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Journal of Applied Mathematics
Volume 2011 (2011), Article ID 127521, 14 pages
http://dx.doi.org/10.1155/2011/127521
Research Article

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

Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu 300, Taiwan

Received 13 July 2011; Accepted 20 October 2011

Academic Editor: Yansheng Liu

Copyright © 2011 Chi-Ming Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the notions of the asymptotic 𝒮𝒦-sequence with respect to the stronger Meir-Keeler cone-type mapping 𝜉int(𝑃){𝜃}[0,1) and the asymptotic 𝒲𝒦-sequence with respect to the weaker Meir-Keeler cone-type mapping 𝜙int(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 𝑓𝐷𝑋 a map. We say 𝑓 is contractive if there exists 𝛼[0,1) such that for all 𝑥,𝑦𝐷,𝑑(𝑓𝑥,𝑓𝑦)𝛼𝑑(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,𝑦𝑋,𝑑𝛼(𝑓𝑥,𝑓𝑦)2[𝑑].(𝑥,𝑓𝑥)+𝑑(𝑦,𝑓𝑦)(1.2)

(Chatterjea [3]) There exists 𝛼[0,1) such that for all 𝑥,𝑦𝑋,𝑑𝛼(𝑓𝑥,𝑓𝑦)2[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 𝑓𝑋𝑋 a map. Suppose there exists a function 𝜙++ 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,𝑦𝑋.(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>0𝛿>0suchthat𝜂𝑑(𝑥,𝑦)<𝜂+𝛿𝑑(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,𝑦𝑋. Then, 𝑓 has a unique fixed point.

Subsequently, some authors worked on this notion of Meir-Keeler contraction (e.g., [710]).

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., [1327]).

Throughout this paper, by we denote the set of all real numbers, while 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, 𝑎,𝑏0, 𝑥,𝑦𝑃𝑎𝑥+𝑏𝑦𝑃,(iii)𝑥𝑃 and 𝑥𝑃𝑥=𝜃.

For given a cone 𝑃𝐸, we can define a partial ordering with respect to 𝑃 by 𝑥𝑦 or 𝑥𝑦 if and only if 𝑦𝑥𝑃 for all 𝑥,𝑦𝐸. The real Banach space 𝐸 equipped with the partial ordered induced by 𝑃 is denoted by (𝐸,). We shall write 𝑥𝑦 to indicate that 𝑥𝑦 but 𝑥𝑦, while 𝑥𝑦 will stand for 𝑦𝑥int(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 𝑒𝑓 and 𝑓𝑔, then 𝑒𝑔.(ii)If 𝑒𝑓 and 𝑓𝑔, then 𝑒𝑔.(iii)If 𝑒𝑓 and 𝑓𝑔, then 𝑒𝑔.(iv)If 𝑎𝑃 and 𝑎𝑒 for each 𝑒int(𝑃), then 𝑎=𝜃.

Proposition 1.5 (see [29]). Suppose 𝑒int(𝑃), 𝜃𝑎𝑛, and 𝑎𝑛𝜃. Then, there exists 𝑛0 such that 𝑎𝑛𝑒 for all 𝑛𝑛0.

The cone 𝑃 is called normal if there exists a real number 𝐾>0 such that for all 𝑥,𝑦𝐸,𝜃𝑥𝑦𝑥𝐾𝑦.(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 {𝑥𝑛} is a sequence such that𝑥1𝑥2𝑥𝑛𝑦,(1.7) for some 𝑦𝐸, then there is 𝑥𝐸 such that 𝑥𝑛𝑥0 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 𝑑𝑋×𝑋(𝐸,) satisfies(i)𝜃𝑑(𝑥,𝑦), for all 𝑥,𝑦𝑋,(ii)𝑑(𝑥,𝑦)=𝜃 if and only if 𝑥=𝑦,(iii)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥), for all 𝑥,𝑦𝑋,(iv)𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧)𝑑(𝑥,𝑧), for all 𝑥,𝑦,𝑧𝑋.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 {𝑥𝑛} be a sequence in 𝑋 and 𝑥𝑋. If for every 𝑐𝐸 with 𝜃𝑐 there is 𝑛0 such that 𝑑𝑥𝑛,𝑥𝑐,𝑛>𝑛0,(1.8) then {𝑥𝑛} is said to be convergent and {𝑥𝑛} converges to 𝑥.

Definition 1.8 (see [11]). Let (𝑋,𝑑) be a cone metric space, and let {𝑥𝑛} be a sequence in 𝑋. We say that {𝑥𝑛} is a Cauchy sequence if for any 𝑐𝐸 with 𝜃𝑐, there is 𝑛0 such that 𝑑𝑥𝑛,𝑥𝑚𝑐,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 {𝑥𝑛} converges to 𝑥 if and only if 𝑑(𝑥𝑛,𝑥)𝜃 as 𝑛. Further, in the case {𝑥𝑛} is a Cauchy sequence if and only if 𝑑(𝑥𝑛,𝑥𝑚)𝜃 as 𝑚,𝑛.

In this paper, we introduce the notions of the asymptotic 𝒮𝒦-sequence with respect to the stronger Meir-Keeler cone-type mapping 𝜉int(𝑃){𝜃}[0,1) and the asymptotic 𝒲𝒦-sequence with respect to the weaker Meir-Keeler cone-type mapping 𝜙int(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 𝒮𝒦-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 𝛽𝑡𝑛1𝑡𝑛0.(2.1) Let 𝑓𝑋𝑋 be a mapping satisfying 𝑑(𝑓𝑥,𝑓𝑦)𝛽(𝑑(𝑥,𝑦))𝑑(𝑥,𝑦),for𝑥,𝑦𝑋,(2.2) where 𝛽𝑆. Then, 𝑓 has a unique fixed point 𝑧𝑋.

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

Definition 2.2. Let (𝑋,𝑑) be a cone metric space with cone 𝑃, and let [𝜉int(𝑃){𝜃}0,1).(2.3) Then, the function 𝜉 is called a stronger Meir-Keeler cone-type mapping, if for each 𝜂int(𝑃) with 𝜂𝜃 there exists 𝛿𝜃 such that for 𝑥,𝑦𝑋 with 𝜂𝑑(𝑥,𝑦)𝛿+𝜂 there exists 𝛾𝜂[0,1) such that 𝜉(𝑑(𝑥,𝑦))<𝛾𝜂.

Example 2.3. Let 𝐸=, 𝑃={𝑥𝐸𝑥𝜃} a normal cone, 𝑋=[0,), and let 𝑑𝑋×𝑋𝐸 be the Euclidean metric. Define 𝜉int(𝑃){𝜃}[0,1) by 𝜉(𝑑(𝑥,𝑦))=𝛾 where 𝛾[0,1), 𝑥,𝑦𝑋, then 𝜉 is a stronger Meir-Keeler cone-type mapping.

Example 2.4. Let 𝐸=, 𝑃={𝑥𝐸𝑥𝜃} a normal cone, 𝑋=[0,), and let 𝑑𝑋×𝑋𝐸 be the Euclidean metric. Define 𝜉int(𝑃){𝜃}[0,1) by 𝜉(𝑑(𝑥,𝑦))=𝑑(𝑥,𝑦)/(𝑑(𝑥,𝑦)+1) for 𝑥,𝑦𝑋, 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, 𝜉int(𝑃){𝜃}[0,1) a stronger Meir-Keeler cone-type mapping, and let 𝑓𝑛𝑛,𝑓𝑛𝑋𝑋(2.4) be a sequence of mappings. Suppose that there exists 𝛼 such that the sequence {𝑓𝑛}𝑛 satisfy that 𝑑𝑓𝛼𝑖𝑥,𝑓𝛼𝑗𝑦𝜉(𝑑(𝑥,𝑦))𝑑(𝑥,𝑦),𝑥,𝑦𝑋,and𝑖,𝑗.(2.5) Then, we call {𝑓𝑛}𝑛 an asymptotic 𝒮𝒦-sequence with respect to this stronger Meir-Keeler cone-type mapping 𝜉.

Example 2.6. Let 𝐸=2 and 𝑃={(𝑥,𝑦)2|𝑥,𝑦𝜃} a normal cone in 𝐸. Let 𝑋=(𝑥,0)2𝑥0(0,𝑦)2,𝑦0(2.6) and we define the mapping 𝑑𝑋×𝑋𝐸 by 9𝑑((𝑥,0),(𝑦,0))=5||||,||||,||||,3𝑥𝑦𝑥𝑦𝑑((0,𝑥),(0,𝑦))=𝑥𝑦5||||,9𝑥𝑦𝑑((𝑥,0),(0,𝑦))=𝑑((0,𝑦),(𝑥,0))=53𝑥+𝑦,𝑥+5𝑦.(2.7) Let the asymptotic 𝒮𝒦-sequence of mappings, {𝑓𝑛}𝑛, 𝑓𝑛𝑋𝑋 be 𝑓𝑛(𝑥,0)=(0,3𝑛𝑓𝑥),𝑛1(0,𝑦)=3𝑛+1,𝑦,0(2.8) and let 𝜉int(𝑃)[0,1) be 1𝜉((𝑥,𝑦))=31𝑑(𝑥,𝑦),if𝑑(𝑥,𝑦)2,𝑑(𝑥,𝑦)1𝑑(𝑥,𝑦)+1,if𝑑(𝑥,𝑦)2.(2.9) Then, 𝜉 is a stronger Meir-Keeler cone-type mapping and for 𝛼=2, and let {𝑓𝑛}𝑛 be an asymptotic 𝒮𝒦-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 𝒮𝒦-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 𝜉int(𝑃){𝜃}[0,1) be a stronger Meir-Keeler cone-type mapping. Suppose 𝑓𝑛𝑛,𝑓𝑛𝑋𝑋(2.10) is an asymptotic 𝒮𝒦-sequence with respect to this stronger Meir-Keeler cone-type mapping 𝜉. Then, {𝑓𝑛}𝑛 has a unique common fixed point in 𝑋.

Proof. Since {𝑓𝑛}𝑛 is an asymptotic 𝒮𝒦-sequence with respect to this stronger Meir-Keeler cone-type mapping 𝜉, there exists 𝛼 such that 𝑑𝑓𝛼𝑖𝑥,𝑓𝛼𝑗𝑦𝜉(𝑑(𝑥,𝑦))𝑑(𝑥,𝑦),𝑥,𝑦𝑋,and𝑖,𝑗.(2.11)
Given 𝑥0𝑋 and we define the sequence {𝑥𝑛} recursively as follows: 𝑥𝑛=𝑓𝛼𝑛𝑥𝑛1,𝑛.(2.12) Hence, for each 𝑛, we have 𝑑𝑥𝑛,𝑥𝑛+1𝑓=𝑑𝛼𝑛𝑥𝑛1,𝑓𝛼𝑛+1𝑥𝑛𝑑𝑥𝜉𝑛1,𝑥𝑛𝑥𝑑𝑛1,𝑥𝑛𝑥𝑑𝑛1,𝑥𝑛.()
Thus, the sequence {𝑑(𝑥𝑛,𝑥𝑛+1)} is descreasing. Regularity of 𝑃 guarantees that the mentioned sequence is convergent. Let lim𝑛𝑑(𝑥𝑛,𝑥𝑛+1)=𝜂0. Then, there exists 𝜅0 such that for all 𝑛𝜅0𝑥𝜂𝑑𝑛,𝑥𝑛+1𝜂+𝛿.(2.13)
For each 𝑛, since 𝜉 is a stronger Meir-Keeler type mapping, for these 𝜂 and 𝛿0 we have that for 𝑥𝜅0+𝑛,𝑥𝜅0+𝑛+1𝑋 with 𝜂𝑑(𝑥𝜅0+𝑛,𝑥𝜅0+𝑛+1)𝛿+𝜂, there exists 𝛾𝜂[0,1) such that 𝜉(𝑑(𝑥𝜅0+𝑛,𝑥𝜅0+𝑛+1))<𝛾𝜂. Thus, by (*), we can deduce 𝑑𝑥𝜅0+𝑛,𝑥𝜅0+𝑛+1𝑑𝑥=𝜉𝜅0+𝑛1,𝑥𝜅0+𝑛𝑥𝑑𝜅0+𝑛1,𝑥𝜅0+𝑛𝛾𝜂𝑥𝑑𝜅0+𝑛1,𝑥𝜅0+𝑛,(2.14) and it follows that for each 𝑛𝑑𝑥𝜅0+𝑛,𝑥𝜅0+𝑛+1𝛾𝜂𝑥𝑑𝜅0+𝑛1,𝑥𝜅0+𝑛𝛾𝑛𝜂𝑥𝑑𝜅0+1,𝑥𝜅0+2.(2.15) So, lim𝑛𝑑𝑥𝜅0+𝑛,𝑥𝜅0+𝑛+1=𝜃,since𝛾𝜂<1.(2.16)
We now claim that lim𝑛𝑑(𝑥𝜅0+𝑛,𝑥𝜅0+𝑚)=𝜃 for 𝑚>𝑛. For 𝑚,𝑛 with 𝑚>𝑛, we have 𝑑𝑥𝜅0+𝑛,𝑥𝜅0+𝑚𝑚1𝑖=𝑛𝑑𝑥𝜅0+𝑖,𝑥𝜅0+𝑖+1𝛾𝜂𝑚11𝛾𝜂𝑑𝑥𝜅0+1,𝑥𝜅0+2,(2.17) and hence 𝑑(𝑥𝑛,𝑥𝑚)𝜃, since 0<𝛾𝜂<1. So {𝑥𝑛} is a Cauchy sequence. Since (𝑋,𝑑) is a complete cone metric space, there exists 𝜈𝑋 such that lim𝑛𝑥𝑛=𝜈.
We next prove that 𝜈 is a unique periodic point of 𝑓𝑗, for all 𝑗. Since for all 𝑗, 𝑑𝜈,𝑓𝛼𝑗𝜈=𝑑𝜈,𝑥𝑛𝑥+𝑑𝑛,𝑓𝛼𝑗𝜈=𝑑𝜈,𝑥𝑛𝑓+𝑑𝛼𝑛𝑥𝑛1,𝑓𝛼𝑗𝜈=𝑑𝜈,𝑥𝑛𝑑𝑥+𝜉𝑛1𝑥,𝜈𝑑𝑛1,𝜈𝑑𝜈,𝑥𝑛+𝛾𝜂𝑥𝑑𝑛1,,𝜈(2.18) we have 𝑑(𝜈,𝑓𝛼𝑗𝜈)𝜃. This implies that 𝜈=𝑓𝛼𝑗𝜈. So, 𝜈 is a periodic point of 𝑓𝑗, for all 𝑗.
Let 𝜇 be another periodic point of 𝑓𝑖, for all 𝑖. Then, 𝑑𝑓(𝜇,𝜈)=𝑑𝛼𝑖𝜇,𝑓𝛼𝑗𝜈𝜉(𝑑(𝜇,𝜈))𝑑(𝜇,𝜈)𝛾𝜂𝑑(𝜇,𝜈).(2.19) Then, 𝜇=𝜈.
Since 𝑓𝑖𝜈=𝑓𝑖(𝑓𝛼𝑖𝜈)=𝑓𝛼𝑗(𝑓𝑖𝜈), we have that 𝑓𝑖𝜈 is also a periodic point of 𝑓𝑖, for all 𝑗. Therefore, 𝜈=𝑓𝑖𝜈, for all 𝑗, that is, 𝜈 is a unique common fixed point of {𝑓𝑛}𝑛.

Example 2.8. It is easy to get that (0,0) is a unique common fixed point of the asymptotic 𝒮𝒦-sequence {𝑓𝑛}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 𝑓𝑛𝑛,𝑓𝑛𝑋𝑋(2.20) satisfy that for some 𝛼, 𝑑𝑓𝛼𝑖𝑥,𝑓𝛼𝑗𝑦𝑐𝑑(𝑥,𝑦),𝑥,𝑦𝑋,and𝑖,𝑗.(2.21) Then, {𝑓𝑛}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 𝑓𝑋𝑋 satisfies that for some 𝛼, 𝑑(𝑓𝛼𝑥,𝑓𝛼𝑦)𝑐𝑑(𝑥,𝑦),𝑥,𝑦𝑋.(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,𝜉𝑖,𝑗[int(𝑃){𝜃}0,1),𝑖,𝑗(2.23) be stronger Meir-Keeler cone-type mappings with sup𝑖,𝑗𝜉𝑖,𝑗(𝑡)𝜉(𝑡)𝑡𝑃.(2.24) Suppose the sequence {𝑓𝑛}𝑛, 𝑓𝑛𝑋𝑋 satisfy that for some 𝛼, 𝑑𝑓𝛼𝑖𝑥,𝑓𝛼𝑗𝑦𝜉𝑖,𝑗(𝑑(𝑥,𝑦))𝑑(𝑥,𝑦),𝑥,𝑦𝑋,𝑖,𝑗.(2.25) Then, we call {𝑓𝑛}𝑛 a generalized asymptotic 𝒮𝒦-sequence with respect to the stronger Meir-Keeler cone-type mappings {𝜉𝑖,𝑗}𝑖,𝑗.

Example 2.12. Let 𝐸=2 and 𝑃={(𝑥,𝑦)2|𝑥,𝑦𝜃} a normal cone in 𝐸. Let 𝑋=(𝑥,0)2𝑥0(0,𝑦)2,𝑦0(2.26) and we define the mapping 𝑑𝑋×𝑋𝐸 by 9𝑑((𝑥,0),(𝑦,0))=5||||,||||,||||,3𝑥𝑦𝑥𝑦𝑑((0,𝑥),(0,𝑦))=𝑥𝑦5||||,9𝑥𝑦𝑑((𝑥,0),(0,𝑦))=𝑑((0,𝑦),(𝑥,0))=53𝑥+𝑦,𝑥+5𝑦.(2.27) Let {𝑓𝑛}𝑛, 𝑓𝑛𝑋𝑋 be 𝑓𝑛(𝑥,0)=(0,2𝑛𝑓𝑥),𝑛1(0,𝑦)=2𝑛+1,𝑦,0(2.28) and let 𝜉𝑖,𝑗,𝜉𝑃[0,1) be 𝜉𝑖,𝑗1(𝑡)=21,if𝑡1,2+14𝑡𝑖+𝑗𝜉3,if𝑡1,(𝑡)=4,if𝑡3,𝑡𝑡+1,if𝑡3.(2.29) Then, {𝜉𝑖,𝑗}𝑖,𝑗 be stronger Meir-Keeler cone-type mappings with sup𝑖,𝑗𝜉𝑖,𝑗(𝑡)𝜉(𝑡)𝑡𝑃,(2.30) and for 𝛼=2, let {𝑓𝑛}𝑛 be a generalized asymptotic 𝒮𝒦-sequence with respect to the stronger Meir-Keeler cone-type mappings {𝜉𝑖,𝑗}𝑖,𝑗.

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,𝜉𝑖,𝑗[int(𝑃){𝜃}0,1),𝑖,𝑗(2.31) be stronger Meir-Keeler cone-type mappings with sup𝑖,𝑗𝜉𝑖,𝑗(𝑡)𝜉(𝑡)𝑡𝑃,(2.32) and let 𝑓𝑛𝑛,𝑓𝑛𝑋𝑋(2.33) be a generalized asymptotic 𝒮𝒦-sequence with respect to the stronger Meir-Keeler cone-type mappings {𝜉𝑖,𝑗}𝑖,𝑗. Then, {𝑓𝑛}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 𝒮𝒦-sequence {𝑓𝑛}𝑛 of Example 2.12.

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

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

Definition 3.1. Let (𝑋,𝑑) be a cone metric space with cone 𝑃, and let 𝜙int(𝑃){𝜃}int(𝑃){𝜃}.(3.1) Then, the function 𝜙 is called a weaker Meir-Keeler cone-type mapping, if for each 𝜂int(𝑃) with 𝜂𝜃 there exists 𝛿𝜃 such that for 𝑥,𝑦𝑋 with 𝜂𝑑(𝑥,𝑦)𝛿+𝜂 there exists 𝑛0 such that 𝜙𝑛0(𝑑(𝑥,𝑦))𝜂.

Example 3.2. Let 𝐸=, 𝑃={𝑥𝐸𝑥𝜃} a normal cone, 𝑋=[0,), and let 𝑑𝑋×𝑋𝐸 be the Euclidean metric. Define 𝜙int(𝑃){𝜃}int(𝑃){𝜃} by 𝜙(𝑑(𝑥,𝑦))=(1/3)𝑑(𝑥,𝑦) for 𝑥,𝑦𝑋, 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, 𝜙int(𝑃){𝜃}int(𝑃){𝜃} be a weaker Meir-Keeler cone-type mapping, and let 𝑓𝑛𝑛,𝑓𝑛𝑋𝑋(3.2) be a sequence of mappings. Suppose that there exists 𝛼 such that the sequence {𝑓𝑛}𝑛 satisfy that 𝑑𝑓𝛼𝑖𝑥,𝑓𝛼𝑗𝑦𝜙(𝑑(𝑥,𝑦)),𝑥,𝑦𝑋,𝑖,𝑗.(3.3) Then, we call {𝑓𝑛}𝑛 an asymptotic 𝒲𝒦-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 𝒲𝒦-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 𝜙int(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 𝑡𝜃,(ii)for 𝑡𝑛int(𝑃){𝜃}, if lim𝑛𝑡𝑛=𝛾𝜃, then lim𝑛𝜙(𝑡𝑛)𝛾,(iii){𝜙𝑛(𝑡)}𝑛 is decreasing.Suppose that 𝑓𝑛𝑛,𝑓𝑛𝑋𝑋(3.4) is an asymptotic 𝒲𝒦-sequence with respect to this weaker Meir-Keeler cone-type mapping 𝜙. Then, {𝑓𝑛}𝑛 has a unique common fixed point in 𝑋.

Proof. Since {𝑓𝑛}𝑛 is an asymptotic 𝒲𝒦-sequence with respect to this weaker Meir-Keeler cone-type mapping 𝜉, there exists 𝛼 such that 𝑑𝑓𝛼𝑖𝑥,𝑓𝛼𝑗𝑦𝜙(𝑑(𝑥,𝑦)),𝑥,𝑦𝑋,𝑖,𝑗.(3.5)
Given 𝑥0𝑋 and we define the sequence {𝑥𝑛} recursively as follows: 𝑥𝑛=𝑓𝛼𝑛𝑥𝑛1,𝑛.(3.6) Hence, for each 𝑛, we have 𝑑𝑥𝑛,𝑥𝑛+1𝑓=𝑑𝛼𝑛𝑥𝑛1,𝑓𝛼𝑛+1𝑥𝑛𝑑𝑥𝜙𝑛1,𝑥𝑛𝑑𝑓=𝜙𝛼𝑛1𝑥𝑛2,𝑓𝛼𝑛𝑥𝑛1𝜙2𝑑𝑥𝑛2,𝑥𝑛1𝜙𝑛𝑑𝑥0,𝑥1.(3.7)
Since {𝜙𝑛(𝑑(𝑥0,𝑥1))}𝑛 is decreasing. Regularity of 𝑃 guarantees that the mentioned sequence is convergent. Let lim𝑛𝜙𝑛(𝑑(𝑥0,𝑥1))=𝜂, 𝜂𝜃. We claim that 𝜂=𝜃. On the contrary, assume that 𝜃𝜂. Then, by the definition of the weaker Meir-Keeler cone-type mapping, there exists 𝛿0 such that for 𝑥0,𝑥1𝑋 with 𝜂𝑑(𝑥0,𝑥1)𝛿+𝜂 there exists 𝑛0 such that 𝜙𝑛0(𝑑(𝑥0,𝑥1))𝜂. Since lim𝑛𝜙𝑛(𝑑(𝑥,𝑓𝑥))=𝜂, there exists 𝑚0 such that 𝜂𝜙𝑚𝑑(𝑥0,𝑥1)𝛿+𝜂, for all 𝑚𝑚0. Thus, we conclude that 𝜙𝑚0+𝑛0(𝑑(𝑥0,𝑥1))𝜂. So, we get a contradiction. So, lim𝑛𝜙𝑛(𝑑(𝑥0,𝑥1))=𝜃, and so lim𝑛𝑑(𝑥𝑛,𝑥𝑛+1)=𝜃.
Next, we let 𝑐𝑚=𝑑(𝑥𝑚,𝑥𝑚+1), and we claim that the following result holds: foreach𝜀𝜃,thereis𝑛0(𝜀)suchthatforall𝑚,𝑛𝑛0𝑑𝑥(𝜀),𝑚,𝑥𝑚+1𝜀.() We will prove (3.7) by contradiction. Suppose that (3.7) is false. Then, there exists some 𝜀𝜃 such that for all 𝑘, there are 𝑚𝑘,𝑛𝑘 with 𝑚𝑘>𝑛𝑘𝑘 satisfying:(1)𝑚𝑘 is even and 𝑛𝑘 is odd,(2)𝑑(𝑥𝑚𝑘,𝑥𝑛𝑘)𝜀,(3)𝑚𝑘 is the smallest even number such that the conditions (1), (2) hold.By (2), we have lim𝑘𝑑(𝑥𝑚𝑘,𝑥𝑛𝑘)=𝜀, and 𝑥𝜀𝑑𝑚𝑘,𝑥𝑛𝑘𝑥𝑑𝑚𝑘,𝑥𝑚𝑘+1𝑥+𝑑𝑚𝑘+1,𝑥𝑛𝑘+1𝑥+𝑑𝑛𝑘+1,𝑥𝑛𝑘𝑥𝑑𝑚𝑘,𝑥𝑚𝑘+1𝑑𝑥+𝜙𝑚𝑘,𝑥𝑛𝑘𝑥+𝑑𝑛𝑘+1,𝑥𝑛𝑘.(3.8) Letting 𝑘. Then, by the condition (ii) of this weaker Meir-Keeler cone-type mapping 𝜙, we have 𝜀𝜃+lim𝑘𝜙𝑑𝑥𝑚𝑘,𝑥𝑛𝑘+𝜃𝜀,(3.9) a contradiction. So, {𝑥𝑛} is a Cauchy sequence. Since (𝑋,𝑑) is a complete cone metric space, there exists 𝜈𝑋 such that lim𝑛𝑥𝑛=𝜈.
We next prove that 𝜈 is a unique periodic point of 𝑓𝑗, for all 𝑗. Since for all 𝑗, 𝑑𝜈,𝑓𝛼𝑗𝜈=𝑑𝜈,𝑥𝑛𝑥+𝑑𝑛,𝑓𝛼𝑗𝜈=𝑑𝜈,𝑥𝑛𝑓+𝑑𝛼𝑛𝑥𝑛1,𝑓𝛼𝑗𝜈=𝑑𝜈,𝑥𝑛𝑑𝑥+𝜙𝑛1,𝜈𝑑𝜈,𝑥𝑛𝑥+𝑑𝑛1,,𝜈(3.10) we have 𝑑(𝜈,𝑓𝛼𝑗𝜈)𝜃. This implies that 𝜈=𝑓𝛼𝑗𝜈. So, 𝜈 is a periodic point of 𝑓𝑗, for all 𝑗.
Let 𝜇 be another periodic point of 𝑓𝑖, for all 𝑖. Then, 𝑑𝑓(𝜇,𝜈)=𝑑𝛼𝑖𝜇,𝑓𝛼𝑗𝜈𝜙(𝑑(𝜇,𝜈))𝑑(𝜇,𝜈).(3.11) Then, 𝜇=𝜈.
Since 𝑓𝑖𝜈=𝑓𝑖(𝑓𝛼𝑖𝜈)=𝑓𝛼𝑗(𝑓𝑖𝜈), we have that 𝑓𝑖𝜈 is also a periodic point of 𝑓𝑖, for all 𝑗. Therefore, 𝜈=𝑓𝑖𝜈, for all 𝑗, that is, 𝜈 is a unique common fixed point of {𝑓𝑛}𝑛.

Acknowledgment

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

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