The Point of Coincidence and Common Fixed Point for Three Mappings in Cone Metric Spaces

Abstract

The aim of this paper is to present the point of coincidence and common fixed point for three mappings in cone metric spaces over normal cone which satisfy a different contractive condition. Our result generalizes the recent related results proved by Stojan Radenović (2009) and Rangamma and Prudhvi (2012).

1. Introduction and Preliminaries

It is well known that the classical contraction mapping principle of Banach is a fundamental result in fixed point theory. Several authors have obtained various extensions and generalizations of Banach’s theorems by considering contractive mappings on different metric spaces. Huang and Zhang [1] have replaced real numbers by ordering Banach space and have defined a cone metric space. They have proved some fixed point theorems of contractive mappings on cone metric spaces. Further generalizations of Huang and Zhang were obtained by Abbas and Jungck [2]. In 2009 Radenović [3] has obtained coincidence point result for two mappings in cone metric spaces which satisfy new contractive conditions. Recently, in this paper we generalized the coincidence point results of Radenović [3] for three maps with different contractive condition.

We recall some definitions and results that will be needed in what follows.

Definition 1. Let 𝐸 be a real Banach space and 𝑃 be a subset of 𝐸. Then 𝑃 is called a cone if(1)𝑃is closed, nonempty and satisfies 𝑃≠{0},(2)𝑎,𝑏∈𝑅, 𝑎,𝑏≥0, and 𝑥,𝑦∈𝑃 imply 𝑎𝑥+𝑏𝑦∈𝑃,(3)𝑥∈𝑃 and −𝑥∈𝑃 imply 𝑥=0.
Given a cone 𝑃⊆𝐸, we define a partial ordering ≤ with respect to 𝑃 by 𝑥≤𝑦 if and only if 𝑦−𝑥∈𝑃. We shall write 𝑥<𝑦 if 𝑥≤𝑦 and 𝑥≠𝑦, while 𝑥≪𝑦 if and only if 𝑦−𝑥∈int𝑃, where int𝑃 is the interior of 𝑃. A cone 𝑃 is called normal if there is a number 𝐾>0 such that, for all 𝑥,𝑦∈𝐸, 0≤𝑥≤𝑦 implies ‖𝑥‖≤𝐾‖𝑦‖. The least positive number satisfying the above inequality is called the normal constant of 𝑃.
In the following we suppose that 𝐸 is a real Banach space and 𝑃 is a cone in 𝐸 with int𝑃≠𝜙 and ≤ is a partial ordering with respect to 𝑃.

Definition 2. Let 𝑋 be a nonempty set. Suppose that the mapping 𝑑∶𝑋×𝑋→𝐸 satisfies(i)0≤𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 and 𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦,(ii)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋,(iii)𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝑋.Then 𝑑 is called a cone metric on 𝑋, and (𝑋,𝑑) is called a cone metric space.

Example 3. Let 𝐸=𝑅², 𝑃={(𝑥,𝑦)∈𝐸∶𝑥,𝑦≥0}⊆𝑅², 𝑋=𝑅², and 𝑑∶𝑋×𝑋→𝐸 be defined by 𝑑(𝑥,𝑦)=𝑑((𝑥₁,𝑥₂),(𝑦₁,𝑦₂))=[max(|𝑥₁−𝑦₁|,|𝑥₂−𝑦₂|),𝛼max(|𝑥₁−𝑦₁|,|𝑥₂−𝑦₂|)], where 𝛼≥0 is a constant; then (𝑋,𝑑) is a cone metric space.

Definition 4. Let (𝑋,𝑑) be a cone metric space, {𝑥_𝑛} be a sequence in 𝑋 and 𝑥∈𝑋. Then {𝑥_𝑛} converges to 𝑥 if for every 𝑐 lies in 𝐸 with 0≪𝑐 there is an 𝑁 such that for all 𝑛>𝑁, 𝑑(𝑥_𝑛,𝑥)≪𝑐. One denotes this by 𝑥_𝑛→𝑥 as 𝑛→∞.

Definition 5. Let (𝑋,𝑑) be a cone metric space, {𝑥_𝑛} be a sequence in 𝑋. If for every 𝑐 lies in 𝐸 with 0≪𝑐 there is an 𝑁 such that for all 𝑛,𝑚>𝑁, 𝑑(𝑥_𝑛,𝑥_𝑚)≪𝑐, then {𝑥_𝑛} is called a Cauchy sequence in 𝑋.

Definition 6. A cone metric space (𝑋,𝑑) is said to be complete if every Cauchy sequence in 𝑋 is convergent in 𝑋.

Lemma 7. Let (𝑋,𝑑) be a cone metric space and 𝑃 be a normal cone. Let {𝑥_𝑛} be a sequence in 𝑋. One has the following. (i){𝑥_𝑛} converges to 𝑥∈𝑋 if and only if 𝑑(𝑥_𝑛,𝑥)→0 as 𝑛→∞.(ii){𝑥_𝑛} is a Cauchy sequence if and only if 𝑑(𝑥_𝑛,𝑥_𝑚)→0 as 𝑛,𝑚→∞.(iii){𝑥_𝑛} converges to 𝑥∈𝑋 and {𝑥_𝑛} converges to 𝑦∈𝑋. Then 𝑥=𝑦.

Definition 8. Let 𝑓 and 𝑔 be self-maps on set 𝑋. If 𝑓𝑥=𝑔𝑥=𝑤 for some 𝑥 in 𝑋, then 𝑥 is called a coincidence point of 𝑓 and 𝑔, and 𝑤 is called a point of coincidence of 𝑓 and 𝑔.

Definition 9. Two self-mappings 𝑓 and 𝑔 of a cone metric space 𝑋 are said to be weakly compatible if 𝑓𝑔𝑥=𝑔𝑓𝑥 whether 𝑓𝑥=𝑔𝑥.

2. Main Result

In this section, we give fixed point theorems for mappings defined on cone metric space with generalized contractive condition.

Theorem 10. Let (𝑋,𝑑) be a cone metric space and 𝑃 be a normal cone with normal constant K. Suppose that the mappings 𝑓,𝑔, and ℎ∶𝑋→𝑋 satisfy the condition 𝑑𝑑𝑑𝑑𝑑+𝑑𝑓𝑥,𝑔𝑦≤𝑎ℎ𝑥,ℎ𝑦+𝑏𝑓𝑥,ℎ𝑥+𝑐𝑔𝑦,ℎ𝑦+𝜆ℎ𝑥,𝑔𝑦𝑓𝑥,ℎ𝑦 (1) for all 𝑥,𝑦∈𝑋, where 𝑎, 𝑏, 𝑐, and 𝜆 are nonnegative real numbers satisfying 𝑎+𝑏+𝑐+2𝜆<1. If the range of ℎ contains range of 𝑓 and also range of 𝑔 and ℎ(𝑋) is a complete subspace of 𝑋, then 𝑓, 𝑔, and ℎ have a unique point of coincidence in 𝑋. Moreover, if (𝑓,ℎ) and (𝑔,ℎ) are weakly compatible, then 𝑓, 𝑔, and ℎ have a unique common fixed point.

Proof. Let 𝑥₀∈𝑋 be an arbitrary point. Since 𝑓(𝑋) and 𝑔(𝑋) are contained in ℎ(𝑋), there exists 𝑥₁∈𝑋 such that 𝑦₀=𝑓𝑥₀=ℎ𝑥₁, and also there exists 𝑥₂∈𝑋 such that 𝑦₁=𝑔𝑥₁=ℎ𝑥₂. Continuing this process, a sequence {𝑦_𝑛} can be chosen such that 𝑦_2𝑛=𝑓𝑥_2𝑛=ℎ𝑥_2𝑛+1 and 𝑦_2𝑛+1=𝑔𝑥_2𝑛+1=ℎ𝑥_2𝑛+2, for 𝑛=0,1,2,…; then 𝑑𝑦_2𝑛,𝑦_2𝑛+1=𝑑𝑓𝑥_2𝑛,𝑔𝑥_2𝑛+1≤𝑎𝑑ℎ𝑥_2𝑛,ℎ𝑥_2𝑛+1+𝑏𝑑𝑓𝑥_2𝑛,ℎ𝑥_2𝑛+𝑐𝑑𝑔𝑥_2𝑛+1,ℎ𝑥_2𝑛+1+𝜆𝑑ℎ𝑥_2𝑛,𝑔𝑥_2𝑛+1+𝑑𝑓𝑥_2𝑛,ℎ𝑥_2𝑛+1=𝑎𝑑𝑦_2𝑛−1,𝑦_2𝑛+𝑏𝑑𝑦_2𝑛,𝑦_2𝑛−1+𝑐𝑑𝑦_2𝑛+1,𝑦_2𝑛+𝜆𝑑𝑦_2𝑛−1,𝑦_2𝑛+1+𝑑𝑦_2𝑛,𝑦_2𝑛≤𝑎𝑑𝑦_2𝑛−1,𝑦_2𝑛+𝑏𝑑𝑦_2𝑛,𝑦_2𝑛−1+𝑐𝑑𝑦_2𝑛+1,𝑦_2𝑛+𝜆𝑑𝑦_2𝑛−1,𝑦_2𝑛+𝑑𝑦_2𝑛,𝑦_2𝑛+1=(𝑎+𝑏+𝜆)𝑑𝑦_2𝑛−1,𝑦_2𝑛+(𝑐+𝜆)𝑑𝑦_2𝑛,𝑦_2𝑛+1. (2) This implies that ‖𝑑(𝑦_2𝑛,𝑦_2𝑛+1)‖≤((𝑎+𝑏+𝜆)/(1−(𝑐+𝜆)))‖𝑑(𝑦_2𝑛−1,𝑦_2𝑛)‖.
Thus 𝑑𝑦_2𝑛,𝑦_2𝑛+1≤𝜂𝑑𝑦_2𝑛−1,𝑦_2𝑛, (3) where 𝜂=(𝑎+𝑏+𝜆)/(1−(𝑐+𝜆))∈[0,1), as 𝑎+𝑏+𝑐+2𝜆<1.
Writing 𝑑_𝑛= ‖𝑑(𝑦_𝑛,𝑦_𝑛+1)‖, we obtain 𝑑_2𝑛≤𝜂𝑑_2𝑛−1. (4)
Again 𝑑𝑦_2𝑛+2,𝑦_2𝑛+1=𝑑𝑓𝑥_2𝑛+2,𝑔𝑥_2𝑛+1≤𝑎𝑑ℎ𝑥_2𝑛+2,ℎ𝑥_2𝑛+1+𝑏𝑑𝑓𝑥_2𝑛+2,ℎ𝑥_2𝑛+2+𝑐𝑑𝑔𝑥_2𝑛+1,ℎ𝑥_2𝑛+1+𝜆𝑑ℎ𝑥_2𝑛+2,𝑔𝑥_2𝑛+1+𝑑𝑓𝑥_2𝑛+2,ℎ𝑥_2𝑛+1=𝑎𝑑𝑦_2𝑛+1,𝑦_2𝑛+𝑏𝑑𝑦_2𝑛+2,𝑦_2𝑛+1+𝑐𝑑𝑦_2𝑛+1,𝑦_2𝑛+𝜆𝑑𝑦_2𝑛+1,𝑦_2𝑛+1+𝑑𝑦_2𝑛+2,𝑦_2𝑛≤𝑎𝑑𝑦_2𝑛+1,𝑦_2𝑛+𝑏𝑑𝑦_2𝑛+2,𝑦_2𝑛+1+𝑐𝑑𝑦_2𝑛+1,𝑦_2𝑛+𝜆𝑑𝑦_2𝑛+2,𝑦_2𝑛+1+𝑑𝑦_2𝑛+1,𝑦_2𝑛=(𝑎+𝑐+𝜆)𝑑𝑦_2𝑛+1,𝑦_2𝑛+(𝑏+𝜆)𝑑𝑦_2𝑛+2,𝑦_2𝑛+1. (5) This implies that ‖𝑑(𝑦_2𝑛+2,𝑦_2𝑛+1)‖≤((𝑎+𝑐+𝜆)/(1−(𝑏+𝜆)))‖𝑑(𝑦_2𝑛+1,𝑦_2𝑛)‖.
Thus 𝑑𝑦_2𝑛+2,𝑦_2𝑛+1≤𝜇𝑑𝑦_2𝑛+1,𝑦_2𝑛, (6) where 𝜇=(𝑎+𝑐+𝜆)/(1−(𝑏+𝜆))∈[0,1), as 𝑎+𝑏+𝑐+2𝜆<1.
Therefore 𝑑_2𝑛+1≤𝜇𝑑_2𝑛. (7)
From (4) and (7) we get 𝑑_2𝑛≤𝜂𝑑_2𝑛−1≤𝜂𝜇𝑑_2𝑛−2≤⋯≤𝜂^𝑛𝜇^𝑛𝑑_0,𝑑_2𝑛+1≤𝜇𝑑_2𝑛≤𝜂𝜇𝑑_2𝑛−1≤⋯≤𝜂^𝑛𝜇^𝑛+1𝑑₀. (8) Therefore 𝑑_2𝑛+𝑑_2𝑛+1≤𝜂^𝑛𝜇^𝑛1+𝜇𝑑₀,𝑑_2𝑛+1+𝑑_2𝑛+2≤𝜂^𝑛𝜇^𝑛+11+𝜂𝑑₀. (9)(10) Now we will show that {𝑦_𝑛} is a Cauchy sequence. By triangle inequality for 𝑚>𝑛, we have 𝑑𝑦_𝑛,𝑦_𝑚≤𝑑𝑦_𝑛,𝑦_𝑛+1+𝑑𝑦_𝑛+1,𝑦_𝑛+2+⋯+𝑑𝑦_𝑚−1,𝑦_𝑚. (11)
Hence, as 𝑃 is normal cone with normal constant 𝐾, 𝑑𝑦_𝑛,𝑦_𝑚≤𝐾𝑑𝑦_𝑛,𝑦_𝑛+1+𝑑𝑦_𝑛+1,𝑦_𝑛+2+⋯+𝑑𝑦_𝑚−1,𝑦_𝑚≤𝐾𝑑𝑦_𝑛,𝑦_𝑛+1+𝑑𝑦_𝑛+1,𝑦_𝑛+2+⋯+𝑑𝑦_𝑚−1,𝑦_𝑚. (12)
If 𝑛 is even, then from (9) and (12) we have 𝑑𝑦_𝑛,𝑦_𝑚≤𝐾𝑑_𝑛+𝑑_𝑛+1+𝑑_𝑛+2+𝑑_𝑛+3𝜂+⋯≤𝐾^𝑛/2𝜇^𝑛/21+𝜇𝑑₀+𝜂^(𝑛+2)/2𝜇^(𝑛+2)/21+𝜇𝑑₀+⋯=𝐾𝜂𝜇^𝑛/21+𝜇𝑑1−𝜂𝜇⁰. (13)
If 𝑛 is odd, then from (10) and (12) we have 𝑑𝑦_𝑛,𝑦_𝑚≤𝐾𝑑_𝑛+𝑑_𝑛+1+𝑑_𝑛+2+𝑑_𝑛+3𝜂+⋯≤𝐾^(𝑛−1)/2𝜇^{(𝑛−1)/2+1}(1+𝜂)𝑑₀+𝜂^(𝑛+1)/2×𝜇^(𝑛+1)/2+11+𝜂𝑑₀+⋯=𝐾𝜂𝜇^(𝑛−1)/21+𝜂𝜇𝑑1−𝜂𝜇⁰. (14)
Since 𝜂<1, 𝜇<1, therefore 𝜂𝜇<1, so in both cases, ‖𝑑(𝑦_𝑛,𝑦_𝑚)‖→0 as 𝑛→∞.
From Lemma 7, it follows that {𝑦_𝑛}={ℎ𝑥_𝑛+1} is a Cauchy sequence. Since ℎ(𝑋) is a complete subspace of 𝑋, there exists 𝑞 in ℎ(𝑋) such that {ℎ𝑥_𝑛+1}→𝑞 as 𝑛→∞; consequently we can find 𝑝 in 𝑋 such that ℎ𝑝=𝑞. We shall show that ℎ𝑝=𝑓𝑝=𝑔𝑝.
Now using contractive condition (1), we can write 𝑑𝑓𝑝,𝑔𝑥_2𝑛+1≤𝑎𝑑ℎ𝑝,ℎ𝑥_2𝑛+1𝑑𝑑+𝑏𝑓𝑝,ℎ𝑝+𝑐𝑔𝑥_2𝑛+1,ℎ𝑥_2𝑛+1+𝜆𝑑ℎ𝑝,𝑔𝑥_2𝑛+1+𝑑𝑓𝑝,ℎ𝑥_2𝑛+1. (15)
Taking 𝑛→∞, we have 𝑑𝑑𝑑𝑑𝑑+𝑑=(𝑑𝑓𝑝,𝑞≤𝑎ℎ𝑝,𝑞+𝑏𝑓𝑝,ℎ𝑝+𝑐𝑞,𝑞+𝜆ℎ𝑝,𝑞𝑓𝑝,𝑞𝑏+𝜆)𝑓𝑝,𝑞,sinceℎ𝑝=𝑞. (16) Hence, 𝑓𝑝=𝑞, since 𝑎+𝑏+𝑐+2𝜆<1 and 𝑎,𝑏,𝑐,𝜆≥0.
Again from (1), we can write 𝑑𝑓𝑥_2𝑛,𝑔𝑝≤𝑎𝑑ℎ𝑥_2𝑛𝑑,ℎ𝑝+𝑏𝑓𝑥_2𝑛,ℎ𝑥_2𝑛𝑑𝑑+𝑐𝑔𝑝,ℎ𝑝+𝜆ℎ𝑥_2𝑛,𝑔𝑝+𝑑𝑓𝑥_2𝑛,ℎ𝑝. (17)
Taking 𝑛→∞, we have 𝑑𝑑𝑑𝑑𝑑+𝑑=(𝑑𝑞,𝑔𝑝≤𝑎𝑞,ℎ𝑝+𝑏𝑞,𝑞+𝑐𝑔𝑝,ℎ𝑝+𝜆𝑞,𝑔𝑝𝑞,ℎ𝑝𝑐+𝜆)𝑔𝑝,𝑞,sinceℎ𝑝=𝑞. (18) Hence, 𝑔𝑝=𝑞, since 𝑎+𝑏+𝑐+2𝜆<1 and 𝑎,𝑏,𝑐,𝜆≥0.
So we get ℎ𝑝=𝑔𝑝=𝑓𝑝=𝑞. (19)
Therefore 𝑝 is a coincidence point of 𝑓, 𝑔, and ℎ.
Now we show that 𝑓, 𝑔, and ℎ have a unique point of coincidence. For this, assume that there exists another point of coincidence 𝑟 in 𝑋 such that 𝑓𝑝₁=𝑔𝑝₁=ℎ𝑝₁=𝑟.
Consider 𝑑𝑔𝑝,𝑔𝑝₁=𝑑𝑓𝑝,𝑔𝑝₁≤𝑎𝑑ℎ𝑝,ℎ𝑝₁+𝑏𝑑𝑓𝑝,ℎ𝑝+𝑐𝑑𝑔𝑝₁,ℎ𝑝₁+𝜆𝑑ℎ𝑝,𝑔𝑝₁+𝑑𝑓𝑝,ℎ𝑝₁=(𝑎+2𝜆)𝑑𝑔𝑝,𝑔𝑝₁. (20) Since 𝑎+𝑏+𝑐+2𝜆<1 and 𝑎,𝑏,𝑐,𝜆≥0, so from (20), 𝑔𝑝=𝑔𝑝₁.
Therefore, 𝑞=𝑓𝑝=ℎ𝑝=𝑝=𝑔𝑝₁=𝑓𝑝₁=ℎ𝑝₁=𝑟, and hence 𝑓, 𝑔, and ℎ have unique point of coincidence in 𝑋.
Now from (1) we have 𝑑=𝑑𝑑𝑑𝑑𝑑+𝑑.𝑓𝑓𝑝,𝑓𝑝𝑓𝑓𝑝,𝑔𝑝≤𝑎ℎ𝑓𝑝,ℎ𝑝+𝑏𝑓𝑓𝑝,ℎ𝑓𝑝+𝑐𝑔𝑝,ℎ𝑝+𝜆ℎ𝑓𝑝,𝑔𝑝𝑓𝑓𝑝,ℎ𝑝 (21)
As (𝑓,ℎ) is weakly compatible, therefore from (19) and (21) we can write 𝑑𝑓𝑓𝑝,𝑓𝑝≤(𝑎+2𝜆)𝑑𝑓𝑓𝑝,𝑓𝑝. (22)
As 𝑎+𝑏+𝑐+2𝜆<1 and 𝑎,𝑏,𝑐,𝜆≥0, so from (22), 𝑓𝑓𝑝=𝑓𝑝.
Therefore, 𝑓𝑞=𝑞. (23)
Also, 𝑞=𝑓𝑝=𝑓𝑓𝑝=𝑓ℎ𝑝=ℎ𝑓𝑝=ℎ𝑞. (24) Again from (1) we have 𝑑=𝑑𝑑𝑑𝑑𝑑+𝑑.𝑔𝑝,𝑔𝑔𝑝𝑓𝑝,𝑔𝑔𝑝≤𝑎ℎ𝑝,ℎ𝑔𝑝+𝑏𝑓𝑝,ℎ𝑝+𝑐𝑔𝑔𝑝,ℎ𝑔𝑝+𝜆ℎ𝑝,𝑔𝑔𝑝𝑓𝑝,ℎ𝑔𝑝 (25)
As (𝑔,ℎ) is weakly compatible, therefore from (19) and (25) we can write 𝑑𝑔𝑝,𝑔𝑔𝑝≤(𝑎+2𝜆)𝑑𝑔𝑝,𝑔𝑔𝑝. (26) As 𝑎+𝑏+𝑐+2𝜆<1 and 𝑎,𝑏,𝑐,𝜆≥0, so from (26), 𝑔𝑔𝑝=𝑔𝑝.
Hence, 𝑔𝑞=𝑞. (27) From (23), (24), and (27), it follows that 𝑞 is common fixed point for 𝑓, 𝑔, and ℎ.
Now we shall prove the uniqueness of common fixed point for 𝑓, 𝑔, and ℎ. Suppose 𝑟 is another common fixed point for 𝑓, 𝑔, and ℎ.
Consider 𝑑𝑑𝑑𝑑𝑑+𝑑=(𝑑.𝑞,𝑟≤𝑎ℎ𝑞,ℎ𝑟+𝑏𝑓𝑞,ℎ𝑞+𝑐𝑔𝑟,ℎ𝑟+𝜆ℎ𝑞,𝑔𝑟𝑓𝑞,ℎ𝑟𝑎+2𝜆)𝑞,𝑟 (28) Therefore, 𝑞=𝑟, since 𝑎+𝑏+𝑐+2𝜆<1 and 𝑎,𝑏,𝑐,𝜆≥0. Thus 𝑓, 𝑔, and ℎ have unique common fixed point in 𝑋.

Remark 11. (i) If we take 𝑏=𝑐=𝜆=0, 𝑎=𝑘 in Theorem 10, then 𝑑𝑓𝑥,𝑔𝑦≤𝑘𝑑ℎ𝑥,ℎ𝑦,where𝑘∈[0,1). (29)
(ii) If we take 𝑎=𝜆=0, 𝑏=𝑐=𝑘 in Theorem 10, then 𝑑𝑑+𝑑,1𝑓𝑥,𝑔𝑦≤𝑘𝑓𝑥,ℎ𝑥𝑔𝑦,ℎ𝑦where𝑘∈0,2. (30)
(iii) If we take 𝑎=𝑏=𝑐=0, 𝜆=𝑘 in Theorem 10, then 𝑑𝑑+𝑑,1𝑓𝑥,𝑔𝑦≤𝑘ℎ𝑥,𝑔𝑦𝑓𝑥,ℎ𝑦where𝑘∈0,2. (31)
From Remark 11, it is clear that Theorem 2.1 in [4] is a special case of Theorem 10 with 𝑎=𝑘 and 𝑏=𝑐=𝜆=0, where 𝑘∈[0,1), and Theorem 2.3 in [4] is a special case of Theorem 10 with 𝑎=𝜆=0 and 𝑏=𝑐=𝑘, where 𝑘∈[0,1/2). Therefore, we can say that Theorem 10 has generalized and unified the main results in [4].
In Theorem 10 if we take 𝑔=𝑓, then as immediate consequence of Theorem 10 we obtain the following corollary.

Corollary 12. Let (𝑋,𝑑) be a cone metric space and 𝑃 be a normal cone with normal constant 𝐾. Suppose that the mappings 𝑓,ℎ∶𝑋→𝑋 satisfy the condition 𝑑𝑑𝑑𝑑𝑑+𝑑,𝑓𝑥,𝑓𝑦≤𝑎ℎ𝑥,ℎ𝑦+𝑏𝑓𝑥,ℎ𝑥+𝑐𝑓𝑦,ℎ𝑦+𝜆ℎ𝑥,𝑓𝑦𝑓𝑥,ℎ𝑦 (32) for all 𝑥,𝑦∈𝑋, where 𝑎, 𝑏, 𝑐, and 𝜆 are nonnegative real numbers satisfying 𝑎+𝑏+𝑐+2𝜆<1. If the range of ℎ contains the range of 𝑓 and ℎ(𝑋) is a complete subspace of 𝑋, then 𝑓 and ℎ have a unique point of coincidence in 𝑋. Moreover, if (𝑓,ℎ) is weakly compatible, then 𝑓 and ℎ have a unique common fixed point.

Remark 13. (i) If we take 𝑏=𝑐=𝜆=0, 𝑎=𝑘 in Corollary 12, then 𝑑𝑓𝑥,𝑓𝑦≤𝑘𝑑ℎ𝑥,ℎ𝑦,where𝑘∈[0,1). (33)
(ii) If we take 𝑎=𝜆=0, 𝑏=𝑐=𝑘 in Corollary 12, then 𝑑𝑑+𝑑,1𝑓𝑥,𝑓𝑦≤𝑘𝑓𝑥,ℎ𝑥𝑓𝑦,ℎ𝑦where𝑘∈0,2. (34)
(iii) If we take 𝑎=𝑏=𝑐=0,𝜆=𝑘 in Corollary 12, then 𝑑𝑑+𝑑,1𝑓𝑥,𝑓𝑦≤𝑘ℎ𝑥,𝑓𝑦𝑓𝑥,ℎ𝑦where𝑘∈0,2. (35)
From Remark 13 it is clear that Theorem 2.3 [3] is a special case of Corollary 12. Therefore we can say that Theorem 10 has generalized and unified the main result of Radenović in [3].
We present now some nontrivial examples that illustrate how general and important is the result given by Theorem 10.

Example 14. Let 𝐸=𝑅², with the norm ‖(𝑥,𝑦)‖=|𝑥|+|𝑦|, be a real Banach space and let 𝑃={(𝑥,𝑦)∈𝐸∶𝑥,𝑦≥0}. If we consider 𝑋={𝛼,𝛽,𝛾,𝛿} and define 𝑑∶𝑋×𝑋→𝐸 by 𝑑=(),𝑑=(),𝑑()()=(),𝑑=(),𝑑=(),𝑑=(),𝑑()()=(),𝛼,𝛽=𝑑𝛽,𝛼0.9,0.9𝛼,𝛾=𝑑𝛾,𝛼0.5,3𝛼,𝛿=𝑑𝛿,𝛼1,2.2𝛽,𝛾=𝑑𝛾,𝛽0.5,3𝛽,𝛿=𝑑𝛿,𝛽1,2.5𝛾,𝛿=𝑑𝛿,𝛾1,3𝛼,𝛼=𝑑𝛽,𝛽=𝑑𝛾,𝛾=𝑑𝛿,𝛿0,0 (36) then (𝑋,𝑑) is a cone metric space. Let 𝑓, 𝑔, and ℎ∶𝑋→𝑋 be defined, respectively, as follows: 𝑓𝛼=𝛽,𝑓𝛽=𝛽,𝑓𝛾=𝛼,𝑓𝛿=𝛽,𝑔𝛼=𝛽,𝑔𝛽=𝛽,𝑔𝛾=𝛿,𝑔𝛿=𝛽,ℎ𝛼=𝛿,ℎ𝛽=𝛽,ℎ𝛾=𝛾,ℎ𝛿=𝛼. (37) Then 𝑓, 𝑔, and ℎ have the properties mentioned in Theorem 10, and also 𝑓, 𝑔, and ℎ satisfy the inequality (1).
Hence the conditions of Theorem 10 are satisfied. Therefore we conclude that 𝑓, 𝑔, and ℎ have unique point of coincidence and also unique common fixed point.
Here it is seen that 𝛽 is unique point of coincidence and also the unique common fixed point of 𝑓, 𝑔, and ℎ.

Remark 15. Example 14 does not satisfy the conditions (29) and (30) at the points 𝑥=𝛾, 𝑦=𝛾 and 𝑥=𝛽, 𝑦=𝛾, respectively. Therefore, we can say that inequalities of Theorems 2.1 and 2.3 of [4] fail at the points 𝑥=𝛾, 𝑦=𝛾 and 𝑥=𝛽, 𝑦=𝛾, respectively. Hence, Theorem 2.1 and Theorem 2.3 of [4] cannot apply to Example 14.

Example 16. Let 𝐸=𝑅, with the norm ‖𝑥‖=|𝑥|, be a real Banach space and let 𝑃={𝑥∈𝐸∶𝑥≥0}. Let 𝑋={0,1,2} and also define 𝑑∶𝑋×𝑋→𝐸 by 𝑑(𝑥,𝑦)=|𝑥−𝑦| for all 𝑥,𝑦∈𝑋.
Then (𝑋,𝑑) is a cone metric space. Let 𝑓,ℎ∶𝑋→𝑋 be defined, respectively, as follows: 𝑓𝑥=2−𝑥,if𝑥≠0,0,if𝑥=0. (38) Also ℎ𝑥=𝑥,for𝑥∈𝑋. (39) Then 𝑓 and ℎ have the properties mentioned in Corollary 12, and also 𝑓 and ℎ satisfy the inequality (32).
Hence the conditions of Corollary 12 are satisfied. Therefore we conclude that 𝑓 and ℎ have unique point of coincidence and also unique common fixed point.
Here it is seen that 0 is unique point of coincidence and also the unique common fixed point of 𝑓 and ℎ.

Remark 17. Example 16 does not satisfy the conditions ((33), (35)), and (34) at the points 𝑥=1, 𝑦=2 and 𝑥=2, 𝑦=0, respectively. Therefore, we can say that inequalities ((2.4), (2.6)) and (2.5) of [3] fail at the points 𝑥=1, 𝑦=2 and 𝑥=2, 𝑦=0, respectively. Hence, Theorem 2.3 of [3] cannot apply to Example 16.

Remark 18. Example 14 does not satisfy the inequality 2.8 of [5] at the point 𝑥=𝛼, 𝑦=𝛾. Therefore, it is clear that Corollary 2.10 of [5] cannot apply to Example 14. Hence Theorem 10 is more general than Corollary 2.10 of [5].