Fixed Point and Common Fixed Point Theorems on Ordered Cone b-Metric Spaces

Abstract

The concept of a cone b-metric space has been introduced recently as a generalization of a b-metric space and a cone metric space in 2011. The aim of this paper is to establish some fixed point and common fixed point theorems on ordered cone b-metric spaces. The proposed theorems expand and generalize several well-known comparable results in the literature to ordered cone b-metric spaces. Some supporting examples are given.

1. Introduction

Fixed point theory has attracted many researchers since 1922 with the admired Banach fixed point theorem. This theorem supplies a method for solving a variety of applied dilemma in mathematical sciences and engineering. A large literature on this subject exists, and this is a very active area of research at present. Banach contraction principle has been generalized in dissimilar directions in different spaces by mathematicians over the years; for more details on this and related topics, we refer to [1–6] and references therein.

In contemporary time, fixed point theory has evolved speedily in partially ordered cone metric spaces; that is, cone metric spaces equipped with a partial ordering, for some new results in ordered metric spaces see [7]. A coming early result in this bearing was constituted by Altun and Durmaz [8] under the condition of normality for cones. Then, Altun et al. [9] generalized the results of Altun and Durmaz [8] by omitting the assumption of normality condition for cones. Afterward, several authors have studied fixed point and common fixed point problems in ordered cone metric spaces; for more details see [10–17].

In 2011, Hussain and Shah [18] presented cone b-metric spaces as a generalization of b-metric spaces and cone metric spaces; for some new results in b-metric spaces see [19]. They not only constructed some topological properties in such spaces but also ameliorated some current results about KKM mappings in the setting of a cone b-metric space. After some time, many authors have been motivated to demonstrate fixed point theorems as well as common fixed point theorems for two or more mappings on cone b-metric spaces by the incipient work of Hussain and Shah [18] (see [20–23] and the references therein).

In [8], Altun and Durmaz proved the following results under the condition of normality for cones.

Theorem 1 (see [8]). Let (𝑋,⊑) be a partially ordered set, suppose that there exists a cone metric 𝑑 in 𝑋 such that the cone metric space (𝑋,𝑑) is complete, and let 𝑃 be a normal cone with normal constant 𝐾. Let 𝑓∶𝑋→𝑋 be a continuous and nondecreasing mapping with respect to ⊑. Suppose that the following three assertions hold: (i) there exists 𝑘∈[0,1) such that 𝑑(𝑓𝑥,𝑓𝑦)⪯𝑘𝑑(𝑥,𝑦) for all 𝑥,𝑦∈𝑋 with 𝑦⊑𝑥;(ii) there exists 𝑥₀∈𝑋 such that 𝑥₀⊑𝑓x₀.Then 𝑓 has a fixed point in 𝑋.

In [9], Altun et al. generalized the above theorem and proved it without normality condition for cones.

Theorem 2 (see [9]). Let (𝑋,⊑) be a partially ordered set and suppose that there exists a cone metric 𝑑 in 𝑋 such that the cone metric space (𝑋,𝑑) is complete over a solid cone 𝑃. Let 𝑓∶𝑋→𝑋 be a continuous and nondecreasing mapping with respect to ⊑. Suppose that the following two assertions hold: (i) there exist 𝑘,𝑙,𝑟∈[0,1) with 𝑘+2𝑙+2𝑟<1 such that 𝑑𝑑𝑑𝑓𝑥,𝑓𝑦⪯𝑘𝑑𝑥,𝑦+𝑙𝑓𝑥,𝑥+𝑑𝑓𝑦,𝑦+𝑟𝑓𝑥,𝑦+𝑑𝑓𝑦,𝑥 (1) for all 𝑥,𝑦∈𝑋 with 𝑦⊑𝑥;(ii) there exists 𝑥₀∈𝑋 such that 𝑥₀⊑𝑓𝑥₀.Then 𝑓 has a fixed point in 𝑋.

Theorem 3 (see [9]). Let (𝑋,⊑) be a partially ordered set and suppose that there exists a cone metric 𝑑 in 𝑋 such that the cone metric space (𝑋,𝑑) is complete over a solid cone 𝑃. Let 𝑓∶𝑋→𝑋 be a nondecreasing mapping with respect to ⊑. Suppose that the following three assertions hold: (i) there exist 𝑘,𝑙,𝑟∈[0,1) with 𝑘+2𝑙+2𝑟<1 such that 𝑑𝑑𝑑𝑓𝑥,𝑓𝑦⪯𝑘𝑑𝑥,𝑦+𝑙𝑓𝑥,𝑥+𝑑𝑓𝑦,𝑦+𝑟𝑓𝑥,𝑦+𝑑𝑓𝑦,𝑥 (2) for all 𝑥,𝑦∈𝑋 with 𝑦⊑𝑥;(ii) there exists 𝑥₀∈𝑋 such that 𝑥₀⊑𝑓𝑥₀;(iii) if an increasing sequence {𝑥_𝑛} converges to 𝑥 in 𝑋, then 𝑥_𝑛⊑𝑥 for all 𝑛. Then 𝑓 has a fixed point in 𝑋.

In the same paper, they also presented the following two common fixed point results in ordered cone metric spaces.

Theorem 4 (see [9]). Let (𝑋,⊑) be a partially ordered set and suppose that there exists a cone metric 𝑑 in 𝑋 such that the cone metric space (𝑋,𝑑) is complete over a solid cone 𝑃. Let 𝑓,𝑔∶𝑋→𝑋 be two weakly increasing mappings with respect to ⊑. Suppose that the following three assertions hold:(i) there exist 𝑘,𝑙,𝑟∈[0,1) with 𝑘+2𝑙+2𝑟<1 such that 𝑑𝑑𝑑𝑓𝑥,𝑔𝑦⪯𝑘𝑑𝑥,𝑦+𝑙𝑥,𝑓𝑥+𝑑𝑦,𝑔𝑦+𝑟𝑦,𝑓𝑥+𝑑𝑥,𝑔𝑦 (3) for all comparative 𝑥,𝑦∈𝑋;(ii)𝑓 or 𝑔 is continuous. Then 𝑓 and 𝑔 have a common fixed point 𝑥^∗∈𝑋.

Theorem 5 (see [9]). Let (𝑋,⊑) be a partially ordered set and suppose that there exists a cone metric 𝑑 in 𝑋 such that the cone metric space (𝑋,𝑑) is complete over a solid cone 𝑃. Let 𝑓,𝑔∶𝑋→𝑋 be two weakly increasing mappings with respect to⊑. Suppose that the following three assertions hold: (i) there exist 𝑘,𝑙,𝑟∈[0,1) with 𝑘+2𝑙+2𝑟<1 such that 𝑑𝑑𝑑𝑓𝑥,𝑔𝑦⪯𝑘𝑑𝑥,𝑦+𝑙𝑥,𝑓𝑥+𝑑𝑦,𝑔𝑦+𝑟𝑦,𝑓𝑥+𝑑𝑥,𝑔𝑦 (4) for all comparative 𝑥,𝑦∈𝑋;(ii) if an increasing sequence {𝑥_𝑛} converges to 𝑥 in 𝑋, then 𝑥_𝑛⊑𝑥 for all n. Then 𝑓 and 𝑔 have a common fixed point 𝑥^∗∈𝑋.

In this paper, we prove some fixed point and common fixed point theorems on ordered cone b-metric spaces. Our results extend and generalize several well-known comparable results in the literature to ordered cone b-metric spaces. Throughout this paper, we do not impose the normality condition for the cones, but the only assumption is that the cone 𝑃 is solid, that is, int 𝑃≠∅.

The following definitions and results shall be needed in the sequel.

Let 𝐸 be a real Banach space and 𝜃 denotes the zero element in 𝐸. A cone 𝑃 is a subset of 𝐸 such that(1)𝑃 is nonempty closed set 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 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 use 𝑥≪𝑦 to indicate that 𝑦−𝑥∈int 𝑃, where int 𝑃 denotes the interior of 𝑃. A cone 𝑃 is called normal if there exists the number 𝐾 such that 𝜃⪯𝑥⪯𝑦⟹‖𝑥‖≤𝐾𝑦, (5) for all 𝑥,𝑦∈𝐸. The least positive number 𝐾 satisfying the above condition is called the normal constant of 𝑃.

Definition 6 (see [18]). Let 𝑋 be a nonempty set and 𝐸 a real Banach space equipped with the partial ordering ⪯ with respect to the cone 𝑃. A vector-valued function 𝑑∶𝑋×𝑋→𝐸 is said to be a cone b-metric function on 𝑋 with the constant 𝑠≥1 if the following conditions are satisfied:(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)⪯𝑠(𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧)) for all 𝑥,𝑦,𝑧∈𝑋. Then pair (𝑋,𝑑) is called a cone b-metric space (or a cone metric type space); we shall use the first mentioned term.

Observe that if 𝑠=1, then the ordinary triangle inequality in a cone metric space is satisfied; however, it does not hold true when 𝑠>1. Thus the class of cone b-metric spaces is effectively larger than that of the ordinary cone metric spaces. That is, every cone metric space is a cone b-metric space, but the converse need not be true. The following examples show the above remarks.

Example 7. Let 𝑋={−1,0,1},𝐸=ℝ²,and𝑃={(𝑥,𝑦)∶𝑥≥0,𝑦≥0}. Define 𝑑∶𝑋×𝑋→𝑃 by 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋,𝑑(𝑥,𝑥)=𝜃,𝑥∈𝑋, and 𝑑(−1,0)=(3,3),𝑑(−1,1)=𝑑(0,1)=(1,1). Then (𝑋,𝑑) is a complete cone b-metric space but the triangle inequality is not satisfied. Indeed, we have that 𝑑(−1,1)+𝑑(1,0)=(1,1)+(1,1)=(2,2)≺(3,3)=𝑑(−1,0). It is not hard to verify that 𝑠=3/2.

Example 8. Let 𝑋=ℝ, 𝐸=ℝ², and𝑃={(𝑥,𝑦)∈𝐸∶𝑥≥0,𝑦≥0}. Define 𝑑∶𝑋×𝑋→𝐸 by 𝑑(𝑥,𝑦)=(|𝑥−𝑦|²,|𝑥−𝑦|²). Then, it is easy to see that (𝑋,𝑑) is a cone b-metric space with the coefficient 𝑠=2. But it is not a cone metric spaces since the triangle inequality is not satisfied.

Definition 9 (see [18]). Let (𝑋,𝑑) be a cone b-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}. One denotes 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.

The following lemma is useful in our work.

Lemma 10 (see [24]). (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. (3) If 𝑎⪯𝑏 and 𝑏≪𝑐, then 𝑎≪𝑐. (4) If 𝜃⪯𝑢≪𝑐 for each 𝜃≪𝑐, then 𝑢=𝜃.

2. Fixed Point Results

In this section, we prove some fixed point theorems on ordered cone b-metric space. We begin with a simple but a useful lemma.

Lemma 11. Let {𝑥_𝑛} be a sequence in a cone b-metric space (𝑋,𝑑) with the coefficient 𝑠≥1 relative to a solid cone 𝑃 such that 𝑑𝑥_𝑛,𝑥_𝑛+1⪯ℎ𝑑𝑥_𝑛−1,𝑥_𝑛, (6) where ℎ∈[0,1/𝑠) and 𝑛=1,2,…. Then {𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑑).

Proof. Let 𝑚>𝑛≥1. It follows that 𝑑𝑥_𝑛,𝑥_𝑚⪯𝑠𝑑𝑥_𝑛,𝑥_𝑛+1+𝑠²𝑑𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑠^𝑚−𝑛𝑑𝑥_𝑚−1,𝑥_𝑚. (7) Now, (6) and 𝑠ℎ<1 imply that 𝑑𝑥_𝑛,𝑥_𝑚⪯𝑠𝑑𝑥_𝑛,𝑥_𝑛+1+𝑠²𝑑𝑥_𝑛+1,𝑥_𝑛+2+⋯+𝑠^𝑚−𝑛𝑑𝑥_𝑚−1,𝑥_𝑚⪯𝑠ℎ^𝑛𝑑𝑥₀,𝑥₁+𝑠²ℎ^𝑛+1𝑑𝑥₀,𝑥₁+⋯+𝑠^𝑚−𝑛ℎ^𝑚−1𝑑𝑥₀,𝑥₁=𝑠ℎ^𝑛+𝑠²ℎ^𝑛+1+⋯+𝑠^𝑚−𝑛ℎ^𝑚−1𝑑𝑥₀,𝑥₁=𝑠ℎ^𝑛1+𝑠ℎ+(𝑠ℎ)²+⋯+(𝑠ℎ)^{𝑚−𝑛−1}𝑑𝑥₀,𝑥₁⪯𝑠ℎ^𝑛𝑑𝑥1−𝑠ℎ⁰,𝑥₁⟶𝜃as𝑛⟶∞. (8) According to Lemma 10(2), and for any 𝑐∈𝐸 with 𝑐≫𝜃, there exists 𝑁₀∈ℕ such that for any 𝑛>𝑁₀, (𝑠ℎ^𝑛/(1−𝑠ℎ))𝑑(𝑥₀,𝑥₁)≪𝑐. Furthermore, from (8) and for any 𝑚>𝑛>𝑁₀, Lemma 10(3) shows that 𝑑𝑥_𝑛,𝑥_𝑚≪𝑐. (9) Hence, by Definition 9(2){𝑥_𝑛} is a Cauchy sequence in 𝑋.

Theorem 12. Let (𝑋,⊑) be a partially ordered set and suppose that there exists a cone b-metric 𝑑 in 𝑋 such that the cone b-metric space (𝑋,𝑑) is complete with the coefficient 𝑠≥1 relative to a solid cone 𝑃. Let 𝑓∶𝑋→𝑋 be a continuous and nondecreasing mapping with respect to ⊑. Suppose that the following three assertions hold: (i) there exist 𝑎_𝑖,𝑖=1,…,5, such that 2𝑠𝑎₁+(𝑠+1)(𝑎₂+𝑎₃)+(𝑠²+𝑠)(𝑎₄+𝑎₅)<2 with ∑⁵_𝑖=1𝑎_𝑖<1, 𝑑𝑓𝑥,𝑓𝑦⪯𝑎₁𝑑𝑥,𝑦+𝑎₂𝑑𝑓𝑥,𝑥+𝑎₃𝑑𝑓𝑦,𝑦+𝑎₄𝑑𝑓𝑥,𝑦+𝑎₅𝑑𝑓𝑦,𝑥 (10) for all 𝑥,𝑦∈𝑋 with 𝑦⊑𝑥; (ii) there exists 𝑥₀∈𝑋 such that 𝑥₀⊑𝑓𝑥₀. Then 𝑓 has a fixed point 𝑥^∗∈𝑋.

Proof. If 𝑥₀=𝑓𝑥₀, then the proof is finished. Suppose that 𝑥₀≠𝑓𝑥₀. Since 𝑥₀⊑𝑓𝑥₀ and 𝑓 is nondecreasing with respect to⊑, we obtain by induction that 𝑥₀⊑𝑓𝑥₀=𝑥₁⊑𝑓¹𝑥₀=𝑥₂⊑⋯⊑𝑓^𝑛−1𝑥₀=𝑥_𝑛⊑𝑓^𝑛𝑥₀=𝑥_𝑛+1⊑⋯. Then we have, 𝑑𝑥_𝑛+1,𝑥_𝑛=𝑑𝑓^𝑛𝑥₀,𝑓^𝑛−1𝑥₀=𝑑𝑓𝑓^𝑛−1𝑥₀,𝑓𝑓^𝑛−2𝑥₀⪯𝑎₁𝑑𝑓^𝑛−1𝑥₀,𝑓^𝑛−2𝑥₀+𝑎₂𝑑𝑓^𝑛−1𝑥₀,𝑓^𝑛−2𝑥₀+𝑎₃𝑑𝑓^𝑛𝑥₀,𝑓^𝑛−1𝑥₀+𝑎₄𝑑𝑓^𝑛𝑥₀,𝑓^𝑛−2𝑥₀+𝑎₅𝑑𝑓^𝑛−1𝑥₀,𝑓^𝑛−1𝑥₀=𝑎₁𝑑𝑥_𝑛,𝑥_𝑛−1+𝑎₂𝑑𝑥_𝑛+1,𝑥_𝑛+𝑎₃𝑑𝑥_𝑛,𝑥_𝑛−1+𝑎₄𝑑𝑥_𝑛+1,𝑥_𝑛−1+𝑎₅𝑑𝑥_𝑛,𝑥_𝑛⪯𝑎₁𝑑𝑥_𝑛,𝑥_𝑛−1+𝑎₂𝑑𝑥_𝑛+1,𝑥_𝑛+𝑎₃𝑑𝑥_𝑛,𝑥_𝑛−1+𝑠𝑎₄𝑑𝑥_𝑛+1,𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑛−1. (11) Then, one can assert that 𝑑𝑥_𝑛+1,𝑥_𝑛⪯𝑎₁+𝑎₃+𝑠𝑎₄𝑑𝑥_𝑛,𝑥_𝑛−1+𝑎₂+𝑠𝑎₄𝑑𝑥_𝑛+1,𝑥_𝑛. (12) On the other hand, we have 𝑑𝑥_𝑛,𝑥_𝑛+1=𝑑𝑓^𝑛−1𝑥₀,𝑓^𝑛𝑥₀=𝑑𝑓𝑓^𝑛−2𝑥₀,𝑓𝑓^𝑛−1𝑥₀⪯𝑎₁𝑑𝑓^𝑛−2𝑥₀,𝑓^𝑛−1𝑥₀+𝑎₂𝑑𝑓^𝑛−2𝑥₀,𝑓^𝑛−1𝑥₀+𝑎₃𝑑𝑓^𝑛−1𝑥₀,𝑓^𝑛𝑥₀+𝑎₄𝑑𝑓^𝑛−1𝑥₀,𝑓^𝑛−1𝑥₀+𝑎₅𝑑𝑓^𝑛𝑥₀,𝑓^𝑛−2𝑥₀=𝑎₁𝑑𝑥_𝑛,𝑥_𝑛−1+𝑎₂𝑑𝑥_𝑛,𝑥_𝑛−1+𝑎₃𝑑𝑥_𝑛+1,𝑥_𝑛+𝑎₄𝑑𝑥_𝑛,𝑥_𝑛+𝑎₅𝑑𝑥_𝑛+1,𝑥_𝑛−1⪯𝑎₁𝑑𝑥_𝑛,𝑥_𝑛−1+𝑎₂𝑑𝑥_𝑛,𝑥_𝑛−1+𝑎₃𝑑𝑥_𝑛+1,𝑥_𝑛+𝑠𝑎₅𝑑𝑥_𝑛+1,𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑛−1. (13) Then, one can assert that 𝑑𝑥_𝑛+1,𝑥_𝑛⪯𝑎₁+𝑎₂+𝑠𝑎₅𝑑𝑥_𝑛,𝑥_𝑛−1+𝑎₃+𝑠𝑎₅𝑑𝑥_𝑛+1,𝑥_𝑛. (14) Adding (12) and (14), we get 𝑑𝑥_𝑛+1,𝑥_𝑛⪯2𝑎₁+𝑎₂+𝑎₃+𝑠𝑎₄+𝑠𝑎₅2−𝑎₂+𝑎₃+𝑠𝑎₄+𝑠𝑎₅𝑑𝑥_𝑛,𝑥_𝑛−1=𝜆𝑑𝑥_𝑛,𝑥_𝑛−1, (15) where 𝜆=(2𝑎₁+𝑎₂+𝑎₃+𝑠𝑎₄+𝑠𝑎₅)/(2−(𝑎₂+𝑎₃+𝑠𝑎₄+𝑠𝑎₅))<1/𝑠. According to Lemma 11, we have {𝑥_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑥^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗. Since 𝑓 is continuous, then 𝑥^∗=lim𝑥_𝑛+1=lim𝑓^𝑛𝑥₀=lim𝑓(𝑓^𝑛−1𝑥₀)=𝑓(lim𝑓^𝑛−1𝑥₀)=𝑓(lim𝑥_𝑛)=𝑓(𝑥^∗). Therefore, 𝑥^∗ is a fixed point of 𝑓.

If we use condition (iii) instead of the continuity of 𝑓 in Theorem 12, we have the following result.

Theorem 13. Let (𝑋,⊑) be a partially ordered set and suppose that there exists a cone b-metric 𝑑 in 𝑋 such that the cone b-metric space (𝑋,𝑑) is complete with the coefficient 𝑠≥1 relative to a solid cone 𝑃. Let 𝑓∶𝑋→𝑋 be a nondecreasing mapping with respect to⊑. Suppose that the following three assertions hold: (i) there exist 𝑎_𝑖,𝑖=1,…,5, such that 2𝑠𝑎₁+(𝑠+1)(𝑎₂+𝑎₃)+(𝑠²+𝑠)(𝑎₄+𝑎₅)<2 with ∑⁵_𝑖=1𝑎_𝑖<1, 𝑑𝑓𝑥,𝑓𝑦⪯𝑎₁𝑑𝑥,𝑦+𝑎₂𝑑𝑓𝑥,𝑥+𝑎₃𝑑𝑓𝑦,𝑦+𝑎₄𝑑𝑓𝑥,𝑦+𝑎₅𝑑𝑓𝑦,𝑥 (16) for all 𝑥,𝑦∈𝑋 with 𝑦⊑𝑥;(ii) there exists 𝑥₀∈𝑋 such that 𝑥₀⊑𝑓𝑥₀; (iii) if an increasing sequence {𝑥_𝑛} converges to 𝑥 in 𝑋, then 𝑥_𝑛⊑𝑥 for all 𝑛.Then 𝑓 has a fixed point 𝑥^∗∈𝑋.

Proof. As in the Theorem 12, we can construct an increasing sequence {𝑥_𝑛} and prove that there exists 𝑥^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗. Now, condition (iii) implies 𝑥_𝑛⊑𝑥^∗ for all 𝑛. Therefore, we can use condition (i) and so 𝑑𝑓𝑥_𝑛,𝑓𝑥^∗⪯𝑎₁𝑑𝑥_𝑛,𝑥^∗+𝑎₂𝑑𝑓𝑥_𝑛,𝑥_𝑛+𝑎₃𝑑𝑓𝑥^∗,𝑥^∗+𝑎₄𝑑𝑓𝑥_𝑛,𝑥^∗+𝑎₅𝑑𝑓𝑥^∗,𝑥_𝑛. (17) Taking 𝑛→∞, we have 𝑑(𝑥^∗,𝑓𝑥^∗)⪯(𝑎₃+𝑎₅)𝑑(𝑥^∗,𝑓𝑥^∗)𝑑(𝑥^∗,𝑓𝑥^∗). Since (𝑎₃+𝑎₅)<1, Lemma 10(1) shows that 𝑑(𝑥^∗,𝑓𝑥^∗)=𝜃; that is, 𝑥^∗=𝑓𝑥^∗. Therefore 𝑥^∗ is a fixed point of 𝑓.

3. Common Fixed Point Results

Now, we give two common fixed point theorems on ordered cone b-metric spaces. We need the following definition.

Definition 14 (see [9]). Let (𝑋,⊑) be a partially ordered set. Two mappings 𝑓,𝑔∶𝑋→𝑋 are said to be weakly increasing if 𝑓𝑥⊑𝑔𝑓𝑥 and 𝑔𝑥⊑𝑓𝑔𝑥 hold for all 𝑥∈𝑋.

Theorem 15. Let (𝑋,⊑) be a partially ordered set and suppose that there exists a cone b-metric 𝑑 in 𝑋 such that the cone b-metric space (𝑋,𝑑) is complete with the coefficient 𝑠≥1 relative to a solid cone 𝑃. Let 𝑓,𝑔∶𝑋→𝑋 be two weakly increasing mappings with respect to⊑. Suppose that the following three assertions hold:(i) there exist 𝑎_𝑖,𝑖=1,…,5, such that 2𝑠𝑎₁+(𝑠+1)(𝑎₂+𝑎₃)+(𝑠²+𝑠)(𝑎₄+𝑎₅)<2 with ∑⁵_𝑖=1𝑎_𝑖<1, 𝑑𝑓𝑥,𝑔𝑦⪯𝑎₁𝑑𝑥,𝑦+𝑎₂𝑑𝑥,𝑓𝑥+𝑎₃𝑑𝑦,𝑔𝑦+𝑎₄𝑑𝑦,𝑓𝑥+𝑎₅𝑑𝑥,𝑔𝑦 (18) for all comparative 𝑥,𝑦∈𝑋;(ii)𝑓 or 𝑔 is continuous. Then 𝑓 and 𝑔 have a common fixed point 𝑥^∗∈𝑋.

Proof. Let 𝑥₀ be an arbitrary point of 𝑋 and define a sequence {𝑥_𝑛} in 𝑋 as follows: 𝑥_2𝑛+1=𝑓𝑥_2𝑛 and 𝑥_2𝑛+2=𝑔𝑥_2𝑛+1 for all 𝑛>0. Note that, since 𝑓 and 𝑔 are weakly increasing, we have 𝑥₁=𝑓𝑥₀⊑𝑔𝑓𝑥₀=𝑔𝑥₁=𝑥₂ and 𝑥₂=𝑔𝑥₁⊑𝑓𝑔𝑥₁=𝑓𝑥₂=𝑥₃, and continuing this process we have 𝑥₁⊑𝑥₂⊑⋯⊑𝑥_𝑛⊑𝑥_𝑛+1⊑⋯. That is, the sequence {𝑥_𝑛} is nondecreasing. Now, since 𝑥_2𝑛 and 𝑥_2𝑛+1 are comparative, we can use the inequality (18), and then we have 𝑑𝑥_2𝑛+1,𝑥_2𝑛+2=𝑑𝑓𝑥_2𝑛,𝑔𝑥_2𝑛+1⪯𝑎₁𝑑𝑥_2𝑛,𝑥_2𝑛+1+𝑎₂𝑑𝑥_2𝑛,𝑓𝑥_2𝑛+𝑎₃𝑑𝑥_2𝑛+1,𝑔𝑥_2𝑛+1+𝑎₄𝑑𝑥_2𝑛+1,𝑓𝑥_2𝑛+𝑎₅𝑑𝑥_2𝑛,𝑔𝑥_2𝑛+1⪯𝑎₁𝑑𝑥_2𝑛,𝑥_2𝑛+1+𝑎₂𝑑𝑥_2𝑛,𝑥_2𝑛+1+𝑎₃𝑑𝑥_2𝑛+1,𝑥_2𝑛+2+𝑎₄𝑑𝑥_2𝑛+1,𝑥_2𝑛+1+𝑎₅𝑑𝑥_2𝑛,𝑥_2𝑛+2⪯𝑎₁+𝑎₂𝑑𝑥_2𝑛,𝑥_2𝑛+1+𝑎₃𝑑𝑥_2𝑛+1,𝑥_2𝑛+2+𝑠𝑎₅𝑑𝑥_2𝑛,𝑥_2𝑛+1+𝑑𝑥_2𝑛+1,𝑥_2𝑛+2=𝑎₁+𝑎₂+𝑠𝑎₅𝑑𝑥_2𝑛,𝑥_2𝑛+1+𝑎₃+𝑠𝑎₅𝑑𝑥_2𝑛+1,𝑥_2𝑛+2. (19) Hence, 1−𝑎₃+𝑠𝑎₅𝑑𝑥_2𝑛+1,𝑥_2𝑛+2⪯𝑎₁+𝑎₂+𝑠𝑎₅𝑑𝑥_2𝑛,𝑥_2𝑛+1. (20)
On the other hand and by symmetry we have 𝑑𝑥_2𝑛+2,𝑥_2𝑛+1=𝑑𝑔𝑥_2𝑛+1,𝑓𝑥_2𝑛⪯𝑎₁𝑑𝑥_2𝑛+1,𝑥_2𝑛+𝑎₂𝑑𝑥_2𝑛+1,𝑔𝑥_2𝑛+1+𝑎₃𝑑𝑥_2𝑛,𝑓𝑥_2𝑛+𝑎₄𝑑𝑥_2𝑛,𝑔𝑥_2𝑛+1+𝑎₅𝑑𝑥_2𝑛+1,𝑓𝑥_2𝑛⪯𝑎₁𝑑𝑥_2𝑛+1,𝑥_2𝑛+𝑎₂𝑑𝑥_2𝑛+1,𝑥_2𝑛+2+𝑎₃𝑑𝑥_2𝑛,𝑥_2𝑛+1+𝑎₄𝑑𝑥_2𝑛,𝑥_2𝑛+2+𝑎₅𝑑𝑥_2𝑛+1,𝑥_2𝑛+1⪯𝑎₁+𝑎₃𝑑𝑥_2𝑛,𝑥_2𝑛+1+𝑎₂𝑑𝑥_2𝑛+1,𝑥_2𝑛+2+𝑠𝑎₄𝑑𝑥_2𝑛,𝑥_2𝑛+1+𝑑𝑥_2𝑛+1,𝑥_2𝑛+2=𝑎₁+𝑎₃+𝑠𝑎₄𝑑𝑥_2𝑛,𝑥_2𝑛+1+𝑎₂+𝑠𝑎₄𝑑𝑥_2𝑛+1,𝑥_2𝑛+2. (21) Hence, 1−𝑎₂+𝑠𝑎₄𝑑𝑥_2𝑛+2,𝑥_2𝑛+1⪯𝑎₁+𝑎₃+𝑠𝑎₄𝑑𝑥_2𝑛,𝑥_2𝑛+1. (22) Adding inequalities (20) and (22), we get 𝑑𝑥_2𝑛+1,𝑥_2𝑛+2⪯2𝑎₁+𝑎₂+𝑎₃+𝑠𝑎₄+𝑠𝑎₅2−𝑎₂+𝑎₃+𝑠𝑎₄+𝑠𝑎₅𝑑𝑥_2𝑛,𝑥_2𝑛+1=𝜆𝑑𝑥_2𝑛,𝑥_2𝑛+1, (23) where 𝜆=(2𝑎₁+𝑎₂+𝑎₃+𝑠𝑎₄+𝑠𝑎₅)/(2−(𝑎₂+𝑎₃+𝑠𝑎₄+𝑠𝑎₅))<1/𝑠. Similarly, it can be shown that 𝑑𝑥_2𝑛+3,𝑥_2𝑛+2⪯𝜆𝑑𝑥_2𝑛+,𝑥_2𝑛+2. (24) Therefore, 𝑑𝑥_𝑛+1,𝑥_𝑛+2⪯𝜆𝑑𝑥_𝑛,𝑥_𝑛+1. (25) According to Lemma 11, we have {𝑥_𝑛} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑥^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗. Suppose that 𝑓 is continuous. Then, 𝑥^∗=lim𝑥_𝑛+1=lim𝑓^𝑛𝑥₀=lim𝑓(𝑓^𝑛−1𝑥₀)=𝑓(lim𝑓^𝑛−1𝑥₀)=𝑓(lim𝑥_𝑛)=𝑓(𝑥^∗). Therefore, 𝑥^∗ is a fixed point of 𝑓. Now, we need to show that 𝑥^∗ is a fixed point of 𝑔. Since 𝑥^∗⊑𝑥^∗, we can use the inequality (18) for 𝑥=𝑦=𝑥^∗. Then we have 𝑑𝑓𝑥^∗,𝑔𝑥^∗⪯𝑎₁𝑑𝑥^∗,𝑥^∗+𝑎₂𝑑𝑥^∗,𝑓𝑥^∗+𝑎₃𝑑𝑥^∗,𝑔𝑥^∗+𝑎₄𝑑𝑥^∗,𝑓𝑥^∗+𝑎₅𝑑𝑥^∗,𝑔𝑥^∗=𝑎₁𝑑𝑥^∗,𝑥^∗+𝑎₂𝑑𝑥^∗,𝑥^∗+𝑎₃𝑑𝑥^∗,𝑔𝑥^∗+𝑎₄𝑑𝑥^∗,𝑥^∗+𝑎₅𝑑𝑥^∗,𝑔𝑥^∗=𝑎₃𝑑𝑥^∗,𝑔𝑥^∗+𝑎₅𝑑𝑥^∗,𝑔𝑥^∗=𝑎₃+𝑎₅𝑑𝑥^∗,𝑔𝑥^∗. (26) Hence, 𝑑𝑥^∗,𝑔𝑥^∗⪯𝑎₃+𝑎₅𝑑𝑥^∗,𝑔𝑥^∗. (27) Since (𝑎₃+𝑎₅)<1, Lemma 10(1) shows that 𝑑(𝑥^∗,𝑔𝑥^∗)=𝜃; that is, 𝑥^∗=𝑔𝑥^∗. Therefore 𝑥^∗ is a fixed point of 𝑔. Therefore, 𝑓 and 𝑔 have a common fixed point. The proof is similar when 𝑔 is a continuous mapping.

Theorem 16. Let (𝑋,⊑) be a partially ordered set and suppose that there exists a cone b-metric 𝑑 in 𝑋 such that the cone b-metric space (𝑋,𝑑) is complete with the coefficient 𝑠≥1 relative to a solid cone 𝑃. Let 𝑓,𝑔∶𝑋→𝑋 be two weakly increasing mappings with respect to⊑. Suppose that the following three assertions hold: (i) there exist 𝑎_𝑖,𝑖=1,…,5 such that 2𝑠𝑎₁+(𝑠+1)(𝑎₂+𝑎₃)+(𝑠²+𝑠)(𝑎₄+𝑎₅)<2 with ∑⁵_𝑖=1𝑎_𝑖<1, 𝑑𝑓𝑥,𝑔𝑦⪯𝑎₁𝑑𝑥,𝑦+𝑎₂𝑑𝑥,𝑓𝑥+𝑎₃𝑑𝑦,𝑔𝑦+𝑎₄𝑑𝑦,𝑓𝑥+𝑎₅𝑑𝑥,𝑔𝑦, (28) for all comparative 𝑥,𝑦∈𝑋; (ii) if an increasing sequence {𝑥_𝑛} converges to 𝑥 in 𝑋, then 𝑥_𝑛⊑𝑥 for all 𝑛. Then 𝑓 and 𝑔 have a common fixed point 𝑥^∗∈𝑋.

Proof. As in Theorem 15, we can construct an increasing sequence {𝑥_𝑛} and prove that there exists 𝑥^∗∈𝑋 such that 𝑥_𝑛→𝑥^∗, also; by the construction of 𝑥_𝑛, 𝑔𝑥_𝑛→𝑥^∗. Now, condition (iii) implies 𝑥_𝑛⊑𝑥^∗ for all 𝑛. Putting 𝑥=𝑥^∗ and 𝑦=𝑥_𝑛 in (28), we get 𝑑𝑓𝑥^∗,𝑔𝑥_𝑛⪯𝑎₁𝑑𝑥^∗,𝑥_𝑛+𝑎₂𝑑𝑥^∗,𝑓𝑥^∗+𝑎₃𝑑𝑥_𝑛,𝑔𝑥_𝑛+𝑎₄𝑑𝑥_𝑛,𝑓𝑥^∗+𝑎₅𝑑𝑥^∗,𝑔𝑥_𝑛=𝑎₁𝑑𝑥_𝑛,𝑥^∗+𝑎₂𝑑𝑓𝑥^∗,𝑥^∗+𝑎₃𝑑𝑥_𝑛,𝑔𝑥_𝑛+𝑎₄𝑑𝑥_𝑛,𝑓𝑥^∗+𝑎₅𝑑𝑔𝑥_𝑛,𝑥^∗⪯𝑎₁𝑑𝑥_𝑛,𝑥^∗+𝑎₂𝑑𝑓𝑥^∗,𝑔𝑥_𝑛+𝑑𝑔𝑥_𝑛,𝑥^∗+𝑎₃𝑑𝑥_𝑛,𝑥^∗+𝑑𝑥^∗,𝑔𝑥_𝑛+𝑎₄𝑑𝑥_𝑛,𝑥^∗+𝑑𝑥^∗,𝑔𝑥_𝑛+𝑑𝑔𝑥_𝑛,𝑓𝑥^∗+𝑎₅𝑑𝑔𝑥_𝑛,𝑥^∗=𝑎₁+𝑎₃+𝑎₄𝑑𝑥_𝑛,𝑥^∗+𝑎₂+𝑎₃+𝑎₄+𝑎₅𝑑𝑔𝑥_𝑛,𝑥^∗+𝑎₂+𝑎₄𝑑𝑓𝑥^∗,𝑔𝑥_𝑛. (29) Hence, 𝑑𝑓𝑥^∗,𝑔𝑥_𝑛⪯𝑎₁+𝑎₃+𝑎₄1−𝑎₂+𝑎₄𝑑𝑥_𝑛,𝑥^∗+𝑎₂+𝑎₃+𝑎₄+𝑎₅1−𝑎₂+𝑎₄𝑑𝑔𝑥_𝑛,𝑥^∗. (30) Since 𝑥_𝑛→𝑥^∗ and 𝑔𝑥_𝑛→𝑥^∗, then by Definition 9(1) and for 𝑐≫𝜃 there exists 𝑁₀∈ℕ such that for all 𝑛>𝑁₀, 𝑑(𝑥_𝑛,𝑥^∗)≪𝑐(1−(𝑎₂+𝑎₄))/2(𝑎₁+𝑎₃+𝑎₄), and 𝑑(𝑔𝑥_𝑛,𝑥^∗)≪𝑐(1−(𝑎₂+𝑎₄))/2(𝑎₂+𝑎₃+𝑎₄+𝑎₅). Then we have 𝑑𝑔𝑥_𝑛,𝑓𝑥^∗=𝑑𝑓𝑥^∗,𝑔𝑥_𝑛⪯𝑎₁+𝑎₃+𝑎₄1−𝑎₂+𝑎₄𝑑𝑥_𝑛,𝑥^∗+𝑎₂+𝑎₃+𝑎₄+𝑎₅1−𝑎₂+𝑎₄𝑑𝑔𝑥_𝑛,𝑥^∗≪𝑎₁+𝑎₃+𝑎₄1−𝑎₂+𝑎₄𝑐1−𝑎₂+𝑎₄2𝑎₁+𝑎₃+𝑎₄+𝑎₂+𝑎₃+𝑎₄+𝑎₅1−𝑎₂+𝑎₄𝑐1−𝑎₂+𝑎₄2𝑎₂+𝑎₃+𝑎₄+𝑎₅=𝑐2+𝑐2=𝑐. (31) Now again, according to Definition 9(1) it follows that 𝑔𝑥_𝑛→𝑓𝑥^∗. It follows that 𝑓𝑥^∗=𝑥^∗. In a similar way and using that 𝑥^∗⊑𝑥^∗, we can prove that 𝑔𝑥^∗=𝑥^∗. Therefore, 𝑓 and 𝑔 have a common fixed point.

Now, we present two examples to illustrate our results. In the first example (the case of a normal cone), the conditions of Theorem 12 are fulfilled, but Theorem 2 of Altun et al. [9, Theorem 12] cannot be applied. In the second example (the case of a nonnormal cone), the conditions of Theorem 12 are fulfilled, but Theorem 3 of Altun et al. [9, Theorem 13] cannot be applied.

Example 17. Let 𝑋=[0,1] endowed with the standard order and 𝐸=ℝ² and let 𝑃={(𝑥,𝑦)∶𝑥,𝑦≥0}. Define 𝑑∶𝑋×𝑋→𝐸 as in Example 8. Define 𝑓∶𝑋→𝑋 by 𝑓(𝑥)=𝑥²/6. Then 𝑓 is a continuous and nondecreasing mapping with respect to ⊑. Then we have 𝑑𝑥𝑓𝑥,𝑓𝑦=𝑑²6,𝑦²6=𝑥²6−𝑦²6²,𝑥²6−𝑦²6²=136𝑥+𝑦²𝑥−𝑦²,𝑥−𝑦²⪯436𝑥−𝑦²,𝑥−𝑦²⪯4𝑑,36𝑥,𝑦 (32) where 𝑎₁=4/36, 𝑎₂=𝑎₃=𝑎₄=𝑎₅=0. It is clear that the conditions of Theorem 12 are satisfied. Therefore, 𝑓 has a fixed point 𝑥=0.

Example 18. Let 𝑋=[0,∞) endowed with the standard order and 𝐸=𝐶¹_ℝ[0,1] with ‖𝑢‖=‖𝑢‖_∞+‖𝑢^′‖_∞,𝑢∈𝐸 and let 𝑃={𝑢∈𝐸∶𝑢(𝑡)≥0on[0,1]}. It is well known that this cone is solid, but it is not normal. Define a cone metric 𝑑∶𝑋×𝑋→𝐸 by 𝑑(𝑥,𝑦)(𝑡)=|𝑥−𝑦|²𝑒^𝑡. Then (𝑋,𝑑) is a complete cone b-metric space with the coefficient 𝑠=2. Let us define 𝑓∶𝑋→𝑋 by 𝑓(𝑥)=𝑥/2. Then 𝑓 is a continuous and nondecreasing mapping with respect to ⊑. Then we have 𝑓 is an increasing mapping; also we have 𝑑(𝑡)=1𝑓𝑥,𝑓𝑦21𝑥−2𝑦²𝑒^𝑡=14𝑥−𝑦²𝑒^𝑡⪯14𝑥−𝑦²𝑒^𝑡+15𝑥2²𝑒^𝑡⪯14𝑑(𝑡)+1𝑥,𝑦5𝑑𝑓𝑥,𝑥(𝑡), (33) where 𝑎₁=1/4, 𝑎₂=1/5, 𝑎₃=𝑎₄=𝑎₅=0. It is clear that the conditions of Theorem 12 are satisfied. Therefore, 𝑓 has a fixed point 𝑥=0.

Acknowledgments

The authors would like to acknowledge the financial support received from Ministry of Higher Education, Malaysia, under the Research Grant no. ERGS/1/2011/STG/UKM/01/13. The authors thank the referee for his/her careful reading of the paper and useful suggestions.