Abstract and Applied Analysis

Abstract and Applied Analysis / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 901792 | https://doi.org/10.1155/2012/901792

Zaid Mohammed Fadail, Abd Ghafur Bin Ahmad, "Common Coupled Fixed Point Theorems of Single-Valued Mapping for c-Distance in Cone Metric Spaces", Abstract and Applied Analysis, vol. 2012, Article ID 901792, 24 pages, 2012. https://doi.org/10.1155/2012/901792

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

Academic Editor: Ngai-Ching Wong
Received30 May 2012
Accepted08 Jul 2012
Published21 Aug 2012

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 ๐‘“ by considering different types of contractive conditions. Some of these works are noted in [2โ€“12].

In [13], Bhaskar and Lakshmikantham introduced the concept of coupled fixed point for a given partially ordered set ๐‘‹. Lakshmikantham and ฤ†iriฤ‡ [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 [16โ€“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 [26โ€“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 (๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹ร—๐‘‹ is said to be a coupled fixed point of the mapping ๐นโˆถ๐‘‹ร—๐‘‹โ†’๐‘‹ if ๐น(๐‘ฅ,๐‘ฆ)=๐‘ฅ and ๐น(๐‘ฆ,๐‘ฅ)=๐‘ฆ.

Definition 1.1. An element (๐‘ฅ,๐‘ฆ)โˆˆ๐‘‹ร—๐‘‹ is called (1)a coupled coincidence point of mappings ๐นโˆถ๐‘‹ร—๐‘‹โ†’๐‘‹ and gโˆถ๐‘‹โ†’๐‘‹ if g๐‘ฅ=๐น(๐‘ฅ,๐‘ฆ) and g๐‘ฆ=๐น(๐‘ฆ,๐‘ฅ), and (g๐‘ฅ,g๐‘ฆ) is called coupled point of coincidence. (2)a common coupled fixed point of mappings ๐นโˆถ๐‘‹ร—๐‘‹โ†’๐‘‹ and gโˆถ๐‘‹โ†’๐‘‹ if ๐‘ฅ=g๐‘ฅ=๐น(๐‘ฅ,๐‘ฆ),๐‘ฆ=g๐‘ฆ=๐น(๐‘ฆ,๐‘ฅ).

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

Definition 1.2. The mappings ๐นโˆถ๐‘‹ร—๐‘‹โ†’๐‘‹ and gโˆถ๐‘‹โ†’๐‘‹ are called w-compatible if g(F(๐‘ฅ,๐‘ฆ))=๐น(g๐‘ฅ,g๐‘ฆ), whenever g๐‘ฅ=๐น(๐‘ฅ,๐‘ฆ) and g๐‘ฆ=๐น(๐‘ฆ,๐‘ฅ).

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 ๐‘ƒ has nonempty interior.

2. Preliminaries

Let ๐ธ be a real Banach space and ๐œƒ denote the zero element in ๐ธ. A cone ๐‘ƒ is a subset of ๐ธ such that (1)๐‘ƒ is nonempty set closed and ๐‘ƒโ‰ {๐œƒ}, (2)if ๐‘Ž,๐‘ are nonnegative real numbers and ๐‘ฅ,๐‘ฆโˆˆ๐‘ƒ, then ๐‘Ž๐‘ฅ+๐‘๐‘ฆโˆˆ๐‘ƒ, (3)๐‘ฅโˆˆ๐‘ƒ and โˆ’๐‘ฅโˆˆ๐‘ƒ implies ๐‘ฅ=๐œƒ.

For any cone ๐‘ƒโŠ‚๐ธ, the partial ordering โชฏ with respect to ๐‘ƒ is defined by ๐‘ฅโชฏ๐‘ฆ if and only if ๐‘ฆโˆ’๐‘ฅโˆˆ๐‘ƒ. The notation of โ‰บ stand for ๐‘ฅโชฏ๐‘ฆ but ๐‘ฅโ‰ ๐‘ฆ. Also, we used ๐‘ฅโ‰ช๐‘ฆ to indicate that ๐‘ฆโˆ’๐‘ฅโˆˆint๐‘ƒ, where int๐‘ƒ denotes the interior of ๐‘ƒ. A cone ๐‘ƒ is called normal if there exists a number ๐พ such that ๐œƒโชฏ๐‘ฅโชฏ๐‘ฆโ‡’โ€–๐‘ฅโ€–โ‰ค๐พโ€–๐‘ฆโ€–,(2.1) for all ๐‘ฅ,๐‘ฆโˆˆ๐ธ. The least positive number ๐พ satisfying the above condition is called the normal constant of ๐‘ƒ.

Definition 2.1 (see [1]). Let ๐‘‹ be a nonempty set and ๐ธ a real Banach space equipped with the partial ordering โชฏ with respect to the cone ๐‘ƒ. Suppose that the mapping ๐‘‘โˆถ๐‘‹ร—๐‘‹โ†’๐ธ satisfies the following condition: (1)๐œƒโชฏ๐‘‘(๐‘ฅ,๐‘ฆ) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹ and ๐‘‘(๐‘ฅ,๐‘ฆ)=๐œƒ if and only if ๐‘ฅ=๐‘ฆ, (2)๐‘‘(๐‘ฅ,๐‘ฆ)=๐‘‘(๐‘ฆ,๐‘ฅ) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, (3)๐‘‘(๐‘ฅ,๐‘ฆ)โชฏ๐‘‘(๐‘ฅ,๐‘ฆ)+๐‘‘(๐‘ฆ,๐‘ง) for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹. Then ๐‘‘ is called a cone metric on ๐‘‹ and (๐‘‹,๐‘‘) is called a cone metric space.

Definition 2.2 (see [1]). Let (๐‘‹,๐‘‘) be a cone metric space and {๐‘ฅ๐‘›} a sequence in ๐‘‹ and ๐‘ฅโˆˆ๐‘‹. (1)For all ๐‘โˆˆ๐ธ with ๐œƒโ‰ช๐‘, if there exists a positive integer ๐‘ such that ๐‘‘(๐‘ฅ๐‘›,๐‘ฅ)โ‰ช๐‘ for all ๐‘›>๐‘, then ๐‘ฅ๐‘› is said to be convergent and ๐‘ฅ is the limit of {๐‘ฅ๐‘›}. We denote this by ๐‘ฅ๐‘›โ†’๐‘ฅ. (2)For all ๐‘โˆˆ๐ธ with ๐œƒโ‰ช๐‘, if there exists a positive integer ๐‘ such that ๐‘‘(๐‘ฅ๐‘›,๐‘ฅ๐‘š)โ‰ช๐‘ for all ๐‘›,๐‘š>๐‘, then {๐‘ฅ๐‘›} is called a Cauchy sequence in ๐‘‹. (3)A cone metric space (๐‘‹,๐‘‘) is called complete if every Cauchy sequence in ๐‘‹ is convergent.

Lemma 2.3 (see [8]). (1)If ๐ธ be a real Banach space with a cone ๐‘ƒ and ๐‘Žโชฏ๐œ†๐‘Ž, where ๐‘Žโˆˆ๐‘ƒ and 0โ‰ค๐œ†<1, then ๐‘Ž=๐œƒ. (2)If ๐‘โˆˆint๐‘ƒ, ๐œƒโชฏ๐‘Ž๐‘› and ๐‘Ž๐‘›โ†’๐œƒ, then there exists a positive integer ๐‘ such that ๐‘Ž๐‘›โ‰ช๐‘ for all ๐‘›โ‰ฅ๐‘.

Next, we give the notation of c-distance on a cone metric space which is a generalization of ๐œ”-distance of Kada et al. [32] with some properties.

Definition 2.4 (see [23]). Let (๐‘‹,๐‘‘) be a cone metric space. A function ๐‘žโˆถ๐‘‹ร—๐‘‹โ†’๐ธ is called a c-distance on ๐‘‹ if the following conditions hold: (q1)๐œƒโชฏ๐‘ž(๐‘ฅ,๐‘ฆ) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, (q2)๐‘ž(๐‘ฅ,๐‘ฆ)โชฏ๐‘ž(๐‘ฅ,๐‘ฆ)+๐‘ž(๐‘ฆ,๐‘ง) for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹, (q3) for each ๐‘ฅโˆˆ๐‘‹ and ๐‘›โ‰ฅ1, if ๐‘ž(๐‘ฅ,๐‘ฆ๐‘›)โชฏ๐‘ข for some ๐‘ข=๐‘ข๐‘ฅโˆˆ๐‘ƒ, then ๐‘ž(๐‘ฅ,๐‘ฆ)โชฏ๐‘ข whenever {๐‘ฆ๐‘›} is a sequence in ๐‘‹ converging to a point ๐‘ฆโˆˆ๐‘‹, (q4) for all ๐‘โˆˆ๐ธ with ๐œƒโ‰ช๐‘, there exists ๐‘’โˆˆ๐ธ with ๐œƒโ‰ช๐‘’ such that ๐‘ž(๐‘ง,๐‘ฅ)โ‰ช๐‘’ and ๐‘ž(๐‘ง,๐‘ฆ)โ‰ช๐‘’ imply ๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ช๐‘.

Example 2.5 (see [23]). Let ๐ธ=โ„ and ๐‘ƒ={๐‘ฅโˆˆ๐ธโˆถ๐‘ฅโ‰ฅ0}. Let ๐‘‹=[0,โˆž) and define a mapping ๐‘‘โˆถ๐‘‹ร—๐‘‹โ†’๐ธ by ๐‘‘(๐‘ฅ,๐‘ฆ)=|๐‘ฅโˆ’๐‘ฆ| for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. Then (๐‘‹,๐‘‘) is a cone metric space. Define a mapping ๐‘žโˆถ๐‘‹ร—๐‘‹โ†’๐ธ by ๐‘ž(๐‘ฅ,๐‘ฆ)=๐‘ฆ for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. Then ๐‘ž is a c-distance on ๐‘‹.

Lemma 2.6 (see [23]). Let (๐‘‹,๐‘‘) be a cone metric space and ๐‘ž is a ๐‘-distance on ๐‘‹. Let {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} be sequences in ๐‘‹ and ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹. Suppose that ๐‘ข๐‘› is a sequences in ๐‘ƒ converging to ๐œƒ. Then the following hold. (1)If ๐‘ž(๐‘ฅ๐‘›,๐‘ฆ)โชฏ๐‘ข๐‘› and ๐‘ž(๐‘ฅ๐‘›,๐‘ง)โชฏ๐‘ข๐‘›, then ๐‘ฆ=๐‘ง. (2)If ๐‘ž(๐‘ฅ๐‘›,๐‘ฆ๐‘›)โชฏ๐‘ข๐‘› and ๐‘ž(๐‘ฅ๐‘›,๐‘ง)โชฏ๐‘ข๐‘›, then {๐‘ฆ๐‘›} converges to ๐‘ง. (3)If ๐‘ž(๐‘ฅ๐‘›,๐‘ฅ๐‘š)โชฏ๐‘ข๐‘› for ๐‘š>๐‘›, then {๐‘ฅ๐‘›}is a Cauchy sequence in ๐‘‹. (4)If ๐‘ž(๐‘ฆ,๐‘ฅ๐‘›)โชฏ๐‘ข๐‘›, then {๐‘ฅ๐‘›} is a Cauchy sequence in ๐‘‹.

Remark 2.7 (see [23]). (1)๐‘ž(๐‘ฅ,๐‘ฆ)=๐‘ž(๐‘ฆ,๐‘ฅ) does not necessarily hold for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. (2)๐‘ž(๐‘ฅ,๐‘ฆ)=๐œƒ is not necessarily equivalent to ๐‘ฅ=๐‘ฆ for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹.

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 (๐‘‹,๐‘‘) be a cone metric space with a cone ๐‘ƒ having nonempty interior and ๐‘ž is a c-distance on ๐‘‹. Let ๐นโˆถ๐‘‹ร—๐‘‹โ†’๐‘‹ and gโˆถ๐‘‹โ†’๐‘‹ be two mappings and suppose that there exists mappings ๐‘˜,๐‘™โˆถ๐‘‹ร—๐‘‹โ†’[0,1) such that the following hold: (a)๐‘˜(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โ‰ค๐‘˜(g๐‘ฅ,g๐‘ฆ) and ๐‘™(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โ‰ค๐‘™(g๐‘ฅ,g๐‘ฆ) for all ๐‘ฅ,๐‘ฆ,๐‘ข,๐‘ฃโˆˆ๐‘‹, (b)๐‘˜(๐‘ฅ,๐‘ฆ)=๐‘˜(๐‘ฆ,๐‘ฅ) and ๐‘™(๐‘ฅ,๐‘ฆ)=๐‘™(๐‘ฆ,๐‘ฅ) for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, (c)(๐‘˜+๐‘™)(๐‘ฅ,๐‘ฆ)<1 for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, (d)๐‘ž(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โชฏ๐‘˜(g๐‘ฅ,g๐‘ฆ)๐‘ž(g๐‘ฅ,g๐‘ข)+๐‘™(g๐‘ฅ,g๐‘ฆ)๐‘ž(g๐‘ฆ,g๐‘ฃ) for all ๐‘ฅ,๐‘ฆ,๐‘ข,๐‘ฃโˆˆ๐‘‹. If ๐น(๐‘‹ร—๐‘‹)โŠ†g(๐‘‹) and g(๐‘‹) is a complete subspace of ๐‘‹, then ๐น and g have a unique coupled point of coincidence (๐‘ข,๐‘ฃ) in ๐‘‹ร—๐‘‹. Further, if ๐‘ข=g๐‘ฅ1=๐น(๐‘ฅ1,๐‘ฆ1) and ๐‘ฃ=g๐‘ฆ1=๐น(๐‘ฆ1,๐‘ฅ1), then ๐‘ž(๐‘ข,๐‘ข)=๐œƒ and ๐‘ž(๐‘ฃ,๐‘ฃ)=๐œƒ. Moreover, if ๐น and g are w-compatible, then ๐น and g have a unique common coupled fixed point and the common coupled fixed point of ๐น and g is of the form (๐‘ข,๐‘ข) for some ๐‘ขโˆˆ๐‘‹.

Proof. Choose ๐‘ฅ0,๐‘ฆ0โˆˆ๐‘‹. Set g๐‘ฅ1=๐น(๐‘ฅ0,๐‘ฆ0),g๐‘ฆ1=๐น(๐‘ฆ0,๐‘ฅ0), this can be done because ๐น(๐‘‹ร—๐‘‹)โŠ†g(๐‘‹). Continuing this process, we obtain two sequences {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} such that g๐‘ฅ๐‘›+1=๐น(๐‘ฅ๐‘›,๐‘ฆ๐‘›),g๐‘ฆ๐‘›+1=๐น(๐‘ฆ๐‘›,๐‘ฅ๐‘›). Then we have ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘ฅ=๐‘ž๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ฅ,๐น๐‘›,๐‘ฆ๐‘›๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฅ๐‘›โˆ’1,g๐‘ฆ๐‘›โˆ’1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ๐‘›โˆ’1,g๐‘ฆ๐‘›โˆ’1๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธ๎€ท๐น๎€ท๐‘ฅ=๐‘˜๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐น๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท๐น๎€ท๐‘ฅ+๐‘™๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐น๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฆ๐‘›โˆ’1,gy๐‘›๎€ธ๎€ทโชฏ๐‘˜g๐‘ฅ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธโ‹ฎ๎€ทโชฏ๐‘˜g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธ.(3.1) Similarly, we have ๐‘ž๎€ทg๐‘ฆ๐‘›,g๐‘ฆ๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘ฆ=๐‘ž๐‘›โˆ’1,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘ฆ,๐น๐‘›,๐‘ฅ๐‘›๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฆ๐‘›โˆ’1,g๐‘ฅ๐‘›โˆ’1๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฆ๐‘›โˆ’1,g๐‘ฅ๐‘›โˆ’1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท๐น๎€ท๐‘ฆ=๐‘˜๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๎€ธ๎€ท๐‘ฅ,๐น๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธ๎€ท๐น๎€ท๐‘ฆ+๐‘™๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๎€ธ๎€ท๐‘ฅ,๐น๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ทโชฏ๐‘˜g๐‘ฆ๐‘›โˆ’2,g๐‘ฅ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฆ๐‘›โˆ’2,g๐‘ฅ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธโ‹ฎ๎€ทโชฏ๐‘˜g๐‘ฆ1,g๐‘ฅ1๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฆ1,g๐‘ฅ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท=๐‘˜g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ.(3.2) Put ๐‘ž๐‘›=๐‘ž(g๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1)+๐‘ž(g๐‘ฆ๐‘›,g๐‘ฆ๐‘›+1). Then we have ๐‘ž๐‘›๎€ท=๐‘žg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ๎€ท+๐‘žg๐‘ฆ๐‘›,g๐‘ฆ๐‘›+1๎€ธ๎€ทโชฏ๐‘˜g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธ๎€ท+๐‘˜g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ=๎€บ๐‘˜๎€ทg๐‘ฅ1,g๐‘ฆ1๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๐‘ž๎€ท๎€ธ๎€ป๎€บg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘žg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›=๎€บ๐‘˜๎€ท๎€ธ๎€ปg๐‘ฅ1,g๐‘ฆ1๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๐‘ž๎€ธ๎€ป๐‘›โˆ’1=โ„Ž๐‘ž๐‘›โˆ’1โ‹ฎโชฏโ„Ž๐‘›โˆ’1๐‘ž1,(3.3) where โ„Ž=๐‘˜(g๐‘ฅ1,g๐‘ฆ1)+๐‘™(g๐‘ฅ1,g๐‘ฆ1)<1.
Let ๐‘š>๐‘›โ‰ฅ1. It follows that ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘š๎€ธ๎€ทโชฏ๐‘žg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ๎€ท+๐‘žg๐‘ฅ๐‘›+1,g๐‘ฅ๐‘›+2๎€ธ๎€ท+โ‹ฏ+๐‘žg๐‘ฅ๐‘šโˆ’1,g๐‘ฅ๐‘š๎€ธ,๐‘ž๎€ทg๐‘ฆ๐‘›,g๐‘ฆ๐‘š๎€ธ๎€ทโชฏ๐‘žg๐‘ฆ๐‘›,g๐‘ฆ๐‘›+1๎€ธ๎€ท+๐‘žg๐‘ฆ๐‘›+1,g๐‘ฅ๐‘›+2๎€ธ๎€ท+โ‹ฏ+๐‘žg๐‘ฆ๐‘šโˆ’1,g๐‘ฆ๐‘š๎€ธ.(3.4) Then we have ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘š๎€ธ๎€ท+๐‘žg๐‘ฆ๐‘›,g๐‘ฆ๐‘š๎€ธโชฏ๐‘ž๐‘›+๐‘ž๐‘›+1+โ‹ฏ+๐‘ž๐‘šโˆ’1โชฏโ„Ž๐‘›๐‘ž1+โ„Ž๐‘›+1๐‘ž1+โ‹ฏ+โ„Ž๐‘šโˆ’1๐‘ž1=๎€ทโ„Ž๐‘›โˆ’1+โ„Ž๐‘›+โ‹ฏ+โ„Ž๐‘šโˆ’2๎€ธ๐‘ž1โชฏโ„Ž๐‘›โˆ’1๐‘ž1โˆ’โ„Ž1.(3.5) Consequently, ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘š๎€ธโชฏโ„Ž๐‘›โˆ’1๐‘ž1โˆ’โ„Ž1,๐‘ž๎€ทg๐‘ฆ๐‘›,g๐‘ฆ๐‘š๎€ธโชฏโ„Ž๐‘›โˆ’1๐‘ž1โˆ’โ„Ž1.(3.6) Thus, Lemma 2.6 (3) shows that {g๐‘ฅ๐‘›} and {g๐‘ฆ๐‘›} are Cauchy sequences in g(๐‘‹). Since g(๐‘‹) is complete, there exists ๐‘ฅโˆ— and ๐‘ฆโˆ— in ๐‘‹ such that g๐‘ฅ๐‘›โ†’g๐‘ฅโˆ— and g๐‘ฆ๐‘›โ†’g๐‘ฆโˆ— as ๐‘›โ†’โˆž. Using q3, we have ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅโˆ—๎€ธโชฏโ„Ž๐‘›โˆ’1๐‘ž1โˆ’โ„Ž1,๐‘ž๎€ทg๐‘ฆ๐‘›,g๐‘ฆโˆ—๎€ธโชฏโ„Ž๐‘›โˆ’1๐‘ž1โˆ’โ„Ž1.(3.7)
On the other hand, ๐‘ž๎€ทg๐‘ฅ๐‘›๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๐น๎€ท๐‘ฅ๎€ธ๎€ธ=๐‘ž๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฅ๐‘›โˆ’1,g๐‘ฆ๐‘›โˆ’1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅโˆ—๎€ธ๎€ท+๐‘™g๐‘ฅ๐‘›โˆ’1,g๐‘ฆ๐‘›โˆ’1๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆโˆ—๎€ธ๎€ทF๎€ท๐‘ฅ=๐‘˜๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐น๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฅ๐‘›โˆ’1,g๐‘ฅโˆ—๎€ธ๎€ท๐น๎€ท๐‘ฅ+๐‘™๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐น๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฆ๐‘›โˆ’1,g๐‘ฆโˆ—๎€ธ๎€ทโชฏ๐‘˜g๐‘ฅ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅโˆ—๎€ธ๎€ท+๐‘™g๐‘ฅ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆโˆ—๎€ธโ‹ฎ๎€ทโชฏ๐‘˜g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅโˆ—๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆโˆ—๎€ธ๎€ทโชฏ๐‘˜g๐‘ฅ1,g๐‘ฆ1๎€ธโ„Ž๐‘›โˆ’2๐‘ž1โˆ’โ„Ž1๎€ท๐‘ฅ+๐‘™1,๐‘ฆ1๎€ธโ„Ž๐‘›โˆ’2๐‘ž1โˆ’โ„Ž1=๎€บ๐‘˜๎€ทg๐‘ฅ1,g๐‘ฆ1๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1โ„Ž๎€ธ๎€ป๐‘›โˆ’2๐‘ž1โˆ’โ„Ž1โ„Ž=โ„Ž๐‘›โˆ’2๐‘ž1โˆ’โ„Ž1=โ„Ž๐‘›โˆ’1๐‘ž1โˆ’โ„Ž1.(3.8) Thus, Lemma 2.6 (1), (3.5), and (3.8) show that g๐‘ฅโˆ—=๐น(๐‘ฅโˆ—,๐‘ฆโˆ—). By similar way, we can prove that g๐‘ฆโˆ—=๐น(๐‘ฆโˆ—,๐‘ฅโˆ—). Therefore, (๐‘ฅโˆ—,๐‘ฆโˆ—) is a coupled coincidence point of ๐น and g.
Suppose that ๐‘ข=g๐‘ฅโˆ—=๐น(๐‘ฅโˆ—,๐‘ฆโˆ—) and ๐‘ฃ=g๐‘ฆโˆ—=๐น(๐‘ฆโˆ—,๐‘ฅโˆ—). Then we have ๐‘ž๎€ท(๐‘ข,๐‘ข)=๐‘žg๐‘ฅโˆ—,g๐‘ฅโˆ—๎€ธ๎€ท๐น๎€ท๐‘ฅ=๐‘žโˆ—,๐‘ฆโˆ—๎€ธ๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฅโˆ—,g๐‘ฆโˆ—๎€ธ๐‘ž๎€ทg๐‘ฅโˆ—,g๐‘ฅโˆ—๎€ธ๎€ท+๐‘™g๐‘ฅโˆ—,g๐‘ฆโˆ—๎€ธ๐‘ž๎€ทg๐‘ฆโˆ—,g๐‘ฆโˆ—๎€ธ๎€ท=๐‘˜(๐‘ข,๐‘ฃ)๐‘ž(๐‘ข,๐‘ข)+๐‘™(๐‘ข,๐‘ฃ)๐‘ž(๐‘ฃ,๐‘ฃ),(3.9)๐‘ž(๐‘ฃ,๐‘ฃ)=๐‘žg๐‘ฆโˆ—,g๐‘ฆโˆ—๎€ธ๎€ท๐น๎€ท๐‘ฆ=๐‘žโˆ—,๐‘ฅโˆ—๎€ธ๎€ท๐‘ฆ,๐นโˆ—,๐‘ฅโˆ—๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฆโˆ—,g๐‘ฅโˆ—๎€ธ๐‘ž๎€ทg๐‘ฆโˆ—,g๐‘ฆโˆ—๎€ธ๎€ท+๐‘™g๐‘ฆโˆ—,g๐‘ฅโˆ—๎€ธ๐‘ž๎€ทg๐‘ฅโˆ—,g๐‘ฅโˆ—๎€ธ=๐‘˜(๐‘ฃ,๐‘ข)๐‘ž(๐‘ฃ,๐‘ฃ)+๐‘™(๐‘ฃ,๐‘ข)๐‘ž(๐‘ข,๐‘ข)=๐‘˜(๐‘ข,๐‘ฃ)๐‘ž(๐‘ฃ,๐‘ฃ)+๐‘™(๐‘ข,๐‘ฃ)๐‘ž(๐‘ข,๐‘ข).(3.10) This implies that =[๐‘˜๐‘ž]=[].๐‘ž(๐‘ข,๐‘ข)+๐‘ž(๐‘ฃ,๐‘ฃ)โชฏ๐‘˜(๐‘ข,๐‘ฃ)๐‘ž(๐‘ข,๐‘ข)+๐‘™(๐‘ข,๐‘ฃ)๐‘ž(๐‘ฃ,๐‘ฃ)+๐‘˜(๐‘ข,๐‘ฃ)๐‘ž(๐‘ฃ,๐‘ฃ)+๐‘™(๐‘ข,๐‘ฃ)๐‘ž(๐‘ข,๐‘ข)(๐‘ข,๐‘ฃ)+๐‘™(๐‘ข,๐‘ฃ)][(๐‘ข,๐‘ข)+๐‘ž(๐‘ฃ,๐‘ฃ)(๐‘˜+๐‘™)(๐‘ข,๐‘ฃ)][๐‘ž(๐‘ข,๐‘ข)+๐‘ž(๐‘ฃ,๐‘ฃ)(3.11) Since (๐‘˜+๐‘™)(๐‘ข,๐‘ฃ)<1, Lemma 2.3 (1) shows that ๐‘ž(๐‘ข,๐‘ข)+๐‘ž(๐‘ฃ,๐‘ฃ)=๐œƒ. But ๐‘ž(๐‘ข,๐‘ข)โ‰ฝ๐œƒ and ๐‘ž(๐‘ฃ,๐‘ฃ)โ‰ฝ๐œƒ. Consequently, ๐‘ž(๐‘ข,๐‘ข)=๐œƒ and ๐‘ž(๐‘ฃ,๐‘ฃ)=๐œƒ.
Finally, suppose there is another coupled point of coincidence (๐‘ข1,๐‘ฃ1) of ๐น and g such that ๐‘ข1=g๐‘ฅ๎…ž=๐น(๐‘ฅ๎…ž,๐‘ฆ๎…ž) and ๐‘ฃ1=g๐‘ฆ๎…ž=๐น(๐‘ฆ๎…ž,๐‘ฅ๎…ž) for some (๐‘ฅ๎…ž,๐‘ฆ๎…ž) in ๐‘‹ร—๐‘‹. Then we have ๐‘ž๎€ท๐‘ข,๐‘ข1๎€ธ๎€ท=๐‘žg๐‘ฅโˆ—,g๐‘ฅ๎…ž๎€ธ๎€ท๐น๎€ท๐‘ฅ=๐‘žโˆ—,๐‘ฆโˆ—๎€ธ๎€ท๐‘ฅ,F๎…ž,๐‘ฆ๎…ž๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฅโˆ—,g๐‘ฆโˆ—๎€ธ๐‘ž๎€ทg๐‘ฅโˆ—,g๐‘ฅ๎…ž๎€ธ๎€ท+๐‘™g๐‘ฅโˆ—,g๐‘ฆโˆ—๎€ธ๐‘ž๎€ทg๐‘ฆโˆ—,g๐‘ฆ๎…ž๎€ธ๎€ท=๐‘˜(๐‘ฃ,๐‘ข)๐‘ž๐‘ฃ,๐‘ฃ1๎€ธ๎€ท+๐‘™(๐‘ฃ,๐‘ข)๐‘ž๐‘ข,๐‘ข1๎€ธ๎€ท=๐‘˜(๐‘ข,๐‘ฃ)๐‘ž๐‘ข,๐‘ข1๎€ธ๎€ท+๐‘™(๐‘ข,๐‘ฃ)๐‘ž๐‘ฃ,๐‘ฃ1๎€ธ,(3.12)
and also, ๐‘ž๎€ท๐‘ฃ,๐‘ฃ1๎€ธ๎€ท=๐‘žg๐‘ฆโˆ—,g๐‘ฆ๎…ž๎€ธ๎€ท๐น๎€ท๐‘ฆ=๐‘žโˆ—,๐‘ฅโˆ—๎€ธ๎€ท๐‘ฆ,F๎…ž,๐‘ฅ๎…ž๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฆโˆ—,g๐‘ฅโˆ—๎€ธ๐‘ž๎€ทg๐‘ฆโˆ—,g๐‘ฆ๎…ž๎€ธ๎€ท+๐‘™g๐‘ฆโˆ—,g๐‘ฅโˆ—๎€ธ๐‘ž๎€ทg๐‘ฅโˆ—,g๐‘ฅ๎…ž๎€ธ๎€ท=๐‘˜(๐‘ฃ,๐‘ข)๐‘ž๐‘ฃ,๐‘ฃ1๎€ธ๎€ท+๐‘™(๐‘ฃ,๐‘ข)๐‘ž๐‘ข,๐‘ข1๎€ธ๎€ท=๐‘˜(๐‘ข,๐‘ฃ)๐‘ž๐‘ฃ,๐‘ฃ1๎€ธ๎€ท+๐‘™(๐‘ข,๐‘ฃ)๐‘ž๐‘ข,๐‘ข1๎€ธ.(3.13) This implies that ๐‘ž๎€ท๐‘ข,๐‘ข1๎€ธ๎€ท+๐‘ž๐‘ฃ,๐‘ฃ1๎€ธ๎€ทโชฏ๐‘˜(๐‘ข,๐‘ฃ)๐‘ž๐‘ข,๐‘ข1๎€ธ๎€ท+๐‘™(๐‘ข,๐‘ฃ)๐‘ž๐‘ฃ,๐‘ฃ1๎€ธ๎€ท+๐‘˜(๐‘ข,๐‘ฃ)๐‘ž๐‘ฃ,๐‘ฃ1๎€ธ๎€ท+๐‘™(๐‘ข,๐‘ฃ)๐‘ž๐‘ข,๐‘ข1๎€ธ=[]๎€บ๐‘ž๎€ท๐‘˜(๐‘ข,๐‘ฃ)+๐‘™(๐‘ข,๐‘ฃ)๐‘ข,๐‘ข1๎€ธ๎€ท+๐‘ž๐‘ฃ,๐‘ฃ1=[]๎€บ๐‘ž๎€ท๎€ธ๎€ป(๐‘˜+๐‘™)(๐‘ข,๐‘ฃ)๐‘ข,๐‘ข1๎€ธ๎€ท+๐‘ž๐‘ฃ,๐‘ฃ1.๎€ธ๎€ป(3.14) Since (๐‘˜+๐‘™)(๐‘ข,๐‘ฃ)<1, Lemma 2.3 (1) shows that ๐‘ž(๐‘ข,๐‘ข1)+๐‘ž(๐‘ฃ,๐‘ฃ1)=๐œƒ. But ๐‘ž(๐‘ข,๐‘ข1)โ‰ฝ๐œƒ and ๐‘ž(๐‘ฃ,๐‘ฃ1)โ‰ฝ๐œƒ. Hence ๐‘ž(๐‘ข,๐‘ข1)=๐œƒ and ๐‘ž(๐‘ฃ,๐‘ฃ1)=๐œƒ. Also we have ๐‘ž(๐‘ข,๐‘ข)=๐œƒ and ๐‘ž(๐‘ฃ,๐‘ฃ)=๐œƒ. Thus, Lemma 2.6 (1) shows that ๐‘ข=๐‘ข1 and ๐‘ฃ=๐‘ฃ1, which implies that (๐‘ข,๐‘ฃ)=(๐‘ข1,๐‘ฃ1). Similarly, we can prove that ๐‘ข=๐‘ฃ1 and ๐‘ฃ=๐‘ข1. Thus, ๐‘ข=๐‘ฃ. Therefore, (๐‘ข,๐‘ข) is the unique coupled point of coincidence. Now, let ๐‘ข=g๐‘ฅโˆ—=๐น(๐‘ฅโˆ—,๐‘ฆโˆ—). Since ๐น and g are w-compatible, then we have ๎€ทg๐‘ข=gg๐‘ฅโˆ—๎€ธ๎€ท๐‘ฅ=g๐นโˆ—,๐‘ฆโˆ—๎€ธ๎€ท=๐นg๐‘ฅโˆ—,g๐‘ฆโˆ—๎€ธ๎€ท=๐นg๐‘ฅโˆ—,g๐‘ฅโˆ—๎€ธ=๐น(๐‘ข,๐‘ข).(3.15) Thus (g๐‘ข,g๐‘ข) is a coupled point of coincidence. The uniqueness of the coupled point of coincidence implies that g๐‘ข=๐‘ข. Therefore, ๐‘ข=g๐‘ข=๐น(๐‘ข,๐‘ข). Hence (๐‘ข,๐‘ข) is the unique common coupled fixed point of ๐น and g.

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

Corollary 3.2. Let (๐‘‹,๐‘‘) be a cone metric space with a cone ๐‘ƒ having nonempty interior and ๐‘ž is a c-distance on ๐‘‹. Suppose the mappings ๐นโˆถ๐‘‹ร—๐‘‹โ†’๐‘‹ and gโˆถ๐‘‹โ†’๐‘‹ satisfy the following contractive condition: ๐‘ž(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โชฏ๐‘˜๐‘ž(g๐‘ฅ,g๐‘ข)+๐‘™๐‘ž(g๐‘ฆ,g๐‘ฃ),(3.16) for all ๐‘ฅ,๐‘ฆ,๐‘ข,๐‘ฃโˆˆ๐‘‹, where ๐‘˜,๐‘™ are nonnegative constants with ๐‘˜+๐‘™<1. If ๐น(๐‘‹ร—๐‘‹)โŠ†g(๐‘‹) and g(๐‘‹) is a complete subspace of ๐‘‹, then ๐น and g have a coupled coincidence point in ๐‘‹. Further, if g๐‘ฅ1=๐น(๐‘ฅ1,๐‘ฆ1) and g๐‘ฆ1=๐น(๐‘ฆ1,๐‘ฅ1), then ๐‘ž(g๐‘ฅ1,g๐‘ฅ1)=๐œƒ and ๐‘ž(g๐‘ฆ1,g๐‘ฆ1)=๐œƒ. Moreover, if ๐น and g are w -compatible, then ๐น and g have a unique common coupled fixed point and the common coupled fixed point of ๐น and g is of the form (๐‘ข,๐‘ข) for some ๐‘ขโˆˆ๐‘‹.

Corollary 3.3. Let (๐‘‹,๐‘‘) be a cone metric space with a cone ๐‘ƒ having nonempty interior and ๐‘ž is a c-distance on ๐‘‹. Suppose the mappings ๐นโˆถ๐‘‹ร—๐‘‹โ†’๐‘‹ and gโˆถ๐‘‹โ†’๐‘‹ satisfy the following contractive condition: ๐‘ž๎€บ๐‘ž๎€ป(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โชฏ๐‘˜(g๐‘ฅ,g๐‘ข)+๐‘ž(g๐‘ฆ,g๐‘ฃ),(3.17) for all ๐‘ฅ,๐‘ฆ,๐‘ข,๐‘ฃโˆˆ๐‘‹, where ๐‘˜โˆˆ[0,1/2) is a constants. If ๐น(๐‘‹ร—๐‘‹)โŠ†g(๐‘‹) and g(๐‘‹) is a complete subspace of ๐‘‹, then ๐น and g have a unique coupled point of coincidence (๐‘ข,๐‘ฃ) in ๐‘‹ร—๐‘‹. Further, if ๐‘ข=g๐‘ฅ1=๐น(๐‘ฅ1,๐‘ฆ1) and ๐‘ฃ=g๐‘ฆ1=๐น(๐‘ฆ1,๐‘ฅ1), then ๐‘ž(๐‘ข,๐‘ข)=๐œƒ and ๐‘ž(๐‘ฃ,๐‘ฃ)=๐œƒ. Moreover, if ๐น and g are w -compatible, then ๐น and g have a unique common coupled fixed point and the common coupled fixed point of ๐น and g is of the form (๐‘ข,๐‘ข) for some ๐‘ขโˆˆ๐‘‹.

Theorem 3.4. Let (๐‘‹,๐‘‘) be a cone metric space with a cone ๐‘ƒ having nonempty interior and ๐‘ž is a c-distance on ๐‘‹. Let ๐นโˆถ๐‘‹ร—๐‘‹โ†’๐‘‹ and gโˆถ๐‘‹โ†’๐‘‹ be two mappings and suppose that there exists mappings ๐‘˜,๐‘™โˆถ๐‘‹ร—๐‘‹โ†’[0,1) such that the following hold: (a)๐‘˜(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โ‰ค๐‘˜(g๐‘ฅ,g๐‘ฆ) and ๐‘™(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โ‰ค๐‘™(g๐‘ฅ,g๐‘ฆ) for all ๐‘ฅ,๐‘ฆ,๐‘ข,๐‘ฃโˆˆ๐‘‹, (b)(๐‘˜+๐‘™)(๐‘ฅ,๐‘ฆ)<1 for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, (c)๐‘ž(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โชฏ๐‘˜(g๐‘ฅ,g๐‘ฆ)๐‘ž(g๐‘ฅ,๐น(๐‘ฅ,๐‘ฆ))+๐‘™(g๐‘ฅ,g๐‘ฆ)๐‘ž(g๐‘ข,๐น(๐‘ข,๐‘ฃ)) for all ๐‘ฅ,๐‘ฆ,๐‘ข,๐‘ฃโˆˆ๐‘‹.
If ๐น(๐‘‹ร—๐‘‹)โŠ†g(๐‘‹) and g(๐‘‹) is a complete subspace of ๐‘‹, then ๐น and g have a unique coupled point of coincidence (๐‘ข,๐‘ฃ) in ๐‘‹ร—๐‘‹. Further, if ๐‘ข=g๐‘ฅ1=๐น(๐‘ฅ1,๐‘ฆ1) and ๐‘ฃ=g๐‘ฆ1=๐น(๐‘ฆ1,๐‘ฅ1), then ๐‘ž(๐‘ข,๐‘ข)=๐œƒ and ๐‘ž(๐‘ฃ,๐‘ฃ)=๐œƒ. Moreover, if ๐น and g are w -compatible, then ๐น and g have a unique common coupled fixed point and the common coupled fixed point of ๐น and g is of the form (๐‘ข,๐‘ข) for some ๐‘ขโˆˆ๐‘‹.

Proof. Choose ๐‘ฅ0,๐‘ฆ0โˆˆ๐‘‹. Set g๐‘ฅ1=๐น(๐‘ฅ0,๐‘ฆ0),g๐‘ฆ1=๐น(๐‘ฆ0,๐‘ฅ0). This can be done because ๐น(๐‘‹ร—๐‘‹)โŠ†g(๐‘‹). Continuing this process, we obtain two sequences {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} such that g๐‘ฅ๐‘›+1=๐น(๐‘ฅ๐‘›,๐‘ฆ๐‘›),โ€‰g๐‘ฆ๐‘›+1=๐น(๐‘ฆ๐‘›,๐‘ฅ๐‘›).
Then we have ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘ฅ=๐‘ž๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ฅ,๐น๐‘›,๐‘ฆ๐‘›๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฅ๐‘›โˆ’1,g๐‘ฆ๐‘›โˆ’1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1๎€ท๐‘ฅ,๐น๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ท๎€ธ๎€ธ+๐‘™g๐‘ฅ๐‘›โˆ’1,g๐‘ฆ๐‘›โˆ’1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›๎€ท๐‘ฅ,๐น๐‘›,๐‘ฆ๐‘›๎€ท๐น๎€ท๐‘ฅ๎€ธ๎€ธ=๐‘˜๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐น๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท๐น๎€ท๐‘ฅ+๐‘™๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐น๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ๎€ทโชฏ๐‘˜g๐‘ฅ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธโ‹ฎ๎€ทโชฏ๐‘˜g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ.(3.18)
Hence ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธโชฏ๐‘˜๎€ทg๐‘ฅ1,g๐‘ฆ1๎€ธ๎€ท1โˆ’๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท=โ„Ž๐‘žg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธโชฏโ„Ž2๐‘ž๎€ทgx๐‘›โˆ’2,g๐‘ฅ๐‘›โˆ’1๎€ธโ‹ฎโชฏโ„Ž๐‘›โˆ’1๐‘ž๎€ทg๐‘ฅ1,g๐‘ฅ2๎€ธ,(3.19) where โ„Ž=๐‘˜(g๐‘ฅ1,g๐‘ฆ1)/(1โˆ’๐‘™(g๐‘ฅ1,g๐‘ฆ1))<1. It follows that ๐‘ž๎€ท๐น๎€ท๐‘ฅ๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ฅ,๐น๐‘›,๐‘ฆ๐‘›๎€ท๎€ธ๎€ธ=๐‘žg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ๎€ทโชฏโ„Ž๐‘žg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ.(3.20)
Similarly, we have ๐‘ž๎€ทg๐‘ฆ๐‘›,g๐‘ฆ๐‘›+1๎€ธโชฏ๐‘˜๎€ทg๐‘ฆ1,g๐‘ฅ1๎€ธ๎€ท1โˆ’๐‘™g๐‘ฆ1,g๐‘ฅ1๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธ๎€ท=๐‘‘๐‘žg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธโชฏ๐‘‘2๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’1๎€ธโ‹ฎโชฏ๐‘‘๐‘›โˆ’1๐‘ž๎€ทg๐‘ฆ1,g๐‘ฆ2๎€ธ,(3.21) where ๐‘‘=๐‘˜(g๐‘ฆ2,g๐‘ฅ2)/(1โˆ’๐‘™(g๐‘ฆ0,g๐‘ฅ0))<1. It follows that ๐‘ž๎€ท๐น๎€ท๐‘ฆ๐‘›โˆ’1,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘ฆ,๐น๐‘›,๐‘ฅ๐‘›๎€ท๎€ธ๎€ธ=๐‘žg๐‘ฆ๐‘›,g๐‘ฆ๐‘›+1๎€ธ๎€ทโชฏd๐‘žg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธ.(3.22)
Let ๐‘š>๐‘›โ‰ฅ1. Then, it follows that ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘š๎€ธ๎€ทโชฏ๐‘žg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ๎€ท+๐‘žg๐‘ฅ๐‘›+1,g๐‘ฅ๐‘›+2๎€ธ๎€ท+โ‹ฏ+๐‘žg๐‘ฅ๐‘šโˆ’1,g๐‘ฅ๐‘š๎€ธโชฏ๎€ทโ„Ž๐‘›โˆ’1+โ„Ž๐‘›+โ‹ฏ+โ„Ž๐‘šโˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ1,g๐‘ฅ2๎€ธโชฏโ„Ž๐‘›โˆ’1๐‘ž๎€ท1โˆ’โ„Žg๐‘ฅ1,g๐‘ฅ2๎€ธ,(3.23) and also, ๐‘ž๎€ทg๐‘ฆ๐‘›,g๐‘ฆ๐‘š๎€ธ๎€ทโชฏ๐‘žg๐‘ฆ๐‘›,g๐‘ฆ๐‘›+1๎€ธ๎€ท+๐‘žg๐‘ฆ๐‘›+1,g๐‘ฆ๐‘›+2๎€ธ๎€ท+โ‹ฏ+๐‘žg๐‘ฆ๐‘šโˆ’1,g๐‘ฆ๐‘š๎€ธโชฏ๎€ท๐‘‘๐‘›โˆ’1+๐‘‘๐‘›+โ‹ฏ+๐‘‘๐‘šโˆ’2๎€ธ๐‘ž๎€ทg๐‘ฆ1,g๐‘ฆ2๎€ธโชฏ๐‘‘๐‘›๐‘ž๎€ท1โˆ’๐‘‘g๐‘ฆ1,g๐‘ฆ2๎€ธ.(3.24) Thus, Lemma 2.6 (3) shows that {g๐‘ฅ๐‘›} and {g๐‘ฆ๐‘›} are Cauchy sequences in g(๐‘‹). Since g(๐‘‹) is complete, there exists ๐‘ฅโˆ— and ๐‘ฆโˆ— in ๐‘‹ such that g๐‘ฅ๐‘›โ†’g๐‘ฅโˆ— and g๐‘ฆ๐‘›โ†’g๐‘ฆโˆ— as ๐‘›โ†’โˆž. Using (q3), we have ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅโˆ—๎€ธโชฏโ„Ž๐‘›โˆ’1๐‘ž๎€ท1โˆ’โ„Žg๐‘ฅ1,g๐‘ฅ2๎€ธ,๐‘ž๎€ท(3.25)g๐‘ฆ๐‘›,g๐‘ฆโˆ—๎€ธโชฏ๐‘‘๐‘›โˆ’1๐‘ž๎€ท1โˆ’๐‘‘g๐‘ฆ1,g๐‘ฆ2๎€ธ.(3.26)
On the other hand and by using (3.20), we have ๐‘ž๎€ทg๐‘ฅ๐‘›๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๐น๎€ท๐‘ฅ๎€ธ๎€ธ=๐‘ž๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธโชฏโ„Ž๐‘žg๐‘ฅ๐‘›โˆ’1,g๐‘ฅโˆ—๎€ธโ„Žโชฏโ„Ž๐‘›โˆ’2๐‘ž๎€ท1โˆ’โ„Žg๐‘ฅ1,g๐‘ฅ2๎€ธ=โ„Ž๐‘›โˆ’1๐‘ž๎€ท1โˆ’โ„Žg๐‘ฅ1,g๐‘ฅ2๎€ธ,(3.27) also by using (3.22), we have ๐‘ž๎€ทg๐‘ฆ๐‘›๎€ท๐‘ฆ,๐นโˆ—,๐‘ฅโˆ—๎€ท๐น๎€ท๐‘ฆ๎€ธ๎€ธ=๐‘ž๐‘›โˆ’1,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ท๐‘ฆ,๐นโˆ—,๐‘ฅโˆ—๎€ท๎€ธ๎€ธโชฏ๐‘‘๐‘žg๐‘ฆ๐‘›โˆ’1,g๐‘ฆโˆ—๎€ธ๐‘‘โชฏ๐‘‘๐‘›โˆ’2๐‘ž๎€ท1โˆ’๐‘‘g๐‘ฆ1,g๐‘ฆ2๎€ธ=๐‘‘๐‘›โˆ’1๐‘ž๎€ท1โˆ’๐‘‘g๐‘ฆ1,g๐‘ฆ2๎€ธ.(3.28) Thus, Lemma 2.6 (1), (3.25), and (3.27) show that g๐‘ฅโˆ—=๐น(๐‘ฅโˆ—,๐‘ฆโˆ—). Again, Lemma 2.6 (1), (3.26), and (3.28) show that g๐‘ฆโˆ—=๐น(๐‘ฆโˆ—,๐‘ฅโˆ—). Therefore, (๐‘ฅโˆ—,๐‘ฆโˆ—) is a coupled coincidence point of ๐น and g.
Suppose that ๐‘ข=g๐‘ฅโˆ—=๐น(๐‘ฅโˆ—,๐‘ฆโˆ—) and ๐‘ฃ=g๐‘ฆโˆ—=๐น(๐‘ฆโˆ—,๐‘ฅโˆ—). Then we have ๐‘ž๎€ท(๐‘ข,๐‘ข)=๐‘žg๐‘ฅโˆ—,g๐‘ฅโˆ—๎€ธ๎€ท๐น๎€ท๐‘ฅ=๐‘žโˆ—,๐‘ฆโˆ—๎€ธ๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฅโˆ—,g๐‘ฆโˆ—๎€ธ๐‘ž๎€ทg๐‘ฅโˆ—๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธ+๐‘™g๐‘ฅโˆ—,g๐‘ฆโˆ—๎€ธ๐‘ž๎€ทg๐‘ฅโˆ—๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—=[]=[]๎€ธ๎€ธ=๐‘˜(๐‘ข,๐‘ฃ)๐‘ž(๐‘ข,๐‘ข)+๐‘™(๐‘ข,๐‘ฃ)๐‘ž(๐‘ข,๐‘ข)๐‘˜(๐‘ข,๐‘ฃ)+๐‘™(๐‘ข,๐‘ฃ)๐‘ž(๐‘ข,๐‘ข)(๐‘˜+๐‘™)(๐‘ข,๐‘ฃ)๐‘ž(๐‘ข,๐‘ข).(3.29) Since (๐‘˜+๐‘™)(๐‘ข,๐‘ฃ)<1, Lemma 2.3 (1) shows that ๐‘ž(๐‘ข,๐‘ข)=๐œƒ. By similar way, we have ๐‘ž(๐‘ฃ,๐‘ฃ)=๐œƒ.
Finally, suppose there is another coupled point of coincidence (๐‘ข1,๐‘ฃ1) of ๐น and g such that ๐‘ข1=g๐‘ฅ๎…ž=๐น(๐‘ฅ๎…ž,๐‘ฆ๎…ž) and ๐‘ฃ1=g๐‘ฆ๎…ž=๐น(๐‘ฆ๎…ž,๐‘ฅ๎…ž) for some (๐‘ฅ๎…ž,๐‘ฆ๎…ž) in ๐‘‹ร—๐‘‹. Then we have ๐‘ž๎€ท๐‘ข,๐‘ข1๎€ธ๎€ท=๐‘žg๐‘ฅโˆ—,g๐‘ฅ๎…ž๎€ธ๎€ท๐น๎€ท๐‘ฅ=๐‘žโˆ—,๐‘ฆโˆ—๎€ธ๎€ท๐‘ฅ,๐น๎…ž,๐‘ฆ๎…ž๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฅโˆ—,g๐‘ฆโˆ—๎€ธ๐‘ž๎€ทg๐‘ฅโˆ—๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธ+๐‘™g๐‘ฅโˆ—,g๐‘ฆโˆ—๎€ธ๐‘ž๎€ทg๐‘ฅ๎…ž๎€ท๐‘ฅ,๐น๎…ž,๐‘ฆ๎…ž๎€ท๐‘ข๎€ธ๎€ธ=๐‘˜(๐‘ข,๐‘ฃ)๐‘ž(๐‘ข,u)+๐‘™(๐‘ข,๐‘ฃ)๐‘ž1,๐‘ข1๎€ธ=๐œƒ,(3.30) and also, ๐‘ž๎€ท๐‘ฃ,๐‘ฃ1๎€ธ๎€ท=๐‘žg๐‘ฆโˆ—,g๐‘ฆ๎…ž๎€ธ๎€ท๐น๎€ท๐‘ฆ=๐‘žโˆ—,๐‘ฅโˆ—๎€ธ๎€ท๐‘ฆ,๐น๎…ž,๐‘ฅ๎…ž๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฆโˆ—,g๐‘ฅโˆ—๎€ธ๐‘ž๎€ทg๐‘ฆโˆ—๎€ท๐‘ฆ,๐นโˆ—,๐‘ฅโˆ—๎€ท๎€ธ๎€ธ+๐‘™g๐‘ฆโˆ—,g๐‘ฅโˆ—๎€ธ๐‘ž๎€ทg๐‘ฆ๎…ž๎€ท๐‘ฆ,๐น๎…ž,๐‘ฅ๎…ž๎€ท๐‘ฃ๎€ธ๎€ธ=๐‘˜(๐‘ฃ,๐‘ข)๐‘ž(๐‘ฃ,๐‘ฃ)+๐‘™(๐‘ฃ,๐‘ข)๐‘ž1,๐‘ฃ1๎€ธ=๐œƒ.(3.31) Also, we have ๐‘ž(๐‘ข,๐‘ข)=๐œƒ and ๐‘ž(๐‘ฃ,๐‘ฃ)=๐œƒ. Thus, Lemma 2.6 (1) shows that ๐‘ข=๐‘ข1 and ๐‘ฃ=๐‘ฃ1, which implies that (๐‘ข,๐‘ฃ)=(๐‘ข1,๐‘ฃ1). Similarly, we can prove that ๐‘ข=๐‘ฃ1 and ๐‘ฃ=๐‘ข1. Thus, ๐‘ข=๐‘ฃ. Therefore, (๐‘ข,๐‘ข) is the unique coupled point of coincidence. Now, let ๐‘ข=g๐‘ฅโˆ—=๐น(๐‘ฅโˆ—,๐‘ฆโˆ—). Since ๐น and g are w-compatible, then we have ๎€ทg๐‘ข=gg๐‘ฅโˆ—๎€ธ๎€ท๐‘ฅ=g๐นโˆ—,๐‘ฆโˆ—๎€ธ๎€ท=๐นg๐‘ฅโˆ—,g๐‘ฆโˆ—๎€ธ๎€ท=๐นg๐‘ฅโˆ—,g๐‘ฅโˆ—๎€ธ=๐น(๐‘ข,๐‘ข).(3.32) Then, (g๐‘ข,g๐‘ข) is a coupled point of coincidence. The uniqueness of the coupled point of coincidence implies that g๐‘ข=๐‘ข. Therefore, ๐‘ข=g๐‘ข=๐น(๐‘ข,๐‘ข). Hence, (๐‘ข,๐‘ข) is the unique common coupled fixed point of ๐น and g.

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

Corollary 3.5. Let (๐‘‹,๐‘‘) be a cone metric space with a cone ๐‘ƒ having nonempty interior and ๐‘ž is a c-distance on ๐‘‹. Suppose the mappings ๐นโˆถ๐‘‹ร—๐‘‹โ†’๐‘‹ and gโˆถ๐‘‹โ†’๐‘‹ satisfy the following contractive condition: ๐‘ž(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โชฏ๐‘˜๐‘ž(g๐‘ฅ,๐น(๐‘ฅ,๐‘ฆ))+๐‘™๐‘ž(g๐‘ข,๐น(๐‘ข,๐‘ฃ)),(3.33) for all ๐‘ฅ,๐‘ฆ,๐‘ข,๐‘ฃโˆˆ๐‘‹, where ๐‘˜,๐‘™ are nonnegative constants with ๐‘˜+๐‘™<1. If ๐น(๐‘‹ร—๐‘‹)โŠ†g(๐‘‹) and g(๐‘‹) is a complete subspace of ๐‘‹, then ๐น and g have a unique coupled point of coincidence (๐‘ข,๐‘ฃ) in ๐‘‹ร—๐‘‹. Further, if ๐‘ข=g๐‘ฅ1=๐น(๐‘ฅ1,๐‘ฆ1), and ๐‘ฃ=g๐‘ฆ1=๐น(๐‘ฆ1,๐‘ฅ1), then ๐‘ž(๐‘ข,๐‘ข)=๐œƒ and ๐‘ž(๐‘ฃ,๐‘ฃ)=๐œƒ. Moreover, if ๐น and g are w -compatible, then ๐น and g have a unique common coupled fixed point and the common coupled fixed point of ๐น and g is of the form (๐‘ข,๐‘ข) for some ๐‘ขโˆˆ๐‘‹.

Corollary 3.6. Let (๐‘‹,๐‘‘) be a cone metric space with a cone ๐‘ƒ having nonempty interior and ๐‘ž is a c-distance on ๐‘‹. Suppose the mappings ๐นโˆถ๐‘‹ร—๐‘‹โ†’๐‘‹ and gโˆถ๐‘‹โ†’๐‘‹ satisfy the following contractive condition: ๐‘ž๎€บ๐‘ž๎€ป(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โชฏ๐‘˜(g๐‘ฅ,๐น(๐‘ฅ,๐‘ฆ))+๐‘ž(g๐‘ข,๐น(u,๐‘ฃ)),(3.34) for all ๐‘ฅ,๐‘ฆ,๐‘ข,๐‘ฃโˆˆ๐‘‹, where ๐‘˜โˆˆ[0,1/2) is constants. If ๐น(๐‘‹ร—๐‘‹)โŠ†g(๐‘‹) and g(๐‘‹) is a complete subspace of ๐‘‹, then ๐น and g have a unique coupled point of coincidence (๐‘ข,๐‘ฃ) in ๐‘‹ร—๐‘‹. Further, if ๐‘ข=g๐‘ฅ1=๐น(๐‘ฅ1,๐‘ฆ1) and ๐‘ฃ=g๐‘ฆ1=๐น(๐‘ฆ1,๐‘ฅ1), then ๐‘ž(๐‘ข,๐‘ข)=๐œƒ and ๐‘ž(๐‘ฃ,๐‘ฃ)=๐œƒ. Moreover, if ๐น and g are w-compatible, then ๐น and g have a unique common coupled fixed point and the common coupled fixed point of ๐น and g is of the form (๐‘ข,๐‘ข) for some ๐‘ขโˆˆ๐‘‹.

Finally, we provide another result with another contractive type.

Theorem 3.7. Let (๐‘‹,๐‘‘) be a cone metric space with a cone ๐‘ƒ having nonempty interior and ๐‘ž is a c-distance on ๐‘‹. Let ๐นโˆถ๐‘‹ร—๐‘‹โ†’๐‘‹ and gโˆถ๐‘‹โ†’๐‘‹ be two mappings and suppose that there exists mappings ๐‘˜,๐‘™,๐‘Ÿโˆถ๐‘‹ร—๐‘‹โ†’[0,1) such that the following hold: (a)๐‘˜(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โ‰ค๐‘˜(g๐‘ฅ,g๐‘ฆ), ๐‘™(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โ‰ค๐‘™(g๐‘ฅ,g๐‘ฆ) andโ€‰โ€‰๐‘Ÿ(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โ‰ค๐‘Ÿ(g๐‘ฅ,g๐‘ฆ) for all ๐‘ฅ,๐‘ฆ,๐‘ข,๐‘ฃโˆˆ๐‘‹, (b)(๐‘˜+2๐‘™+๐‘Ÿ)(๐‘ฅ,๐‘ฆ)<1 for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, (c)(1โˆ’๐‘Ÿ(g๐‘ฅ,g๐‘ฆ))๐‘ž(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โชฏ๐‘˜(g๐‘ฅ,g๐‘ฆ)๐‘ž(g๐‘ฅ,๐น(๐‘ฅ,๐‘ฆ))+๐‘™(g๐‘ฅ,g๐‘ฆ)๐‘ž(g๐‘ฅ,๐น(๐‘ข,๐‘ฃ)) for all ๐‘ฅ,๐‘ฆ,๐‘ข,๐‘ฃโˆˆ๐‘‹.
If ๐น(๐‘‹ร—๐‘‹)โŠ†g(๐‘‹) and g(๐‘‹) is a complete subspace of ๐‘‹, then ๐น and g have a unique coupled point of coincidence (๐‘ข,๐‘ฃ) in ๐‘‹ร—๐‘‹. Further, if ๐‘ข=g๐‘ฅ1=๐น(๐‘ฅ1,๐‘ฆ1) and ๐‘ฃ=g๐‘ฆ1=๐น(๐‘ฆ1,๐‘ฅ1), then ๐‘ž(๐‘ข,๐‘ข)=๐œƒ and ๐‘ž(๐‘ฃ,๐‘ฃ)=๐œƒ. Moreover, if ๐น and g are w-compatible, then ๐น and g have a unique common coupled fixed point and the common coupled fixed point of ๐น and g is of the form (๐‘ข,๐‘ข) for some ๐‘ขโˆˆ๐‘‹.

Proof. Choose ๐‘ฅ0,๐‘ฆ0โˆˆ๐‘‹. Set g๐‘ฅ1=๐น(๐‘ฅ0,๐‘ฆ0),g๐‘ฆ1=๐น(๐‘ฆ0,๐‘ฅ0) this can be done because ๐น(๐‘‹ร—๐‘‹)โŠ†g(๐‘‹). Continuing this process, we obtain to sequences {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} such that g๐‘ฅn+1=๐น(๐‘ฅ๐‘›,๐‘ฆ๐‘›),g๐‘ฆ๐‘›+1=๐น(๐‘ฆ๐‘›,๐‘ฅ๐‘›). Observe that (1โˆ’๐‘Ÿ(g๐‘ฅ,g๐‘ฆ))๐‘ž(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โชฏ๐‘˜(g๐‘ฅ,g๐‘ฆ)๐‘ž(g๐‘ฅ,๐น(๐‘ฅ,๐‘ฆ))+๐‘™(g๐‘ฅ,g๐‘ฆ)๐‘ž(g๐‘ฅ,๐น(๐‘ข,๐‘ฃ)),(3.35) equivalently ๐‘ž(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ))โชฏ๐‘˜(g๐‘ฅ,g๐‘ฆ)๐‘ž(g๐‘ฅ,๐น(๐‘ฅ,๐‘ฆ))+๐‘™(g๐‘ฅ,g๐‘ฆ)๐‘ž(g๐‘ฅ,๐น(๐‘ข,๐‘ฃ))+๐‘Ÿ(g๐‘ฅ,g๐‘ฆ)๐‘ž(๐น(๐‘ฅ,๐‘ฆ),๐น(๐‘ข,๐‘ฃ)).(3.36) Then, we have ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘ฅ=๐‘ž๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ฅ,๐น๐‘›,๐‘ฆ๐‘›๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฅ๐‘›โˆ’1,g๐‘ฆ๐‘›โˆ’1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1๎€ท๐‘ฅ,๐น๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ท๎€ธ๎€ธ+๐‘™g๐‘ฅ๐‘›โˆ’1,g๐‘ฆ๐‘›โˆ’1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1๎€ท๐‘ฅ,๐น๐‘›,๐‘ฆ๐‘›๎€ท๎€ธ๎€ธ+๐‘Ÿg๐‘ฅ๐‘›โˆ’1,g๐‘ฆ๐‘›โˆ’1๎€ธ๐‘ž๎€ท๐น๎€ท๐‘ฅ๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ฅ,๐น๐‘›,๐‘ฆ๐‘›๎€ท๐น๎€ท๐‘ฅ๎€ธ๎€ธ=๐‘˜๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐น๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท๐น๎€ท๐‘ฅ+๐‘™๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐น๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›+1๎€ธ๎€ท๐น๎€ท๐‘ฅ+๐‘Ÿ๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐น๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ๎€ทโชฏ๐‘˜g๐‘ฅ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›+1๎€ธ๎€ท+๐‘Ÿg๐‘ฅ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธโ‹ฎ๎€ทโชฏ๐‘˜g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›+1๎€ธ๎€ท+๐‘Ÿg๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ๎€ทโชฏ๐‘˜g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ท๐‘ฅ๐‘›โˆ’1,๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ท๐‘ฅ๐‘›โˆ’1,๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ๎€ท+๐‘Ÿg๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ.(3.37) Hence ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธโชฏ๐‘˜๎€ทg๐‘ฅ1,g๐‘ฆ1๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๎€ท1โˆ’๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๎€ทโˆ’๐‘Ÿg๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท=โ„Ž๐‘žg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธโชฏโ„Ž2๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’2,g๐‘ฅ๐‘›โˆ’1๎€ธโ‹ฎโชฏโ„Ž๐‘›โˆ’1๐‘ž๎€ทg๐‘ฅ1,g๐‘ฅ2๎€ธ,(3.38) whereโ€‰โ„Ž=(๐‘˜(g๐‘ฅ1,g๐‘ฆ1)+๐‘™(g๐‘ฅ1,g๐‘ฆ1))/(1โˆ’๐‘™(g๐‘ฅ1,g๐‘ฆ1)โˆ’๐‘Ÿ(g๐‘ฅ1,g๐‘ฆ1))<1.
Similarly, we have ๐‘ž๎€ทg๐‘ฆ๐‘›,g๐‘ฆ๐‘›+1๎€ธโชฏ๐‘˜๎€ทg๐‘ฆ1,g๐‘ฅ1๎€ธ๎€ท+๐‘™g๐‘ฆ1,g๐‘ฅ1๎€ธ๎€ท1โˆ’๐‘™g๐‘ฆ1,g๐‘ฅ1๎€ธ๎€ทโˆ’๐‘Ÿg๐‘ฆ1,g๐‘ฅ1๎€ธ๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธ๎€ท=๐‘‘๐‘žg๐‘ฆ๐‘›โˆ’1,g๐‘ฆ๐‘›๎€ธโชฏ๐‘‘2๐‘ž๎€ทg๐‘ฆ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’1๎€ธโ‹ฎโชฏ๐‘‘๐‘›โˆ’1๐‘ž๎€ทg๐‘ฆ1,g๐‘ฆ2๎€ธ,(3.39) where ๐‘‘=(๐‘˜(g๐‘ฆ1,g๐‘ฅ1)+๐‘™(g๐‘ฆ1,g๐‘ฅ1))/(1โˆ’๐‘™(g๐‘ฆ1,g๐‘ฅ1)โˆ’๐‘Ÿ(g๐‘ฆ1,g๐‘ฅ1))<1.
Let ๐‘š>๐‘›โ‰ฅ1. Then, it follows that ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅ๐‘š๎€ธ๎€ทโชฏ๐‘žg๐‘ฅ๐‘›,g๐‘ฅ๐‘›+1๎€ธ๎€ท+๐‘žg๐‘ฅ๐‘›+1,g๐‘ฅ๐‘›+2๎€ธ๎€ท+โ‹ฏ+๐‘žg๐‘ฅ๐‘šโˆ’1,g๐‘ฅ๐‘š๎€ธโชฏ๎€ทโ„Ž๐‘›โˆ’1+โ„Ž๐‘›+โ‹ฏ+โ„Ž๐‘šโˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ1,g๐‘ฅ2๎€ธโชฏโ„Ž๐‘›๐‘ž๎€ท1โˆ’โ„Žg๐‘ฅ1,g๐‘ฅ2๎€ธ,๐‘ž๎€ทg๐‘ฆ๐‘›,g๐‘ฆ๐‘š๎€ธ๎€ทโชฏ๐‘žg๐‘ฆ๐‘›,g๐‘ฆ๐‘›+1๎€ธ๎€ท+๐‘žg๐‘ฆ๐‘›+1,g๐‘ฆ๐‘›+2๎€ธ๎€ท+โ‹ฏ+๐‘žg๐‘ฆ๐‘šโˆ’1,g๐‘ฆ๐‘š๎€ธโชฏ๎€ท๐‘‘๐‘›โˆ’1+๐‘‘๐‘›+โ‹ฏ+๐‘‘๐‘šโˆ’2๎€ธ๐‘ž๎€ทg๐‘ฆ1,g๐‘ฆ2๎€ธโชฏ๐‘‘๐‘›๐‘ž๎€ท1โˆ’๐‘‘g๐‘ฆ1,g๐‘ฆ2๎€ธ.(3.40) Thus, Lemma 2.6 (3) shows that {g๐‘ฅ๐‘›} and {g๐‘ฆ๐‘›} are Cauchy sequences in g(๐‘‹). Since g(๐‘‹) is complete, there exists ๐‘ฅโˆ—,๐‘ฆโˆ—โˆˆ๐‘‹ such that g๐‘ฅ๐‘›โ†’g๐‘ฅโˆ— and g๐‘ฆ๐‘›โ†’g๐‘ฆโˆ— as ๐‘›โ†’โˆž. Using (q3), we have ๐‘ž๎€ทg๐‘ฅ๐‘›,g๐‘ฅโˆ—๎€ธโชฏโ„Ž๐‘›๐‘ž๎€ท1โˆ’โ„Žg๐‘ฅ1,g๐‘ฅ2๎€ธ,๐‘ž๎€ท(3.41)g๐‘ฆ๐‘›,g๐‘ฆโˆ—๎€ธโชฏ๐‘‘๐‘›๐‘ž๎€ท1โˆ’๐‘‘g๐‘ฆ1,g๐‘ฆ2๎€ธ.(3.42)
On the other hand, ๐‘ž๎€ทg๐‘ฅ๐‘›๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๐น๎€ท๐‘ฅ๎€ธ๎€ธ=๐‘ž๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฅ๐‘›โˆ’1,g๐‘ฆ๐‘›โˆ’1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1๎€ท๐‘ฅ,๐น๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ท๎€ธ๎€ธ+๐‘™g๐‘ฅ๐‘›โˆ’1,g๐‘ฆ๐‘›โˆ’1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธ+๐‘Ÿg๐‘ฅ๐‘›โˆ’1,g๐‘ฆ๐‘›โˆ’1๎€ธ๐‘ž๎€ท๐น๎€ท๐‘ฅ๐‘›โˆ’1,๐‘ฆ๐‘›โˆ’1๎€ธ๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๐น๎€ท๐‘ฅ๎€ธ๎€ธ=๐‘˜๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐น๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฅ๐‘›โˆ’1,๐‘ฅ๐‘›๎€ธ๎€ท๐น๎€ท๐‘ฅ+๐‘™๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐น๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฅ๐‘›โˆ’1๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๐น๎€ท๐‘ฅ๎€ธ๎€ธ+๐‘Ÿ๐‘›โˆ’2,๐‘ฆ๐‘›โˆ’2๎€ธ๎€ท๐‘ฆ,๐น๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’2๐‘ž๎€ท๎€ธ๎€ธg๐‘ฅ๐‘›๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฅ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1๎€ท๐‘ฅ,Fโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธ+๐‘Ÿg๐‘ฅ๐‘›โˆ’2,g๐‘ฆ๐‘›โˆ’2๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—โ‹ฎ๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธ+๐‘Ÿg๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธโชฏ๐‘˜g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—๎€ท๎€ธ๎€ธ+๐‘Ÿg๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—.๎€ธ๎€ธ(3.43) Then, ๐‘ž๎€ทg๐‘ฅ๐‘›๎€ท๐‘ฅ,๐นโˆ—,๐‘ฆโˆ—โชฏ๐‘˜๎€ท๎€ธ๎€ธg๐‘ฅ1,g๐‘ฆ1๎€ธ๎€ท+๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๎€ท1โˆ’๐‘™g๐‘ฅ1,g๐‘ฆ1๎€ธ๎€ทโˆ’๐‘Ÿg๐‘ฅ1,g๐‘ฆ1๎€ธ๐‘ž๎€ทg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธ๎€ท=โ„Ž๐‘žg๐‘ฅ๐‘›โˆ’1,g๐‘ฅ๐‘›๎€ธโชฏโ„Žโ„Ž๐‘›โˆ’2๐‘ž๎€ทg๐‘ฅ1,g๐‘ฅ2๎€ธ=โ„Ž๐‘›โˆ’1๐‘ž๎€ทg๐‘ฅ1,g๐‘ฅ2