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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 901792, 24 pages
http://dx.doi.org/10.1155/2012/901792
Research Article

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

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

Received 30 May 2012; Accepted 8 July 2012

Academic Editor: Ngai-Ching Wong

Copyright © 2012 Zaid Mohammed Fadail and Abd Ghafur Bin Ahmad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [212].

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 [1623].

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 [2631].

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 𝐹𝑋×𝑋𝑋 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 𝐹𝑋×𝑋𝑋 and g𝑋𝑋 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 𝐹𝑋×𝑋𝑋 and g𝑋𝑋 if 𝑥=g𝑥=𝐹(𝑥,𝑦),𝑦=g𝑦=𝐹(𝑦,𝑥).

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

Definition 1.2. The mappings 𝐹𝑋×𝑋𝑋 and g𝑋𝑋 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,𝑦𝑃, then 𝑎𝑥+𝑏𝑦𝑃, (3)𝑥𝑃 and 𝑥𝑃 implies 𝑥=𝜃.

For any cone 𝑃𝐸, the partial ordering with respect to 𝑃 is defined by 𝑥𝑦 if and only if 𝑦𝑥𝑃. The notation of stand for 𝑥𝑦 but 𝑥𝑦. Also, we used 𝑥𝑦 to indicate that 𝑦𝑥int𝑃, where int𝑃 denotes the interior of 𝑃. A cone 𝑃 is called normal if there exists a number 𝐾 such that 𝜃𝑥𝑦𝑥𝐾𝑦,(2.1) for all 𝑥,𝑦𝐸. 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 𝑑𝑋×𝑋𝐸 satisfies the following condition: (1)𝜃𝑑(𝑥,𝑦) for all 𝑥,𝑦𝑋 and 𝑑(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦, (2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦𝑋, (3)𝑑(𝑥,𝑦)𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧𝑋. 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 {𝑥𝑛} a sequence in 𝑋 and 𝑥𝑋. (1)For all 𝑐𝐸 with 𝜃𝑐, if there exists a positive integer 𝑁 such that 𝑑(𝑥𝑛,𝑥)𝑐 for all 𝑛>𝑁, then 𝑥𝑛 is said to be convergent and 𝑥 is the limit of {𝑥𝑛}. We denote this by 𝑥𝑛𝑥. (2)For all 𝑐𝐸 with 𝜃𝑐, if there exists a positive integer 𝑁 such that 𝑑(𝑥𝑛,𝑥𝑚)𝑐 for all 𝑛,𝑚>𝑁, then {𝑥𝑛} 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 𝑎𝜆𝑎, where 𝑎𝑃 and 0𝜆<1, then 𝑎=𝜃. (2)If 𝑐int𝑃, 𝜃𝑎𝑛 and 𝑎𝑛𝜃, then there exists a positive integer 𝑁 such that 𝑎𝑛𝑐 for all 𝑛𝑁.

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 𝑞𝑋×𝑋𝐸 is called a c-distance on 𝑋 if the following conditions hold: (q1)𝜃𝑞(𝑥,𝑦) for all 𝑥,𝑦𝑋, (q2)𝑞(𝑥,𝑦)𝑞(𝑥,𝑦)+𝑞(𝑦,𝑧) for all 𝑥,𝑦,𝑧𝑋, (q3) for each 𝑥𝑋 and 𝑛1, if 𝑞(𝑥,𝑦𝑛)𝑢 for some 𝑢=𝑢𝑥𝑃, then 𝑞(𝑥,𝑦)𝑢 whenever {𝑦𝑛} is a sequence in 𝑋 converging to a point 𝑦𝑋, (q4) for all 𝑐𝐸 with 𝜃𝑐, there exists 𝑒𝐸 with 𝜃𝑒 such that 𝑞(𝑧,𝑥)𝑒 and 𝑞(𝑧,𝑦)𝑒 imply 𝑑(𝑥,𝑦)𝑐.

Example 2.5 (see [23]). Let 𝐸= and 𝑃={𝑥𝐸𝑥0}. Let 𝑋=[0,) and define a mapping 𝑑𝑋×𝑋𝐸 by 𝑑(𝑥,𝑦)=|𝑥𝑦| for all 𝑥,𝑦𝑋. Then (𝑋,𝑑) is a cone metric space. Define a mapping 𝑞𝑋×𝑋𝐸 by 𝑞(𝑥,𝑦)=𝑦 for all 𝑥,𝑦𝑋. 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 {𝑥𝑛} and {𝑦𝑛} be sequences in 𝑋 and 𝑥,𝑦,𝑧𝑋. Suppose that 𝑢𝑛 is a sequences in 𝑃 converging to 𝜃. Then the following hold. (1)If 𝑞(𝑥𝑛,𝑦)𝑢𝑛 and 𝑞(𝑥𝑛,𝑧)𝑢𝑛, then 𝑦=𝑧. (2)If 𝑞(𝑥𝑛,𝑦𝑛)𝑢𝑛 and 𝑞(𝑥𝑛,𝑧)𝑢𝑛, then {𝑦𝑛} converges to 𝑧. (3)If 𝑞(𝑥𝑛,𝑥𝑚)𝑢𝑛 for 𝑚>𝑛, then {𝑥𝑛}is a Cauchy sequence in 𝑋. (4)If 𝑞(𝑦,𝑥𝑛)𝑢𝑛, then {𝑥𝑛} 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,𝑦𝑋. (2)𝑞(𝑥,𝑦)=𝜃 is not necessarily equivalent to 𝑥=𝑦 for all 𝑥,𝑦𝑋.

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 𝐹𝑋×𝑋𝑋 and g𝑋𝑋 be two mappings and suppose that there exists mappings 𝑘,𝑙𝑋×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,𝑣𝑋, (b)𝑘(𝑥,𝑦)=𝑘(𝑦,𝑥) and 𝑙(𝑥,𝑦)=𝑙(𝑦,𝑥) for all 𝑥,𝑦𝑋, (c)(𝑘+𝑙)(𝑥,𝑦)<1 for all 𝑥,𝑦𝑋, (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,𝑣𝑋. 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 𝑢𝑋.

Proof. Choose 𝑥0,𝑦0𝑋. 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 {𝑥𝑛} and {𝑦𝑛} such that g𝑥𝑛+1=𝐹(𝑥𝑛,𝑦𝑛),g𝑦𝑛+1=𝐹(𝑦𝑛,𝑥𝑛). Then we have 𝑞g𝑥𝑛,g𝑥𝑛+1𝐹𝑥=𝑞𝑛1,𝑦𝑛1𝑥,𝐹𝑛,𝑦𝑛𝑘g𝑥𝑛1,g𝑦𝑛1𝑞g𝑥𝑛1,g𝑥𝑛+𝑙g𝑥𝑛1,g𝑦𝑛1𝑞g𝑦𝑛1,g𝑦𝑛𝐹𝑥=𝑘𝑛2,𝑦𝑛2𝑦,𝐹𝑛2,𝑥𝑛2𝑞g𝑥𝑛1,g𝑥𝑛𝐹𝑥+𝑙𝑛2,𝑦𝑛2𝑦,𝐹𝑛2,𝑥𝑛2𝑞g𝑦𝑛1,gy𝑛𝑘g𝑥𝑛2,g𝑦𝑛2𝑞g𝑥𝑛1,g𝑥𝑛+𝑙g𝑥𝑛2,g𝑦𝑛2𝑞g𝑦𝑛1,g𝑦𝑛𝑘g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛+𝑙g𝑥1,g𝑦1𝑞g𝑦𝑛1,g𝑦𝑛.(3.1) Similarly, we have 𝑞g𝑦𝑛,g𝑦𝑛+1𝐹𝑦=𝑞𝑛1,𝑥𝑛1𝑦,𝐹𝑛,𝑥𝑛𝑘g𝑦𝑛1,g𝑥𝑛1𝑞g𝑦𝑛1,g𝑦𝑛+𝑙g𝑦𝑛1,g𝑥𝑛1𝑞g𝑥𝑛1,g𝑥𝑛𝐹𝑦=𝑘𝑛2,𝑥𝑛2𝑥,𝐹𝑛2,𝑦𝑛2𝑞g𝑦𝑛1,g𝑦𝑛𝐹𝑦+𝑙𝑛2,𝑥𝑛2𝑥,𝐹𝑛2,𝑦𝑛2𝑞g𝑥𝑛1,g𝑥𝑛𝑘g𝑦𝑛2,g𝑥𝑛2𝑞g𝑦𝑛1,g𝑦𝑛+𝑙g𝑦𝑛2,g𝑥𝑛2𝑞g𝑥𝑛1,g𝑥𝑛𝑘g𝑦1,g𝑥1𝑞g𝑦𝑛1,g𝑦𝑛+𝑙g𝑦1,g𝑥1𝑞g𝑥𝑛1,g𝑥𝑛=𝑘g𝑥1,g𝑦1𝑞g𝑦𝑛1,g𝑦𝑛+𝑙g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛.(3.2) Put 𝑞𝑛=𝑞(g𝑥𝑛,g𝑥𝑛+1)+𝑞(g𝑦𝑛,g𝑦𝑛+1). Then we have 𝑞𝑛=𝑞g𝑥𝑛,g𝑥𝑛+1+𝑞g𝑦𝑛,g𝑦𝑛+1𝑘g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛+𝑙g𝑥1,g𝑦1𝑞g𝑦𝑛1,g𝑦𝑛+𝑘g𝑥1,g𝑦1𝑞g𝑦𝑛1,g𝑦𝑛+𝑙g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛=𝑘g𝑥1,g𝑦1+𝑙g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛+𝑞g𝑦𝑛1,g𝑦𝑛=𝑘g𝑥1,g𝑦1+𝑙g𝑥1,g𝑦1𝑞𝑛1=𝑞𝑛1𝑛1𝑞1,(3.3) where =𝑘(g𝑥1,g𝑦1)+𝑙(g𝑥1,g𝑦1)<1.
Let 𝑚>𝑛1. It follows that 𝑞g𝑥𝑛,g𝑥𝑚𝑞g𝑥𝑛,g𝑥𝑛+1+𝑞g𝑥𝑛+1,g𝑥𝑛+2++𝑞g𝑥𝑚1,g𝑥𝑚,𝑞g𝑦𝑛,g𝑦𝑚𝑞g𝑦𝑛,g𝑦𝑛+1+𝑞g𝑦𝑛+1,g𝑥𝑛+2++𝑞g𝑦𝑚1,g𝑦𝑚.(3.4) Then we have 𝑞g𝑥𝑛,g𝑥𝑚+𝑞g𝑦𝑛,g𝑦𝑚𝑞𝑛+𝑞𝑛+1++𝑞𝑚1𝑛𝑞1+𝑛+1𝑞1++𝑚1𝑞1=𝑛1+𝑛++𝑚2𝑞1𝑛1𝑞11.(3.5) Consequently, 𝑞g𝑥𝑛,g𝑥𝑚𝑛1𝑞11,𝑞g𝑦𝑛,g𝑦𝑚𝑛1𝑞11.(3.6) Thus, Lemma 2.6 (3) shows that {g𝑥𝑛} and {g𝑦𝑛} are Cauchy sequences in g(𝑋). Since g(𝑋) is complete, there exists 𝑥 and 𝑦 in 𝑋 such that g𝑥𝑛g𝑥 and g𝑦𝑛g𝑦 as 𝑛. Using q3, we have 𝑞g𝑥𝑛,g𝑥𝑛1𝑞11,𝑞g𝑦𝑛,g𝑦𝑛1𝑞11.(3.7)
On the other hand, 𝑞g𝑥𝑛𝑥,𝐹,𝑦𝐹𝑥=𝑞𝑛1,𝑦𝑛1𝑥,𝐹,𝑦𝑘g𝑥𝑛1,g𝑦𝑛1𝑞g𝑥𝑛1,g𝑥+𝑙g𝑥𝑛1,g𝑦𝑛1𝑞g𝑦𝑛1,g𝑦F𝑥=𝑘𝑛2,𝑦𝑛2𝑦,𝐹𝑛2,𝑥𝑛2𝑞g𝑥𝑛1,g𝑥𝐹𝑥+𝑙𝑛2,𝑦𝑛2𝑦,𝐹𝑛2,𝑥𝑛2𝑞g𝑦𝑛1,g𝑦𝑘g𝑥𝑛2,g𝑦𝑛2𝑞g𝑥𝑛1,g𝑥+𝑙g𝑥𝑛2,g𝑦𝑛2𝑞g𝑦𝑛1,g𝑦𝑘g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥+𝑙g𝑥1,g𝑦1𝑞g𝑦𝑛1,g𝑦𝑘g𝑥1,g𝑦1𝑛2𝑞11𝑥+𝑙1,𝑦1𝑛2𝑞11=𝑘g𝑥1,g𝑦1+𝑙g𝑥1,g𝑦1𝑛2𝑞11=𝑛2𝑞11=𝑛1𝑞11.(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)=𝑞g𝑥,g𝑥𝐹𝑥=𝑞,𝑦𝑥,𝐹,𝑦𝑘g𝑥,g𝑦𝑞g𝑥,g𝑥+𝑙g𝑥,g𝑦𝑞g𝑦,g𝑦=𝑘(𝑢,𝑣)𝑞(𝑢,𝑢)+𝑙(𝑢,𝑣)𝑞(𝑣,𝑣),(3.9)𝑞(𝑣,𝑣)=𝑞g𝑦,g𝑦𝐹𝑦=𝑞,𝑥𝑦,𝐹,𝑥𝑘g𝑦,g𝑥𝑞g𝑦,g𝑦+𝑙g𝑦,g𝑥𝑞g𝑥,g𝑥=𝑘(𝑣,𝑢)𝑞(𝑣,𝑣)+𝑙(𝑣,𝑢)𝑞(𝑢,𝑢)=𝑘(𝑢,𝑣)𝑞(𝑣,𝑣)+𝑙(𝑢,𝑣)𝑞(𝑢,𝑢).(3.10) This implies that =[𝑘𝑞]=[].𝑞(𝑢,𝑢)+𝑞(𝑣,𝑣)𝑘(𝑢,𝑣)𝑞(𝑢,𝑢)+𝑙(𝑢,𝑣)𝑞(𝑣,𝑣)+𝑘(𝑢,𝑣)𝑞(𝑣,𝑣)+𝑙(𝑢,𝑣)𝑞(𝑢,𝑢)(𝑢,𝑣)+𝑙(𝑢,𝑣)][(𝑢,𝑢)+𝑞(𝑣,𝑣)(𝑘+𝑙)(𝑢,𝑣)][𝑞(𝑢,𝑢)+𝑞(𝑣,𝑣)(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 𝑞𝑢,𝑢1=𝑞g𝑥,g𝑥𝐹𝑥=𝑞,𝑦𝑥,F,𝑦𝑘g𝑥,g𝑦𝑞g𝑥,g𝑥+𝑙g𝑥,g𝑦𝑞g𝑦,g𝑦=𝑘(𝑣,𝑢)𝑞𝑣,𝑣1+𝑙(𝑣,𝑢)𝑞𝑢,𝑢1=𝑘(𝑢,𝑣)𝑞𝑢,𝑢1+𝑙(𝑢,𝑣)𝑞𝑣,𝑣1,(3.12)
and also, 𝑞𝑣,𝑣1=𝑞g𝑦,g𝑦𝐹𝑦=𝑞,𝑥𝑦,F,𝑥𝑘g𝑦,g𝑥𝑞g𝑦,g𝑦+𝑙g𝑦,g𝑥𝑞g𝑥,g𝑥=𝑘(𝑣,𝑢)𝑞𝑣,𝑣1+𝑙(𝑣,𝑢)𝑞𝑢,𝑢1=𝑘(𝑢,𝑣)𝑞𝑣,𝑣1+𝑙(𝑢,𝑣)𝑞𝑢,𝑢1.(3.13) This implies that 𝑞𝑢,𝑢1+𝑞𝑣,𝑣1𝑘(𝑢,𝑣)𝑞𝑢,𝑢1+𝑙(𝑢,𝑣)𝑞𝑣,𝑣1+𝑘(𝑢,𝑣)𝑞𝑣,𝑣1+𝑙(𝑢,𝑣)𝑞𝑢,𝑢1=[]𝑞𝑘(𝑢,𝑣)+𝑙(𝑢,𝑣)𝑢,𝑢1+𝑞𝑣,𝑣1=[]𝑞(𝑘+𝑙)(𝑢,𝑣)𝑢,𝑢1+𝑞𝑣,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=gg𝑥𝑥=g𝐹,𝑦=𝐹g𝑥,g𝑦=𝐹g𝑥,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 𝐹𝑋×𝑋𝑋 and g𝑋𝑋 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,𝑣𝑋, 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 𝑢𝑋.

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 𝐹𝑋×𝑋𝑋 and g𝑋𝑋 satisfy the following contractive condition: 𝑞𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))𝑘(g𝑥,g𝑢)+𝑞(g𝑦,g𝑣),(3.17) for all 𝑥,𝑦,𝑢,𝑣𝑋, 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 𝑢𝑋.

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 𝐹𝑋×𝑋𝑋 and g𝑋𝑋 be two mappings and suppose that there exists mappings 𝑘,𝑙𝑋×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,𝑣𝑋, (b)(𝑘+𝑙)(𝑥,𝑦)<1 for all 𝑥,𝑦𝑋, (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,𝑣𝑋.
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 𝑢𝑋.

Proof. Choose 𝑥0,𝑦0𝑋. 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 {𝑥𝑛} and {𝑦𝑛} such that g𝑥𝑛+1=𝐹(𝑥𝑛,𝑦𝑛), g𝑦𝑛+1=𝐹(𝑦𝑛,𝑥𝑛).
Then we have 𝑞g𝑥𝑛,g𝑥𝑛+1𝐹𝑥=𝑞𝑛1,𝑦𝑛1𝑥,𝐹𝑛,𝑦𝑛𝑘g𝑥𝑛1,g𝑦𝑛1𝑞g𝑥𝑛1𝑥,𝐹𝑛1,𝑦𝑛1+𝑙g𝑥𝑛1,g𝑦𝑛1𝑞g𝑥𝑛𝑥,𝐹𝑛,𝑦𝑛𝐹𝑥=𝑘𝑛2,𝑦𝑛2𝑦,𝐹𝑛2,𝑥𝑛2𝑞g𝑥𝑛1,g𝑥𝑛𝐹𝑥+𝑙𝑛2,𝑦𝑛2𝑦,𝐹𝑛2,𝑥𝑛2𝑞g𝑥𝑛,g𝑥𝑛+1𝑘g𝑥𝑛2,g𝑦𝑛2𝑞g𝑥𝑛1,g𝑥𝑛+𝑙g𝑥𝑛2,g𝑦𝑛2𝑞g𝑥𝑛,g𝑥𝑛+1𝑘g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛+𝑙g𝑥1,g𝑦1𝑞g𝑥𝑛,g𝑥𝑛+1.(3.18)
Hence 𝑞g𝑥𝑛,g𝑥𝑛+1𝑘g𝑥1,g𝑦11𝑙g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛=𝑞g𝑥𝑛1,g𝑥𝑛2𝑞gx𝑛2,g𝑥𝑛1𝑛1𝑞g𝑥1,g𝑥2,(3.19) where =𝑘(g𝑥1,g𝑦1)/(1𝑙(g𝑥1,g𝑦1))<1. It follows that 𝑞𝐹𝑥𝑛1,𝑦𝑛1𝑥,𝐹𝑛,𝑦𝑛=𝑞g𝑥𝑛,g𝑥𝑛+1𝑞g𝑥𝑛1,g𝑥𝑛.(3.20)
Similarly, we have 𝑞g𝑦𝑛,g𝑦𝑛+1𝑘g𝑦1,g𝑥11𝑙g𝑦1,g𝑥1𝑞g𝑦𝑛1,g𝑦𝑛=𝑑𝑞g𝑦𝑛1,g𝑦𝑛𝑑2𝑞g𝑦𝑛2,g𝑦𝑛1𝑑𝑛1𝑞g𝑦1,g𝑦2,(3.21) where 𝑑=𝑘(g𝑦2,g𝑥2)/(1𝑙(g𝑦0,g𝑥0))<1. It follows that 𝑞𝐹𝑦𝑛1,𝑥𝑛1𝑦,𝐹𝑛,𝑥𝑛=𝑞g𝑦𝑛,g𝑦𝑛+1d𝑞g𝑦𝑛1,g𝑦𝑛.(3.22)
Let 𝑚>𝑛1. Then, it follows that 𝑞g𝑥𝑛,g𝑥𝑚𝑞g𝑥𝑛,g𝑥𝑛+1+𝑞g𝑥𝑛+1,g𝑥𝑛+2++𝑞g𝑥𝑚1,g𝑥𝑚𝑛1+𝑛++𝑚2𝑞g𝑥1,g𝑥2𝑛1𝑞1g𝑥1,g𝑥2,(3.23) and also, 𝑞g𝑦𝑛,g𝑦𝑚𝑞g𝑦𝑛,g𝑦𝑛+1+𝑞g𝑦𝑛+1,g𝑦𝑛+2++𝑞g𝑦𝑚1,g𝑦𝑚𝑑𝑛1+𝑑𝑛++𝑑𝑚2𝑞g𝑦1,g𝑦2𝑑𝑛𝑞1𝑑g𝑦1,g𝑦2.(3.24) Thus, Lemma 2.6 (3) shows that {g𝑥𝑛} and {g𝑦𝑛} are Cauchy sequences in g(𝑋). Since g(𝑋) is complete, there exists 𝑥 and 𝑦 in 𝑋 such that g𝑥𝑛g𝑥 and g𝑦𝑛g𝑦 as 𝑛. Using (q3), we have 𝑞g𝑥𝑛,g𝑥𝑛1𝑞1g𝑥1,g𝑥2,𝑞(3.25)g𝑦𝑛,g𝑦𝑑𝑛1𝑞1𝑑g𝑦1,g𝑦2.(3.26)
On the other hand and by using (3.20), we have 𝑞g𝑥𝑛𝑥,𝐹,𝑦𝐹𝑥=𝑞𝑛1,𝑦𝑛1𝑥,𝐹,𝑦𝑞g𝑥𝑛1,g𝑥𝑛2𝑞1g𝑥1,g𝑥2=𝑛1𝑞1g𝑥1,g𝑥2,(3.27) also by using (3.22), we have 𝑞g𝑦𝑛𝑦,𝐹,𝑥𝐹𝑦=𝑞𝑛1,𝑥𝑛1𝑦,𝐹,𝑥𝑑𝑞g𝑦𝑛1,g𝑦𝑑𝑑𝑛2𝑞1𝑑g𝑦1,g𝑦2=𝑑𝑛1𝑞1𝑑g𝑦1,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)=𝑞g𝑥,g𝑥𝐹𝑥=𝑞,𝑦𝑥,𝐹,𝑦𝑘g𝑥,g𝑦𝑞g𝑥𝑥,𝐹,𝑦+𝑙g𝑥,g𝑦𝑞g𝑥𝑥,𝐹,𝑦=[]=[]=𝑘(𝑢,𝑣)𝑞(𝑢,𝑢)+𝑙(𝑢,𝑣)𝑞(𝑢,𝑢)𝑘(𝑢,𝑣)+𝑙(𝑢,𝑣)𝑞(𝑢,𝑢)(𝑘+𝑙)(𝑢,𝑣)𝑞(𝑢,𝑢).(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 𝑞𝑢,𝑢1=𝑞g𝑥,g𝑥𝐹𝑥=𝑞,𝑦𝑥,𝐹,𝑦𝑘g𝑥,g𝑦𝑞g𝑥𝑥,𝐹,𝑦+𝑙g𝑥,g𝑦𝑞g𝑥𝑥,𝐹,𝑦𝑢=𝑘(𝑢,𝑣)𝑞(𝑢,u)+𝑙(𝑢,𝑣)𝑞1,𝑢1=𝜃,(3.30) and also, 𝑞𝑣,𝑣1=𝑞g𝑦,g𝑦𝐹𝑦=𝑞,𝑥𝑦,𝐹,𝑥𝑘g𝑦,g𝑥𝑞g𝑦𝑦,𝐹,𝑥+𝑙g𝑦,g𝑥𝑞g𝑦𝑦,𝐹,𝑥𝑣=𝑘(𝑣,𝑢)𝑞(𝑣,𝑣)+𝑙(𝑣,𝑢)𝑞1,𝑣1=𝜃.(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=gg𝑥𝑥=g𝐹,𝑦=𝐹g𝑥,g𝑦=𝐹g𝑥,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 𝐹𝑋×𝑋𝑋 and g𝑋𝑋 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,𝑣𝑋, 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 𝑢𝑋.

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 𝐹𝑋×𝑋𝑋 and g𝑋𝑋 satisfy the following contractive condition: 𝑞𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))𝑘(g𝑥,𝐹(𝑥,𝑦))+𝑞(g𝑢,𝐹(u,𝑣)),(3.34) for all 𝑥,𝑦,𝑢,𝑣𝑋, 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 𝑢𝑋.

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 𝐹𝑋×𝑋𝑋 and g𝑋𝑋 be two mappings and suppose that there exists mappings 𝑘,𝑙,𝑟𝑋×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,𝑣𝑋, (b)(𝑘+2𝑙+𝑟)(𝑥,𝑦)<1 for all 𝑥,𝑦𝑋, (c)(1𝑟(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,𝑣𝑋.
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 𝑢𝑋.

Proof. Choose 𝑥0,𝑦0𝑋. 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 {𝑥𝑛} and {𝑦𝑛} such that g𝑥n+1=𝐹(𝑥𝑛,𝑦𝑛),g𝑦𝑛+1=𝐹(𝑦𝑛,𝑥𝑛). Observe that (1𝑟(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(g𝑥,g𝑦)𝑞(g𝑥,𝐹(𝑥,𝑦))+𝑙(g𝑥,g𝑦)𝑞(g𝑥,𝐹(𝑢,𝑣))+𝑟(g𝑥,g𝑦)𝑞(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣)).(3.36) Then, we have 𝑞g𝑥𝑛,g𝑥𝑛+1𝐹𝑥=𝑞𝑛1,𝑦𝑛1𝑥,𝐹𝑛,𝑦𝑛𝑘g𝑥𝑛1,g𝑦𝑛1𝑞g𝑥𝑛1𝑥,𝐹𝑛1,𝑦𝑛1+𝑙g𝑥𝑛1,g𝑦𝑛1𝑞g𝑥𝑛1𝑥,𝐹𝑛,𝑦𝑛+𝑟g𝑥𝑛1,g𝑦𝑛1𝑞𝐹𝑥𝑛1,𝑦𝑛1𝑥,𝐹𝑛,𝑦𝑛𝐹𝑥=𝑘𝑛2,𝑦𝑛2𝑦,𝐹𝑛2,𝑥𝑛2𝑞g𝑥𝑛1,g𝑥𝑛𝐹𝑥+𝑙𝑛2,𝑦𝑛2𝑦,𝐹𝑛2,𝑥𝑛2𝑞g𝑥𝑛1,g𝑥𝑛+1𝐹𝑥+𝑟𝑛2,𝑦𝑛2𝑦,𝐹𝑛2,𝑥𝑛2𝑞g𝑥𝑛,g𝑥𝑛+1𝑘g𝑥𝑛2,g𝑦𝑛2𝑞g𝑥𝑛1,g𝑥𝑛+𝑙g𝑥𝑛2,g𝑦𝑛2𝑞g𝑥𝑛1,g𝑥𝑛+1+𝑟g𝑥𝑛2,g𝑦𝑛2𝑞g𝑥𝑛,g𝑥𝑛+1𝑘g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛+𝑙g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛+1+𝑟g𝑥1,g𝑦1𝑞g𝑥𝑛,g𝑥𝑛+1𝑘g𝑥1,g𝑦1𝑞𝑥𝑛1,𝑥𝑛+𝑙g𝑥1,g𝑦1𝑞𝑥𝑛1,𝑥𝑛+𝑙g𝑥1,g𝑦1𝑞g𝑥𝑛,g𝑥𝑛+1+𝑟g𝑥1,g𝑦1𝑞g𝑥𝑛,g𝑥𝑛+1.(3.37) Hence 𝑞g𝑥𝑛,g𝑥𝑛+1𝑘g𝑥1,g𝑦1+𝑙g𝑥1,g𝑦11𝑙g𝑥1,g𝑦1𝑟g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛=𝑞g𝑥𝑛1,g𝑥𝑛2𝑞g𝑥𝑛2,g𝑥𝑛1𝑛1𝑞g𝑥1,g𝑥2,(3.38) where =(𝑘(g𝑥1,g𝑦1)+𝑙(g𝑥1,g𝑦1))/(1𝑙(g𝑥1,g𝑦1)𝑟(g𝑥1,g𝑦1))<1.
Similarly, we have 𝑞g𝑦𝑛,g𝑦𝑛+1𝑘g𝑦1,g𝑥1+𝑙g𝑦1,g𝑥11𝑙g𝑦1,g𝑥1𝑟g𝑦1,g𝑥1𝑞g𝑦𝑛1,g𝑦𝑛=𝑑𝑞g𝑦𝑛1,g𝑦𝑛𝑑2𝑞g𝑦𝑛2,g𝑦𝑛1𝑑𝑛1𝑞g𝑦1,g𝑦2,(3.39) where 𝑑=(𝑘(g𝑦1,g𝑥1)+𝑙(g𝑦1,g𝑥1))/(1𝑙(g𝑦1,g𝑥1)𝑟(g𝑦1,g𝑥1))<1.
Let 𝑚>𝑛1. Then, it follows that 𝑞g𝑥𝑛,g𝑥𝑚𝑞g𝑥𝑛,g𝑥𝑛+1+𝑞g𝑥𝑛+1,g𝑥𝑛+2++𝑞g𝑥𝑚1,g𝑥𝑚𝑛1+𝑛++𝑚2𝑞g𝑥1,g𝑥2𝑛𝑞1g𝑥1,g𝑥2,𝑞g𝑦𝑛,g𝑦𝑚𝑞g𝑦𝑛,g𝑦𝑛+1+𝑞g𝑦𝑛+1,g𝑦𝑛+2++𝑞g𝑦𝑚1,g𝑦𝑚𝑑𝑛1+𝑑𝑛++𝑑𝑚2𝑞g𝑦1,g𝑦2𝑑𝑛𝑞1𝑑g𝑦1,g𝑦2.(3.40) Thus, Lemma 2.6 (3) shows that {g𝑥𝑛} and {g𝑦𝑛} are Cauchy sequences in g(𝑋). Since g(𝑋) is complete, there exists 𝑥,𝑦𝑋 such that g𝑥𝑛g𝑥 and g𝑦𝑛g𝑦 as 𝑛. Using (q3), we have 𝑞g𝑥𝑛,g𝑥𝑛𝑞1g𝑥1,g𝑥2,𝑞(3.41)g𝑦𝑛,g𝑦𝑑𝑛𝑞1𝑑g𝑦1,g𝑦2.(3.42)
On the other hand, 𝑞g𝑥𝑛𝑥,𝐹,𝑦𝐹𝑥=𝑞𝑛1,𝑦𝑛1𝑥,𝐹,𝑦𝑘g𝑥𝑛1,g𝑦𝑛1𝑞g𝑥𝑛1𝑥,𝐹𝑛1,𝑦𝑛1+𝑙g𝑥𝑛1,g𝑦𝑛1𝑞g𝑥𝑛1𝑥,𝐹,𝑦+𝑟g𝑥𝑛1,g𝑦𝑛1𝑞𝐹𝑥𝑛1,𝑦𝑛1𝑥,𝐹,𝑦𝐹𝑥=𝑘𝑛2,𝑦𝑛2𝑦,𝐹𝑛2,𝑥𝑛2𝑞g𝑥𝑛1,𝑥𝑛𝐹𝑥+𝑙𝑛2,𝑦𝑛2𝑦,𝐹𝑛2,𝑥𝑛2𝑞g𝑥𝑛1𝑥,𝐹,𝑦𝐹𝑥+𝑟𝑛2,𝑦𝑛2𝑦,𝐹𝑛2,𝑥𝑛2𝑞g𝑥𝑛𝑥,𝐹,𝑦𝑘g𝑥𝑛2,g𝑦𝑛2𝑞g𝑥𝑛1,g𝑥𝑛+𝑙g𝑥𝑛2,g𝑦𝑛2𝑞g𝑥𝑛1𝑥,F,𝑦+𝑟g𝑥𝑛2,g𝑦𝑛2𝑞g𝑥𝑛𝑥,𝐹,𝑦𝑘g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛+𝑙g𝑥1,g𝑦1𝑞g𝑥𝑛1𝑥,𝐹,𝑦+𝑟g𝑥1,g𝑦1𝑞g𝑥𝑛𝑥,𝐹,𝑦𝑘g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛+𝑙g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛+𝑙g𝑥1,g𝑦1𝑞g𝑥𝑛𝑥,𝐹,𝑦+𝑟g𝑥1,g𝑦1𝑞g𝑥𝑛𝑥,𝐹,𝑦.(3.43) Then, 𝑞g𝑥𝑛𝑥,𝐹,𝑦𝑘g𝑥1,g𝑦1+𝑙g𝑥1,g𝑦11𝑙g𝑥1,g𝑦1𝑟g𝑥1,g𝑦1𝑞g𝑥𝑛1,g𝑥𝑛=𝑞g𝑥𝑛1,g𝑥𝑛𝑛2𝑞g𝑥1,g𝑥2=𝑛1𝑞g𝑥1,g𝑥2𝑛1𝑞1g𝑥1,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)=𝑞g𝑥,g𝑥𝐹𝑥=𝑞,𝑦𝑥,𝐹,𝑦𝑘g𝑥,g𝑦𝑞g𝑥𝑥,𝐹,𝑦+𝑙g𝑥,g𝑦𝑞g𝑥𝑥,𝐹,𝑦+𝑟g𝑥,g𝑦𝑞𝐹𝑥,𝑦𝑥,𝐹,𝑦=[]=[]=𝑘(𝑢,𝑣)𝑞(𝑢,𝑢)+𝑙(𝑢,𝑣)𝑞(𝑢,𝑢)+𝑟(𝑢,𝑣)𝑞(𝑢,𝑢)=𝑘(𝑢,𝑣)𝑞(𝑢,𝑢)+𝑙(𝑢,𝑣)𝑞(𝑢,𝑢)+𝑙(𝑢,𝑣)𝑞(𝑢,𝑢)+𝑟(𝑢,𝑣)𝑞(𝑢,𝑢)𝑘(𝑢,𝑣)+𝑙(𝑢,𝑣)+𝑙(𝑢,𝑣)+𝑟(𝑢,𝑣)𝑞(𝑢,𝑢)(𝑘+2𝑙+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 𝑞𝑢,𝑢1=𝑞g𝑥,g𝑥𝐹𝑥=𝑞,𝑦𝑥,𝐹,𝑦𝑘g𝑥,g𝑦𝑞g𝑥𝑥,𝐹,𝑦+𝑙g𝑥,g𝑦𝑞g𝑥𝑥,𝐹,𝑦+𝑟g𝑥,g𝑦q𝐹𝑥,𝑦𝑥,𝐹,𝑦=𝑘(𝑢,𝑣)𝑞(𝑢,𝑢)+𝑙(𝑢,𝑣)𝑞𝑢,𝑢1+r(𝑢,𝑣)𝑞𝑢,𝑢1=𝑙(𝑢,𝑣)𝑞𝑢,𝑢1+𝑟(𝑢,𝑣)𝑞𝑢,𝑢1k(𝑢𝑣)𝑞𝑢,𝑢1+𝑙(𝑢,𝑣)𝑞𝑢,𝑢1+𝑙(𝑢,𝑣)𝑞𝑢,𝑢1+𝑟(𝑢,𝑣)𝑞𝑢,𝑢1=[]𝑞𝑘(𝑢,𝑣))+𝑙(𝑢,𝑣)+𝑙(𝑢,𝑣)+𝑟(𝑢,𝑣)𝑢,𝑢1=[]𝑞(𝑘+2𝑙+𝑟)(𝑢,𝑣)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=gg𝑥𝑥=g𝐹,𝑦=𝐹g𝑥,g𝑦=𝐹g𝑥,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