New Fixed Point Results 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 results in literature for c-distance in cone metric spaces by replacing the constants in contractive conditions with functions. Some supporting examples are given.

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 [1–8]. 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 [9] 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 [10–15].

In [16], Cho et al. introduced a new concept of the c-distance in cone metric spaces (also see [17]) and proved some fixed point theorems in ordered cone metric spaces. This is more general than the classical Banach contraction mapping principle. Then, Sintunavarat et al. [18] extended and developed the Banach contraction theorem on c-distance of Cho et al. [16]. They gave some illustrative examples of main results. Their results improve, generalize, and unify the results of Cho et al. [16] and some results of the fundamental metrical fixed point theorems in the literature, for some new results for c-distance see [19–24].

In [21], Fadail et al. proved the following theorems for c-distance in cone metric spaces.

Theorem 1.1. Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a c-distance on 𝑋. Suppose the mapping 𝑓∶𝑋→𝑋 satisfies the following contractive condition: 𝑞𝑓𝑥,𝑓𝑦⪯𝑘𝑞𝑥,𝑦, (1.1) 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.

Theorem 1.2. Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a c-distance on 𝑋. Suppose the mapping 𝑓∶𝑋→𝑋 is continuous and satisfies the following contractive condition: 𝑞𝑓𝑥,𝑓𝑦⪯𝑘𝑞𝑥,𝑦+𝑙𝑞𝑥,𝑓𝑥+𝑟𝑞𝑦,𝑓𝑦, (1.2) for all 𝑥,𝑦∈𝑋, where 𝑘,𝑙,𝑟 are none negative 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.

Theorem 1.3. Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a c-distance on 𝑋. Suppose the mapping 𝑓∶𝑋→𝑋 satisfies the folowing contractive condition: (1−𝑟)𝑞𝑓𝑥,𝑓𝑦⪯𝑘𝑞𝑥,𝑓𝑦+𝑙𝑞𝑥,𝑓𝑥, (1.3) for all 𝑥,𝑦∈𝑋, where 𝑘,𝑙,𝑟 are none negative 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.

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 results extend and develop some theorems on c-distance of Fadail et al. [21]. In this paper, we do not impose the normality condition for the cones; the only assumption is that the cone 𝑃 is solid, that is, int𝑃≠∅.

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)𝑃is 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‖≤𝐾𝑦, (2.1) for all 𝑥,𝑦∈𝐸. The least positive number 𝐾 satisfying the above condition is called the normal constant of 𝑃. It is clear that 𝐾≥1.

Definition 2.1 (see [9]). 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 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 [9]). Let (𝑋,𝑑) be a cone metric space, {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 [25]). (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. [26] with some properties.

Definition 2.4 (see [16]). 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 [16]). 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 [16]). 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 [16]). (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 generalize some fixed point results from [21] by replacing the constants in contractive conditions with functions.

Theorem 3.1. Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a c-distance on 𝑋. Let 𝑓∶𝑋→𝑋 be a mapping and suppose that there exists mapping 𝑘∶𝑋→[0,1) such that the following hold:(a)𝑘(𝑓𝑥)≤𝑘(𝑥) for all 𝑥∈𝑋,(b)𝑞(𝑓𝑥,𝑓𝑦)⪯𝑘(𝑥)𝑞(𝑥,𝑦) for all 𝑥,𝑦∈𝑋. 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. Then we have 𝑞𝑥_𝑛,𝑥_𝑛+1=𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛⪯𝑘𝑥_𝑛−1𝑞𝑥_𝑛−1,𝑥_𝑛=𝑘𝑓𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛⪯𝑘𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛⋯⪯𝑘𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛⪯𝑘²𝑥₀𝑞𝑥_𝑛−2,𝑥_𝑛−1⪯𝑘𝑥⪯⋯_0𝑛𝑞𝑥₀,𝑥₁. (3.1)
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⪯𝑘𝑥₀^𝑛+𝑘𝑥₀^𝑛+1+⋯+𝑘𝑥₀^𝑚−1𝑞𝑥₀,𝑥₁⪯𝑘𝑥₀^𝑛1−𝑘𝑥₀𝑞𝑥₀,𝑥₁. (3.2) Thus, Lemma 2.6(3) 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^∗⪯𝑘𝑥₀^𝑛1−𝑘𝑥₀𝑞𝑥₀,𝑥₁. (3.3)
On the other hand, we have𝑞𝑥_𝑛,𝑓𝑥^∗=𝑞𝑓𝑥_𝑛−1,𝑓𝑥^∗⪯𝑘𝑥_𝑛−1𝑞𝑥_𝑛−1,𝑥^∗=𝑘𝑓𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥^∗⪯𝑘𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥^∗⋯⪯𝑘𝑥₀𝑞𝑥_𝑛−1,𝑥^∗⪯𝑘𝑥₀𝑘𝑥₀^𝑛−11−𝑘𝑥₀𝑞𝑥₀,𝑥₁=𝑘𝑥₀^𝑛1−𝑘𝑥₀𝑞𝑥₀,𝑥₁. (3.4) By Lemma 2.6(3), (3.3), and (3.4), we have 𝑥^∗=𝑓𝑥^∗. Thus, 𝑥^∗ is a fixed point of 𝑓.
Suppose that 𝑣=𝑓𝑣, then we have 𝑞(𝑣,𝑣)=𝑞(𝑓𝑣,𝑓𝑣)⪯𝑘(𝑣)𝑞(𝑣,𝑣). Since 𝑘(𝑣)<1, Lemma 2.3(1) shows that 𝑞(𝑣,𝑣)=𝜃.
Finally, suppose there is another fixed point 𝑦^∗ of 𝑓, then we have: 𝑞(𝑥^∗,𝑦^∗)=𝑞(𝑓𝑥^∗,𝑓𝑦^∗)⪯𝑘(𝑥^∗)𝑞(𝑥^∗,𝑦^∗). Since 𝑘(𝑥^∗)<1, Lemma 2.3(1) shows that 𝑞(𝑥^∗,𝑦^∗)=𝜃, and also we have 𝑞(𝑥^∗,𝑥^∗)=𝜃, hence by Lemma 2.6(1), 𝑥^∗=𝑦^∗. Therefore, the fixed point is unique.

In the above theorem, if 𝑘(𝑥) is constant, then we have the following corollary.

Corollary 3.2 ([21, theorem 3.1]). Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a c-distance on 𝑋. Suppose the mapping 𝑓∶𝑋→𝑋 satisfies the following contractive condition: 𝑞𝑓𝑥,𝑓𝑦⪯𝑘𝑞𝑥,𝑦, (3.5) 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.

Theorem 3.3. Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a c-distance on 𝑋. Let 𝑓∶𝑋→𝑋 be a continuous mapping and suppose that there exists mapping 𝑘,𝑙,𝑟∶𝑋→[0,1) such that the following hold:(a)𝑘(𝑓𝑥)≤𝑘(𝑥), 𝑙(𝑓𝑥)≤𝑙(𝑥), 𝑟(𝑓𝑥)≤𝑟(𝑥) for all 𝑥∈𝑋, (b)(𝑘+𝑙+𝑟)(𝑥)<1 for all 𝑥∈𝑋, (c)𝑞(𝑓𝑥,𝑓𝑦)⪯𝑘(𝑥)𝑞(𝑥,𝑦)+𝑙(𝑥)𝑞(𝑥,𝑓𝑥)+𝑟(𝑥)𝑞(𝑦,𝑓𝑦) for all 𝑥,𝑦∈𝑋. 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. Then we have 𝑞𝑥_𝑛,𝑥_𝑛+1=𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛⪯𝑘𝑥_𝑛−1𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥_𝑛−1𝑞𝑥_𝑛−1,𝑓𝑥_𝑛−1+𝑟𝑥_𝑛−1𝑞𝑥_𝑛,𝑓𝑥_𝑛=𝑘𝑓𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑓𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑓𝑥_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑥_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+1⋯⪯𝑘𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑙𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑥₀𝑞𝑥_𝑛,𝑥_𝑛+1. (3.6) So 𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑥₀+𝑙𝑥₀1−𝑟𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛=ℎ𝑞𝑥_𝑛−1,𝑥_𝑛⪯ℎ²𝑞𝑥_𝑛−2,𝑥_𝑛−1⪯⋯⪯ℎ^𝑛𝑞𝑥₀,𝑥₁, (3.7) where ℎ=(𝑘(𝑥₀)+𝑙(𝑥₀))/(1−𝑟(𝑥₀))<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^𝑛𝑥₀+ℎ^𝑛+1𝑥₀+⋯+ℎ^𝑚−1𝑥₀𝑞𝑥₀,𝑥₁⪯ℎ^𝑛𝑥₀1−ℎ𝑥₀𝑞𝑥₀,𝑥₁. (3.8) Thus, Lemma 2.6(3) 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 𝑞(𝑣,𝑣)=𝑞(𝑓𝑣,𝑓𝑣)⪯𝑘(𝑥₀)𝑞(𝑣,𝑣)+𝑙(𝑥₀)𝑞(𝑣,𝑓𝑣)+𝑟(𝑥₀)𝑞(𝑣,𝑓𝑣)=(𝑘+𝑙+𝑟)(𝑥₀)𝑞(𝑣,𝑣). Since (𝑘+𝑙+𝑟)(𝑥₀)<1, Lemma 2.3(1) shows that 𝑞(𝑣,𝑣)=𝜃.
Finally, suppose there is another fixed point 𝑦^∗ of 𝑓, then we have 𝑞𝑥^∗,𝑦^∗=𝑞𝑓𝑥^∗,𝑓𝑦^∗⪯𝑘𝑥^∗𝑞𝑥^∗,𝑦^∗+𝑙𝑥^∗𝑞𝑥^∗,𝑓𝑥^∗+𝑟𝑥^∗𝑞𝑦^∗,𝑓𝑦^∗=𝑘𝑥^∗𝑞𝑥^∗,𝑦^∗+𝑙𝑥^∗𝑞𝑥^∗,𝑥^∗+𝑟𝑥^∗𝑞𝑦^∗,𝑦^∗=𝑘𝑥^∗𝑞𝑥^∗,𝑦^∗⪯𝑘𝑥^∗𝑞𝑥^∗,𝑦^∗+𝑙𝑥^∗𝑞𝑥^∗,𝑦^∗+𝑟𝑥^∗𝑞𝑥^∗,𝑦^∗=(𝑘+𝑙+𝑟)𝑥^∗𝑞𝑥^∗,𝑦^∗. (3.9) Since (𝑘+𝑙+𝑟)(𝑥₀)<1, Lemma 2.3(1) shows that 𝑞(𝑥^∗,𝑦^∗)=𝜃, and also we have 𝑞(𝑥^∗,𝑥^∗)=𝜃, hence by Lemma 2.6(1), 𝑥^∗=𝑦^∗. Therefore, the fixed point is unique.

In Theorem 3.3, if 𝑘(𝑥),𝑙(𝑥), and 𝑟(𝑥) are constant, then we have the following corollary.

Corollary 3.4 ([21, theorem 3.3]). Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a c-distance on 𝑋. Suppose the mapping 𝑓∶𝑋→𝑋 is continuous and satisfies the following contractive condition: 𝑞𝑓𝑥,𝑓𝑦⪯𝑘𝑞𝑥,𝑦+𝑙𝑞𝑥,𝑓𝑥+𝑟𝑞𝑦,𝑓𝑦, (3.10) for all 𝑥,𝑦∈𝑋, where 𝑘,𝑙,𝑟 are none negative 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.

Theorem 3.5. Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a c-distance on 𝑋. Let 𝑓∶𝑋→𝑋 be a mapping and suppose that there exists mapping 𝑘,𝑙,𝑟∶𝑋→[0,1) such that the following hold:(a)𝑘(𝑓𝑥)≤𝑘(𝑥), 𝑙(𝑓𝑥)≤𝑙(𝑥), 𝑟(𝑓𝑥)≤𝑟(𝑥) for all 𝑥∈𝑋, (b)(2𝑘+𝑙+𝑟)(𝑥)<1 for all 𝑥∈𝑋, (c)(1−𝑟(𝑥))𝑞(𝑓𝑥,𝑓𝑦)⪯𝑘(𝑥)𝑞(𝑥,𝑓𝑦)+𝑙(𝑥)𝑞(𝑥,𝑓𝑥) for all 𝑥,𝑦∈𝑋. 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(𝑥))𝑞𝑓𝑥,𝑓𝑦⪯𝑘(𝑥)𝑞𝑥,𝑓𝑦+𝑙(𝑥)𝑞𝑥,𝑓𝑥, (3.11) equivalently 𝑞𝑓𝑥,𝑓𝑦⪯𝑘(𝑥)𝑞𝑥,𝑓𝑦+𝑙(𝑥)𝑞𝑥,𝑓𝑥+𝑟(𝑥)𝑞𝑓𝑥,𝑓𝑦. (3.12) Then we have 𝑞𝑥_𝑛,𝑥_𝑛+1=𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛⪯𝑘𝑥_𝑛−1𝑞𝑥_𝑛−1,𝑓𝑥_𝑛+𝑙𝑥_𝑛−1𝑞𝑥_𝑛−1,𝑓𝑥_𝑛−1+𝑟𝑥_𝑛−1𝑞𝑓𝑥_𝑛−1,𝑓𝑥_𝑛=𝑘𝑓𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+1+𝑙𝑓𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑓𝑥_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+1+𝑙𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑥_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+1⋯⪯𝑘𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛+1+𝑙𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑥₀𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑘𝑥₀𝑞𝑥_𝑛,𝑥_𝑛+1+𝑙𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑥₀𝑞𝑥_𝑛,𝑥_𝑛+1. (3.13) So 𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑥₀+𝑙𝑥₀1−𝑘𝑥₀−𝑟𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛=ℎ𝑞𝑥_𝑛−1,𝑥_𝑛⪯ℎ²𝑞𝑥_𝑛−2,𝑥_𝑛−1⪯⋯⪯ℎ^𝑛𝑞𝑥₀,𝑥₁, (3.14) where ℎ=(𝑘(𝑥₀)+𝑙(𝑥₀))/(1−𝑘(𝑥₀)−𝑟(𝑥₀))<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. (3.15) Thus, Lemma 2.6(3) 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. (3.16)
On the other hand, we have 𝑞𝑥_𝑛,𝑓𝑥^∗=𝑞𝑓𝑥_𝑛−1,𝑓𝑥^∗⪯𝑘𝑥_𝑛−1𝑞𝑥_𝑛−1,𝑓𝑥^∗+𝑙𝑥_𝑛−1𝑞𝑥_𝑛−1,𝑓𝑥_𝑛−1+𝑟𝑥_𝑛−1𝑞𝑓𝑥_𝑛−1,𝑓𝑥^∗=𝑘𝑓𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑓𝑥^∗+𝑙𝑓𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑓𝑥_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑓𝑥^∗+𝑙𝑥_𝑛−2𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑥_𝑛−2𝑞𝑥_𝑛,𝑥_𝑛+1⋯⪯𝑘𝑥₀𝑞𝑥_𝑛−1,𝑓𝑥^∗+𝑙𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑥₀𝑞𝑥_𝑛,𝑥_𝑛+1⪯𝑘𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑘𝑥₀𝑞𝑥_𝑛,𝑓𝑥^∗+𝑙𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛+𝑟𝑥₀𝑞𝑥_𝑛,𝑓𝑥^∗. (3.17) So 𝑞𝑥_𝑛,𝑓𝑥^∗⪯𝑘𝑥₀+𝑙𝑥₀1−𝑘𝑥₀−𝑟𝑥₀𝑞𝑥_𝑛−1,𝑥_𝑛⪯𝑘𝑥₀+𝑙𝑥₀1−𝑘𝑥₀−𝑟𝑥₀ℎ^𝑛−1𝑞𝑥₀,𝑥₁=ℎℎ^𝑛−1𝑞𝑥₀,𝑥₁=ℎ^𝑛𝑞𝑥₀,𝑥₁⪯ℎ^𝑛𝑞𝑥1−ℎ⁰,𝑥₁. (3.18) By Lemma 2.6(1), (3.16), and (3.18), we have 𝑥^∗=𝑓𝑥^∗. Thus, 𝑥^∗ is a fixed point of 𝑓.
Suppose that 𝑣=𝑓𝑣, then we have 𝑞()(𝑣)𝑞(𝑣)𝑞(𝑣)𝑞(𝑣)𝑞()(𝑣)𝑞()(𝑣)𝑞()(𝑣)𝑞()(𝑣)𝑞()(𝑣)𝑞()(𝑣)𝑞()=(𝑣)𝑞().𝑣,𝑣=𝑞𝑓𝑣,𝑓𝑣⪯𝑘𝑣,𝑓𝑣+𝑙𝑣,𝑓𝑣+𝑟𝑣,𝑓𝑣=𝑘𝑣,𝑣+𝑙𝑣,𝑣+𝑟𝑣,𝑣⪯𝑘𝑣,𝑣+𝑘𝑣,𝑣+𝑙𝑣,𝑣+𝑟𝑣,𝑣2𝑘+𝑙+𝑟)(𝑣,𝑣 (3.19) Since (2𝑘+𝑙+𝑟)(𝑣)<1, Lemma 2.3(1) shows that 𝑞(𝑣,𝑣)=𝜃.
Finally, suppose there is another fixed point 𝑦^∗ of 𝑓, then we have 𝑞𝑥^∗,𝑦^∗=𝑞𝑓𝑥^∗,𝑓𝑦^∗⪯𝑘𝑥^∗𝑞𝑥^∗,𝑓𝑦^∗+𝑙𝑥^∗𝑞𝑥^∗,𝑓𝑥^∗+𝑟𝑥^∗𝑞𝑓𝑥^∗,𝑓𝑦^∗⪯𝑘𝑥^∗𝑞𝑥^∗,𝑓𝑦^∗+𝑘𝑥^∗𝑞𝑥^∗,𝑓𝑦^∗+𝑙𝑥^∗𝑞𝑥^∗,𝑓𝑥^∗+𝑟𝑥^∗𝑞𝑓𝑥^∗,𝑓𝑦^∗=𝑘𝑥^∗𝑞𝑥^∗,𝑦^∗+𝑘𝑥^∗𝑞𝑥^∗,𝑦^∗+𝑙𝑥^∗𝑞𝑥^∗,𝑥^∗+𝑟𝑥^∗𝑞𝑥^∗,𝑦^∗=𝑘𝑥^∗𝑞𝑥^∗,𝑦^∗+𝑘𝑥^∗𝑞𝑥^∗,𝑦^∗+𝑟𝑥^∗𝑞𝑥^∗,𝑦^∗⪯𝑘𝑥^∗𝑞𝑥^∗,𝑦^∗+𝑘𝑥^∗𝑞𝑥^∗,𝑦^∗+𝑙𝑥^∗𝑞𝑥^∗,𝑦^∗+𝑟𝑥^∗𝑞𝑥^∗,𝑦^∗=(2𝑘+𝑙+𝑟)𝑥^∗𝑞𝑥^∗,𝑦^∗. (3.20) Since (2𝑘+𝑙+𝑟)(𝑥^∗)<1, Lemma 2.3(1) shows that 𝑞(𝑥^∗,𝑦^∗)=𝜃 and also we have 𝑞(𝑥^∗,𝑥^∗)=𝜃, hence by Lemma 2.6(1), 𝑥^∗=𝑦^∗. Therefore, the fixed point is unique.

In Theorem 3.5, if 𝑘(𝑥),𝑙(𝑥), and 𝑟(𝑥) are constants, then we have the following corollary.

Corollary 3.6 ([21, Theorem 3.5]). Let (𝑋,𝑑) be a complete cone metric space and 𝑞 is a c-distance on 𝑋. Suppose the mapping 𝑓∶𝑋→𝑋 satisfies the following contractive condition: (1−𝑟)𝑞𝑓𝑥,𝑓𝑦⪯𝑘𝑞𝑥,𝑓𝑦+𝑙𝑞𝑥,𝑓𝑥, (3.21) for all 𝑥,𝑦∈𝑋, where 𝑘,𝑙,𝑟 are none negative 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.

Example 3.7. 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. Then (𝑋,𝑑) is a complete 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 mapping 𝑓∶𝑋→𝑋 by 𝑓𝑥=𝑥²/4 for all 𝑥∈𝑋. Take 𝑘(𝑥)=(𝑥+2)/4, 𝑥∈𝑋. Observe that(a)𝑘(𝑓𝑥)=((𝑥²/4)+2)/4=(1/4)((𝑥²/4)+2)≤(1/4)(𝑥+2)=𝑘(𝑥) for all 𝑥∈𝑋. (b)For all 𝑥∈𝑋, we have 𝑞=𝑦𝑓𝑥,𝑓𝑦=𝑓𝑦₂4⪯𝑦2⪯𝑥4+12𝑦=14()𝑦(𝑥)𝑦(𝑥)𝑞.𝑥+2=𝑘=𝑘𝑥,𝑦 (3.22)Therefore, the conditions of Theorem 3.1 are satisfied. Hence 𝑓 has a unique fixed point 𝑥=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 OUP-UKM-FST-2012. Third author 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.