Common Fixed Point Theorems of the Asymptotic Sequences in Ordered Cone Metric Spaces

Abstract

We introduce the notions of the asymptotic 𝒮ℳ𝒦-sequence with respect to the stronger Meir-Keeler cone-type mapping 𝜉∶int(𝑃)∪{𝜃}→[0,1) and the asymptotic 𝒲ℳ𝒦-sequence with respect to the weaker Meir-Keeler cone-type mapping 𝜙∶int(𝑃)∪{𝜃}→int(𝑃)∪{𝜃} and prove some common fixed point theorems for these two asymptotic sequences in cone metric spaces with regular cone 𝑃. Our results generalize some recent results.

1. Introduction and Preliminaries

Let (𝑋,𝑑) be a metric space, 𝐷 a subset of 𝑋, and 𝑓∶𝐷→𝑋 a map. We say 𝑓 is contractive if there exists 𝛼∈[0,1) such that for all 𝑥,𝑦∈𝐷,𝑑(𝑓𝑥,𝑓𝑦)≤𝛼⋅𝑑(𝑥,𝑦).(1.1) The well-known Banach’s fixed point theorem asserts that if 𝐷=𝑋, 𝑓 is contractive and (𝑋,𝑑) is complete, then 𝑓 has a unique fixed point in 𝑋. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, Kannan [2] and Chatterjea [3] introduced two conditions that can replace (1.1) in Banach’s theorem.

(Kannan [2]) There exists 𝛼∈[0,1) such that for all 𝑥,𝑦∈𝑋,𝑑(𝑓𝑥,𝑓𝑦)≤𝛼2[𝑑(𝑥,𝑓𝑥)+𝑑(𝑦,𝑓𝑦)].(1.2)

(Chatterjea [3]) There exists 𝛼∈[0,1) such that for all 𝑥,𝑦∈𝑋,𝑑(𝑓𝑥,𝑓𝑦)≤𝛼2[𝑑(𝑥,𝑓𝑦)+𝑑(𝑦,𝑓𝑥)].(1.3)

After these three conditions, many papers have been written generalizing some of the conditions (1.1), (1.2), and (1.3). In 1969, Boyd and Wong [4] showed the following fixed point theorem.

Theorem 1.1 (see [4]). Let (𝑋,𝑑) be a complete metric space and 𝑓∶𝑋→𝑋 a map. Suppose there exists a function 𝜙∶ℝ⁺→ℝ⁺ satisfying 𝜙(0)=0, 𝜙(𝑡)<𝑡 for all 𝑡>0 and 𝜙 is right upper semicontinuous such that 𝑑(𝑓𝑥,𝑓𝑦)≤𝜙(𝑑(𝑥,𝑦))∀𝑥,𝑦∈𝑋.(1.4) Then, 𝑓 has a unique fixed point in 𝑋.

Later, Meir-Keeler [5], using a result of Chu and Diaz [6], extended Boyd-Wong’s result to mappings satisfying the following more general condition:∀𝜂>0∃𝛿>0suchthat𝜂≤𝑑(𝑥,𝑦)<𝜂+𝛿⟹𝑑(𝑓𝑥,𝑓𝑦)<𝜂,(1.5) and Meir-Keeler proved the following very interesting fixed point theorem which is a generalization of the Banach contraction principle.

Theorem 1.2 (Meir-Keeler [5]). Let (𝑋,𝑑) be a complete metric space and let 𝑓 be a Meir-Keeler contraction, that is, for every 𝜂>0, there exists 𝛿>0 such that 𝑑(𝑥,𝑦)<𝜂+𝛿 implies 𝑑(𝑓𝑥,𝑓𝑦)<𝜂 for all 𝑥,𝑦∈𝑋. Then, 𝑓 has a unique fixed point.

Subsequently, some authors worked on this notion of Meir-Keeler contraction (e.g., [7–10]).

Huang and Zhang [11] introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space, and they showed some fixed point theorems of contractive type mappings on cone metric spaces. The category of cone metric spaces is larger than metric spaces. Subsequently, many authors like Abbas and Jungck [12] have generalized the results of Huang and Zhang [11] and studied the existence of common fixed points of a pair of self-mappings satisfying a contractive type condition in the framework of normal cone metric spaces. However, authors like Rezapour and Hamlbarani [13] studied the existence of common fixed points of a pair of self and nonself mappings satisfying a contractive type condition in the situation in which the cone does not need to be normal. Many authors studied this subject, and many results on fixed point theory are proved (see, e.g., [13–27]).

Throughout this paper, by ℝ we denote the set of all real numbers, while ℕ is the set of all natural numbers, and we initiate our discussion by introducing some preliminaries and notations.

Definition 1.3 (see [11]). Let 𝐸 be a real Banach space and 𝑃 a nonempty subset of 𝐸. 𝑃≠{𝜃}, where 𝜃 denotes the zero element of 𝐸, is called a cone if and only if(i)𝑃 is closed,(ii)𝑎,𝑏∈ℝ, 𝑎,𝑏≥0, 𝑥,𝑦∈𝑃⇒𝑎𝑥+𝑏𝑦∈𝑃,(iii)𝑥∈𝑃 and −𝑥∈𝑃⇒𝑥=𝜃.

For given a cone 𝑃⊂𝐸, we can define a partial ordering with respect to 𝑃 by 𝑥≼𝑦 or 𝑥≽𝑦 if and only if 𝑦−𝑥∈𝑃 for all 𝑥,𝑦∈𝐸. The real Banach space 𝐸 equipped with the partial ordered induced by 𝑃 is denoted by (𝐸,≼). We shall write 𝑥≺𝑦 to indicate that 𝑥≼𝑦 but 𝑥≠𝑦, while 𝑥≼𝑦 will stand for 𝑦−𝑥∈int(𝑃), where int(𝑃) denotes the interior of 𝑃.

Proposition 1.4 (see [28]). Suppose 𝑃 is a cone in a real Bancah space 𝐸. Then,(i)If 𝑒≼𝑓 and 𝑓≪𝑔, then 𝑒≪𝑔.(ii)If 𝑒≪𝑓 and 𝑓≼𝑔, then 𝑒≪𝑔.(iii)If 𝑒≪𝑓 and 𝑓≪𝑔, then 𝑒≪𝑔.(iv)If 𝑎∈𝑃 and 𝑎≼𝑒 for each 𝑒∈int(𝑃), then 𝑎=𝜃.

Proposition 1.5 (see [29]). Suppose 𝑒∈int(𝑃), 𝜃≼𝑎_𝑛, and 𝑎_𝑛→𝜃. Then, there exists 𝑛₀∈ℕ such that 𝑎_𝑛≪𝑒 for all 𝑛≥𝑛₀.

The cone 𝑃 is called normal if there exists a real number 𝐾>0 such that for all 𝑥,𝑦∈𝐸,𝜃≼𝑥≼𝑦⟹‖𝑥‖≤𝐾‖𝑦‖.(1.6)

The least positive number 𝐾 satisfying above is called the normal constant of 𝑃.

The cone 𝑃 is called regular if every increasing sequence which is bounded from above is convergent, that is, if {𝑥_𝑛} is a sequence such that𝑥₁≼𝑥₂≼⋯≼𝑥_𝑛≼⋯≼𝑦,(1.7) for some 𝑦∈𝐸, then there is 𝑥∈𝐸 such that ‖𝑥_𝑛−𝑥‖→0 as 𝑛→∞. Equivalently, the cone 𝑃 is regular if and only if every decreasing sequence which is bounded from below is convergent. It is well known that a regular cone is a normal cone.

Definition 1.6 (see [11]). Let 𝑋 be a nonempty set, 𝐸 a real Banach space, and 𝑃 a cone in 𝐸. Suppose the mapping 𝑑∶𝑋×𝑋→(𝐸,≼) satisfies(i)𝜃≼𝑑(𝑥,𝑦), for all 𝑥,𝑦∈𝑋,(ii)𝑑(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦,(iii)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥), for all 𝑥,𝑦∈𝑋,(iv)𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧)≽𝑑(𝑥,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 1.7 (see [11]). Let (𝑋,𝑑) be a cone metric space, and let {𝑥_𝑛} be a sequence in 𝑋 and 𝑥∈𝑋. If for every 𝑐∈𝐸 with 𝜃≪𝑐 there is 𝑛₀∈ℕ such that 𝑑𝑥_𝑛,𝑥≪𝑐,∀𝑛>𝑛₀,(1.8) then {𝑥_𝑛} is said to be convergent and {𝑥_𝑛} converges to 𝑥.

Definition 1.8 (see [11]). Let (𝑋,𝑑) be a cone metric space, and let {𝑥_𝑛} be a sequence in 𝑋. We say that {𝑥_𝑛} is a Cauchy sequence if for any 𝑐∈𝐸 with 𝜃≪𝑐, there is 𝑛₀∈ℕ such that 𝑑𝑥_𝑛,𝑥_𝑚≪𝑐,∀𝑛,𝑚>𝑛₀.(1.9)

Definition 1.9 (see [11]). Let (𝑋,𝑑) be a cone metric space. If every Cauchy sequence is convergent in 𝑋, then 𝑋 is called a complete cone metric space.

Remark 1.10 (see [11]). If 𝑃 is a normal cone, then {𝑥_𝑛} converges to 𝑥 if and only if 𝑑(𝑥_𝑛,𝑥)→𝜃 as 𝑛→∞. Further, in the case {𝑥_𝑛} is a Cauchy sequence if and only if 𝑑(𝑥_𝑛,𝑥_𝑚)→𝜃 as 𝑚,𝑛→∞.

In this paper, we introduce the notions of the asymptotic 𝒮ℳ𝒦-sequence with respect to the stronger Meir-Keeler cone-type mapping 𝜉∶int(𝑃)∪{𝜃}→[0,1) and the asymptotic 𝒲ℳ𝒦-sequence with respect to the weaker Meir-Keeler cone-type mapping 𝜙∶int(𝑃)∪{𝜃}→int(𝑃)∪{𝜃} and prove some common fixed point theorems for these two asymptotic sequences in cone metric spaces with regular cone 𝑃.

2. Common Fixed Point Theorems for the Asymptotic 𝒮ℳ𝒦-Sequences

In 1973, Geraghty [30] introduced the following generalization of Banach’s contraction principle.

Theorem 2.1 (see [30]). Let (𝑋,𝑑) be a complete metric space, and let 𝑆 denote the class of the functions 𝛽∶[0,∞)→[0,1) which satisfy the condition 𝛽𝑡_𝑛⟶1⟹𝑡_𝑛⟶0.(2.1) Let 𝑓∶𝑋→𝑋 be a mapping satisfying 𝑑(𝑓𝑥,𝑓𝑦)≤𝛽(𝑑(𝑥,𝑦))⋅𝑑(𝑥,𝑦),for𝑥,𝑦∈𝑋,(2.2) where 𝛽∈𝑆. Then, 𝑓 has a unique fixed point 𝑧∈𝑋.

In this section, we first introduce the notions of the stronger Meir-Keeler cone-type mapping 𝜉∶int(𝑃)∪{𝜃}→[0,1) and the asymptotic 𝒮ℳ𝒦-sequence with respect to this stronger Meir-Keeler cone-type mapping 𝜉, and we next prove some common fixed point theorems for the asymptotic 𝒮ℳ𝒦-sequence in cone metric spaces.

Definition 2.2. Let (𝑋,𝑑) be a cone metric space with cone 𝑃, and let 𝜉∶int(𝑃)∪{𝜃}⟶[0,1).(2.3) Then, the function 𝜉 is called a stronger Meir-Keeler cone-type mapping, if for each 𝜂∈int(𝑃) with 𝜂≫𝜃 there exists 𝛿≫𝜃 such that for 𝑥,𝑦∈𝑋 with 𝜂≼𝑑(𝑥,𝑦)≪𝛿+𝜂 there exists 𝛾_𝜂∈[0,1) such that 𝜉(𝑑(𝑥,𝑦))<𝛾_𝜂.

Example 2.3. Let 𝐸=ℝ, 𝑃={𝑥∈𝐸∶𝑥≽𝜃} a normal cone, 𝑋=[0,∞), and let 𝑑∶𝑋×𝑋→𝐸 be the Euclidean metric. Define 𝜉∶int(𝑃)∪{𝜃}→[0,1) by 𝜉(𝑑(𝑥,𝑦))=𝛾 where 𝛾∈[0,1), 𝑥,𝑦∈𝑋, then 𝜉 is a stronger Meir-Keeler cone-type mapping.

Example 2.4. Let 𝐸=ℝ, 𝑃={𝑥∈𝐸∶𝑥≽𝜃} a normal cone, 𝑋=[0,∞), and let 𝑑∶𝑋×𝑋→𝐸 be the Euclidean metric. Define 𝜉∶int(𝑃)∪{𝜃}→[0,1) by 𝜉(𝑑(𝑥,𝑦))=‖𝑑(𝑥,𝑦)‖/(‖𝑑(𝑥,𝑦)‖+1) for 𝑥,𝑦∈𝑋, then 𝜉 is a stronger Meir-Keeler cone-type mapping.

Definition 2.5. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃, 𝜉∶int(𝑃)∪{𝜃}→[0,1) a stronger Meir-Keeler cone-type mapping, and let 𝑓_𝑛_𝑛∈ℕ,𝑓_𝑛∶𝑋⟶𝑋(2.4) be a sequence of mappings. Suppose that there exists 𝛼∈ℕ such that the sequence {𝑓_𝑛}_𝑛∈ℕ satisfy that 𝑑𝑓^𝛼𝑖𝑥,𝑓^𝛼𝑗𝑦≼𝜉(𝑑(𝑥,𝑦))⋅𝑑(𝑥,𝑦),∀𝑥,𝑦∈𝑋,and𝑖,𝑗∈ℕ.(2.5) Then, we call {𝑓_𝑛}_𝑛∈ℕ an asymptotic 𝒮ℳ𝒦-sequence with respect to this stronger Meir-Keeler cone-type mapping 𝜉.

Example 2.6. Let 𝐸=ℝ² and 𝑃={(𝑥,𝑦)∈ℝ²|𝑥,𝑦≽𝜃} a normal cone in 𝐸. Let 𝑋=(𝑥,0)∈ℝ²∣𝑥≥0∪(0,𝑦)∈ℝ²∣𝑦≥0,(2.6) and we define the mapping 𝑑∶𝑋×𝑋→𝐸 by 𝑑((𝑥,0),(𝑦,0))=9𝑑((0,𝑥),(0,𝑦))=||5𝑥||−𝑥−𝑦||𝑦,||3,||𝑥−𝑦||,𝑑((𝑥,0),(0,𝑦))=𝑑((0,𝑦),(𝑥,0))=5||9𝑥−𝑦||,5𝑥+𝑦,𝑥+35𝑦.(2.7) Let the asymptotic 𝒮ℳ𝒦-sequence of mappings, {𝑓_𝑛}_𝑛∈ℕ, 𝑓_𝑛∶𝑋→𝑋 be 𝑓𝑓_𝑛(𝑥,0)=(0,3^𝑛𝑥),_𝑛(0,𝑦)=13^𝑛+1𝑦,0,(2.8) and let 𝜉∶int(𝑃)→[0,1) be 𝜉((𝑥,𝑦))=⎧⎪⎨⎪⎩13‖‖𝑑𝑑((𝑥𝑥,,𝑦𝑦))‖‖,if𝑑(𝑥,𝑦)≼12,‖𝑑(𝑥,𝑦)‖+1,if𝑑(𝑥,𝑦)≫12.(2.9) Then, 𝜉 is a stronger Meir-Keeler cone-type mapping and for 𝛼=2, and let {𝑓_𝑛}_𝑛∈ℕ be an asymptotic 𝒮ℳ𝒦-sequence with respect to this stronger Meir-Keeler cone-type mapping 𝜉.

Now, we will prove the following common fixed point theorem of the asymptotic 𝒮ℳ𝒦-sequence with respect to this stronger Meir-Keeler cone-type mapping for cone metric spaces with regular cone.

Theorem 2.7. Let (𝑋,𝑑) be a complete cone metric space, 𝑃 a regular cone in 𝐸, and let 𝜉∶int(𝑃)∪{𝜃}→[0,1) be a stronger Meir-Keeler cone-type mapping. Suppose 𝑓_𝑛_𝑛∈ℕ,𝑓_𝑛∶𝑋⟶𝑋(2.10) is an asymptotic 𝒮ℳ𝒦-sequence with respect to this stronger Meir-Keeler cone-type mapping 𝜉. Then, {𝑓_𝑛}_𝑛∈ℕ has a unique common fixed point in 𝑋.

Proof. Since {𝑓_𝑛}_𝑛∈ℕ is an asymptotic 𝒮ℳ𝒦-sequence with respect to this stronger Meir-Keeler cone-type mapping 𝜉, there exists 𝛼∈ℕ such that 𝑑𝑓^𝛼𝑖𝑥,𝑓^𝛼𝑗𝑦≼𝜉(𝑑(𝑥,𝑦))⋅𝑑(𝑥,𝑦),∀𝑥,𝑦∈𝑋,and𝑖,𝑗∈ℕ.(2.11)
Given 𝑥₀∈𝑋 and we define the sequence {𝑥_𝑛} recursively as follows: 𝑥_𝑛=𝑓^𝛼𝑛𝑥_𝑛−1,∀𝑛∈ℕ.(2.12) Hence, for each 𝑛∈ℕ, we have 𝑑𝑥_𝑛,𝑥_𝑛+1=≼𝜉𝑑𝑑𝑓^𝛼𝑛𝑥𝑥_{𝑛−1𝑛−1},𝑓^𝛼𝑛+1𝑥_𝑛≪𝑑𝑥_𝑛−1,𝑥_𝑛⋅𝑑𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛.(∗)
Thus, the sequence {𝑑(𝑥_𝑛,𝑥_𝑛+1)} is descreasing. Regularity of 𝑃 guarantees that the mentioned sequence is convergent. Let lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=𝜂≥0. Then, there exists 𝜅₀∈ℕ such that for all 𝑛≥𝜅₀𝜂≼𝑑𝑥_𝑛,𝑥_𝑛+1≪𝜂+𝛿.(2.13)
For each 𝑛∈ℕ, since 𝜉 is a stronger Meir-Keeler type mapping, for these 𝜂 and 𝛿≫0 we have that for 𝑥_𝜅0+𝑛,𝑥_𝜅0+𝑛+1∈𝑋 with 𝜂≼𝑑(𝑥_𝜅0+𝑛,𝑥_𝜅0+𝑛+1)≪𝛿+𝜂, there exists 𝛾_𝜂∈[0,1) such that 𝜉(𝑑(𝑥_𝜅0+𝑛,𝑥_𝜅0+𝑛+1))<𝛾_𝜂. Thus, by (*), we can deduce 𝑑𝑥_𝜅0+𝑛,𝑥_𝜅0+𝑛+1≪=𝜉𝛾_𝜂𝑑𝑥_{𝜅0+𝑛−1},𝑥_𝜅0+𝑛⋅𝑑𝑥_{𝜅0+𝑛−1},𝑥_𝜅0+𝑛⋅𝑑𝑥_{𝜅0+𝑛−1},𝑥_𝜅0+𝑛,(2.14) and it follows that for each 𝑛∈ℕ𝑑𝑥_𝜅0+𝑛,𝑥_𝜅0+𝑛+1≪≪≪𝛾𝛾⋯_𝜂𝑛𝜂⋅𝑑𝑥_{𝜅0+𝑛−1},𝑥_𝜅0+𝑛⋅𝑑𝑥_𝜅0+1,𝑥_𝜅0+2.(2.15) So, lim_𝑛→∞𝑑𝑥_𝜅0+𝑛,𝑥_𝜅0+𝑛+1=𝜃,since𝛾_𝜂<1.(2.16)
We now claim that lim_𝑛→∞𝑑(𝑥_𝜅0+𝑛,𝑥_𝜅0+𝑚)=𝜃 for 𝑚>𝑛. For 𝑚,𝑛∈ℕ with 𝑚>𝑛, we have 𝑑𝑥_𝜅0+𝑛,𝑥_𝜅0+𝑚≼^𝑚−1_𝑖=𝑛𝑑𝑥_𝜅0+𝑖,𝑥_𝜅0+𝑖+1≺𝛾^𝜂𝑚−11−𝛾_𝜂𝑑𝑥_𝜅0+1,𝑥_𝜅0+2,(2.17) and hence 𝑑(𝑥_𝑛,𝑥_𝑚)→𝜃, since 0<𝛾_𝜂<1. So {𝑥_𝑛} is a Cauchy sequence. Since (𝑋,𝑑) is a complete cone metric space, there exists 𝜈∈𝑋 such that lim_𝑛→∞𝑥_𝑛=𝜈.
We next prove that 𝜈 is a unique periodic point of 𝑓_𝑗, for all 𝑗∈ℕ. Since for all 𝑗∈ℕ, 𝑑𝜈,𝑓^𝛼𝑗𝜈==𝑑𝑑𝜈𝜈,,𝑥𝑥_𝑛𝑛+𝑑𝑥_𝑛,𝑓^𝛼𝑗𝜈=𝑑𝜈,𝑥_𝑛+𝑑𝑓^𝛼𝑛𝑥_𝑛−1,𝑓^𝛼𝑗𝜈≪𝑑𝜈,𝑥_𝑛+𝜉𝑑𝑥_𝑛−1,𝜈⋅𝑑𝑥_𝑛−1,𝜈+𝛾_𝜂⋅𝑑𝑥_𝑛−1,𝜈,(2.18) we have 𝑑(𝜈,𝑓^𝛼𝑗𝜈)→𝜃. This implies that 𝜈=𝑓^𝛼𝑗𝜈. So, 𝜈 is a periodic point of 𝑓_𝑗, for all 𝑗∈ℕ.
Let 𝜇 be another periodic point of 𝑓_𝑖, for all 𝑖∈ℕ. Then, 𝑑(𝜇,𝜈)=𝑑𝑓^𝛼𝑖𝜇,𝑓^𝛼𝑗𝜈≼𝜉(𝑑(𝜇,𝜈))⋅𝑑(𝜇,𝜈)≪𝛾_𝜂𝑑(𝜇,𝜈).(2.19) Then, 𝜇=𝜈.
Since 𝑓_𝑖𝜈=𝑓_𝑖(𝑓^𝛼𝑖𝜈)=𝑓^𝛼𝑗(𝑓_𝑖𝜈), we have that 𝑓_𝑖𝜈 is also a periodic point of 𝑓_𝑖, for all 𝑗∈ℕ. Therefore, 𝜈=𝑓_𝑖𝜈, for all 𝑗∈ℕ, that is, 𝜈 is a unique common fixed point of {𝑓_𝑛}_𝑛∈ℕ.

Example 2.8. It is easy to get that (0,0) is a unique common fixed point of the asymptotic 𝒮ℳ𝒦-sequence {𝑓_𝑛}_𝑛∈ℕ of Example 2.6.

If the stronger Meir-Keeler cone-type mapping 𝜉(𝑡)=𝑐 for some 𝑐∈[0,1), then we are easy to get the following corollaries.

Corollary 2.9. Let (𝑋,𝑑) be a complete cone metric space, 𝑃 a regular cone of a real Banach space 𝐸, and let 𝑐∈[0,1). Suppose the sequence of mappings 𝑓_𝑛_𝑛∈ℕ,𝑓_𝑛∶𝑋⟶𝑋(2.20) satisfy that for some 𝛼∈ℕ, 𝑑𝑓^𝛼𝑖𝑥,𝑓^𝛼𝑗𝑦≼𝑐⋅𝑑(𝑥,𝑦),∀𝑥,𝑦∈𝑋,and𝑖,𝑗∈ℕ.(2.21) Then, {𝑓_𝑛}_𝑛∈ℕ has a unique common fixed point in 𝑋.

Corollary 2.10 (see [11]). Let (𝑋,𝑑) be a complete cone metric space, 𝑃 a regular cone of a real Banach space 𝐸, and let 𝑐∈[0,1). Suppose the mapping 𝑓∶𝑋→𝑋 satisfies that for some 𝛼∈ℕ, 𝑑(𝑓^𝛼𝑥,𝑓^𝛼𝑦)≼𝑐⋅𝑑(𝑥,𝑦),∀𝑥,𝑦∈𝑋.(2.22) Then, 𝑓 has a unique fixed point in 𝑋.

Definition 2.11. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃, and let 𝜉,𝜉_𝑖,𝑗∶int(𝑃)∪{𝜃}→[0,1),∀𝑖,𝑗∈ℕ(2.23) be stronger Meir-Keeler cone-type mappings with sup_{𝑖,𝑗∈ℕ}𝜉_𝑖,𝑗(𝑡)≤𝜉(𝑡)∀𝑡∈𝑃.(2.24) Suppose the sequence {𝑓_𝑛}_𝑛∈ℕ, 𝑓_𝑛∶𝑋→𝑋 satisfy that for some 𝛼∈ℕ, 𝑑𝑓^𝛼𝑖𝑥,𝑓^𝛼𝑗𝑦≼𝜉_𝑖,𝑗(𝑑(𝑥,𝑦))⋅𝑑(𝑥,𝑦),∀𝑥,𝑦∈𝑋,𝑖,𝑗∈ℕ.(2.25) Then, we call {𝑓_𝑛}_𝑛∈ℕ a generalized asymptotic 𝒮ℳ𝒦-sequence with respect to the stronger Meir-Keeler cone-type mappings {𝜉_𝑖,𝑗}_{𝑖,𝑗∈ℕ}.

Example 2.12. Let 𝐸=ℝ² and 𝑃={(𝑥,𝑦)∈ℝ²|𝑥,𝑦≽𝜃} a normal cone in 𝐸. Let 𝑋=(𝑥,0)∈ℝ²∣𝑥≥0∪(0,𝑦)∈ℝ²∣𝑦≥0,(2.26) and we define the mapping 𝑑∶𝑋×𝑋→𝐸 by 𝑑((𝑥,0),(𝑦,0))=9𝑑((0,𝑥),(0,𝑦))=||5𝑥||−𝑥−𝑦||𝑦,||3,||𝑥−𝑦||,𝑑((𝑥,0),(0,𝑦))=𝑑((0,𝑦),(𝑥,0))=5||9𝑥−𝑦||,5𝑥+𝑦,𝑥+35𝑦.(2.27) Let {𝑓_𝑛}_𝑛∈ℕ, 𝑓_𝑛∶𝑋→𝑋 be 𝑓𝑓_𝑛(𝑥,0)=(0,2^𝑛𝑥),_𝑛(0,𝑦)=12^𝑛+1𝑦,0,(2.28) and let 𝜉_𝑖,𝑗,𝜉∶𝑃→[0,1) be 𝜉_𝑖,𝑗(𝑡)=⎧⎪⎨⎪⎩211,if𝑡≼1,2+4‖𝑡1‖^𝑖+𝑗𝜉(𝑡)=⎧⎪⎨⎪⎩3,if𝑡≫1,4,‖𝑡‖if𝑡≼3,‖𝑡‖+1,if𝑡≫3.(2.29) Then, {𝜉_𝑖,𝑗}_{𝑖,𝑗∈ℕ} be stronger Meir-Keeler cone-type mappings with sup_{𝑖,𝑗∈ℕ}𝜉_𝑖,𝑗(𝑡)≼𝜉(𝑡)∀𝑡∈𝑃,(2.30) and for 𝛼=2, let {𝑓_𝑛}_𝑛∈ℕ be a generalized asymptotic 𝒮ℳ𝒦-sequence with respect to the stronger Meir-Keeler cone-type mappings {𝜉_𝑖,𝑗}_{𝑖,𝑗∈ℕ}.

Follows Theorem 3.4, we are easy to conclude the following results.

Theorem 2.13. Let (𝑋,𝑑) be a complete cone metric space, 𝑃 a regular cone of a real Banach space 𝐸, let 𝜉,𝜉_𝑖,𝑗∶int(𝑃)∪{𝜃}→[0,1),∀𝑖,𝑗∈ℕ(2.31) be stronger Meir-Keeler cone-type mappings with sup_{𝑖,𝑗∈ℕ}𝜉_𝑖,𝑗(𝑡)≼𝜉(𝑡)∀𝑡∈𝑃,(2.32) and let 𝑓_𝑛_𝑛∈ℕ,𝑓_𝑛∶𝑋⟶𝑋(2.33) be a generalized asymptotic 𝒮ℳ𝒦-sequence with respect to the stronger Meir-Keeler cone-type mappings {𝜉_𝑖,𝑗}_{𝑖,𝑗∈ℕ}. Then, {𝑓_𝑛}_𝑛∈ℕ has a unique common fixed point in 𝑋.

Example 2.14. It is easy to get that (0,0) is a unique common fixed point of the generalized 𝒮ℳ𝒦-sequence {𝑓_𝑛}_𝑛∈ℕ of Example 2.12.

3. Common Fixed Point Theorems for the Asymptotic 𝒲ℳ𝒦-Sequences

In this section, we first introduce the notions of the weaker Meir-Keeler cone-type mapping 𝜙∶int(𝑃)∪{𝜃}→int(𝑃)∪{𝜃} and the asymptotic 𝒲ℳ𝒦-sequence with respect to this weaker Meir-Keeler cone-type mapping 𝜙, and we next prove some common fixed point theorems for the asymptotic 𝒲ℳ𝒦-sequence in cone metric spaces.

Definition 3.1. Let (𝑋,𝑑) be a cone metric space with cone 𝑃, and let 𝜙∶int(𝑃)∪{𝜃}⟶int(𝑃)∪{𝜃}.(3.1) Then, the function 𝜙 is called a weaker Meir-Keeler cone-type mapping, if for each 𝜂∈int(𝑃) with 𝜂≫𝜃 there exists 𝛿≫𝜃 such that for 𝑥,𝑦∈𝑋 with 𝜂≼𝑑(𝑥,𝑦)≪𝛿+𝜂 there exists 𝑛₀∈ℕ such that 𝜙^𝑛0(𝑑(𝑥,𝑦))≪𝜂.

Example 3.2. Let 𝐸=ℝ, 𝑃={𝑥∈𝐸∶𝑥≽𝜃} a normal cone, 𝑋=[0,∞), and let 𝑑∶𝑋×𝑋→𝐸 be the Euclidean metric. Define 𝜙∶int(𝑃)∪{𝜃}→int(𝑃)∪{𝜃} by 𝜙(𝑑(𝑥,𝑦))=(1/3)𝑑(𝑥,𝑦) for 𝑥,𝑦∈𝑋, then 𝜙 is a weaker Meir-Keeler cone-type mapping.

Definition 3.3. Let (𝑋,𝑑) be a cone metric space with a cone 𝑃, 𝜙∶int(𝑃)∪{𝜃}→int(𝑃)∪{𝜃} be a weaker Meir-Keeler cone-type mapping, and let 𝑓_𝑛_𝑛∈ℕ,𝑓_𝑛∶𝑋→𝑋(3.2) be a sequence of mappings. Suppose that there exists 𝛼∈ℕ such that the sequence {𝑓_𝑛}_𝑛∈ℕ satisfy that 𝑑𝑓^𝛼𝑖𝑥,𝑓^𝛼𝑗𝑦≼𝜙(𝑑(𝑥,𝑦)),∀𝑥,𝑦∈𝑋,𝑖,𝑗∈ℕ.(3.3) Then, we call {𝑓_𝑛}_𝑛∈ℕ an asymptotic 𝒲ℳ𝒦-sequence with respect to this weaker Meir-Keeler cone-type mapping 𝜉.

Now, we will prove the following common fixed point theorem of the asymptotic 𝒲ℳ𝒦-sequence with respect to this weaker Meir-Keeler cone-type mapping for cone metric spaces with regular cone.

Theorem 3.4. Let (𝑋,𝑑) be a complete cone metric space, 𝑃 a regular cone in 𝐸, and let 𝜙∶int(𝑃)∪{𝜃}→int(𝑃)∪{𝜃} be a weaker Meir-Keeler cone-type mapping, and 𝜙 also satisfies the following conditions:(i)𝜙(𝜃)=𝜃; 𝜙(𝑡)≪𝑡 for all 𝑡≫𝜃,(ii)for 𝑡_𝑛∈int(𝑃)∪{𝜃}, if lim_𝑛→∞𝑡_𝑛=𝛾≫𝜃, then lim_𝑛→∞𝜙(𝑡_𝑛)≪𝛾,(iii){𝜙^𝑛(𝑡)}_𝑛∈ℕ is decreasing.Suppose that 𝑓_𝑛_𝑛∈ℕ,𝑓_𝑛∶𝑋⟶𝑋(3.4) is an asymptotic 𝒲ℳ𝒦-sequence with respect to this weaker Meir-Keeler cone-type mapping 𝜙. Then, {𝑓_𝑛}_𝑛∈ℕ has a unique common fixed point in 𝑋.

Proof. Since {𝑓_𝑛}_𝑛∈ℕ is an asymptotic 𝒲ℳ𝒦-sequence with respect to this weaker Meir-Keeler cone-type mapping 𝜉, there exists 𝛼∈ℕ such that 𝑑𝑓^𝛼𝑖𝑥,𝑓^𝛼𝑗𝑦≼𝜙(𝑑(𝑥,𝑦)),∀𝑥,𝑦∈𝑋,𝑖,𝑗∈ℕ.(3.5)
Given 𝑥₀∈𝑋 and we define the sequence {𝑥_𝑛} recursively as follows: 𝑥_𝑛=𝑓^𝛼𝑛𝑥_𝑛−1,∀𝑛∈ℕ.(3.6) Hence, for each 𝑛∈ℕ, we have 𝑑𝑥_𝑛,𝑥_𝑛+1=≼𝑑𝜙𝑓𝑑^𝛼𝑛𝑥𝑥_{𝑛−1𝑛−1},𝑓^𝛼𝑛+1𝑥_𝑛=𝜙𝑑𝑓_𝛼𝑛−1,𝑥_𝑛≼𝜙₂𝑥_𝑛−2,𝑓^𝛼𝑛𝑥_𝑛−1≼≼𝜙⋯_𝑛𝑑𝑥_𝑛−2,𝑥_𝑛−1𝑑𝑥₀,𝑥₁.(3.7)
Since {𝜙^𝑛(𝑑(𝑥₀,𝑥₁))}_𝑛∈ℕ is decreasing. Regularity of 𝑃 guarantees that the mentioned sequence is convergent. Let lim_𝑛→∞𝜙^𝑛(𝑑(𝑥₀,𝑥₁))=𝜂, 𝜂≥𝜃. We claim that 𝜂=𝜃. On the contrary, assume that 𝜃≪𝜂. Then, by the definition of the weaker Meir-Keeler cone-type mapping, there exists 𝛿≫0 such that for 𝑥₀,𝑥₁∈𝑋 with 𝜂≼𝑑(𝑥₀,𝑥₁)≪𝛿+𝜂 there exists 𝑛₀∈ℕ such that 𝜙^𝑛0(𝑑(𝑥₀,𝑥₁))≪𝜂. Since lim_𝑛→∞𝜙^𝑛(𝑑(𝑥,𝑓𝑥))=𝜂, there exists 𝑚₀∈ℕ such that 𝜂≼𝜙^𝑚𝑑(𝑥₀,𝑥₁)≪𝛿+𝜂, for all 𝑚≥𝑚₀. Thus, we conclude that 𝜙^𝑚0+𝑛0(𝑑(𝑥₀,𝑥₁))≪𝜂. So, we get a contradiction. So, lim_𝑛→∞𝜙^𝑛(𝑑(𝑥₀,𝑥₁))=𝜃, and so lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=𝜃.
Next, we let 𝑐_𝑚=𝑑(𝑥_𝑚,𝑥_𝑚+1), and we claim that the following result holds: foreach𝜀≫𝜃,thereis𝑑𝑛₀𝑥(𝜀)∈ℕsuchthatforall𝑚,𝑛≥𝑛₀(𝜀),_𝑚,𝑥_𝑚+1≪𝜀.(∗∗) We will prove (3.7) by contradiction. Suppose that (3.7) is false. Then, there exists some 𝜀≫𝜃 such that for all 𝑘∈ℕ, there are 𝑚_𝑘,𝑛_𝑘∈ℕ with 𝑚_𝑘>𝑛_𝑘≥𝑘 satisfying:(1)𝑚_𝑘 is even and 𝑛_𝑘 is odd,(2)𝑑(𝑥_𝑚𝑘,𝑥_𝑛𝑘)≽𝜀,(3)𝑚_𝑘 is the smallest even number such that the conditions (1), (2) hold.By (2), we have lim_𝑘→∞𝑑(𝑥_𝑚𝑘,𝑥_𝑛𝑘)=𝜀, and 𝜀≼≼𝑑𝑑𝑥𝑥_{𝑚𝑘𝑚𝑘},𝑥_𝑛𝑘≼𝑑𝑥_𝑚𝑘,𝑥_𝑚𝑘+1+𝑑𝑥_𝑚𝑘+1,𝑥_𝑛𝑘+1+𝑑𝑥_𝑛𝑘+1,𝑥_𝑛𝑘,𝑥_𝑚𝑘+1+𝜙𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘+𝑑𝑥_𝑛𝑘+1,𝑥_𝑛𝑘.(3.8) Letting 𝑘→∞. Then, by the condition (ii) of this weaker Meir-Keeler cone-type mapping 𝜙, we have 𝜀≼𝜃+lim_𝑘→∞𝜙𝑑𝑥_𝑚𝑘,𝑥_𝑛𝑘+𝜃≪𝜀,(3.9) a contradiction. So, {𝑥_𝑛} is a Cauchy sequence. Since (𝑋,𝑑) is a complete cone metric space, there exists 𝜈∈𝑋 such that lim_𝑛→∞𝑥_𝑛=𝜈.
We next prove that 𝜈 is a unique periodic point of 𝑓_𝑗, for all 𝑗∈ℕ. Since for all 𝑗∈ℕ, 𝑑𝜈,𝑓^𝛼𝑗𝜈==𝑑𝑑𝜈𝜈,,𝑥𝑥_𝑛𝑛+𝑑𝑥_𝑛,𝑓^𝛼𝑗𝜈=𝑑𝜈,𝑥_𝑛+𝑑𝑓^𝛼𝑛𝑥_𝑛−1,𝑓^𝛼𝑗𝜈≪𝑑𝜈,𝑥_𝑛+𝜙𝑑𝑥_𝑛−1,𝜈+𝑑𝑥_𝑛−1,𝜈,(3.10) we have 𝑑(𝜈,𝑓^𝛼𝑗𝜈)→𝜃. This implies that 𝜈=𝑓^𝛼𝑗𝜈. So, 𝜈 is a periodic point of 𝑓_𝑗, for all 𝑗∈ℕ.
Let 𝜇 be another periodic point of 𝑓_𝑖, for all 𝑖∈ℕ. Then, 𝑑(𝜇,𝜈)=𝑑𝑓^𝛼𝑖𝜇,𝑓^𝛼𝑗𝜈≼𝜙(𝑑(𝜇,𝜈))≪𝑑(𝜇,𝜈).(3.11) Then, 𝜇=𝜈.
Since 𝑓_𝑖𝜈=𝑓_𝑖(𝑓^𝛼𝑖𝜈)=𝑓^𝛼𝑗(𝑓_𝑖𝜈), we have that 𝑓_𝑖𝜈 is also a periodic point of 𝑓_𝑖, for all 𝑗∈ℕ. Therefore, 𝜈=𝑓_𝑖𝜈, for all 𝑗∈ℕ, that is, 𝜈 is a unique common fixed point of {𝑓_𝑛}_𝑛∈ℕ.

Acknowledgment

This research is supported by the National Science Council of the Republic of China.