Coupled Fixed Point Theorems of Single-Valued Mapping for c-Distance in Cone Metric Spaces

Abstract

A new concept of the c-distance in cone metric space has been introduced recently in 2011. The aim of this paper is to extend and generalize some coupled fixed-point theorems on c-distance in cone metric space. Some examples are given.

1. Introduction

In 2007, Huang and Zhang [1] introduced the concept of cone metric space where each pair of points is assigned to a member of a real Banach space with a cone. Then, several authors have studied the existence and uniqueness of the fixed point and common fixed point for self-map 𝑓 by considered different types of contractive conditions. Some of these works are noted in [2–12].

In [13], Bhaskar and Lakshmikantham introduced the concept of coupled fixed point for a given partially ordered set 𝑋. Lakshmikantham and Ćirić [14] proved some more coupled fixed-point theorems in partially ordered set.

In [15], Sabetghadam et al. considered the corresponding definition of coupled fixed point for the mapping in complete cone metric space and proved some coupled fixed point theorems. Then, several authors have studied the existence and uniqueness of the coupled fixed point and coupled common fixed point by considered different types of contractive conditions. Some of these works are noted in [16–23].

Recently, Cho et al. [23] introduced a new concept of the c-distance in cone metric spaces (also see [24]) and proved some fixed-point theorems in ordered cone metric spaces. This is more general than the classical Banach contraction mapping principle. Sintunavarat et al. [25] extended and developed the Banach contraction theorem on c-distance of Cho et al. [23]. Wang and Guo [24] proved some common fixed point theorems for this new distance. Several authors have studied on the generalized distance in cone metric space. Some of this works are noted in [26–29].

In [28], Cho et al. proved some coupled fixed point theorems in ordered cone metric spaces by using the concept of c-distance.

Recall the following definition.

Definition 1.1 (see [15]). Let (𝑋,𝑑) be a cone metric space. An element (𝑥,𝑦)∈𝑋×𝑋 is said to be a coupled fixed point of the mapping 𝐹∶𝑋×𝑋→𝑋 if 𝐹(𝑥,𝑦)=𝑥 and 𝐹(𝑦,𝑥)=𝑦.

The following theorems are the main results given in [15].

Theorem 1.2 (see [15]). Let (𝑋,𝑑) be a complete cone metric space. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition for all 𝑥,𝑦,𝑢,𝑣∈𝑋: 𝑑𝐹𝑥,𝑦,𝐹(𝑢,𝑣)⪯𝑘𝑑(𝑥,𝑢)+𝑙𝑑𝑦,𝑣, (1.1) where 𝑘,𝑙 are nonnegative constants with 𝑘+𝑙<1. Then 𝐹 has a unique coupled fixed point.

Theorem 1.3 (see [15]). Let (𝑋,𝑑) be a complete cone metric space. Suppose the mapping 𝐹∶𝑋×X→𝑋 satisfies the following contractive condition for all 𝑥,𝑦,𝑢,𝑣∈𝑋: 𝑑𝐹𝑥,𝑦,𝐹(𝑢,𝑣)⪯𝑘𝑑𝐹𝑥,𝑦,𝑥+𝑙𝑑(𝐹(𝑢,𝑣),𝑢), (1.2) where 𝑘,𝑙 are nonnegative constants with 𝑘+𝑙<1. Then 𝐹 has a unique coupled fixed point.

In this paper we proved some coupled fixed point results for c-distance in cone metric space. Our theorems extend and develop some theorems of Sabetghadam et al. [15] on c-distance of Cho et al. [23] in cone metric space.

2. Preliminaries

Let 𝐸 be a real Banach space and 𝜃 denote to the zero element in 𝐸. A cone 𝑃 is a subset of 𝐸 such that (1)𝑃 is nonempty set closed and 𝑃≠{𝜃}, (2)if 𝑎,𝑏 are nonnegative real numbers and 𝑥,𝑦∈𝑃 then 𝑎𝑥+𝑏𝑦∈𝑃, and(3)𝑥∈𝑃 and −𝑥∈𝑃 implies 𝑥=𝜃.For any cone 𝑃⊂𝐸, the partial ordering ⪯ with respect to 𝑃 is defined by 𝑥⪯𝑦 if and only if 𝑦−𝑥∈𝑃. The notation of ≺ stands for 𝑥⪯𝑦 but 𝑥≠𝑦. Also, we used 𝑥≪𝑦 to indicate that 𝑦−𝑥∈int𝑃, where int𝑃 denotes the interior of 𝑃. A cone 𝑃 is called normal if there exists a number 𝐾 such that 𝜃⪯𝑥⪯𝑦⟹‖𝑥‖≤𝐾𝑦 (2.1) for all 𝑥,𝑦∈𝐸. The least positive number 𝐾 satisfying the above condition is called the normal constant of 𝑃.

Definition 2.1 (see [1]). Let 𝑋 be a nonempty set and 𝐸 be a real Banach space equipped with the partial ordering ⪯ with respect to the cone 𝑃. Suppose that the mapping 𝑑∶𝑋×𝑋→𝐸 satisfies the following condition: (1)𝜃⪯𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦,(2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋, and (3)𝑑(𝑥,𝑦)⪯𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋. Then 𝑑 is called a cone metric on 𝑋 and (𝑋,𝑑) is called a cone metric space.

Definition 2.2 (see [1]). Let (𝑋,𝑑) be a cone metric space and {𝑥_𝑛} be a sequence in 𝑋 and 𝑥∈𝑋.
One has the following:(1)for all 𝑐∈𝐸 with 𝜃≪𝑐, if there exists a positive integer 𝑁 such that 𝑑(𝑥_𝑛,𝑥)≪𝑐 for all 𝑛>𝑁, then 𝑥_𝑛 is said to be convergent and 𝑥 is the limit of {𝑥_𝑛}. We denote this by 𝑥_𝑛→𝑥, (2)for all 𝑐∈𝐸 with 𝜃≪𝑐, if there exists a positive integer 𝑁 such that 𝑑(𝑥_𝑛,𝑥_𝑚)≪𝑐 for all 𝑛,𝑚>𝑁 then {𝑥_𝑛} is called a Cauchy sequence in 𝑋, and (3)a cone metric space (𝑋,𝑑) is called complete if every Cauchy sequence in 𝑋 is convergent.

Lemma 2.3 (see [8]). (1) If 𝐸 be a real Banach space with a cone 𝑃 and 𝑎⪯𝜆𝑎 where 𝑎∈𝑃 and 0≤𝜆<1, then 𝑎=𝜃.
(2) If 𝑐∈int𝑃, 𝜃⪯𝑎_𝑛 and 𝑎_𝑛→𝜃, then there exists a positive integer 𝑁 such that 𝑎_𝑛≪𝑐 for all 𝑛≥𝑁.

Next we give the notation of c-distance on a cone metric space which is a generalization of 𝜔-distance of Kada et al. [30] with some properties.

Definition 2.4 (see [23]). Let (𝑋,𝑑) be a cone metric space. A function 𝑞∶𝑋×𝑋→𝐸 is called a c-distance on 𝑋 if the following conditions hold: (q1)𝜃⪯𝑞(𝑥,𝑦) for all 𝑥,𝑦∈𝑋, (q2)𝑞(𝑥,𝑦)⪯𝑞(𝑥,𝑦)+𝑞(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋, (q3) for each 𝑥∈𝑋 and 𝑛≥1, if 𝑞(𝑥,𝑦_𝑛)⪯𝑢 for some 𝑢=𝑢_𝑥∈𝑃, then 𝑞(𝑥,𝑦)⪯𝑢 whenever {𝑦_𝑛} is a sequence in 𝑋 converging to a point 𝑦∈𝑋, and (q4) for all 𝑐∈𝐸 with 𝜃≪𝑐, there exists 𝑒∈𝐸 with 𝜃≪𝑒 such that 𝑞(𝑧,𝑥)≪𝑒 and 𝑞(𝑧,𝑦)≪𝑒 imply 𝑑(𝑥,𝑦)≪𝑐.

Example 2.5 (see [23]). Let 𝐸=ℝ and 𝑃=(𝑥∈𝐸∶𝑥≥0). Let 𝑋=[0,∞) and define a mapping 𝑑∶𝑋×𝑋→𝐸 by 𝑑(𝑥,𝑦)=|𝑥−𝑦| for all 𝑥,𝑦∈𝑋. Then (𝑋,𝑑) is a cone metric space. Define a mapping 𝑞∶𝑋×𝑋→𝐸 by 𝑞(𝑥,𝑦)=𝑦 for all 𝑥,𝑦∈𝑋. Then 𝑞 is a c-distance on 𝑋.

Lemma 2.6 (see [23]). Let (𝑋,𝑑) be a cone metric space and 𝑞 is a 𝑐-distance on 𝑋. Let {𝑥_𝑛} and {𝑦_𝑛} be sequences in 𝑋 and 𝑥,𝑦,𝑧∈𝑋. Suppose that 𝑢_𝑛 is a sequences in 𝑃 converging to 𝜃. Then the following hold.(1)If q(x_n,y)⪯u_n and q(x_n,z)⪯u_n, then y=z. (2)If q(x_n,y_n)⪯u_n and q(x_n,z)⪯u_n, then {y_n} converges to z. (3)If q(x_n,x_m)⪯u_n for m>n, then {x_n} is a Cauchy sequence in X. (4)If q(y,x_n)⪯u_n, then {x_n} is a Cauchy sequence in X.

Remark 2.7 (see [23]). (1) 𝑞(𝑥,𝑦)=𝑞(𝑦,𝑥) does not necessarily for all 𝑥,𝑦∈𝑋.
(2) 𝑞(𝑥,𝑦)=𝜃 is not necessarily equivalent to 𝑥=𝑦 for all 𝑥,𝑦∈𝑋.

3. Main Results

In this section we prove some coupled fixed point theorems using c-distance in cone metric space. In whole paper cone metric space is over nonnormal cone with nonempty interior.

Theorem 3.1. Let (𝑋,𝑑) be a complete cone metric space, and 𝑞 is a c -distance on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping and suppose that there exists mappings 𝑘,𝑙∶𝑋×𝑋→[0,1) such that the following hold:(a)𝑘(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑘(𝑥,𝑦) and 𝑙(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑙(𝑥,𝑦) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, (b)𝑘(𝑥,𝑦)=𝑘(𝑦,𝑥) and 𝑙(𝑥,𝑦)=𝑙(𝑦,𝑥) for all 𝑥,𝑦∈𝑋, (c)(𝑘+𝑙)(𝑥,𝑦)<1 for all 𝑥,𝑦∈𝑋, (d)𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘(𝑥,𝑦)𝑞(𝑥,𝑢)+𝑙(𝑥,𝑦)𝑞(𝑦,𝑣) for all 𝑥,𝑦,𝑢,𝑣∈𝑋.Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Proof. Choose 𝑥₀,𝑦₀∈𝑋. Set 𝑥₁=𝐹(𝑥₀,𝑦₀),𝑦₁=𝐹(𝑦₀,𝑥₀),𝑥₂=𝐹(𝑥₁,𝑦₁),𝑦₂=𝐹(𝑦₁,𝑥₁),…,𝑥_𝑛+1=𝐹(𝑥_𝑛,𝑦_𝑛),𝑦_𝑛+1=𝐹(𝑦_𝑛,𝑥_𝑛). Then we have the following: 𝑞𝑥_𝑛,𝑥_𝑛+1=𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,F𝑥_𝑛,𝑦_𝑛⪯𝑘𝑥_𝑛−1,𝑦_𝑛−1𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥_𝑛−1,𝑦_𝑛−1𝑞𝑦_𝑛−1,𝑦_𝑛=𝑘𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑦_𝑛−1,𝑦_𝑛⪯𝑘𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑦_𝑛−1,𝑦_𝑛⋮⪯𝑘𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥₀,𝑦₀𝑞𝑦_𝑛−1,𝑦_𝑛. (3.1)
And similarly 𝑞𝑦_𝑛,𝑦_𝑛+1=𝑞𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝐹𝑦_𝑛,𝑥_𝑛⪯𝑘𝑦_𝑛−1,𝑥_𝑛−1𝑞𝑦_𝑛−1,𝑦_𝑛+𝑙𝑦_𝑛−1,𝑥_𝑛−1𝑞𝑥_𝑛−1,𝑥_𝑛=𝑘𝐹𝑦_𝑛−2,𝑥_𝑛−2,𝐹𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑦_𝑛−1,𝑦_𝑛+𝑙𝐹𝑦_𝑛−2,𝑥_𝑛−2,𝐹𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛⪯𝑘𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑦_𝑛−1,𝑦_𝑛+𝑙𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛⋮⪯𝑘𝑦₀,𝑥₀𝑞𝑦_𝑛−1,𝑦_𝑛+𝑙𝑦₀,𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛=𝑘𝑥₀,𝑦₀𝑞𝑦_𝑛−1,𝑦_𝑛+𝑙𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛. (3.2) Put, 𝑞_𝑛=𝑞(𝑥_𝑛,𝑥_𝑛+1)+𝑞(𝑦_𝑛,𝑦_𝑛+1). Then we have 𝑞_𝑛=𝑞𝑥_𝑛,𝑥_𝑛+1+𝑞𝑦_𝑛,𝑦_𝑛+1⪯𝑘𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥₀,𝑦₀𝑞𝑦_𝑛−1,𝑦_𝑛+𝑘𝑥₀,𝑦₀𝑞𝑦_𝑛−1,𝑦_𝑛+𝑙𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛=𝑘𝑥₀,𝑦₀+𝑙𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑞𝑦_𝑛−1,𝑦_𝑛=𝑘𝑥₀,𝑦₀+𝑙𝑥₀,𝑦_{0𝑞𝑛−1}=ℎ𝑞_𝑛−1⋮⪯ℎ^𝑛𝑞₀, (3.3) where ℎ=𝑘(𝑥₀,𝑦₀)+𝑙(𝑥₀,𝑦₀)<1.
Let 𝑚>𝑛≥1. It follows that 𝑞𝑥_𝑛,𝑥_𝑚⪯𝑞𝑥_𝑛,𝑥_𝑛+1+𝑞𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑞𝑥_𝑚−1,𝑥_𝑚,𝑞𝑦_𝑛,𝑦_𝑚⪯𝑞𝑦_𝑛,𝑦_𝑛+1+𝑞𝑦_𝑛+1,𝑥_𝑦+2+⋯+𝑞𝑦_𝑦−1,𝑦_𝑚. (3.4) Then we have 𝑞𝑥_𝑛,𝑥_𝑚+𝑞𝑦_𝑛,𝑦_𝑚⪯𝑞_𝑛+𝑞_𝑛+1+⋯+𝑞_𝑚−1⪯ℎ^𝑛𝑞₀+ℎ^𝑛+1𝑞₀+⋯+ℎ^𝑚−1𝑞₀=ℎ^𝑛+ℎ^𝑛+1+⋯+ℎ^𝑚−1𝑞₀⪯ℎ^𝑛𝑞1−ℎ⁰. (3.5)
From (3.5) we have 𝑞𝑥_𝑛,𝑥_𝑚⪯ℎ^𝑛𝑞1−ℎ⁰, (3.6) and also 𝑞𝑦_𝑛,𝑦_𝑚⪯ℎ^𝑛𝑞1−ℎ⁰. (3.7) Thus, Lemma 2.6(3) shows that {𝑥_𝑛} and {𝑦_𝑛} are Cauchy sequences in 𝑋. Since 𝑋 is complete, there exists 𝑥^∗ and 𝑦^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗ and 𝑦_𝑛→𝑦^∗ as 𝑛→∞. By (q3) we have the following: 𝑞𝑥_𝑛,𝑥^∗⪯ℎ^𝑛𝑞1−ℎ⁰, (3.8) and also 𝑞𝑦_𝑛,𝑦^∗⪯ℎ^𝑛𝑞1−ℎ⁰. (3.9)
On the other hand, 𝑞𝑥_𝑛,𝐹𝑥^∗,𝑦^∗=𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥^∗,𝑦^∗⪯𝑘𝑥_𝑛−1,𝑦_𝑛−1𝑞𝑥_𝑛−1,𝑥^∗+𝑙𝑥_𝑛−1,𝑦_𝑛−1𝑞𝑦_𝑛−1,𝑦^∗=𝑘𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥^∗+𝑙𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑦_𝑛−1,𝑦^∗⪯𝑘𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑥_𝑛−1,𝑥^∗+𝑙𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑦_𝑛−1,𝑦^∗⋮⪯𝑘𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥^∗+𝑙𝑥₀,𝑦₀𝑞𝑦_𝑛−1,𝑦^∗⪯𝑘𝑥₀,𝑦₀ℎ^𝑛−1𝑞1−ℎ⁰+𝑙𝑥₀,𝑦₀ℎ^𝑛−1𝑞1−ℎ⁰=𝑘𝑥₀,𝑦₀+𝑙𝑥₀,𝑦₀ℎ^𝑛−1𝑞1−ℎ⁰ℎ=ℎ^𝑛−1𝑞1−ℎ⁰=ℎ^𝑛𝑞1−ℎ⁰. (3.10) By Lemma 2.6 (1), (3.8), and (3.10), we have 𝑥^∗=𝐹(𝑥^∗,𝑦^∗). By similar way we have 𝑦^∗=𝐹(𝑦^∗,𝑥^∗). Therefore (𝑥^∗,𝑦^∗) is a coupled fixed point of 𝐹.
Suppose that 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then we have 𝑞𝑥₁,𝑥₁=𝑞𝐹𝑥₁,𝑦₁,𝐹𝑥₁,𝑦₁⪯𝑘𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁+𝑙𝑥₁,𝑦₁𝑞𝑦₁,𝑦₁, (3.11) and also 𝑞𝑦₁,𝑦₁=𝑞𝐹𝑦₁,𝑥₁,𝐹𝑦₁,𝑥₁⪯𝑘𝑦₁,𝑥₁𝑞𝑦₁,𝑦₁+𝑙𝑦₁,𝑥₁𝑞𝑥₁,𝑥₁=𝑘𝑥₁,𝑦₁𝑞𝑦₁,𝑦₁+𝑙𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁ (3.12) which implies that 𝑞𝑥₁,𝑥₁+𝑞𝑦₁,𝑦₁⪯𝑘𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁+𝑘𝑥₁,𝑦₁𝑞𝑦₁,𝑦₁+𝑙𝑥₁,𝑦₁𝑞𝑦₁,𝑦₁+𝑙𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁=𝑘𝑥₁,𝑦₁+𝑙𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁+𝑞𝑦₁,𝑦₁=(𝑘+𝑙)𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁+𝑞𝑦₁,𝑦₁. (3.13) Since (𝑘+𝑙)(𝑥₁,𝑦₁)<1, Lemma 2.3 (1) shows that 𝑞(𝑥₁,𝑥₁)+𝑞(𝑦₁,𝑦₁)=𝜃. But 𝑞(𝑥₁,𝑥₁)≽𝜃 and 𝑞(𝑦₁,𝑦₁)≽𝜃, hence 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃.
Finally, suppose that there is another coupled fixed point (𝑥^′,𝑦^′) then we have 𝑞𝑥^∗,𝑥^′=𝑞𝐹𝑥^∗,𝑦^∗,𝐹𝑥^′,𝑦^′⪯𝑘𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑙𝑥^∗,𝑦^∗𝑞𝑦^∗,𝑦^′ (3.14) and also 𝑞𝑦^∗,𝑦^′=𝑞𝐹𝑦^∗,𝑥^∗,𝐹𝑦^′,𝑥^′⪯𝑘𝑦^∗,𝑥^∗𝑞𝑦^∗,𝑦^′+𝑙𝑦^∗,𝑥^∗𝑞𝑥^∗,𝑥^′=𝑘𝑥^∗,𝑦^∗𝑞𝑦^∗,𝑦^′+𝑙𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′, (3.15) which implies that 𝑞𝑥^∗,𝑥^′+𝑞𝑦^∗,𝑦^′⪯𝑘𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑘𝑥^∗,𝑦^∗𝑞𝑦^∗,𝑦^′+𝑙𝑥^∗,𝑦^∗𝑞𝑦^∗,𝑦^′+𝑙𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′=𝑘𝑥^∗,𝑦^∗+𝑙𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑞𝑦^∗,𝑦^′=(𝑘+𝑙)𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑞𝑦^∗,𝑦^′. (3.16) Since (𝑘+𝑙)(𝑥^∗,𝑦^∗)<1, Lemma 2.3 (1) shows that 𝑞(𝑥^∗,𝑥^′)+𝑞(𝑦^∗,𝑦^′)=𝜃. But 𝑞(𝑥^∗,𝑥^′)≽𝜃 and 𝑞(𝑦^∗,𝑦^′)≽𝜃. Hence 𝑞(𝑥^∗,𝑥^′)=𝜃 and 𝑞(𝑦^∗,𝑦^′)=𝜃. Also we have 𝑞(𝑥^∗,𝑥^∗)=𝜃 and 𝑞(𝑦^∗,𝑦^∗)=𝜃. Hence Lemma 2.6 part 1 shows that 𝑥^∗=𝑥^′ and 𝑦^∗=𝑦^′, which implies that (𝑥^∗,𝑦^∗)=(𝑥^′,𝑦^′). Similarly, we prove that 𝑥^∗=𝑦^′ and 𝑦^∗=𝑥^′. Hence, 𝑥^∗=𝑦^∗. Therefore, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

From above Theorem we have the following corollaries.

Corollary 3.2. Let (𝑋,𝑑) be a complete cone metric space, and 𝑞 is a c -distance on 𝑋. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition: 𝑞𝐹𝑥,𝑦,𝐹(𝑢,𝑣)⪯𝑘𝑞(𝑥,𝑢)+𝑙𝑞𝑦,𝑣, (3.17) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘,𝑙 are nonnegative constants with 𝑘+𝑙<1. Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁) then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Corollary 3.3. Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a c -distance on 𝑋. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition: 𝑞𝐹𝑥,𝑦,𝐹(𝑢,𝑣)⪯𝑘𝑞(𝑥,𝑢)+𝑞𝑦,𝑣, (3.18) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘∈[0,1/2). Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Theorem 3.4. Let (𝑋,𝑑) be a complete cone metric space, and 𝑞 is a c -distance on 𝑋. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 is continuous, and suppose that there exists mappings 𝑘,𝑙,𝑟∶𝑋×𝑋→[0,1) such that the following hold: (a)𝑘(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑘(𝑥,𝑦), 𝑙(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑙(𝑥,𝑦)𝑟(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑟(𝑥,𝑦) for all 𝑥,𝑦,𝑢,𝑣∈𝑋,(b)(𝑘+𝑙+𝑟)(𝑥,𝑦)<1 for all 𝑥,𝑦∈𝑋, and (c)𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘(𝑥,𝑦)𝑞(𝑥,𝑢)+𝑙(𝑥,𝑦)𝑞(𝑥,𝐹(𝑥,𝑦))+𝑟(𝑥,𝑦)𝑞(𝑢,𝐹(𝑢,𝑣)) for all 𝑥,𝑦,𝑢,𝑣∈𝑋.Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Proof. Choose 𝑥₀,𝑦₀∈𝑋. Set 𝑥₁=𝐹(𝑥₀,𝑦₀),𝑦₁=𝐹(𝑦₀,𝑥₀),𝑥₂=𝐹(𝑥₁,𝑦₁),𝑦₂=𝐹(𝑦₁,𝑥₁),…,𝑥_𝑛+1=𝐹(𝑥_𝑛,𝑦_𝑛),𝑦_𝑛+1=𝐹(𝑦_𝑛,𝑥_𝑛). Then we have the following: 𝑞𝑥_𝑛,𝑥_𝑛+1=𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛⪯𝑘𝑥_𝑛−1,𝑦_𝑛−1𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥_𝑛−1,𝑦_𝑛−1𝑞𝑥_𝑛−1,𝐹𝑥_𝑛−1,𝑦_𝑛−1+𝑟𝑥_𝑛−1,𝑦_𝑛−1𝑞𝑥_𝑛,𝐹𝑥_𝑛,𝑦_𝑛=𝑘𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+1⋮⪯𝑘𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑥₀,𝑦₀𝑞𝑥_𝑛,𝑥_𝑛+1. (3.19) Then, we have 𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑥₀,𝑦₀+𝑙𝑥₀,𝑦₀1−𝑟𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛=ℎ𝑞𝑥_𝑛−1,𝑥_𝑛⪯ℎ²𝑞𝑥_𝑛−2,𝑥_𝑛−1⋮⪯ℎ^𝑛𝑞𝑥₀,𝑥₁, (3.20) where ℎ=(𝑘(𝑥₀,𝑦₀)+𝑙(𝑥₀,𝑦₀))/(1−𝑟(𝑥₀,𝑦₀))<1.
Similarly we have 𝑞𝑦_𝑛,𝑦_𝑛+1⪯𝑘𝑦₀,𝑥₀+𝑙𝑦₀,𝑥₀1−𝑙𝑦₀,𝑥₀−𝑟𝑦₀,𝑥₀𝑞𝑦_𝑛−1,𝑦_𝑛=𝑑𝑞𝑦_𝑛−1,𝑦_𝑛⪯𝑑²𝑞𝑦_𝑛−2,𝑦_𝑛−1⋮⪯𝑑^𝑛𝑞𝑦₀,𝑦₁, (3.21) where 𝑑=(𝑘(𝑦₀,𝑥₀)+𝑙(𝑦₀,𝑥₀))/(1−𝑙(𝑦₀,𝑥₀)−𝑟(𝑦₀,𝑥₀))<1.
Let 𝑚>𝑛≥1. Then it follows that 𝑞𝑥_𝑛,𝑥_𝑚⪯𝑞𝑥_𝑛,𝑥_𝑛+1+𝑞𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑞𝑥_𝑚−1,𝑥_𝑚⪯ℎ^𝑛+ℎ^𝑛+1+⋯+ℎ^𝑚−1𝑞𝑥₀,𝑥₁⪯ℎ^𝑛𝑞𝑥1−ℎ⁰,𝑥₁ (3.22) and also 𝑞𝑦_𝑛,𝑦_𝑚⪯𝑞𝑦_𝑛,𝑦_𝑛+1+𝑞𝑦_𝑛+1,𝑦_𝑛+2+⋯+𝑞𝑦_𝑚−1,𝑦_𝑚⪯𝑑^𝑛+𝑑^𝑛+1+⋯+𝑑^𝑚−1𝑞𝑦₀,𝑦₁⪯𝑑^𝑛𝑞𝑦1−𝑑⁰,𝑦₁. (3.23) Thus, Lemma 2.6 (3) shows that {𝑥_𝑛} and {𝑦_𝑛} are Cauchy sequences in 𝑋. Since 𝑋 is complete, there exists 𝑥^∗ and 𝑦^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗ and 𝑦_𝑛→𝑦^∗ as 𝑛→∞. Since 𝐹 is continuous, then 𝑥^∗=lim𝑥_𝑛+1=lim𝐹(𝑥_𝑛,𝑦_𝑛)=𝐹(lim𝑥_𝑛,lim𝑦_𝑛)=𝐹(𝑥^∗,𝑦^∗). Similarly, 𝑦^∗=𝐹(𝑦^∗,𝑥^∗). Therefore, (𝑥^∗,𝑦^∗) is a coupled fixed point of 𝐹.
Suppose that 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then we have 𝑞𝑥₁,𝑥₁=𝑞𝐹𝑥₁,𝑦₁,𝐹𝑥₁,𝑦₁⪯𝑘𝑥₁,𝑥₁𝑞𝑥₁,𝑥₁+𝑙𝑥₁,𝑥₁𝑞𝑥₁,𝐹𝑥₁,𝑦₁+𝑟𝑥₁,𝑥₁𝑞𝑥₁,𝐹𝑥₁,𝑦₁=𝑘𝑥₁,𝑥₁𝑞𝑥₁,𝑥₁+𝑙𝑥₁,𝑥₁𝑞𝑥₁,𝑥₁+𝑟𝑥₁,𝑥₁𝑞𝑥₁,𝑥₁=(𝑘+𝑙+𝑟)𝑥₁,𝑥₁𝑞𝑥₁,𝑥₁. (3.24) Since (𝑘+𝑙+𝑟)(𝑥₁,𝑥₁)<1, Lemma 2.3 (1) shows that 𝑞(𝑥₁,𝑥₁)=𝜃. By similar way, 𝑞(𝑦₁,𝑦₁)=𝜃.
Finally, suppose that there is another coupled fixed point (𝑥^′,𝑦^′), then we have 𝑞𝑥^∗,𝑥^′=𝑞𝐹𝑥^∗,𝑦^∗,𝐹𝑥^′,𝑦^′⪯𝑘𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑙𝑥^∗,𝑦^∗𝑞𝑥^∗,𝐹𝑥^∗,𝑦^∗+𝑟𝑥^∗,𝑦^∗𝑞𝑥^′,𝐹𝑥^′,𝑦^′=𝑘𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑙𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^∗+𝑟𝑥^∗,𝑦^∗𝑞𝑥^′,𝑥^′=𝑘𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′⪯𝑘𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑙𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑟𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′=(𝑘+𝑙+𝑟)𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′. (3.25) Since (𝑘+𝑙+𝑟)(𝑥^∗,𝑦^∗)<1, Lemma 2.3 (1) shows that 𝑞(𝑥^∗,𝑥^′)=𝜃. Also we have 𝑞(𝑥^∗,𝑥^∗)=𝜃. Hence Lemma 2.6 (1) show that 𝑥^∗=𝑥^′. By similar way we have 𝑦^∗=𝑦^′ which implies that (𝑥^∗,𝑦^∗)=(𝑥^′,𝑦^′). Similarly, we prove that 𝑥^∗=𝑦^′ and 𝑦^∗=𝑥^′. Hence, 𝑥^∗=𝑦^∗. Therefore, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

From the above theorem, we have the following corollaries.

Corollary 3.5. Let (𝑋,𝑑) be a complete cone metric space, and 𝑞 is a c -distance on 𝑋. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 is continuous, and suppose that there exist mappings 𝑙,𝑟∶𝑋×𝑋→[0,1) such that the following hold: (a)𝑙(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑙(𝑥,𝑦) and 𝑟(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑟(𝑥,𝑦) for all 𝑥,𝑦,𝑢,𝑣∈𝑋,(b)(𝑙+𝑟)(𝑥,𝑦)<1 for all 𝑥,𝑦∈𝑋, and(c)𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑙(𝑥,𝑦)𝑞(𝑥,𝐹(𝑥,𝑦))+𝑟(𝑥,𝑦)𝑞(𝑢,𝐹(𝑢,𝑣)) for all 𝑥,𝑦,𝑢,𝑣∈𝑋.Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Corollary 3.6. Let (𝑋,𝑑) be a complete cone metric space, and 𝑞 is a c -distance on 𝑋. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 is continuous and satisfies the following contractive condition: 𝑞𝐹𝑥,𝑦,𝐹(𝑢,𝑣)⪯𝑘𝑞(𝑥,𝑢)+𝑙𝑞𝑥,𝐹𝑥,𝑦+𝑟𝑞(𝑢,𝐹(𝑢,𝑣)), (3.26) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘,𝑙,𝑟 are nonnegative constants with 𝑘+𝑙+𝑟<1. Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Corollary 3.7. Let (𝑋,𝑑) be a complete cone metric space, and 𝑞 is a c -distance on 𝑋. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 is continuous and satisfies the following contractive condition for all: 𝑞𝐹𝑥,𝑦,𝐹(𝑢,𝑣)⪯𝑙𝑞𝑥,𝐹𝑥,𝑦+𝑟𝑞(𝑢,𝐹(𝑢,𝑣)), (3.27) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑙,𝑟 are nonnegative constants with 𝑙+𝑟<1. Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁) then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Corollary 3.8. Let (𝑋,𝑑) be a complete cone metric space, and 𝑞 is a c -distance on 𝑋. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 is continuous and satisfies the following contractive condition for all: 𝑞𝐹𝑥,y,𝐹(𝑢,𝑣)⪯𝑙𝑞𝑥,𝐹𝑥,𝑦+𝑞(𝑢,𝐹(𝑢,𝑣)), (3.28) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑙∈[0,1/2) is a constant. Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Finally, we provide another result without condition (b) in Theorem 3.1, and we do not require that 𝐹 is continuous.

Theorem 3.9. Let (𝑋,𝑑) be a complete cone metric space, and 𝑞 is a c -distance on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping, and suppose that there exists mappings 𝑘,𝑙,𝑟∶𝑋×𝑋→[0,1) such that the following hold: (a)𝑘(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑘(𝑥,𝑦), 𝑙(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑙(𝑥,𝑦) and 𝑟(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑟(𝑥,𝑦) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, (b)(𝑘+2𝑙+𝑟)(𝑥,𝑦)<1 for all 𝑥,𝑦∈𝑋, and(c)(1−𝑟(𝑥,𝑦))𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘(𝑥,𝑦)𝑞(𝑥,𝐹(𝑥,𝑦))+𝑙(𝑥,𝑦)𝑞(𝑥,𝐹(𝑢,𝑣)) for all 𝑥,𝑦,𝑢,𝑣∈𝑋.Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Proof. Choose 𝑥₀,𝑦₀∈𝑋. Set 𝑥₁=𝐹(𝑥₀,𝑦₀),𝑦₁=𝐹(𝑦₀,𝑥₀),𝑥₂=𝐹(𝑥₁,𝑦₁),𝑦₂=𝐹(𝑦₁,𝑥₁),…,𝑥_𝑛+1=𝐹(𝑥_𝑛,𝑦_𝑛),𝑦_𝑛+1=𝐹(𝑦_𝑛,𝑥_𝑛). Observe that 1−𝑟𝑥,𝑦𝑞𝐹𝑥,𝑦,𝐹(𝑢,𝑣)⪯𝑘𝑥,𝑦𝑞𝑥,𝐹𝑥,𝑦+𝑙𝑥,𝑦𝑞(𝑥,𝐹(𝑢,𝑣)), (3.29) equivalently 𝑞𝐹𝑥,𝑦,𝐹(𝑢,𝑣)⪯𝑘𝑥,𝑦𝑞𝑥,𝐹𝑥,𝑦+𝑙𝑥,𝑦𝑞(𝑥,𝐹(𝑢,𝑣))+𝑟𝑥,𝑦𝑞𝐹𝑥,𝑦,𝐹(𝑢,𝑣). (3.30) Then we have the following: 𝑞𝑥_𝑛,𝑥_𝑛+1=𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛⪯𝑘𝑥_𝑛−1,𝑦_𝑛−1𝑞𝑥_𝑛−1,𝐹𝑥_𝑛−1,𝑦_𝑛−1+𝑙𝑥_𝑛−1,𝑦_𝑛−1𝑞𝑥_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛+𝑟𝑥_𝑛−1,𝑦_𝑛−1𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛=𝑘𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+1+𝑟𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+1+𝑟𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+1⋮⪯𝑘𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛+1+𝑟𝑥₀,𝑦₀𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥₀,𝑦₀𝑞𝑥_𝑛,𝑥_𝑛+1+𝑟𝑥₀,𝑦₀𝑞𝑥_𝑛,𝑥_𝑛+1. (3.31) Then, we have 𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑥₀,𝑦₀+𝑙𝑥₀,𝑦₀1−𝑙𝑥₀,𝑦₀−𝑟𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛=ℎ𝑞𝑥_𝑛−1,𝑥_𝑛⪯ℎ²𝑞𝑥_𝑛−2,𝑥_𝑛−1⋮⪯ℎ^𝑛𝑞𝑥₀,𝑥₁, (3.32) where ℎ=(𝑘(𝑥₀,𝑦_𝑜)+𝑙(𝑥₀,𝑦_𝑜))/(1−𝑙(𝑥₀,𝑦_𝑜)−𝑟(𝑥₀,𝑦_𝑜))<1.
Similarly we have 𝑞𝑦_𝑛,𝑦_𝑛+1⪯𝑘𝑦₀,𝑥₀+𝑙𝑦₀,𝑥₀1−𝑙𝑦₀,𝑥₀−𝑟𝑦₀,𝑥₀𝑞𝑦_𝑛−1,𝑦_𝑛=𝑑𝑞𝑦_𝑛−1,𝑦_𝑛⪯𝑑²𝑞𝑦_𝑛−2,𝑦_𝑛−1⋮⪯𝑑^𝑛𝑞𝑦₀,𝑦₁, (3.33) where 𝑑=(𝑘(𝑦₀,𝑥₀)+𝑙(𝑦₀,𝑥₀))/(1−𝑙(𝑦₀,𝑥₀)−𝑟(𝑦₀,𝑥₀))<1.
Let 𝑚>𝑛≥1. Then it follows that 𝑞𝑥_𝑛,𝑥_𝑚⪯𝑞𝑥_𝑛,𝑥_𝑛+1+𝑞𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑞𝑥_𝑚−1,𝑥_𝑚⪯ℎ^𝑛+ℎ^𝑛+1+⋯+ℎ^𝑚−1𝑞𝑥₀,𝑥₁⪯ℎ^𝑛𝑞𝑥1−ℎ⁰,𝑥₁ (3.34) and also 𝑞𝑦_𝑛,𝑦_𝑚⪯𝑞𝑦_𝑛,𝑦_𝑛+1+𝑞𝑦_𝑛+1,𝑦_𝑛+2+⋯+𝑞𝑦_𝑚−1,𝑦_𝑚⪯𝑑^𝑛+𝑑^𝑛+1+⋯+𝑑^𝑚−1𝑞𝑦₀,𝑦₁⪯𝑑^𝑛𝑞𝑦1−𝑑⁰,𝑦₁. (3.35) Thus, Lemma 2.6 (3) shows that {𝑥_𝑛} and {𝑦_𝑛} are Cauchy sequences in 𝑋. Since 𝑋 is complete, there exists 𝑥^∗ and 𝑦^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗ and 𝑦_𝑛→𝑦^∗ as 𝑛→∞. By (q3) we have 𝑞𝑥_𝑛,𝑥^∗⪯ℎ^𝑛𝑞𝑥1−ℎ⁰,𝑥₁,𝑞𝑦_𝑛,𝑦^∗⪯𝑑^𝑛𝑞𝑦1−𝑑⁰,𝑦₁. (3.36)(3.37)
On the other hand, we have 𝑞𝑥_𝑛,𝐹𝑥^∗,𝑦^∗=𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥^∗,𝑦^∗⪯𝑘𝑥_𝑛−1,𝑦_𝑛−1𝑞𝑥_𝑛−1,𝐹𝑥_𝑛−1,𝑦_𝑛−1+𝑙𝑥_𝑛−1,𝑦_𝑛−1𝑞𝑥_𝑛−1,𝐹𝑥^∗,𝑦^∗+𝑟𝑥_𝑛−1,𝑦_𝑛−1𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥^∗,𝑦^∗=𝑘𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝐹𝑥_n−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑥_𝑛−1,𝐹𝑥^∗,𝑦^∗+𝑟𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞𝑥_𝑛,𝐹𝑥^∗,𝑦^∗⪯𝑘𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑥_𝑛−1,𝐹𝑥^∗,𝑦^∗+𝑟𝑥_𝑛−2,𝑦_𝑛−2𝑞𝑥_𝑛,𝐹𝑥^∗,𝑦^∗⋮⪯𝑘𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝐹𝑥^∗,𝑦^∗+𝑟𝑥₀,𝑦₀𝑞𝑥_𝑛,𝐹𝑥^∗,𝑦^∗⪯𝑘𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥₀,𝑦₀𝑞𝑥_𝑛,𝐹𝑥^∗,𝑦^∗+𝑟𝑥₀,𝑦₀𝑞𝑥_𝑛,𝐹𝑥^∗,𝑦^∗. (3.38) Then, we have 𝑞𝑥_𝑛,𝐹𝑥^∗,𝑦^∗⪯𝑘𝑥₀,𝑦₀+𝑙𝑥₀,𝑦₀1−𝑙𝑥₀,𝑦₀−𝑟𝑥₀,𝑦₀𝑞𝑥_𝑛−1,𝑥_𝑛=ℎ𝑞𝑥_𝑛−1,𝑥_𝑛⪯ℎℎ^𝑛−1𝑞𝑥₀,𝑥₁=ℎ^𝑛𝑞𝑥₀,𝑥₁⪯ℎ^𝑛𝑞𝑥1−ℎ⁰,𝑥₁. (3.39) By Lemma 2.6 (1), (3.36), and (3.39), we have 𝑥^∗=𝐹(𝑥^∗,𝑦^∗).
By similar way we have 𝑞𝑦_𝑛,𝐹𝑦^∗,𝑥^∗⪯𝑑^𝑛𝑞𝑦1−𝑑⁰,𝑦₁. (3.40) By Lemma 2.6 (1), (3.37), and (3.40), we have 𝑦^∗=𝐹(𝑦^∗,𝑥^∗). Therefore, (𝑥^∗,𝑦^∗) is a coupled fixed point of 𝐹.
Suppose that 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then we have 𝑞𝑥₁,𝑥₁=𝑞𝐹𝑥₁,𝑦₁,𝐹𝑥₁,𝑦₁⪯𝑘𝑥₁,𝑦₁𝑞𝑥₁,𝐹𝑥₁,𝑦₁+𝑙𝑥₁,𝑦₁𝑞𝑥₁,𝐹𝑥₁,𝑦₁+𝑟𝑥₁,𝑦₁𝑞𝐹𝑥₁,𝑦₁,𝐹𝑥₁,𝑦₁=𝑘𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁+𝑙𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁+𝑟𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁⪯𝑘𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁+𝑙𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁+𝑙𝑥₁,y₁𝑞𝑥₁,𝑥₁+𝑟𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁=(𝑘+2𝑙+𝑟)𝑥₁,𝑦₁𝑞𝑥₁,𝑥₁. (3.41) Since (𝑘+2𝑙+𝑟)(𝑥₁,𝑦₁)<1, Lemma 2.3 (1) shows that 𝑞(𝑥₁,𝑥₁)=𝜃. By similar way, 𝑞(𝑦₁,𝑦₁)=𝜃.
Finally, suppose that there is another coupled fixed point (𝑥^′,𝑦^′), then we have 𝑞𝑥^∗,𝑥^′=𝑞𝐹𝑥^∗,𝑦^∗,𝐹𝑥^′,𝑦^′⪯𝑘𝑥^∗,𝑦^∗𝑞𝑥^∗,𝐹𝑥^∗,𝑦^∗+𝑙𝑥^∗,𝑦^∗𝑞𝑥^∗,𝐹𝑥^′,𝑦^′+𝑟𝑥^∗,𝑦^∗𝑞𝐹𝑥^∗,𝑦^∗,𝐹𝑥^′,𝑦^′=𝑘𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^∗+𝑙𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑟𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′=𝑙𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑟𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′⪯𝑘𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑙𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑙𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′+𝑟𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′=(𝑘+2𝑙+𝑟)𝑥^∗,𝑦^∗𝑞𝑥^∗,𝑥^′. (3.42) Since (𝑘+2𝑙+𝑟)(𝑥^∗,𝑦^∗)<1, Lemma 2.3 (1) shows that 𝑞(𝑥^∗,𝑥^′)=𝜃. Also we have 𝑞(𝑥^∗,𝑥^∗)=𝜃. Hence Lemma 2.6 (1) shows that 𝑥^∗=𝑥^′. By similar way we have 𝑦^∗=𝑦^′ which implies that (𝑥^∗,𝑦^∗)=(𝑥^′,𝑦^′). Similarly, we prove that 𝑥^∗=𝑦^′ and 𝑦^∗=𝑥^′. Hence, 𝑥^∗=𝑦^∗. Therefore the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

From the above theorem, we have the following corollaries.

Corollary 3.10. Let (𝑋,𝑑) be a complete cone metric space, and 𝑞 is a c -distance on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping, and suppose that there exists mappings 𝑘,𝑙∶𝑋×𝑋→[0,1) such that the following hold: (a)𝑘(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑘(𝑥,𝑦) and 𝑙(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑙(𝑥,𝑦) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, (b)(𝑘+2𝑙)(𝑥,𝑦)<1 for all 𝑥,𝑦∈𝑋, and(c)𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘(𝑥,𝑦)𝑞(𝑥,𝐹(𝑥,𝑦))+𝑙(𝑥,𝑦)𝑞(𝑥,𝐹(𝑢,𝑣)) for all 𝑥,𝑦,𝑢,𝑣∈𝑋.Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Corollary 3.11. Let (𝑋,𝑑) be a complete cone metric space, and 𝑞 is a c -distance on 𝑋. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition: (1−𝑟)𝑞𝐹𝑥,𝑦,𝐹(𝑢,𝑣)⪯𝑘𝑞𝑥,𝐹𝑥,𝑦+𝑙𝑞(𝑥,𝐹(𝑢,𝑣)), (3.43) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘,𝑙,𝑟 are nonnegative constants with 𝑘+2𝑙+𝑟<1. Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Corollary 3.12. Let (𝑋,𝑑) be a complete cone metric space, and 𝑞 is a c -distance on 𝑋. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition: 𝑞𝐹𝑥,𝑦,𝐹(𝑢,𝑣)⪯𝑘𝑞𝑥,𝐹𝑥,𝑦+𝑙𝑞(𝑥,𝐹(𝑢,𝑣)), (3.44) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘,𝑙 are nonnegative constants with 𝑘+2𝑙<1. Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈X×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Corollary 3.13. Let (𝑋,𝑑) be a complete cone metric space, and 𝑞 is a c -distance on 𝑋. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition: 𝑞𝐹𝑥,𝑦,𝐹(𝑢,𝑣)⪯𝑘𝑞𝑥,𝐹𝑥,𝑦+𝑞(𝑥,𝐹(𝑢,𝑣)), (3.45) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘∈[0,1/3) is a constant. Then 𝐹 has a coupled fixed point (𝑥^∗,𝑦^∗)∈𝑋×𝑋. Further, if 𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑥₁,𝑥₁)=𝜃 and 𝑞(𝑦₁,𝑦₁)=𝜃. Moreover, the coupled fixed point is unique and is of the form (𝑥^∗,𝑥^∗) for some 𝑥^∗∈𝑋.

Example 3.14. Consider Example 2.5. Define the mapping 𝐹∶𝑋×𝑋→𝑋 by 𝐹(𝑥,𝑦)=(𝑥+𝑦)/4 for all (𝑥,𝑦)∈𝑋×𝑋. Then we have, 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))=𝐹(𝑢,𝑣)=(𝑢+𝑣)/4=𝑢/4+𝑣/4⪯(3/7)𝑢+(3/7)𝑣=𝑘𝑞(𝑥,𝑢)+𝑙𝑞(𝑦,𝑣) with 𝑘=𝑙=3/7, and 𝑘+𝑙=6/7<1. Therefore, the conditions of Theorem 3.1 are satisfied, and then 𝐹 has a unique coupled fixed point (𝑥,𝑦)=(0,0) and 𝐹(0,0)=0 with 𝑞(0,0)=0.

Example 3.15. Consider Example 2.5. Define the mapping 𝐹∶𝑋×𝑋→𝑋 by 𝐹(7/8,7/8)=1/4 and 𝐹(𝑥,𝑦)=(𝑥+𝑦)/4 for all (𝑥,𝑦)∈𝑋×𝑋 with (𝑥,𝑦)≠(7/8,7/8). Since 𝑑(𝐹(1,1),𝐹(7/8,7/8))=𝑑(1,7/8)+𝑑(1,7/8), there is not 𝑘,𝑙∈[0,1) such that 𝑑(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘𝑑(𝑥,𝑢)+𝑙𝑑(𝑦,𝑣) for all 𝑥,𝑦,𝑢,𝑣 in 𝑋. Since Theorem 1.2 of Sabetghadam et al. [15, Theorem  2.2] cannot applied to this example on cone metric space. To check this example on c-distance, we have (1)If (𝑥,𝑦)=(𝑢,𝑣)=(7/8,7/8), then we have 𝑞(𝐹(7/8,7/8),𝐹(7/8,7/8))=𝐹(7/8,7/8)=1/4⪯𝑘(7/8)+𝑙(7/8)=𝑘𝑞(7/8,7/8)+𝑙𝑞(7/8,7/8) with 𝑘=𝑙=3/7, and 𝑘+𝑙=6/7<1.(2)If (𝑥,𝑦)≠(𝑢,𝑣)≠(7/8,7/8), then we have 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))=F(𝑢,𝑣)=(𝑢+𝑣)/4=𝑢/4+𝑣/4⪯(3/7)𝑢+(3/7)𝑣=𝑘𝑞(𝑥,𝑢)+𝑙𝑞(𝑦,𝑣) with 𝑘=𝑙=3/7, and 𝑘+𝑙=6/7<1.(3)If (𝑥,𝑦)=(7/8,7/8),(𝑢,𝑣)≠(7/8,7/8), then we have 𝑞(𝐹(7/8,7/8),𝐹(𝑢,𝑣))=𝐹(𝑢,𝑣)=(𝑢+𝑣)/4=𝑢/4+𝑣/4⪯(3/7)𝑢+(3/7)𝑣=𝑘𝑞(7/8,u)+𝑙𝑞(7/8,𝑣) with 𝑘=𝑙=3/7, and 𝑘+𝑙=6/7<1.(4)If (𝑥,𝑦)≠(7/8,7/8),(𝑢,𝑣)=(7/8,7/8), then we have 𝑞(𝐹(𝑥,𝑦),𝐹(7/8,7/8))=𝐹(7/8,7/8)=1/4⪯𝑘(7/8)+𝑙(7/8)=𝑘𝑞(𝑥,7/8)+𝑙𝑞(𝑦,7/8) with 𝑘=𝑙=3/7, and 𝑘+𝑙=6/7<1.Hence 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘𝑞(𝑥,𝑢)+𝑙𝑞(𝑦,𝑣) for all 𝑥,𝑦,𝑢,𝑣 with 𝑘=𝑙=3/7, and 𝑘+𝑙=6/7<1. Therefore, the conditions of Theorem 3.1 are satisfied, and then 𝐹 has a unique coupled fixed point (𝑥,𝑦)=(0,0) and 𝐹(0,0)=0 with 𝑞(0,0)=0.

Remark 3.16. In Example 3.14, it is easy to see that 𝑑(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘𝑑(𝑥,𝑢)+𝑙𝑑(𝑦,𝑣) for all 𝑥,𝑦,𝑢,𝑣 in 𝑋 with 𝑘=𝑙=1/4. Therefore, the condition of Theorem 1.2 of Sabetghadam et al. [15, Theorem  2.2] is satisfied, and then 𝐹 has a unique coupled fixed point (𝑥,𝑦)=(0,0), 𝐹(0,0)=0 with 𝑞(0,0)=0.

Example 3.17. Let 𝐸=ℝ² and 𝑃={(𝑥,𝑦)∈𝐸∶𝑥,𝑦≥0}. Let 𝑋=[0,1) and define a mapping 𝑑∶𝑋×𝑋→𝐸 by 𝑑(𝑥,𝑦)=(|𝑥−𝑦|,|𝑥−𝑦|) for all 𝑥,𝑦∈𝑋. Then (𝑋,𝑑) is a complete cone metric space, see [15]. Define a mapping 𝑞∶𝑋×𝑋→𝐸 by 𝑞(𝑥,𝑦)=(𝑦,𝑦) for all 𝑥,𝑦∈𝑋. Then 𝑞 is a c-distance on 𝑋. In fact (q1)–(q3) are immediate. Let 𝑐∈𝐸 with 𝜃≪𝑐 and put 𝑒=𝑐/2. If 𝑞(𝑧,𝑥)≪𝑒 and 𝑞(𝑧,𝑦)≪𝑒, then we have 𝑑=,⪯=()+()𝑥,𝑦𝑥−𝑦𝑥−𝑦𝑥+𝑦,𝑥+𝑦𝑥,𝑥𝑦,𝑦=𝑞𝑧,𝑥+𝑞𝑧,𝑦≪𝑒+𝑒=𝑐. (3.46) This shows that (q4) holds. Therefore, 𝑞 is a c-distance on 𝑋. Define the mapping 𝐹∶𝑋×𝑋→𝑋 by 𝐹(𝑥,𝑦)=(𝑥+𝑦)/4 for all (𝑥,𝑦)∈𝑋×𝑋. Then we have, 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))=(𝐹(𝑢,𝑣),𝐹(𝑢,𝑣))=((𝑢+𝑣)/4,(𝑢+𝑣)/4) = (𝑢/4,𝑢/4)+(𝑣/4,𝑣/4)=(1/4)(𝑢, 𝑢)+(1/4)(𝑣,𝑣)⪯(3/7)(𝑢,𝑢) + (3/7)(𝑣,𝑣)=𝑘𝑞(𝑥,𝑢)+𝑙𝑞(𝑦,𝑣) with 𝑘=𝑙=3/7, and 𝑘+𝑙=6/7<1. Therefore, the conditions of Theorem 3.1 are satisfied, and then 𝐹 has a unique coupled fixed point (𝑥,𝑦)=(0,0) and 𝐹(0,0)=0 with 𝑞(0,0)=0.

Acknowledgments

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research Grant no. OUP-UKM-FST-2012. The authors thank the referee for his/her careful reading of the manuscript and useful suggestions.