal{B}(z,v,y),\mathcal{B}(w,u,x))(\mathcal{B}(x,u,w),\mathcal{B}(y,v,z)).\end{array}$$

(3.24)

From (3.23) and (3.24), we have

(3.25)

for all $x,y,z,u,v,w\in X$ with $x⪰u⪰w$ and $y⪯v⪯z$, where

$$\delta (a,b)=\frac{1}{2}[\vartheta (a,b)+\vartheta (b,a)]$$

for all $a,b\in int(P)\cup \{0_E\}$. It is easy to verify that $\delta \in \mathrm{\Delta }$. If we apply Theorems 3.1 and 3.3, we know that F has a coupled fixed point. □

Corollary 3.8 Let$(X,⪯)$be a partially ordered set and$\mathcal{B}$be a cone ball-metric induced by the cone metric d on X with a regular cone P such that$(X,\mathcal{B})$is a complete cone ball-metric space. Let$F:X\times X\to X$be a mapping having the mixed monotone property on X. Suppose that there exists$k\in [0,1)$such that

$$\mathcal{B}(F(x,y),F(u,v),F(w,z)){\le }_E\frac{k}{2}(\mathcal{B}(x,u,w)+\mathcal{B}(y,v,z))$$

(3.26)

for all$x,y,z,u,v,w\in X$for which$x⪰u⪰w$and$y⪯v⪯z$. If there exists$x_0,y_0\in X$such that

$$x_0⪯F(x_0,y_0)\mathit{\text{and}}{y}_0⪰F(y_0,x_0)$$

and either

• (a) F is continuous or

• (b) X has the following property:

• (i) if a nondecreasing sequence $\{x_n\}$ converges to x, then $x_n⪯x$ for all $n\in \mathbb{N}$,

• (ii) if a nonincreasing sequence $\{y_n\}$ converges to y, then $y⪰y_n$ for all $n\in \mathbb{N}$,

then F has a coupled fixed point.

Let Ξ denote the class of functions $\xi :int(P)\cup \{0_E\}\to [0,1)$ which satisfies the following condition:

For any sequence $\{a_n\}$ in $int(P)\cup \{0_E\}$,

$$\xi (a_n)\to 1⟹a_n\to 0_E.$$

Theorem 3.9 Let$(X,⪯)$be a partially ordered set and$\mathcal{B}$be a cone ball-metric induced by the cone metric d on X with a regular cone P such that$(X,\mathcal{B})$is a complete cone ball-metric space. Let$F:X\times X\to X$be a mapping having the mixed monotone property on X. Suppose that there exists$\xi \in \mathrm{\Xi }$such that

(3.27)

for all $x,y,z,u,v,w\in X$ for which $x⪰u⪰w$ and $y⪯v⪯z$ If there exists $x_0,y_0\in X$ such that

$$x_0⪯F(x_0,y_0)\mathit{\text{and}}{y}_0⪰F(y_0,x_0)$$

and either

• (a) F is continuous or

• (b) X has the following property:

• (i) if a nondecreasing sequence $\{x_n\}$ converges to x, then $x_n⪯x$ for all $n\in \mathbb{N}$,

• (ii) if a nonincreasing sequence $\{y_n\}$ converges to y, then $y⪰y_n$ for all $n\in \mathbb{N}$,

then F has a coupled fixed point.

Proof If we taking $\delta (a,b)=\xi (a+b)$ for all $a,b\in int(P)\cup \{0_E\}$ in Theorem 3.1, then, from (a), we get the conclusion. Also, if we take $\delta (a,b)=\xi (a+b)$ for all $a,b\in int(P)\cup \{0_E\}$ in Theorem 3.3, then, from (b), we obtain the conclusion. □

Theorem 3.10 Let$(X,⪯)$be a partially ordered set and$\mathcal{B}$be a cone ball-metric induced by the cone metric d on X with a regular cone P such that$(X,\mathcal{B})$is a complete cone ball-metric space. Let$F:X\times X\to X$be a mapping having the mixed monotone property on X and such that$F(x,y)⪯F(y,x)$, whenever$x⪯y$ . Suppose that there exists$\delta \in \mathrm{\Delta }$such that

(3.28)

for all$x,y,z,u,v,w\in X$for which$w⪯u⪯x\prec y⪯v⪯z$. If there exists$x_0,y_0\in X$such that

$$x_0⪯y_0,x_0⪯F(x_0,y_0)\mathit{\text{and}}{y}_0⪰F(y_0,x_0)$$

and either

• (a) F is continuous or

• (b) X has the following property:

• (i) if a nondecreasing sequence $\{x_n\}$ converges to x, then $x_n⪯x$ for all $n\in \mathbb{N}$,

• (ii) if a nonincreasing sequence $\{y_n\}$ converges to y, then $y⪰y_n$ for all $n\in \mathbb{N}$,

then F has a coupled fixed point.

Proof From the assumption, there exist $x_0,y_0\in X$ such that $x_0⪯F(x_0,y_0)$ and $y_0⪰F(y_0,x_0)$. Now, we define $x_1,y_1\in X$ such that $x_1=F(x_0,y_0)⪰x_0$ and $y_1=F(y_0,x_0)⪯y_0$. Further, the fact $x_0⪯y_0$, we get $F(x_0,y_0)⪯F(y_0,x_0)$. Thus, we now have

$$x_0⪯x_1=F(x_0,y_0)⪯F(y_0,x_0)=y_1⪯y_0.$$

Let $x_2=F(x_1,y_1)$ and $y_2=F(y_1,x_1)$. From the fact that $x_1⪯y_1$ and the mixed monotone property, we have

$$x_1⪯x_2⪯y_2⪯y_1.$$

Continuing this procedure, we have two sequences $\{x_n\}$ and $\{y_n\}$ such that

$$x_{n+1}=F(x_n,y_n),y_{n+1}=F(y_n,x_n)$$

and

$$x_n⪯x_{n+1}=F(x_n,y_n)⪯F(y_n,x_n)=y_{n+1}⪯y_n$$

for all $n\ge 0$. If there exists a nonnegative integer k such that $x_k=y_k=c$ (say), then we have

$$c⪯F(c,c)⪯F(c,c)⪯c,$$

that is, $c=F(c,c)$. Therefore, $(c,c)$ is a coupled fixed point of F.

Therefore, we assume that

$$x_n\prec y_n$$

(3.29)

for all $n\ge 0$. In view of (3.29), for all $n\ge 0$, the inequality (3.28) holds with

The rest of the proof can be completed by repeating the same steps given in Theorem 3.1 and Theorem 3.3. This completes the proof. □

Example 3.11 Let $E=\mathbb{R}$ be a usual norm space with a regular cone $P=\{z\in \mathbb{R}|z\ge 0\}$ and $X=\mathbb{N}\cup \{0\}$ be a cone metric space with a metric $d(x,y)=|x-y|$ for all $x,y\in X$. Then $(X,d)$ is a complete cone metric space. Therefore, we get a cone ball metric $\mathcal{B}:X\times X\times X\to E$ such that

$$\mathcal{B}(x,y,z)=max\{x,y,z\}-min\{x,y,z\}$$

for all $x,y,z\in X$ and so $(X,\mathcal{B})$ is a complete cone ball-metric space.

Let a partial order ⪯ on X be defined as follows: For $x,y\in X$,

$$x⪯y\text{ holds if }x{\ge }_{E}y.$$

Let $F:X\times X\to X$ be defined by

$$F(x,y)=\{\begin{array}{cc}1;\hfill & \text{if }x\prec y,\hfill \\ 0;\hfill & \text{if otherwise}.\hfill \end{array}$$

Let $w⪯u⪯x\prec y⪯v⪯z$ hold, then we have $w{\ge }_{E}u{\ge }_{E}x{>}_{E}y{\ge }_{E}v{\ge }_{E}z$. Therefore, we have

$$F(x,y)=F(u,v)=F(w,z)=1$$

and

$$F(y,x)=F(v,u)=F(z,w)=0.$$

So the left side of (3.28) is

$$\mathcal{B}(1,1,1)+\mathcal{B}(0,0,0)=0$$

and then (3.28) is satisfied for all $\delta \in \mathrm{\Delta }$. Thus, Theorem 3.10 is applicable to this example with $x_0=81$ and $y_0=0$. Therefore, F has a coupled fixed points that is a point $(0,0)$ and $(1,0)$.

Remark 3.12 Example 3.11 is not applied by Theorems 3.1, 3.3, and 3.9. This is evident by the fact that the inequality (3.1), (3.19), and (3.27) are not satisfied when $w=u=x=y=3$, $v=0$ and $z=1$. Moreover, the coupled fixed point is not unique.

4 Uniqueness of coupled fixed point in cone ball-metric spaces

In this section, we study the necessary condition for the uniqueness of a coupled fixed point. If $(X,⪯)$ is a partially ordered set, then we endow the product of $X\times X$ with the following partial order relation: for any $(x,y),(u,v)\in X\times X$,

$$(x,y)⋖(u,v)⟺x⪯u\text{and}y⪰v.$$

Theorem 4.1 In addition to the hypotheses in Theorem 3.1, suppose that, for any$(x,y),(z,t)\in X\times X$, there exists a point$(u,v)\in X\times X$which is comparable to$(x,y)$and$(z,t)$. Then F has a unique coupled fixed point.

Proof By Theorem 3.1, we get F has a coupled fixed point $(x,y)$, that is,

$$x=F(x,y)\text{and}y=F(y,x).$$

We may assume that $(z,t)$ are another coupled fixed points of F and so

$$z=F(z,t)\text{and}t=F(t,z).$$

Next, we prove that $x=z$ and $y=t$. By assumption, there exists $(u,v)\in X\times X$ which is comparable to $(x,y)$ and $(z,t)$. We put $u_0=u$ and $v_0=v$ and construct two sequences $\{u_n\}$ and $\{v_n\}$ by

$$u_n=F(u_{n-1},v_{n-1})\text{and}{v}_n=F(v_{n-1},u_{n-1})$$

for all $n\in \mathbb{N}$. Since $(u,v)$ is comparable with $(x,y)$, we may assume that $(u_0,v_0)=(u,v)⋖(x,y)$. It easy to see that $(u_n,v_n)⋖(x,y)$ for all $n\in \mathbb{N}$. From (3.1), we have

(4.1)

This implies that $\{\mathcal{B}(x,x,u_n)+\mathcal{B}(y,y,v_n)\}$ is a decreasing sequence and so

$$\mathcal{B}(x,x,u_n)+\mathcal{B}(y,y,v_n)\to \mathcal{B}$$

for some $\mathcal{B}{\ge }_E{0}_E$.

Now, we show that $\mathcal{B}=0_E$. We may assume that $\mathcal{B}{>}_E{0}_E$. By the similar method as in the proof of Theorem 3.1, we can conclude that

$$\delta (\mathcal{B}(x,x,u_{n-1}),\mathcal{B}(v_{n-1},y,y))\to 1.$$

Since $\delta \in \mathrm{\Delta }$, we get $\mathcal{B}(x,x,u_{n-1})\to 0_E$ and $\mathcal{B}(v_{n-1},y,y)\to 0_E$. Therefore, we have

$$\mathcal{B}(x,x,u_{n-1})+\mathcal{B}(v_{n-1},y,y)\to 0_E,$$

which is a contradiction. Thus, we have $\mathcal{B}(x,x,u_n)+\mathcal{B}(v_n,y,y)\to 0_E$ as $n\to \mathrm{\infty }$. Similarly, one can prove $\mathcal{B}(z,z,u_n)+\mathcal{B}(v_n,t,t)\to 0_E$ as $n\to \mathrm{\infty }$.

Finally, we have

$$\begin{array}{rcl}\mathcal{B}(z,x,x)+\mathcal{B}(y,t,t)& {\le }_E& \mathcal{B}(z,u_n,u_n)+\mathcal{B}(u_n,x,x)+\mathcal{B}(y,v_n,v_n)+\mathcal{B}(v_n,t,t)\\ =& \mathcal{B}(z,z,u_n)+\mathcal{B}(x,x,u_n)+\mathcal{B}(y,y,v_n)+\mathcal{B}(v_n,t,t).\end{array}$$

Taking $n\to \mathrm{\infty }$ in above inequalities, we have $\mathcal{B}(z,x,x)+\mathcal{B}(y,t,t)=0_E$, that is, $z=x$ and $y=t$.

For the case when $(u_0,v_0)=(u,v)⋗(x,y)$ is similar. This completes the proof. □

Theorem 4.2 In addition to the hypotheses in Theorem 3.3, suppose that, for any$(x,y),(z,t)\in X\times X$, there exists a point$(u,v)\in X\times X$which is comparable to$(x,y)$and$(z,t)$. Then F has a unique coupled fixed point.

Proof By the similar method given in the prove of Theorem 4.1 and by applying Theorem 3.3, we can get the conclusion. □

Open problems:

• In our theorems, can the mixed monotone property be replaced by a more general property (see the work of Sintunavarat et al.[36])?

• In our theorems, can the mixed monotone property be replaced by another property (see the work of Ðorić et al.[16])?

• Can the coupled fixed-point theorems in this paper be extended to coupled best proximity point theorems (see the work of Sintunavarat et al.[37])?

• Can the main results in this paper be extended to multivalued case of coupled fixed point?

• Can the concept of cone ball-metric be extended to another distance?