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

Abstract

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

1. Introduction

The concept of cone metric spaces is a generalization of metric spaces, where each pair of points is assigned to a member of a real Banach space with a cone, for new results on cone metric spaces see [1–6]. This cone naturally induces a partial order in the Banach spaces. The concept of cone metric space was introduced in the work of Huang and Zhang [7], where they also established the Banach contraction mapping principle in this space. Then, several authors have studied fixed point problems in cone metric spaces. Some of these works are noted in [8–13].

In [14], Cho et al. introduced a new concept of the c-distance in cone metric spaces and proved some fixed point theorems in ordered cone metric spaces. This is more general than the classical Banach contraction mapping principle.

In [15], Sintunavarat et al. extended and developed the Banach contraction theorem on c-distance of Cho et al. [14]. They gave some illustrative examples of the main results. Their results improve, generalize, and unify the results of Cho et al. [14] and some results of the fundamental metrical fixed point theorems in the literature.

In this paper we proved some fixed point theorems for c-distance in cone metric space. These theorems extend and develop some theorems in literature on c-distance of Cho et al. [14] in cone metric space.

The following theorems are the main results given in [7, 14, 16].

Theorem 1.1 (see [16]). Let (𝑋,𝑑) be a complete cone metric space. Suppose that the mapping 𝑓∶𝑋→𝑋 satisfies the contractive condition: 𝑑𝑓𝑥,𝑓𝑦⪯𝑘𝑑𝑥,𝑦, (1.1) for all 𝑥,𝑦∈𝑋, where 𝑘∈[0,1) is a constant. Then 𝑓 has a unique fixed point in 𝑋 and for any 𝑥∈𝑋, iterative sequence {𝑓^𝑛𝑥} converges to the fixed point.

Theorem 1.2 (see [7]). Let (𝑋,𝑑) be a complete cone metric space and 𝑃 be a normal cone with normal constant 𝐾. Suppose that the mapping 𝑓∶𝑋→𝑋 satisfies the contractive condition: 𝑑𝑓𝑥,𝑓𝑦⪯𝑘𝑑𝑓𝑥,𝑥+𝑑𝑓𝑦,𝑦, (1.2) for all 𝑥,𝑦∈𝑋, where 𝑘∈[0,1/2) is a constant. Then 𝑓 has a unique fixed point in 𝑋 and for any 𝑥∈𝑋, iterative sequence {𝑓^𝑛𝑥} converges to the fixed point.

Theorem 1.3 (see [7]). Let (𝑋,𝑑) be a complete cone metric space and 𝑃 be a normal cone with normal constant 𝐾. Suppose that the mapping 𝑓∶𝑋→𝑋 satisfies the contractive condition: 𝑑𝑓𝑥,𝑓𝑦⪯𝑘𝑑𝑓𝑥,𝑦+𝑑𝑓𝑦,𝑥, (1.3) for all 𝑥,𝑦∈𝑋, where 𝑘∈[0,1/2) is a constant. Then 𝑓 has a unique fixed point in 𝑋 and for any 𝑥∈𝑋, iterative sequence {𝑓^𝑛𝑥} converges to the fixed point.

Theorem 1.4 (see [14]). Let (𝑋,⊑) be a partially ordered set and suppose that (𝑋,𝑑) is a complete cone metric space. Let 𝑞 is a c-distance on 𝑋 and 𝑓∶𝑋→𝑋 be a continous and nondecreasing mapping with respect to ⊑. Suppose that the following two assertions hold: (1)there exist 𝛼,𝛽,𝛾>0 with 𝛼+𝛽+𝛾<1 such that 𝑞𝑓𝑥,𝑓𝑦⪯𝛼𝑞𝑥,𝑦+𝛽𝑞𝑥,𝑓𝑥+𝛾𝑞𝑦,𝑓𝑦, (1.4) for all 𝑥,𝑦∈𝑋 with 𝑦⊑𝑥,(2) there exists 𝑥₀∈𝑋 such that 𝑥₀⊑𝑓𝑥₀.Then 𝑓 has a fixed point 𝑥^∗∈𝑋. If 𝑣=𝑓𝑣 then 𝑞(𝑣,𝑣)=𝜃.

2. Preliminaries

Let 𝐸 be a real Banach space and 𝜃 denote to the zero element in 𝐸. A cone 𝑃 is a subset of 𝐸 such that(1)𝑃 is nonempty set closed and 𝑃≠{𝜃},(2) if 𝑎,𝑏 are nonnegative real numbers and 𝑥,𝑦∈𝑃 then 𝑎𝑥+𝑏𝑦∈𝑃,(3)𝑥∈𝑃 and −𝑥∈𝑃 imply that 𝑥=𝜃.

For any cone 𝑃⊂𝐸, the partial ordering ⪯ with respect to 𝑃 is defined by 𝑥⪯𝑦 if and only if 𝑦−𝑥∈𝑃. The notation of ≺ stands for 𝑥⪯𝑦 but 𝑥≠𝑦. Also, we used 𝑥≪𝑦 to indicate that 𝑦−𝑥∈int𝑃, where int𝑃 denotes the interior of 𝑃. A cone 𝑃 is called normal if there exists a number 𝐾 such that 𝜃⪯𝑥⪯𝑦⟹‖𝑥‖≤𝐾𝑦, (2.1) for all 𝑥,𝑦∈𝐸. The least positive number 𝐾 satisfying the above condition is called the normal constant of 𝑃.

Definition 2.1 (see [7]). Let 𝑋 be a nonempty set and 𝐸 be a real Banach space equipped with the partial ordering ⪯ with respect to the cone 𝑃. Suppose that the mapping 𝑑∶𝑋×𝑋→𝐸 satisfies the following conditions: (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 [7]). Let (𝑋,𝑑) be a cone metric space, {𝑥_𝑛} be 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 [17]). (1)If 𝐸 is 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. [18] with some properties.

Definition 2.4 (see [14]). Let (𝑋,𝑑) is 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 [14]). 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 [14]). 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 [14]). (1)𝑞(𝑥,𝑦)=𝑞(𝑦,𝑥) does not necessarily for all 𝑥,𝑦∈𝑋.(2)𝑞(𝑥,𝑦)=𝜃 is not necessarily equivalent to 𝑥=𝑦 for all 𝑥,𝑦∈𝑋.

3. Main Results

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

Theorem 3.1. Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a 𝑐-distance on 𝑋. Suppose that the mapping 𝑓∶𝑋→𝑋 satisfies the contractive condition: 𝑞𝑓𝑥,𝑓𝑦⪯𝑘𝑞𝑥,𝑦, (3.1) for all 𝑥,𝑦∈𝑋, where 𝑘∈[0,1) is a constant. Then 𝑓 has a fixed point 𝑥^∗∈𝑋 and for any 𝑥∈𝑋, iterative sequence {𝑓^𝑛𝑥} converges to the fixed point. If 𝑣=𝑓𝑣 then 𝑞(𝑣,𝑣)=𝜃. The fixed point is unique.

Proof. Choose 𝑥₀∈𝑋. Set 𝑥₁=𝑓𝑥₀,𝑥₂=𝑓𝑥₁=𝑓²𝑥₀,…,𝑥_𝑛+1=𝑓𝑥_𝑛=𝑓^𝑛+1𝑥₀. We have: 𝑞𝑥_𝑛,𝑥_𝑛+1=𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛⪯𝑘𝑞𝑥_𝑛−1,𝑥_𝑛⪯𝑘²𝑞𝑥_𝑛−2,𝑥_𝑛−1⪯⋯⪯𝑘^𝑛𝑞𝑥₀,𝑥₁. (3.2) Let 𝑚>𝑛≥1. Then it follows that 𝑞𝑥_𝑛,𝑥_𝑚⪯𝑞𝑥_𝑛,𝑥_𝑛+1+𝑞𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑞𝑥_𝑚−1,𝑥_𝑚⪯𝑘^𝑛+𝑘^𝑛+1+⋯+𝑘^𝑚−1𝑞𝑥₀,𝑥₁⪯𝑘^𝑛𝑞𝑥1−𝑘⁰,𝑥₁. (3.3)
Thus, Lemma 2.6 shows that {𝑥_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑥^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗ as 𝑛→∞. By (q3) we have: 𝑞𝑥_𝑛,𝑥^∗⪯𝑘^𝑛𝑞𝑥1−𝑘⁰,𝑥₁. (3.4)
On the other hand, 𝑞𝑥_𝑛,𝑓𝑥^∗=𝑞𝑓𝑥_𝑛−1,𝑓𝑥^∗⪯𝑘𝑞𝑥_𝑛−1,𝑥^∗𝑘⪯𝑘^𝑛−1𝑞𝑥1−𝑘⁰,𝑥₁=𝑘^𝑛𝑞𝑥1−𝑘⁰,𝑥₁. (3.5)
By Lemma 2.6 part 1, (3.4) and (3.5), we have 𝑥^∗=𝑓𝑥^∗. Thus, 𝑥^∗ is a fixed point of 𝑓.
Suppose that 𝑣=𝑓𝑣, then we have the following: 𝑞(𝑣,𝑣)=𝑞(𝑓𝑣,𝑓𝑣)⪯𝑘𝑞(𝑣,𝑣). Since 𝑘<1, Lemma 2.3 show that 𝑞(𝑣,𝑣)=𝜃.
Finally suppose there is another fixed point 𝑦^∗ of 𝑓, then we have the following: 𝑞(𝑥^∗,𝑦^∗)=𝑞(𝑓𝑥^∗,𝑓𝑦^∗)⪯𝑘𝑞(𝑥^∗,𝑦^∗). Since 𝑘<1, Lemma 2.3 show that 𝑞(𝑥^∗,𝑦^∗)=𝜃 and also we have 𝑞(𝑥^∗,𝑥^∗)=𝜃. Hence by Lemma 2.6 part 1, 𝑥^∗=𝑦^∗. Therefore the fixed point is unique.

Corollary 3.2. Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a c-distance on 𝑋. Suppose that the mapping 𝑓∶𝑋→𝑋 satisfies the contractive condition: 𝑞𝑓^𝑛𝑥,𝑓^𝑛𝑦⪯𝑘𝑑𝑥,𝑦, (3.6) for all 𝑥,𝑦∈𝑋, where 𝑘∈[0,1) is a constant. Then 𝑓 has a unique fixed point 𝑥^∗∈𝑋. If 𝑣=𝑓𝑣 then 𝑞(𝑣,𝑣)=𝜃.

Proof. From Theorem 3.1  𝑓^𝑛 has a unique fixed point 𝑥^∗. But 𝑓^𝑛(𝑓𝑥^∗)=𝑓(𝑓^𝑛𝑥^∗)=𝑓(𝑥^∗), so 𝑓(𝑥^∗) is also a fixed point of 𝑓^𝑛. Hence 𝑥^∗=𝑓𝑥^∗. Thus, 𝑥^∗ is a fixed point of 𝑓. Since the fixed point of 𝑓 is also fixed point of 𝑓^𝑛, the fixed point of 𝑓 is unique.
Suppose that 𝑣=𝑓𝑣. From above the fixed point of 𝑓 is also fixed point of 𝑓^𝑛, then we have the following: 𝑞(𝑣,𝑣)=𝑞(𝑓𝑣,𝑓𝑣)=𝑞(𝑓^𝑛𝑣,𝑓^𝑛𝑣)⪯𝑘𝑞(𝑣,𝑣). Since 𝑘<1, Lemma 2.3 show that 𝑞(𝑣,𝑣)=𝜃.

The following result is generalized from Theorem 1.4. We prove a fixed point theorem and we do not require that 𝑋 is a partially ordered set.

Theorem 3.3. Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a 𝑐-distance on 𝑋. Suppose that the mapping 𝑓∶𝑋→𝑋 is continuous and satisfies the contractive condition: 𝑞𝑓𝑥,𝑓𝑦⪯𝑘𝑞𝑥,𝑦+𝑙𝑞𝑥,𝑓𝑥+𝑟𝑞𝑦,𝑓𝑦, (3.7) for all 𝑥,𝑦∈𝑋, where 𝑘, 𝑙, 𝑟 are nonnegative real numbers such that 𝑘+𝑙+𝑟<1. Then 𝑓 has a fixed point 𝑥^∗∈𝑋 and for any 𝑥∈𝑋, iterative sequence {𝑓^𝑛𝑥} converges to the fixed point. If 𝑣=𝑓𝑣 then 𝑞(𝑣,𝑣)=𝜃. The fixed point is unique.

Proof. Choose 𝑥₀∈𝑋. Set 𝑥₁=𝑓𝑥₀,𝑥₂=𝑓𝑥₁=𝑓²𝑥₀,…,𝑥_𝑛+1=𝑓𝑥_𝑛=𝑓^𝑛+1𝑥₀. We have the following: 𝑞𝑥_𝑛,𝑥_𝑛+1=𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛⪯𝑘𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑞𝑥_𝑛−1,𝑓𝑥_𝑛−1+𝑟𝑞𝑥_𝑛,𝑓𝑥_𝑛=𝑘𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑞𝑥_𝑛,𝑥_𝑛+1. (3.8) So 𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘+𝑙𝑞𝑥1−𝑟^𝑛−1,𝑥_𝑛=ℎ𝑞𝑥_𝑛−1,𝑥_𝑛, (3.9) where ℎ=(𝑘+𝑙)/(1−𝑟)<1.
Let 𝑚>𝑛≥1. Then it follows that 𝑞𝑥_𝑛,𝑥_𝑚⪯𝑞𝑥_𝑛,𝑥_𝑛+1+𝑞𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑞𝑥_𝑚−1,𝑥_𝑚⪯ℎ^𝑛+ℎ^𝑛+1+⋯+ℎ^𝑚−1𝑞𝑥₀,𝑥₁⪯ℎ^𝑛𝑞𝑥1−ℎ⁰,𝑥₁. (3.10)
Thus, Lemma 2.6 shows that {𝑥_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑥^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗ as 𝑛→∞. Since 𝑓 is continuous, then 𝑥^∗=lim𝑥_𝑛+1=lim𝑓(𝑥_𝑛)=𝑓(lim𝑥_𝑛)=𝑓(𝑥^∗). Therefore 𝑥^∗ is a fixed point of 𝑓.
Suppose that 𝑣=𝑓𝑣, then we have 𝑞(𝑣,𝑣)=𝑞𝑓𝑣,𝑓𝑣⪯𝑘𝑞(𝑣,𝑣)+𝑙𝑞𝑣,𝑓𝑣+𝑟𝑞𝑣,𝑓𝑣=(𝑘+𝑙+𝑟)𝑞(𝑣,𝑣), (3.11) since 𝑘+𝑙+𝑟<1, Lemma 2.3 show that 𝑞(𝑣,𝑣)=𝜃.
Finally, suppose that, there is another fixed point 𝑦^∗ of 𝑓, then we have the following: 𝑞𝑥^∗,𝑦^∗=𝑞𝑓𝑥^∗,𝑓𝑦^∗⪯𝑘𝑞𝑥^∗,𝑦^∗+𝑙𝑞𝑥^∗,𝑓𝑥^∗+𝑟𝑞𝑦^∗,𝑓𝑦^∗=𝑘𝑞𝑥^∗,𝑦^∗+𝑙𝑞𝑥^∗,𝑥^∗+𝑟𝑞𝑦^∗,𝑦^∗=𝑘𝑞𝑥^∗,𝑦^∗⪯𝑘𝑞𝑥^∗,𝑦^∗+𝑙𝑞𝑥^∗,𝑦^∗+𝑟𝑞𝑥^∗,𝑦^∗=(𝑘+𝑙+𝑟)𝑞𝑥^∗,𝑦^∗. (3.12) Since 𝑘+𝑙+𝑟<1<1, Lemma 2.3 shows that 𝑞(𝑥^∗,𝑦^∗)=𝜃 and also we have 𝑞(𝑥^∗,𝑥^∗)=𝜃. Hence by Lemma 2.6 part 1, 𝑥^∗=𝑦^∗. Therefore the fixed point is unique.

If 𝑘=0 and 𝑟=𝑙, then we have the following result.

Corollary 3.4. Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a c-distance on 𝑋. Suppose that the mapping 𝑓∶𝑋→𝑋 is continuous and satisfies the contractive condition: 𝑞𝑓𝑥,𝑓𝑦⪯𝑙𝑞𝑥,𝑓𝑥+𝑞𝑦,𝑓𝑦, (3.13) for all 𝑥,𝑦∈𝑋,where 𝑙∈[0,1/2) is a constant. Then 𝑓 has a fixed point 𝑥^∗∈𝑋 and for any 𝑥∈𝑋, iterative sequence {𝑓^𝑛𝑥} converges to the fixed point. If 𝑣=𝑓𝑣 then 𝑞(𝑣,𝑣)=𝜃. The fixed point is unique.

Finally, we provide another result and we do not require that 𝑓 is continuous.

Theorem 3.5. Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a 𝑐-distance on 𝑋. Suppose that the mapping 𝑓∶𝑋→𝑋 satisfies the contractive condition: (1−𝑟)𝑞𝑓𝑥,𝑓𝑦⪯𝑘𝑞𝑥,𝑓𝑦+𝑙𝑞𝑥,𝑓𝑥, (3.14) for all 𝑥,𝑦∈𝑋, where 𝑘,𝑙,𝑟 are nonnegative real numbers such that 2𝑘+𝑙+𝑟<1. Then 𝑓 has a fixed point 𝑥^∗∈𝑋 and for any 𝑥∈𝑋, iterative sequence {𝑓^𝑛𝑥} converges to the fixed point. If 𝑣=𝑓𝑣 then 𝑞(𝑣,𝑣)=𝜃. The fixed point is unique.

Proof. Choose 𝑥₀∈𝑋. Set 𝑥₁=𝑓𝑥₀,𝑥₂=𝑓𝑥₁=𝑓²𝑥₀,…,𝑥_𝑛+1=𝑓𝑥_𝑛=𝑓^𝑛+1𝑥₀. Observe that (1−𝑟)𝑞𝑓𝑥,𝑓𝑦⪯𝑘𝑞𝑥,𝑓𝑦+𝑙𝑞𝑥,𝑓𝑥, (3.15) equivalently 𝑞𝑓𝑥,𝑓𝑦⪯𝑘𝑞𝑥,𝑓𝑦+𝑙𝑞𝑥,𝑓𝑥+𝑟𝑞𝑓𝑥,𝑓𝑦. (3.16) Then we have: 𝑞𝑥_𝑛,𝑥_𝑛+1=𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛⪯𝑘𝑞𝑥_𝑛−1,𝑓𝑥_𝑛+𝑙𝑞𝑥_𝑛−1,𝑓𝑥_𝑛−1+𝑟𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛=𝑘𝑞𝑥_𝑛−1,𝑥_𝑛+1+𝑙𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑞𝑥_𝑛−1,𝑥_𝑛+𝑘𝑞𝑥_𝑛,𝑥_𝑛+1+𝑙𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑞𝑥_𝑛,𝑥_𝑛+1. (3.17) So, 𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘+𝑙𝑞𝑥1−𝑘−𝑟^𝑛−1,𝑥_𝑛=ℎ𝑞𝑥_𝑛−1,𝑥_𝑛, (3.18) where ℎ=(𝑘+𝑙)/(1−𝑘−𝑟)<1.
Let 𝑚>𝑛≥1. Then it follows that 𝑞𝑥_𝑛,𝑥_𝑚⪯𝑞𝑥_𝑛,𝑥_𝑛+1+𝑞𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑞𝑥_𝑚−1,𝑥_𝑚⪯ℎ^𝑛+ℎ^𝑛+1+⋯+ℎ^𝑚−1𝑞𝑥₀,𝑥₁⪯ℎ^𝑛𝑞𝑥1−ℎ⁰,𝑥₁. (3.19)
Thus, Lemma 2.6 shows that {𝑥_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑥^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗ as 𝑛→∞.
By (q3) we have: 𝑞𝑥_𝑛,𝑥^∗⪯ℎ^𝑛𝑞𝑥1−ℎ⁰,𝑥₁. (3.20)
On the other hand, 𝑞𝑥_𝑛,𝑓𝑥^∗=𝑞𝑓𝑥_𝑛−1,𝑓𝑥^∗⪯𝑘𝑞𝑥_𝑛−1,𝑓𝑥^∗+𝑙𝑞𝑥_𝑛−1,𝑓𝑥_𝑛−1+𝑟𝑞𝑓𝑥_𝑛−1,𝑓𝑥^∗=𝑘𝑞𝑥_𝑛−1,𝑓𝑥^∗+𝑙𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑞𝑥_𝑛−1,𝑥_𝑛+𝑘𝑞𝑥_𝑛,𝑓𝑥^∗+𝑙𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑞𝑥_𝑛,𝑓𝑥^∗. (3.21) So, 𝑞𝑥_𝑛,𝑓𝑥^∗⪯𝑘+𝑙𝑞𝑥1−𝑘−𝑟^𝑛−1,𝑥_𝑛⪯𝑘+𝑙ℎ1−𝑘−𝑟^𝑛−1𝑞𝑥₀,𝑥₁=ℎℎ^𝑛−1𝑞𝑥₀,𝑥₁=ℎ^𝑛𝑞𝑥₀,𝑥₁⪯ℎ^𝑛𝑞𝑥1−ℎ⁰,𝑥₁. (3.22)
By Lemma 2.6 part 1, (3.20) and (3.22), we have 𝑥^∗=𝑓𝑥^∗. Thus, 𝑥^∗ is a fixed point of 𝑓.
Suppose that 𝑣=𝑓𝑣, then we have 𝑞()()()()()()()()=()𝑞().𝑣,𝑣=𝑞𝑓𝑣,𝑓𝑣⪯𝑘𝑞𝑣,𝑓𝑣+𝑙𝑞𝑣,𝑓𝑣+𝑟𝑞𝑣,𝑓𝑣=𝑘𝑞𝑣,𝑣+𝑙𝑞𝑣,𝑣+𝑟𝑞𝑣,𝑣⪯𝑘𝑞𝑣,𝑣+𝑘𝑞𝑣,𝑣+𝑙𝑞𝑣,𝑣+𝑟𝑞𝑣,𝑣2𝑘+𝑙+𝑟𝑣,𝑣 (3.23) Since 2𝑘+𝑙+𝑟<1, Lemma 2.3 shows that 𝑞(𝑣,𝑣)=𝜃.
Finally, suppose that, there is another fixed point 𝑦^∗ of 𝑓, then we have 𝑞𝑥^∗,𝑦^∗=𝑞𝑓𝑥^∗,𝑓𝑦^∗⪯𝑘𝑞𝑥^∗,𝑓𝑦^∗+𝑙𝑞𝑥^∗,𝑓𝑥^∗+𝑟𝑞𝑓𝑥^∗,𝑓𝑦^∗⪯𝑘𝑞𝑥^∗,𝑓𝑦^∗+𝑘𝑞𝑥^∗,𝑓𝑦^∗+𝑙𝑞𝑥^∗,𝑓𝑥^∗+𝑟𝑞𝑓𝑥^∗,𝑓𝑦^∗=𝑘𝑞𝑥^∗,𝑦^∗+𝑘𝑞𝑥^∗,𝑦^∗+𝑙𝑞𝑥^∗,𝑥^∗+𝑟𝑞𝑥^∗,𝑦^∗=𝑘𝑞𝑥^∗,𝑦^∗+𝑘𝑞𝑥^∗,𝑦^∗+𝑟𝑞𝑥^∗,𝑦^∗⪯𝑘𝑞𝑥^∗,𝑦^∗+𝑘𝑞𝑥^∗,𝑦^∗+𝑙𝑞𝑥^∗,𝑦^∗+𝑟𝑞𝑥^∗,𝑦^∗=(2𝑘+𝑙+𝑟)𝑞𝑥^∗,𝑦^∗. (3.24)
Since 2𝑘+𝑙+𝑟<1, Lemma 2.3 shows that 𝑞(𝑥^∗,𝑦^∗)=𝜃 and also we have 𝑞(𝑥^∗,𝑥^∗)=𝜃. Hence by Lemma 2.6 part 1, 𝑥^∗=𝑦^∗. Therefore the fixed point is unique.

Example 3.6. Consider Example 2.5. Define the mapping 𝑓∶𝑋→𝑋 by 𝑓(3/4)=1/4 and 𝑓𝑥=𝑥/2 for all 𝑥∈𝑋 with 𝑥≠3/4. Since 𝑑(𝑓(1),𝑓(3/4))=𝑑(1,3/4), there is not 𝑘∈[0,1) such that 𝑑(𝑓𝑥,𝑓𝑦)⪯𝑘𝑑(𝑥,𝑦). Since Theorem  2.3 of Rezapour and Hamlbarani [16] cannot be applied to this example on cone metric space. To check this example on c-distance we have:(1)If 𝑥=𝑦=3/4, then we have the following.𝑞𝑓343,𝑓43=𝑓4=143⪯𝑘43=𝑘𝑞4,342with𝑘=3. (3.25)(2) If 𝑥≠𝑦≠3/4, then we have 𝑞=𝑦𝑓𝑥,𝑓𝑦22⪯𝑘𝑞𝑥,𝑦with𝑘=3. (3.26)(3) If 𝑥=3/4,𝑦≠3/4, then we have 𝑞𝑓34=𝑦,𝑓𝑦23⪯𝑘𝑞42,𝑦with𝑘=3. (3.27)(4) If 𝑥≠3/4,𝑦=3/4, then we have 𝑞3𝑓𝑥,𝑓43=𝑓4=143⪯𝑘43=𝑘𝑞𝑥,42with𝑘=3. (3.28)Hence 𝑞(𝑓𝑥,𝑓𝑦)⪯𝑘𝑞(𝑥,𝑦) for all 𝑥∈𝑋. Therefore, the condition of Theorem 3.1 are satisfied and then 𝑓 has a unique fixed point 𝑥=0, 𝑓(0)=0 with 𝑞(0,0)=0.

Acknowledgments

The authors Zaid Mohammed Fadail and Abd Ghafur Bin Ahmad would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research Grant ERGS/1/2011/STG/UKM/01/13. Zoran Golubović is thankful to the Ministry of Science and Technological Development of Serbia. The authors thank the referee for his/her careful reading of the paper and useful suggestions.