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

Abstract

The existence and uniqueness of the common coupled fixed point in cone metric spaces have been studied by considering different types of contractive conditions. A new concept of the c-distance in cone metric space has been recently introduced in 2011. Then, coupled fixed point results for contraction-type mappings in ordered cone metric spaces and cone metric spaces have been considered. In this paper, some common coupled fixed point results on c-distance in cone metric spaces are obtained. Some supporting 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. Subsequently, several authors have studied the existence and uniqueness of the fixed point and common fixed point for self-map 𝑓 by considering 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. Subsequently, several authors have studied the existence and uniqueness of the coupled fixed point and common coupled fixed point by considering 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. Subsequently, several authors have studied on the generalized distance in cone metric space. Some of this works are noted in [26–31].

In [30], Fadail and Ahmad proved some coupled fixed point theorems in cone metric spaces by using the concept of c-distance.

Recall that 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.

Definition 1.1. An element (𝑥,𝑦)∈𝑋×𝑋 is called (1)a coupled coincidence point of mappings 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 if g𝑥=𝐹(𝑥,𝑦) and g𝑦=𝐹(𝑦,𝑥), and (g𝑥,g𝑦) is called coupled point of coincidence. (2)a common coupled fixed point of mappings 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 if 𝑥=g𝑥=𝐹(𝑥,𝑦),𝑦=g𝑦=𝐹(𝑦,𝑥).

Abbas et al. [20] introduced the following definition.

Definition 1.2. The mappings 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 are called w-compatible if g(F(𝑥,𝑦))=𝐹(g𝑥,g𝑦), whenever g𝑥=𝐹(𝑥,𝑦) and g𝑦=𝐹(𝑦,𝑥).

The aim of this paper is to continue the study of common coupled fixed points of mappings but now for c-distance in cone metric space. Our theorems extend and develop some theorems in literature to c-distance in cone metric spaces. In this paper, we do not impose the normality condition for the cones, the only assumption is that cone 𝑃 has nonempty interior.

2. Preliminaries

Let 𝐸 be a real Banach space and 𝜃 denote 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, (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 ≺ stand 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‖≤𝐾‖𝑦‖,(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 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, (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} a sequence in 𝑋 and 𝑥∈𝑋. (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. (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. [32] 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, (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 hold 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 common coupled fixed point results using c-distance in cone metric space. Also, we generalize the contractive conditions in literature by replacing the constants with functions.

Theorem 3.1. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃 having nonempty interior and 𝑞 is a c-distance on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 be two mappings 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(g𝑥,g𝑦) and 𝑙(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑙(g𝑥,g𝑦) 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(g𝑥,g𝑦)𝑞(g𝑥,g𝑢)+𝑙(g𝑥,g𝑦)𝑞(g𝑦,g𝑣) for all 𝑥,𝑦,𝑢,𝑣∈𝑋. If 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋, then 𝐹 and g have a unique coupled point of coincidence (𝑢,𝑣) in 𝑋×𝑋. Further, if 𝑢=g𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑣=g𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Moreover, if 𝐹 and g are w-compatible, then 𝐹 and g have a unique common coupled fixed point and the common coupled fixed point of 𝐹 and g is of the form (𝑢,𝑢) for some 𝑢∈𝑋.

Proof. Choose 𝑥₀,𝑦₀∈𝑋. Set g𝑥₁=𝐹(𝑥₀,𝑦₀),g𝑦₁=𝐹(𝑦₀,𝑥₀), this can be done because 𝐹(𝑋×𝑋)⊆g(𝑋). Continuing this process, we obtain two sequences {𝑥_𝑛} and {𝑦_𝑛} such that g𝑥_𝑛+1=𝐹(𝑥_𝑛,𝑦_𝑛),g𝑦_𝑛+1=𝐹(𝑦_𝑛,𝑥_𝑛). Then we have 𝑞g𝑥_𝑛,g𝑥_𝑛+1=⪯𝑞𝑘𝐹g𝑥𝑥_{𝑛−1𝑛−1},𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛+𝑙g𝑥,g_𝑛−1𝑦_𝑛−1𝑞g𝑥_𝑛−1,g𝑥_𝑛=𝑘𝐹𝑥_𝑛−2,g𝑦_𝑛−1𝑞g𝑦_𝑛−1,g𝑦_𝑛+𝑙𝐹𝑥,𝑦_{𝑛−2𝑛−2},𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞g𝑥_𝑛−1,g𝑥_𝑛⪯𝑘g𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞g𝑦_𝑛−1,gy_𝑛+𝑙g𝑥,g_𝑛−2𝑦_𝑛−2𝑞g𝑥_𝑛−1,g𝑥_𝑛⋮⪯𝑘g𝑥₁,g𝑦_𝑛−2𝑞g𝑦_𝑛−1,g𝑦_𝑛,g𝑦₁𝑞g𝑥_𝑛−1,g𝑥_𝑛+𝑙g𝑥₁,g𝑦₁𝑞g𝑦_𝑛−1,g𝑦_𝑛.(3.1) Similarly, we have 𝑞g𝑦_𝑛,g𝑦_𝑛+1=⪯𝑞𝑘𝐹g𝑦𝑦_{𝑛−1𝑛−1},𝑥_𝑛−1,𝐹𝑦_𝑛,𝑥_𝑛=𝑘𝐹𝑦,g_𝑛−2𝑥_𝑛−1𝑞g𝑦_𝑛−1,g𝑦_𝑛+𝑙g𝑦_𝑛−1,g𝑥_𝑛−1𝑞g𝑥_𝑛−1,g𝑥_𝑛+𝑙𝐹𝑦,𝑥_{𝑛−2𝑛−2},𝐹𝑥_𝑛−2,𝑦_𝑛−2𝑞g𝑦_𝑛−1,g𝑦_𝑛⪯𝑘g𝑦_𝑛−2,𝑥_𝑛−2,𝐹𝑥_𝑛−2,𝑦_𝑛−2𝑞g𝑥_𝑛−1,g𝑥_𝑛⋮⪯𝑘g𝑦₁,g𝑥_𝑛−2𝑞g𝑦_𝑛−1,g𝑦_𝑛+𝑙g𝑦_𝑛−2,g𝑥_𝑛−2𝑞g𝑥_𝑛−1,g𝑥_𝑛=𝑘g𝑥₁,g𝑥₁𝑞g𝑦_𝑛−1,g𝑦_𝑛+𝑙g𝑦₁,g𝑥₁𝑞g𝑥_𝑛−1,g𝑥_𝑛,g𝑦₁𝑞g𝑦_𝑛−1,g𝑦_𝑛+𝑙g𝑥₁,g𝑦₁𝑞g𝑥_𝑛−1,g𝑥_𝑛.(3.2) Put 𝑞_𝑛=𝑞(g𝑥_𝑛,g𝑥_𝑛+1)+𝑞(g𝑦_𝑛,g𝑦_𝑛+1). Then we have 𝑞_𝑛=⪯𝑞𝑘gg𝑥𝑥_𝑛1,g𝑥_𝑛+1+𝑞g𝑦_𝑛,g𝑦_𝑛+1+𝑘g,𝑥g₁𝑦₁𝑞g𝑥_𝑛−1,g𝑥_𝑛+𝑙g𝑥₁,g𝑦₁𝑞g𝑦_𝑛−1,g𝑦_𝑛=𝑘g𝑥₁,g𝑦₁𝑞g𝑦_𝑛−1,g𝑦_𝑛+𝑙g𝑥₁,g𝑦₁𝑞g𝑥_𝑛−1,g𝑥_𝑛=𝑘g𝑥₁,g𝑦₁+𝑙g𝑥₁,g𝑦₁𝑞g𝑥_𝑛−1,g𝑥_𝑛+𝑞g𝑦_𝑛−1,g𝑦_𝑛,g𝑦₁+𝑙g𝑥₁,g𝑦₁𝑞_𝑛−1=ℎ𝑞_𝑛−1⋮⪯ℎ_𝑛−1𝑞₁,(3.3) where ℎ=𝑘(g𝑥₁,g𝑦₁)+𝑙(g𝑥₁,g𝑦₁)<1.
Let 𝑚>𝑛≥1. It follows that 𝑞𝑞gg𝑥𝑦_𝑛𝑛,g𝑥_𝑚⪯𝑞g𝑥_𝑛,g𝑥_𝑛+1+𝑞g𝑥_𝑛+1,g𝑥_𝑛+2+⋯+𝑞g𝑥_𝑚−1,g𝑥_𝑚,,g𝑦_𝑚⪯𝑞g𝑦_𝑛,g𝑦_𝑛+1+𝑞g𝑦_𝑛+1,g𝑥_𝑛+2+⋯+𝑞g𝑦_𝑚−1,g𝑦_𝑚.(3.4) Then we have 𝑞g𝑥_𝑛,g𝑥_𝑚+𝑞g𝑦_𝑛,g𝑦_𝑚⪯𝑞_𝑛+𝑞_𝑛+1+⋯+𝑞_𝑚−1⪯ℎ^𝑛𝑞₁+ℎ^𝑛+1𝑞₁+⋯+ℎ^𝑚−1𝑞₁=ℎ^𝑛−1+ℎ^𝑛+⋯+ℎ^𝑚−2𝑞₁⪯ℎ^𝑛−11−ℎ𝑞¹.(3.5) Consequently, 𝑞g𝑥_𝑛,g𝑥_𝑚⪯ℎ^𝑛−11−ℎ𝑞¹𝑞g𝑦_𝑛,,g𝑦_𝑚⪯ℎ^𝑛−11−ℎ𝑞¹.(3.6) Thus, Lemma 2.6 (3) shows that {g𝑥_𝑛} and {g𝑦_𝑛} are Cauchy sequences in g(𝑋). Since g(𝑋) is complete, there exists 𝑥^∗ and 𝑦^∗ in 𝑋 such that g𝑥_𝑛→g𝑥^∗ and g𝑦_𝑛→g𝑦^∗ as 𝑛→∞. Using q3, we have 𝑞g𝑥_𝑛,g𝑥^∗⪯ℎ^𝑛−11−ℎ𝑞¹𝑞g𝑦_𝑛,,g𝑦^∗⪯ℎ^𝑛−11−ℎ𝑞¹.(3.7)
On the other hand, 𝑞g𝑥_𝑛,𝐹𝑥^∗,𝑦^∗=⪯𝑘𝑞𝐹g𝑥𝑥_{𝑛−1𝑛−1},𝑦_𝑛−1,𝐹𝑥^∗,𝑦^∗+𝑙g𝑥,g_𝑛−1𝑦_𝑛−1𝑞g𝑥_𝑛−1,g𝑥^∗=𝑘F𝑥_𝑛−2,g𝑦_𝑛−1𝑞g𝑦_𝑛−1,g𝑦^∗+𝑙𝐹𝑥,𝑦_{𝑛−2𝑛−2},𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞g𝑥_𝑛−1,g𝑥^∗⪯𝑘g𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞g𝑦_𝑛−1,g𝑦^∗+𝑙g𝑥,g_𝑛−2𝑦_𝑛−2𝑞g𝑥_𝑛−1,g𝑥^∗⋮⪯𝑘g𝑥₁,g𝑦_𝑛−2𝑞g𝑦_𝑛−1,g𝑦^∗⪯𝑘g𝑥₁,g𝑦₁𝑞g𝑥_𝑛−1,g𝑥^∗+𝑙g𝑥₁,g𝑦₁𝑞g𝑦_𝑛−1,g𝑦^∗,g𝑦₁ℎ^𝑛−21−ℎ𝑞¹+𝑙𝑥₁,𝑦₁ℎ^𝑛−21−ℎ𝑞¹=𝑘g𝑥₁,g𝑦₁+𝑙g𝑥₁,g𝑦₁ℎ^𝑛−21−ℎ𝑞¹=ℎℎ^𝑛−21−ℎ𝑞¹=ℎ^𝑛−11−ℎ𝑞¹.(3.8) Thus, Lemma 2.6 (1), (3.5), and (3.8) show that g𝑥^∗=𝐹(𝑥^∗,𝑦^∗). By similar way, we can prove that g𝑦^∗=𝐹(𝑦^∗,𝑥^∗). Therefore, (𝑥^∗,𝑦^∗) is a coupled coincidence point of 𝐹 and g.
Suppose that 𝑢=g𝑥^∗=𝐹(𝑥^∗,𝑦^∗) and 𝑣=g𝑦^∗=𝐹(𝑦^∗,𝑥^∗). Then we have 𝑞(𝑢,𝑢)==𝑞𝑞g𝐹𝑥^∗𝑥,g_∗𝑥^∗⪯𝑘g𝑥_∗,𝑦^∗,𝐹𝑥^∗,𝑦^∗𝑞(𝑣,𝑣)==𝑞𝑘(𝑢g,𝑦𝑣_∗),𝑞g𝑦(𝑢^∗,𝑢𝑞)+g𝑥𝑙(^∗𝑢,,g𝑣𝑥)^∗𝑞(+𝑣,𝑙𝑣)g,𝑥^∗,g𝑦^∗𝑞g𝑦^∗,g𝑦^∗(3.9)=𝑞𝐹𝑦,g_∗𝑦^∗⪯𝑘g𝑦_∗,𝑥^∗,𝐹𝑦^∗,𝑥^∗==𝑘𝑘((𝑣𝑢,,𝑣𝑢)),𝑞𝑞g𝑥((𝑣𝑣^∗,,𝑣𝑣𝑞))+g+𝑦𝑙𝑙^∗(,𝑣(𝑢g,,𝑦𝑢𝑣^∗))𝑞𝑞+(𝑢(𝑢,𝑙,𝑢𝑢)g)𝑦.^∗,g𝑥^∗𝑞g𝑥^∗,g𝑥^∗(3.10) This implies that 𝑞(𝑢,𝑢)+𝑞(𝑣,𝑣)⪯==𝑘[[𝑘((𝑘𝑢(,𝑢+𝑣,𝑣)𝑙)𝑞)(+(𝑢𝑢,𝑙,𝑣𝑢()𝑢)],+[𝑣𝑞)𝑙(]𝑢([𝑢,𝑞,𝑢(𝑣)𝑢)+,𝑞𝑢𝑞()𝑣+(,𝑣𝑣,𝑞)𝑣(+)𝑣],.𝑘𝑣)(]𝑢,𝑣)𝑞(𝑣,𝑣)+𝑙(𝑢,𝑣)𝑞(𝑢,𝑢)(3.11) Since (𝑘+𝑙)(𝑢,𝑣)<1, Lemma 2.3 (1) shows that 𝑞(𝑢,𝑢)+𝑞(𝑣,𝑣)=𝜃. But 𝑞(𝑢,𝑢)≽𝜃 and 𝑞(𝑣,𝑣)≽𝜃. Consequently, 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃.
Finally, suppose there is another coupled point of coincidence (𝑢₁,𝑣₁) of 𝐹 and g such that 𝑢₁=g𝑥^=𝐹(𝑥^,𝑦^) and 𝑣₁=g𝑦^=𝐹(𝑦^,𝑥^) for some (𝑥^,𝑦^) in 𝑋×𝑋. Then we have 𝑞𝑢,𝑢₁==𝑞𝑞g𝐹𝑥^∗𝑥,g_∗𝑥^⪯𝑘g𝑥_∗,𝑦^∗,F𝑥^,𝑦^=𝑘(𝑣,𝑢),𝑞g𝑦^∗𝑣,𝑣𝑞₁g𝑥^∗,g𝑥^+𝑙g𝑥^∗,g𝑦^∗𝑞g𝑦^∗,g𝑦^=𝑘(𝑢,𝑣)𝑞𝑢,𝑢₁+𝑙(𝑣,𝑢)𝑞𝑢,𝑢₁+𝑙(𝑢,𝑣)𝑞𝑣,𝑣₁,(3.12)
and also, 𝑞𝑣,𝑣₁==𝑞𝑞𝐹g𝑦^∗𝑦,g_∗𝑦^⪯𝑘g𝑦_∗,𝑥^∗,F𝑦^,𝑥^=𝑘(𝑣,𝑢),𝑞g𝑥^∗𝑣,𝑣𝑞₁g𝑦^∗,g𝑦^+𝑙g𝑦^∗,g𝑥^∗𝑞g𝑥^∗,g𝑥^=𝑘(𝑢,𝑣)𝑞𝑣,𝑣₁+𝑙(𝑣,𝑢)𝑞𝑢,𝑢₁+𝑙(𝑢,𝑣)𝑞𝑢,𝑢₁.(3.13) This implies that 𝑞𝑢,𝑢₁+𝑞𝑣,𝑣₁=⪯𝑘[𝑘(𝑢(,𝑢𝑣,𝑣)𝑞)+𝑢𝑙,(𝑢𝑢₁,𝑣+)]𝑙𝑞(𝑢,𝑢𝑣,)𝑢𝑞₁𝑣,𝑣₁+𝑘(𝑢,𝑣)𝑞𝑣,𝑣₁+𝑙(𝑢,𝑣)𝑞𝑢,𝑢₁=[(𝑘+𝑙)(𝑢,𝑣)]𝑞𝑢,𝑢₁+𝑞𝑣,𝑣₁+𝑞𝑣,𝑣₁.(3.14) Since (𝑘+𝑙)(𝑢,𝑣)<1, Lemma 2.3 (1) shows that 𝑞(𝑢,𝑢₁)+𝑞(𝑣,𝑣₁)=𝜃. But 𝑞(𝑢,𝑢₁)≽𝜃 and 𝑞(𝑣,𝑣₁)≽𝜃. Hence 𝑞(𝑢,𝑢₁)=𝜃 and 𝑞(𝑣,𝑣₁)=𝜃. Also we have 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Thus, Lemma 2.6 (1) shows that 𝑢=𝑢₁ and 𝑣=𝑣₁, which implies that (𝑢,𝑣)=(𝑢₁,𝑣₁). Similarly, we can prove that 𝑢=𝑣₁ and 𝑣=𝑢₁. Thus, 𝑢=𝑣. Therefore, (𝑢,𝑢) is the unique coupled point of coincidence. Now, let 𝑢=g𝑥^∗=𝐹(𝑥^∗,𝑦^∗). Since 𝐹 and g are w-compatible, then we have g𝑢=gg𝑥^∗=g𝐹𝑥^∗,𝑦^∗=𝐹g𝑥^∗,g𝑦^∗=𝐹g𝑥^∗,g𝑥^∗=𝐹(𝑢,𝑢).(3.15) Thus (g𝑢,g𝑢) is a coupled point of coincidence. The uniqueness of the coupled point of coincidence implies that g𝑢=𝑢. Therefore, 𝑢=g𝑢=𝐹(𝑢,𝑢). Hence (𝑢,𝑢) is the unique common coupled fixed point of 𝐹 and g.

The following corollaries can be obtained as consequences of this theorem.

Corollary 3.2. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃 having nonempty interior and 𝑞 is a c-distance on 𝑋. Suppose the mappings 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 satisfy the following contractive condition: 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘𝑞(g𝑥,g𝑢)+𝑙𝑞(g𝑦,g𝑣),(3.16) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘,𝑙 are nonnegative constants with 𝑘+𝑙<1. If 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋, then 𝐹 and g have a coupled coincidence point in 𝑋. Further, if g𝑥₁=𝐹(𝑥₁,𝑦₁) and g𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(g𝑥₁,g𝑥₁)=𝜃 and 𝑞(g𝑦₁,g𝑦₁)=𝜃. Moreover, if 𝐹 and g are w -compatible, then 𝐹 and g have a unique common coupled fixed point and the common coupled fixed point of 𝐹 and g is of the form (𝑢,𝑢) for some 𝑢∈𝑋.

Corollary 3.3. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃 having nonempty interior and 𝑞 is a c-distance on 𝑋. Suppose the mappings 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 satisfy the following contractive condition: 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘𝑞(g𝑥,g𝑢)+𝑞(g𝑦,g𝑣),(3.17) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘∈[0,1/2) is a constants. If 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋, then 𝐹 and g have a unique coupled point of coincidence (𝑢,𝑣) in 𝑋×𝑋. Further, if 𝑢=g𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑣=g𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Moreover, if 𝐹 and g are w -compatible, then 𝐹 and g have a unique common coupled fixed point and the common coupled fixed point of 𝐹 and g is of the form (𝑢,𝑢) for some 𝑢∈𝑋.

Theorem 3.4. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃 having nonempty interior and 𝑞 is a c-distance on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 be two mappings 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(g𝑥,g𝑦) and 𝑙(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑙(g𝑥,g𝑦) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, (b)(𝑘+𝑙)(𝑥,𝑦)<1 for all 𝑥,𝑦∈𝑋, (c)𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘(g𝑥,g𝑦)𝑞(g𝑥,𝐹(𝑥,𝑦))+𝑙(g𝑥,g𝑦)𝑞(g𝑢,𝐹(𝑢,𝑣)) for all 𝑥,𝑦,𝑢,𝑣∈𝑋.
If 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋, then 𝐹 and g have a unique coupled point of coincidence (𝑢,𝑣) in 𝑋×𝑋. Further, if 𝑢=g𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑣=g𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Moreover, if 𝐹 and g are w -compatible, then 𝐹 and g have a unique common coupled fixed point and the common coupled fixed point of 𝐹 and g is of the form (𝑢,𝑢) for some 𝑢∈𝑋.

Proof. Choose 𝑥₀,𝑦₀∈𝑋. Set g𝑥₁=𝐹(𝑥₀,𝑦₀),g𝑦₁=𝐹(𝑦₀,𝑥₀). This can be done because 𝐹(𝑋×𝑋)⊆g(𝑋). Continuing this process, we obtain two sequences {𝑥_𝑛} and {𝑦_𝑛} such that g𝑥_𝑛+1=𝐹(𝑥_𝑛,𝑦_𝑛), g𝑦_𝑛+1=𝐹(𝑦_𝑛,𝑥_𝑛).
Then we have 𝑞g𝑥_𝑛,g𝑥_𝑛+1=⪯𝑘𝑞𝐹g𝑥𝑥_{𝑛−1𝑛−1},𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛+𝑙g𝑥,g_𝑛−1𝑦_𝑛−1𝑞g𝑥_𝑛−1,𝐹𝑥_𝑛−1,𝑦_𝑛−1=𝑘𝐹𝑥_𝑛−2,g𝑦_𝑛−1𝑞g𝑥_𝑛,𝐹𝑥_𝑛,𝑦_𝑛+𝑙𝐹𝑥,𝑦_{𝑛−2𝑛−2},𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞g𝑥_𝑛−1,g𝑥_𝑛⪯𝑘g𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞g𝑥_𝑛,g𝑥_𝑛+1+𝑙g𝑥,g_𝑛−2𝑦_𝑛−2𝑞g𝑥_𝑛−1,g𝑥_𝑛⋮⪯𝑘g𝑥₁,g𝑦_𝑛−2𝑞g𝑥_𝑛,g𝑥_𝑛+1,g𝑦₁𝑞g𝑥_𝑛−1,g𝑥_𝑛+𝑙g𝑥₁,g𝑦₁𝑞g𝑥_𝑛,g𝑥_𝑛+1.(3.18)
Hence 𝑞g𝑥_𝑛,g𝑥_𝑛+1⪯𝑘g𝑥¹,g𝑦₁1−𝑙g𝑥₁,g𝑦₁𝑞g𝑥^𝑛−1=ℎ𝑞g𝑥_𝑛−1,g𝑥_𝑛⪯ℎ₂,g𝑥_𝑛⋮⪯ℎ𝑞_𝑛−1gx_𝑛−2,g𝑥_𝑛−1𝑞g𝑥₁,g𝑥₂,(3.19) where ℎ=𝑘(g𝑥₁,g𝑦₁)/(1−𝑙(g𝑥₁,g𝑦₁))<1. It follows that 𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛=⪯𝑞ℎ𝑞g𝑥g_𝑛𝑥,g𝑥_{𝑛+1𝑛−1},g𝑥_𝑛.(3.20)
Similarly, we have 𝑞g𝑦_𝑛,g𝑦_𝑛+1⪯𝑘g𝑦¹,g𝑥₁1−𝑙g𝑦₁,g𝑥₁𝑞g𝑦^𝑛−1=𝑑𝑞g𝑦_𝑛−1,g𝑦_𝑛⪯𝑑₂,g𝑦_𝑛⋮⪯𝑑𝑞_𝑛−1g𝑦_𝑛−2,g𝑦_𝑛−1𝑞g𝑦₁,g𝑦₂,(3.21) where 𝑑=𝑘(g𝑦₂,g𝑥₂)/(1−𝑙(g𝑦₀,g𝑥₀))<1. It follows that 𝑞𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝐹𝑦_𝑛,𝑥_𝑛=⪯𝑞d𝑞gg𝑦_𝑛𝑦,g𝑦_{𝑛+1𝑛−1},g𝑦_𝑛.(3.22)
Let 𝑚>𝑛≥1. Then, it follows that 𝑞g𝑥_𝑛,g𝑥_𝑚⪯⪯𝑞ℎg𝑥_𝑛𝑛−1,g𝑥_𝑛+1+𝑞g𝑥_𝑛+1,g𝑥_𝑛+2+⋯+𝑞g𝑥_𝑚−1,g𝑥_𝑚⪯ℎ_𝑛−1+ℎ^𝑛+⋯+ℎ^𝑚−2𝑞g𝑥₁,g𝑥₂1−ℎ𝑞g𝑥¹,g𝑥₂,(3.23) and also, 𝑞g𝑦_𝑛,g𝑦_𝑚⪯⪯𝑞𝑑g𝑦_𝑛𝑛−1,g𝑦_𝑛+1+𝑞g𝑦_𝑛+1,g𝑦_𝑛+2+⋯+𝑞g𝑦_𝑚−1,g𝑦_𝑚⪯𝑑_𝑛+𝑑^𝑛+⋯+𝑑^𝑚−2𝑞g𝑦₁,g𝑦₂1−𝑑𝑞g𝑦¹,g𝑦₂.(3.24) Thus, Lemma 2.6 (3) shows that {g𝑥_𝑛} and {g𝑦_𝑛} are Cauchy sequences in g(𝑋). Since g(𝑋) is complete, there exists 𝑥^∗ and 𝑦^∗ in 𝑋 such that g𝑥_𝑛→g𝑥^∗ and g𝑦_𝑛→g𝑦^∗ as 𝑛→∞. Using (q3), we have 𝑞g𝑥_𝑛,g𝑥^∗⪯ℎ^𝑛−11−ℎ𝑞g𝑥¹𝑞g𝑦_𝑛,g𝑥₂,(3.25),g𝑦^∗⪯𝑑^𝑛−11−𝑑𝑞g𝑦¹,g𝑦₂.(3.26)
On the other hand and by using (3.20), we have 𝑞g𝑥_𝑛,𝐹𝑥^∗,𝑦^∗=⪯ℎ𝑞𝑞𝐹g𝑥𝑥_{𝑛−1𝑛−1},𝑦_𝑛−1,𝐹𝑥^∗,𝑦^∗⪯ℎℎ_𝑛−2,g𝑥^∗1−ℎ𝑞g𝑥¹=ℎ_𝑛−1,g𝑥₂1−ℎ𝑞g𝑥¹,g𝑥₂,(3.27) also by using (3.22), we have 𝑞g𝑦_𝑛,𝐹𝑦^∗,𝑥^∗=⪯𝑑𝑞𝑞𝐹g𝑦𝑦_{𝑛−1𝑛−1},𝑥_𝑛−1,𝐹𝑦^∗,𝑥^∗⪯𝑑𝑑_𝑛−2,g𝑦^∗1−𝑑𝑞g𝑦¹=𝑑_𝑛−1,g𝑦₂1−𝑑𝑞g𝑦¹,g𝑦₂.(3.28) Thus, Lemma 2.6 (1), (3.25), and (3.27) show that g𝑥^∗=𝐹(𝑥^∗,𝑦^∗). Again, Lemma 2.6 (1), (3.26), and (3.28) show that g𝑦^∗=𝐹(𝑦^∗,𝑥^∗). Therefore, (𝑥^∗,𝑦^∗) is a coupled coincidence point of 𝐹 and g.
Suppose that 𝑢=g𝑥^∗=𝐹(𝑥^∗,𝑦^∗) and 𝑣=g𝑦^∗=𝐹(𝑦^∗,𝑥^∗). Then we have 𝑞(𝑢,𝑢)==𝑞𝑞g𝐹𝑥^∗𝑥,g_∗𝑥^∗⪯𝑘g𝑥_∗,𝑦^∗,𝐹𝑥^∗,𝑦^∗===𝑘[[𝑘((𝑘𝑢(,𝑢+𝑣,𝑣)𝑙,)𝑞)g(+𝑦(𝑢𝑢^∗,𝑙,𝑣𝑢()𝑞𝑢)],+𝑞𝑣g()𝑥𝑙𝑢](^∗,𝑞𝑢,𝑢,𝐹()𝑢𝑣.,)𝑢𝑥𝑞)^∗(,𝑢𝑦,^∗𝑢)+𝑙g𝑥^∗,g𝑦^∗𝑞g𝑥^∗,𝐹𝑥^∗,𝑦^∗(3.29) Since (𝑘+𝑙)(𝑢,𝑣)<1, Lemma 2.3 (1) shows that 𝑞(𝑢,𝑢)=𝜃. By similar way, we have 𝑞(𝑣,𝑣)=𝜃.
Finally, suppose there is another coupled point of coincidence (𝑢₁,𝑣₁) of 𝐹 and g such that 𝑢₁=g𝑥^=𝐹(𝑥^,𝑦^) and 𝑣₁=g𝑦^=𝐹(𝑦^,𝑥^) for some (𝑥^,𝑦^) in 𝑋×𝑋. Then we have 𝑞𝑢,𝑢₁==𝑞𝑞𝐹g𝑥^∗𝑥,g_∗𝑥^⪯𝑘g𝑥_∗,𝑦^∗,𝐹𝑥^,𝑦^=𝑘(𝑢,𝑣),𝑞g𝑦(𝑢^∗,u𝑞)+g𝑥𝑙^∗(𝑢,𝐹,𝑣)𝑥𝑞^∗,𝑢𝑦^∗₁+𝑙g𝑥^∗,g𝑦^∗𝑞g𝑥^,𝐹𝑥^,𝑦^=𝜃,,𝑢₁(3.30) and also, 𝑞𝑣,𝑣₁==𝑞𝑞𝐹g𝑦^∗𝑦,g_∗𝑦^⪯𝑘g𝑦_∗,𝑥^∗,𝐹𝑦^,𝑥^=𝑘(𝑣,𝑢),𝑞g𝑥(𝑣^∗,𝑣𝑞)+g𝑦𝑙^∗(,𝑣𝐹,𝑢)𝑦𝑞^∗,𝑣𝑥^∗₁+𝑙g𝑦^∗,g𝑥^∗𝑞g𝑦^,𝐹𝑦^,𝑥^=𝜃.,𝑣₁(3.31) Also, we have 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Thus, Lemma 2.6 (1) shows that 𝑢=𝑢₁ and 𝑣=𝑣₁, which implies that (𝑢,𝑣)=(𝑢₁,𝑣₁). Similarly, we can prove that 𝑢=𝑣₁ and 𝑣=𝑢₁. Thus, 𝑢=𝑣. Therefore, (𝑢,𝑢) is the unique coupled point of coincidence. Now, let 𝑢=g𝑥^∗=𝐹(𝑥^∗,𝑦^∗). Since 𝐹 and g are w-compatible, then we have g𝑢=gg𝑥^∗=g𝐹𝑥^∗,𝑦^∗=𝐹g𝑥^∗,g𝑦^∗=𝐹g𝑥^∗,g𝑥^∗=𝐹(𝑢,𝑢).(3.32) Then, (g𝑢,g𝑢) is a coupled point of coincidence. The uniqueness of the coupled point of coincidence implies that g𝑢=𝑢. Therefore, 𝑢=g𝑢=𝐹(𝑢,𝑢). Hence, (𝑢,𝑢) is the unique common coupled fixed point of 𝐹 and g.

The following corollaries can be obtained as consequences of Theorem 3.4.

Corollary 3.5. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃 having nonempty interior and 𝑞 is a c-distance on 𝑋. Suppose the mappings 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 satisfy the following contractive condition: 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘𝑞(g𝑥,𝐹(𝑥,𝑦))+𝑙𝑞(g𝑢,𝐹(𝑢,𝑣)),(3.33) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘,𝑙 are nonnegative constants with 𝑘+𝑙<1. If 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋, then 𝐹 and g have a unique coupled point of coincidence (𝑢,𝑣) in 𝑋×𝑋. Further, if 𝑢=g𝑥₁=𝐹(𝑥₁,𝑦₁), and 𝑣=g𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Moreover, if 𝐹 and g are w -compatible, then 𝐹 and g have a unique common coupled fixed point and the common coupled fixed point of 𝐹 and g is of the form (𝑢,𝑢) for some 𝑢∈𝑋.

Corollary 3.6. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃 having nonempty interior and 𝑞 is a c-distance on 𝑋. Suppose the mappings 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 satisfy the following contractive condition: 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘𝑞(g𝑥,𝐹(𝑥,𝑦))+𝑞(g𝑢,𝐹(u,𝑣)),(3.34) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘∈[0,1/2) is constants. If 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋, then 𝐹 and g have a unique coupled point of coincidence (𝑢,𝑣) in 𝑋×𝑋. Further, if 𝑢=g𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑣=g𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Moreover, if 𝐹 and g are w-compatible, then 𝐹 and g have a unique common coupled fixed point and the common coupled fixed point of 𝐹 and g is of the form (𝑢,𝑢) for some 𝑢∈𝑋.

Finally, we provide another result with another contractive type.

Theorem 3.7. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃 having nonempty interior and 𝑞 is a c-distance on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 be two mappings 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(g𝑥,g𝑦), 𝑙(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑙(g𝑥,g𝑦) and  𝑟(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑟(g𝑥,g𝑦) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, (b)(𝑘+2𝑙+𝑟)(𝑥,𝑦)<1 for all 𝑥,𝑦∈𝑋, (c)(1−𝑟(g𝑥,g𝑦))𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘(g𝑥,g𝑦)𝑞(g𝑥,𝐹(𝑥,𝑦))+𝑙(g𝑥,g𝑦)𝑞(g𝑥,𝐹(𝑢,𝑣)) for all 𝑥,𝑦,𝑢,𝑣∈𝑋.
If 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋, then 𝐹 and g have a unique coupled point of coincidence (𝑢,𝑣) in 𝑋×𝑋. Further, if 𝑢=g𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑣=g𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Moreover, if 𝐹 and g are w-compatible, then 𝐹 and g have a unique common coupled fixed point and the common coupled fixed point of 𝐹 and g is of the form (𝑢,𝑢) for some 𝑢∈𝑋.

Proof. Choose 𝑥₀,𝑦₀∈𝑋. Set g𝑥₁=𝐹(𝑥₀,𝑦₀),g𝑦₁=𝐹(𝑦₀,𝑥₀) this can be done because 𝐹(𝑋×𝑋)⊆g(𝑋). Continuing this process, we obtain to sequences {𝑥_𝑛} and {𝑦_𝑛} such that g𝑥_n+1=𝐹(𝑥_𝑛,𝑦_𝑛),g𝑦_𝑛+1=𝐹(𝑦_𝑛,𝑥_𝑛). Observe that (1−𝑟(g𝑥,g𝑦))𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘(g𝑥,g𝑦)𝑞(g𝑥,𝐹(𝑥,𝑦))+𝑙(g𝑥,g𝑦)𝑞(g𝑥,𝐹(𝑢,𝑣)),(3.35) equivalently 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘+(g𝑙𝑥(g,𝑥g𝑦,g)𝑦𝑞)(𝑞g𝑥(g,𝑥𝐹,𝐹(𝑥,(𝑢𝑦,))𝑣))+𝑟(g𝑥,g𝑦)𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣)).(3.36) Then, we have 𝑞g𝑥_𝑛,g𝑥_𝑛+1=⪯𝑞𝑘𝐹g𝑥𝑥_{𝑛−1𝑛−1},𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛+𝑙g𝑥,g_𝑛−1𝑦_𝑛−1𝑞g𝑥_𝑛−1,𝐹𝑥_𝑛−1,𝑦_𝑛−1+𝑟g𝑥_𝑛−1,g𝑦_𝑛−1𝑞g𝑥_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛=𝑘𝐹𝑥_𝑛−2,g𝑦_𝑛−1𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛+𝑙𝐹𝑥,𝑦_{𝑛−2𝑛−2},𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞g𝑥_𝑛−1,g𝑥_𝑛+𝑟𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞g𝑥_𝑛−1,g𝑥_𝑛+1⪯𝑘g𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞g𝑥_𝑛,g𝑥_𝑛+1+𝑙g𝑥,g_𝑛−2𝑦_𝑛−2𝑞g𝑥_𝑛−1,g𝑥_𝑛+𝑟g𝑥_𝑛−2,g𝑦_𝑛−2𝑞g𝑥_𝑛−1,g𝑥_𝑛+1⋮⪯𝑘g𝑥₁,g𝑦_𝑛−2𝑞g𝑥_𝑛,g𝑥_𝑛+1+𝑟g,𝑥g₁𝑦₁𝑞g𝑥_𝑛−1,g𝑥_𝑛+𝑙g𝑥₁,g𝑦₁𝑞g𝑥_𝑛−1,g𝑥_𝑛+1⪯𝑘g𝑥₁,g𝑦₁𝑞g𝑥_𝑛,g𝑥_𝑛+1+𝑙g𝑥,g₁𝑦₁𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙g𝑥₁,g𝑦₁𝑞𝑥_𝑛−1,𝑥_𝑛,g𝑦₁𝑞g𝑥_𝑛,g𝑥_𝑛+1+𝑟g𝑥₁,g𝑦₁𝑞g𝑥_𝑛,g𝑥_𝑛+1.(3.37) Hence 𝑞g𝑥_𝑛,g𝑥_𝑛+1⪯𝑘g𝑥¹,g𝑦₁+𝑙g𝑥₁,g𝑦₁1−𝑙g𝑥₁,g𝑦₁−𝑟g𝑥₁,g𝑦₁𝑞g𝑥^𝑛−1=ℎ𝑞g𝑥_𝑛−1,g𝑥_𝑛⪯ℎ₂,g𝑥_𝑛⋮⪯ℎ𝑞_𝑛−1g𝑥_𝑛−2,g𝑥_𝑛−1𝑞g𝑥₁,g𝑥₂,(3.38) where ℎ=(𝑘(g𝑥₁,g𝑦₁)+𝑙(g𝑥₁,g𝑦₁))/(1−𝑙(g𝑥₁,g𝑦₁)−𝑟(g𝑥₁,g𝑦₁))<1.
Similarly, we have 𝑞g𝑦_𝑛,g𝑦_𝑛+1⪯𝑘g𝑦¹,g𝑥₁+𝑙g𝑦₁,g𝑥₁1−𝑙g𝑦₁,g𝑥₁−𝑟g𝑦₁,g𝑥₁𝑞g𝑦^𝑛−1=𝑑𝑞g𝑦_𝑛−1,g𝑦_𝑛⪯𝑑₂,g𝑦_𝑛⋮⪯𝑑𝑞_𝑛−1g𝑦_𝑛−2,g𝑦_𝑛−1𝑞g𝑦₁,g𝑦₂,(3.39) where 𝑑=(𝑘(g𝑦₁,g𝑥₁)+𝑙(g𝑦₁,g𝑥₁))/(1−𝑙(g𝑦₁,g𝑥₁)−𝑟(g𝑦₁,g𝑥₁))<1.
Let 𝑚>𝑛≥1. Then, it follows that 𝑞g𝑥_𝑛,g𝑥_𝑚⪯⪯𝑞ℎg𝑥_𝑛𝑛−1,g𝑥_𝑛+1+𝑞g𝑥_𝑛+1,g𝑥_𝑛+2+⋯+𝑞g𝑥_𝑚−1,g𝑥_𝑚⪯ℎ_𝑛+ℎ^𝑛+⋯+ℎ^𝑚−2𝑞g𝑥₁,g𝑥₂1−ℎ𝑞g𝑥¹𝑞g𝑦_𝑛,g𝑥₂,,g𝑦_𝑚⪯⪯𝑞𝑑g𝑦_𝑛𝑛−1,g𝑦_𝑛+1+𝑞g𝑦_𝑛+1,g𝑦_𝑛+2+⋯+𝑞g𝑦_𝑚−1,g𝑦_𝑚⪯𝑑_𝑛+𝑑^𝑛+⋯+𝑑^𝑚−2𝑞g𝑦₁,g𝑦₂1−𝑑𝑞g𝑦¹,g𝑦₂.(3.40) Thus, Lemma 2.6 (3) shows that {g𝑥_𝑛} and {g𝑦_𝑛} are Cauchy sequences in g(𝑋). Since g(𝑋) is complete, there exists 𝑥^∗,𝑦^∗∈𝑋 such that g𝑥_𝑛→g𝑥^∗ and g𝑦_𝑛→g𝑦^∗ as 𝑛→∞. Using (q3), we have 𝑞g𝑥_𝑛,g𝑥^∗⪯ℎ^𝑛1−ℎ𝑞g𝑥¹𝑞g𝑦_𝑛,g𝑥₂,(3.41),g𝑦^∗⪯𝑑^𝑛1−𝑑𝑞g𝑦¹,g𝑦₂.(3.42)
On the other hand, 𝑞g𝑥_𝑛,𝐹𝑥^∗,𝑦^∗=⪯𝑘𝑞𝐹g𝑥𝑥_{𝑛−1𝑛−1},𝑦_𝑛−1,𝐹𝑥^∗,𝑦^∗+𝑙g𝑥,g_𝑛−1𝑦_𝑛−1𝑞g𝑥_𝑛−1,𝐹𝑥_𝑛−1,𝑦_𝑛−1+𝑟g𝑥_𝑛−1,g𝑦_𝑛−1𝑞g𝑥_𝑛−1,𝐹𝑥^∗,𝑦^∗=𝑘𝐹𝑥_𝑛−2,g𝑦_𝑛−1𝑞𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥^∗,𝑦^∗+𝑙𝐹𝑥,𝑦_{𝑛−2𝑛−2},𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞g𝑥_𝑛−1,𝑥_𝑛+𝑟𝐹𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞g𝑥_𝑛−1,𝐹𝑥^∗,𝑦^∗⪯𝑘g𝑥_𝑛−2,𝑦_𝑛−2,𝐹𝑦_𝑛−2,𝑥_𝑛−2𝑞g𝑥_𝑛,𝐹𝑥^∗,𝑦^∗+𝑙g𝑥,g_𝑛−2𝑦_𝑛−2𝑞g𝑥_𝑛−1,g𝑥_𝑛+𝑟g𝑥_𝑛−2,g𝑦_𝑛−2𝑞g𝑥_𝑛−1,F𝑥^∗,𝑦^∗⋮⪯𝑘g𝑥₁,g𝑦_𝑛−2𝑞g𝑥_𝑛,𝐹𝑥^∗,𝑦^∗+𝑙g𝑥,g₁𝑦₁𝑞g𝑥_𝑛−1,g𝑥_𝑛+𝑟g𝑥₁,g𝑦₁𝑞g𝑥_𝑛−1,𝐹𝑥^∗,𝑦^∗⪯𝑘g𝑥₁,g𝑦₁𝑞g𝑥_𝑛,𝐹𝑥^∗,𝑦^∗+𝑙g𝑥,g₁𝑦₁𝑞g𝑥_𝑛−1,g𝑥_𝑛+𝑙g𝑥₁,g𝑦₁𝑞g𝑥_𝑛−1,g𝑥_𝑛+𝑟g𝑥₁,g𝑦₁𝑞g𝑥_𝑛,𝐹𝑥^∗,𝑦^∗,g𝑦₁𝑞g𝑥_𝑛,𝐹𝑥^∗,𝑦^∗.(3.43) Then, 𝑞g𝑥_𝑛,𝐹𝑥^∗,𝑦^∗⪯𝑘g𝑥¹,g𝑦₁+𝑙g𝑥₁,g𝑦₁1−𝑙g𝑥₁,g𝑦₁−𝑟g𝑥₁,g𝑦₁𝑞g𝑥^𝑛−1=ℎ𝑞g𝑥_𝑛−1,g𝑥_𝑛⪯ℎℎ_𝑛−2,g𝑥_𝑛=ℎ_𝑛−1𝑞g𝑥₁,g𝑥₂⪯ℎ_𝑛−1𝑞g𝑥₁,g𝑥₂1−ℎ𝑞g𝑥¹,g𝑥₂.(3.44) Thus, Lemma 2.6 (1), (3.41), and (3.44) show that g𝑥^∗=𝐹(𝑥^∗,𝑦^∗). Similarly, we can prove that g𝑦^∗=𝐹(𝑦^∗,𝑥^∗). Therefore, (𝑥^∗,𝑦^∗) is a coupled coincidence point of 𝐹 and g. Hence, (g𝑥^∗,g𝑦^∗) is the coupled point of coincidence.
Suppose that 𝑢=g𝑥^∗=𝐹(𝑥^∗,𝑦^∗) and 𝑣=g𝑦^∗=𝐹(𝑦^∗,𝑥^∗). Then, we have 𝑞(𝑢,𝑢)==𝑞𝑞g𝐹𝑥^∗𝑥,g_∗𝑥^∗⪯𝑘g𝑥_∗,𝑦^∗,𝐹𝑥^∗,𝑦^∗+𝑟g,𝑥g_∗𝑦^∗𝑞g𝑥^∗,𝐹𝑥^∗,𝑦^∗+𝑙g𝑥^∗,g𝑦^∗𝑞g𝑥^∗,𝐹𝑥^∗,𝑦^∗====𝑘𝑘[[𝑘(((𝑘𝑢𝑢(,,𝑢+𝑣𝑣,𝑣))2𝑞𝑞)𝑙++((,𝑢𝑢g𝑙,,𝑟𝑦𝑢𝑢)(^∗𝑢))(𝑢,++𝑞𝑣,𝑣)𝑙𝑙)+𝐹((]𝑢𝑢𝑞,,𝑙𝑣𝑣((𝑥𝑢𝑢))^∗,,𝑞𝑞,𝑢𝑣𝑦(())^∗𝑢𝑢.+,,𝑢𝑢,𝑟))𝐹(++𝑢,𝑟𝑙𝑥𝑣(^∗()𝑢𝑢,],,𝑦𝑞𝑣𝑣^∗)()𝑢𝑞𝑞,((𝑢𝑢𝑢),,𝑢𝑢))+𝑟(𝑢,𝑣)𝑞(𝑢,𝑢)(3.45) Since (𝑘+2𝑙+𝑟)(𝑢,𝑣)<1, Lemma 2.3(1) shows that 𝑞(𝑢,𝑢)=𝜃. Similarly, we have 𝑞(𝑣,𝑣)=𝜃.
Finally, suppose there is another coupled point of coincidence (𝑢₁,𝑣₁) of 𝐹 and g such that 𝑢₁=g𝑥^=𝐹(𝑥^,𝑦^) and 𝑣₁=g𝑦^=𝐹(𝑦^,𝑥^) for some (𝑥^,𝑦^) in 𝑋×𝑋. Then, we have 𝑞𝑢,𝑢₁==𝑞𝑞𝐹g𝑥^∗𝑥,g_∗𝑥^⪯𝑘g𝑥_∗,𝑦^∗,𝐹𝑥^,𝑦^+𝑟g,𝑥g_∗𝑦^∗𝑞g𝑥^∗,𝐹𝑥^∗,𝑦^∗+𝑙g𝑥^∗,g𝑦^∗𝑞g𝑥^∗,𝐹𝑥^,𝑦^=𝑘(𝑢,𝑣)𝑞(,𝑢g,𝑦𝑢^∗)+q𝑙𝐹(𝑢,𝑣𝑥)^∗𝑞,𝑦^∗𝑢,𝑢,𝐹₁𝑥^,𝑦^=𝑙(𝑢,𝑣)𝑞𝑢,𝑢₁+r(𝑢,𝑣)𝑞𝑢,𝑢₁⪯k(𝑢𝑣)𝑞𝑢,𝑢₁+𝑟(𝑢,𝑣)𝑞𝑢,𝑢₁=[𝑘(𝑢,𝑣))+𝑙(𝑢+,𝑣𝑙)(+𝑢,𝑙𝑣()𝑢𝑞,𝑣)𝑢,+𝑢₁𝑟(𝑢+,𝑣𝑙)(]𝑢𝑞,𝑣)𝑢𝑞,𝑢₁𝑢,𝑢₁+𝑟(𝑢,𝑣)𝑞𝑢,𝑢₁=[(𝑘+2𝑙+𝑟)(𝑢,𝑣)]𝑞𝑢,𝑢₁.(3.46) Since (𝑘+2𝑙+𝑟)(𝑢,𝑣)<1, Lemma 2.3 (1) shows that 𝑞(𝑢,𝑢₁)=𝜃. By similar way, 𝑞(𝑣,𝑣₁)=𝜃. Also, we have 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Thus, Lemma 2.6 (1) shows that 𝑢=𝑢₁ and 𝑣=𝑣₁, which implies that (𝑢,𝑣)=(𝑢₁,𝑣₁). Similarly, we can prove that 𝑢=𝑣₁ and 𝑣=𝑢₁. Thus, 𝑢=𝑣. Therefore, (𝑢,𝑢) is the unique coupled point of coincidence. Now, let 𝑢=g𝑥^∗=𝐹(𝑥^∗,𝑦^∗). Since 𝐹 and g are w-compatible, then we have g𝑢=gg𝑥^∗=g𝐹𝑥^∗,𝑦^∗=𝐹g𝑥^∗,g𝑦^∗=𝐹g𝑥^∗,g𝑥^∗=𝐹(𝑢,𝑢).(3.47) Then, (g𝑢,g𝑢) is a coupled point of coincidence. The uniqueness of the coupled point of coincidence implies that g𝑢=𝑢. Therefore, 𝑢=g𝑢=𝐹(𝑢,𝑢). Hence, (𝑢,𝑢) is the unique common coupled fixed point of 𝐹 and g.

The following corollaries can be obtained as consequences of Theorem 3.7.

Corollary 3.8. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃 having nonempty interior and 𝑞 is a c-distance on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 be two mappings 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(g𝑥,g𝑦) and 𝑙(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))≤𝑙(g𝑥,g𝑦), (b)(𝑘+2𝑙)(𝑥,𝑦)<1, (c)𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘(g𝑥,g𝑦)𝑞(g𝑥,𝐹(𝑥,𝑦))+𝑙(g𝑥,g𝑦)𝑞(g𝑥,𝐹(𝑢,𝑣)). for all 𝑥,𝑦,𝑢,𝑣∈𝑋. If 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋, then 𝐹 and g have a unique coupled point of coincidence (𝑢,𝑣) in 𝑋×𝑋. Further, if 𝑢=g𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑣=g𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Moreover, if 𝐹 and g are w-compatible, then 𝐹 and g have a unique common coupled fixed point and the common coupled fixed point of 𝐹 and g is of the form (𝑢,𝑢) for some 𝑢∈𝑋.

Corollary 3.9. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃 having nonempty interior and 𝑞 is a c-distance on 𝑋. Suppose the mappings 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 satisfy the following contractive condition: (1−𝑟)𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘𝑞(g𝑥,𝐹(𝑥,𝑦))+𝑙𝑞(g𝑥,𝐹(𝑢,𝑣)),(3.48) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘,𝑙,𝑟 are nonnegative constants with 𝑘+2𝑙+𝑟<1. If 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋, then 𝐹 and g have a unique coupled point of coincidence (𝑢,𝑣) in 𝑋×𝑋. Further, if 𝑢=g𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑣=g𝑦₁=𝐹(𝑦₁,𝑥₁) then 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Moreover, if 𝐹 and g are w-compatible, then 𝐹 and g have a unique common coupled fixed point and the common coupled fixed point of 𝐹 and g is of the form (𝑢,𝑢) for some 𝑢∈𝑋.

Corollary 3.10. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃 having nonempty interior and 𝑞 is a c-distance on 𝑋. Suppose the mappings 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 satisfy the following contractive condition: 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘𝑞(g𝑥,𝐹(𝑥,𝑦))+𝑙𝑞(g𝑥,𝐹(𝑢,𝑣)),(3.49) for all 𝑥,𝑦,𝑢,v∈𝑋, where 𝑘,𝑙 are nonnegative constants with 𝑘+2𝑙<1. If 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋, then 𝐹 and g have a unique coupled point of coincidence (𝑢,𝑣) in 𝑋×𝑋. Further, if 𝑢=g𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑣=g𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Moreover, if 𝐹 and g are w-compatible, then 𝐹 and g have a unique common coupled fixed point and the common coupled fixed point of 𝐹 and g is of the form (𝑢,𝑢) for some 𝑢∈𝑋.

Corollary 3.11. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃 having nonempty interior and 𝑞 is a c-distance on 𝑋. Suppose the mappings 𝐹∶𝑋×𝑋→𝑋 and g∶𝑋→𝑋 satisfy the following contractive condition: 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))⪯𝑘𝑞(g𝑥,𝐹(x,𝑦))+𝑞(g𝑥,𝐹(𝑢,𝑣)),(3.50) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘∈[0,1/3) is a constants. If 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subset of 𝑋, then 𝐹 and g have a unique coupled point of coincidence (𝑢,𝑣) in 𝑋×𝑋. Further, if 𝑢=g𝑥₁=𝐹(𝑥₁,𝑦₁) and 𝑣=g𝑦₁=𝐹(𝑦₁,𝑥₁), then 𝑞(𝑢,𝑢)=𝜃 and 𝑞(𝑣,𝑣)=𝜃. Moreover, if 𝐹 and g are w-compatible, then 𝐹 and g have a unique common coupled fixed point and the common coupled fixed point of 𝐹 and g is of the form (𝑢,𝑢) for some 𝑢∈𝑋.

4. Some Examples

Example 4.1. Let 𝐸=ℝ and 𝑃={𝑥∈𝐸∶𝑥≥0}. Let 𝑋=[0,1] and define a mapping 𝑑∶𝑋×𝑋→𝐸 by 𝑑(𝑥,𝑦)=|𝑥−𝑦| for all 𝑥,𝑦∈𝑋. Clearly, (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. Define the mappings 𝐹∶𝑋×𝑋→𝑋 by 𝐹(𝑥,𝑦)=(𝑥𝑦)²/16 and g∶𝑋→𝑋 by g𝑥=𝑥/2 for all 𝑥,𝑦∈𝑋. Obviously that 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋. Take 𝑘(𝑥,𝑦)=𝑙(𝑥,𝑦)=(𝑥𝑦+1)/8, where 𝑥,𝑦∈𝑋. Observe the following: (a)𝑘(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))=𝑙(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣)) =(((𝑥𝑦)²(𝑢𝑣)²/16²)+1)/8 =(1/8)(((𝑥𝑦)²(𝑢𝑣)²/16²)+1) ≤(1/8)((𝑥𝑦)²/16+1) ≤(1/8)(𝑥𝑦/16+1) ≤(1/8)(𝑥𝑦/4+1) =𝑘(g𝑥,g𝑦) 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(𝑥,𝑦)+𝑙(𝑥,𝑦)=(𝑥𝑦+1)/4<1 for all 𝑥,𝑦∈𝑋.(d)For all 𝑥,𝑦,𝑢,𝑣∈𝑋, we have 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))==𝐹(𝑢𝑣(𝑢),₂𝑣)⪯𝑢𝑣16⪯1𝑢6+𝑣=𝑢16=116+1𝑣6⪯82𝑢𝑥𝑦+18𝑣2=3𝑥2𝑦/+4182𝑢+𝑥32𝑦+18𝑣2=(𝑥𝑦/84)++1812𝑢+𝑥𝑦8/4+81𝑣2=𝑘(g𝑥8,g𝑦)𝑞2𝑢(g+𝑥,(g𝑥𝑢𝑦)/4+8)𝑙+(g1𝑥𝑣2,g𝑦)𝑞(g𝑦,g𝑣).(4.1)Clearly that 𝐹 and g are w-compatible. Therefore, 𝐹 and g satisfy all the conditions of Theorem 3.1. Hence, 𝐹 and g have a unique common coupled fixed point (𝑢,𝑢)=(0,0) and g(0)=𝐹(0,0)=0 with 𝑞(0,0)=0.

Example 4.2. Consider Example 2.5. Define the mappings 𝐹∶𝑋×𝑋→𝑋 by 𝐹(𝑥,𝑦)=(𝑥+𝑦)/4 and g∶𝑋→𝑋 by g𝑥=𝑥 for all 𝑥,𝑦∈𝑋. Obviously that 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋. Then, we have 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))=𝐹(𝑢,𝑣)=(𝑢+𝑣)/4⪯(3/7)(𝑢+𝑣)=𝑘[𝑞(g𝑥,g𝑢)+𝑞(g𝑦,g𝑣)] with 𝑘=3/7∈[0,1/2). Clearly that 𝐹 and g are w-compatible. Therefore, 𝐹 and g satisfy all the conditions of Corollary 3.3. Hence, 𝐹 and g have a unique common coupled fixed point (𝑢,𝑢)=(0,0) and g(0)=𝐹(0,0)=0 with 𝑞(0,0)=0.

Example 4.3. 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,𝑦)≪=⪯===𝑞𝑐𝑒((.𝑥𝑥||(+𝑥𝑧,+𝑥,𝑒−𝑥)𝑦)𝑦+,||+𝑥,(||𝑦+𝑞𝑥,(𝑦−𝑦𝑧)),𝑦𝑦||)(4.2) This show, that (q4) holds. Therefore, 𝑞 is a c-distance on 𝑋. Define the mappings 𝐹∶𝑋×𝑋→𝑋 by 𝐹(𝑥,𝑦)=(𝑥+𝑦)/16 and g∶𝑋→𝑋 by g𝑥=𝑥/2 for all 𝑥,𝑦∈𝑋. Obviously that 𝐹(𝑋×𝑋)⊆g(𝑋) and g(𝑋) is a complete subspace of 𝑋. Then we have 𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))==(𝐹𝑢(+𝑢,𝑣𝑣),𝐹(𝑢,𝑣))=𝑢16,𝑢1+6𝑣=𝑢1/62,1𝑢6+1𝑣6,1𝑣6=18,𝑢8/2+𝑣8/2,𝑣8/2⪯182𝑢,2𝑢+81𝑣2,𝑣2=142𝑢,2𝑢+41𝑣2,𝑣2=𝑘4𝑞(2𝑢g𝑥,2𝑢,g𝑢)++𝑞𝑣2(,g𝑣2𝑦,g𝑣),(4.3) with 𝑘=1/4∈[0,1/2). Clearly that 𝐹 and g are w-compatible. Therefore, 𝐹 and g satisfy all the conditions of Corollary 3.3. Hence, 𝐹 and g have a unique common coupled fixed point (𝑢,𝑢)=(0,0) and g(0)=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 paper and useful suggestions.