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Journal of Applied Mathematics
Volumeย 2011, Article IDย 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: Yanshengย Liu

Copyright ยฉ 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 ๐’ฎโ„ณ๐’ฆ-sequence with respect to the stronger Meir-Keeler cone-type mapping ๐œ‰โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’[0,1) and the asymptotic ๐’ฒโ„ณ๐’ฆ-sequence with respect to the weaker Meir-Keeler cone-type mapping ๐œ™โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’int(๐‘ƒ)โˆช{๐œƒ} and prove some common fixed point theorems for these two asymptotic sequences in cone metric spaces with regular cone ๐‘ƒ. Our results generalize some recent results.

1. Introduction and Preliminaries

Let (๐‘‹,๐‘‘) be a metric space, ๐ท a subset of ๐‘‹, and ๐‘“โˆถ๐ทโ†’๐‘‹ a map. We say ๐‘“ is contractive if there exists ๐›ผโˆˆ[0,1) such that for all ๐‘ฅ,๐‘ฆโˆˆ๐ท,๐‘‘(๐‘“๐‘ฅ,๐‘“๐‘ฆ)โ‰ค๐›ผโ‹…๐‘‘(๐‘ฅ,๐‘ฆ).(1.1) The well-known Banachโ€™s fixed point theorem asserts that if ๐ท=๐‘‹, ๐‘“ is contractive and (๐‘‹,๐‘‘) is complete, then ๐‘“ has a unique fixed point in ๐‘‹. 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 Banachโ€™s theorem.

(Kannan [2]) There exists ๐›ผโˆˆ[0,1) such that for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹,๐‘‘๐›ผ(๐‘“๐‘ฅ,๐‘“๐‘ฆ)โ‰ค2[๐‘‘].(๐‘ฅ,๐‘“๐‘ฅ)+๐‘‘(๐‘ฆ,๐‘“๐‘ฆ)(1.2)

(Chatterjea [3]) There exists ๐›ผโˆˆ[0,1) such that for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹,๐‘‘๐›ผ(๐‘“๐‘ฅ,๐‘“๐‘ฆ)โ‰ค2[๐‘‘].(๐‘ฅ,๐‘“๐‘ฆ)+๐‘‘(๐‘ฆ,๐‘“๐‘ฅ)(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 (๐‘‹,๐‘‘) be a complete metric space and ๐‘“โˆถ๐‘‹โ†’๐‘‹ a map. Suppose there exists a function ๐œ™โˆถโ„+โ†’โ„+ satisfying ๐œ™(0)=0, ๐œ™(๐‘ก)<๐‘ก for all ๐‘ก>0 and ๐œ™ is right upper semicontinuous such that ๐‘‘(๐‘“๐‘ฅ,๐‘“๐‘ฆ)โ‰ค๐œ™(๐‘‘(๐‘ฅ,๐‘ฆ))โˆ€๐‘ฅ,๐‘ฆโˆˆ๐‘‹.(1.4) Then, ๐‘“ has a unique fixed point in ๐‘‹.

Later, Meir-Keeler [5], using a result of Chu and Diaz [6], extended Boyd-Wongโ€™s result to mappings satisfying the following more general condition:โˆ€๐œ‚>0โˆƒ๐›ฟ>0suchthat๐œ‚โ‰ค๐‘‘(๐‘ฅ,๐‘ฆ)<๐œ‚+๐›ฟโŸน๐‘‘(๐‘“๐‘ฅ,๐‘“๐‘ฆ)<๐œ‚,(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 (๐‘‹,๐‘‘) be a complete metric space and let ๐‘“ be a Meir-Keeler contraction, that is, for every ๐œ‚>0, there exists ๐›ฟ>0 such that ๐‘‘(๐‘ฅ,๐‘ฆ)<๐œ‚+๐›ฟ implies ๐‘‘(๐‘“๐‘ฅ,๐‘“๐‘ฆ)<๐œ‚ for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹. Then, ๐‘“ has a unique fixed point.

Subsequently, some authors worked on this notion of Meir-Keeler contraction (e.g., [7โ€“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., [13โ€“27]).

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

Definition 1.3 (see [11]). Let ๐ธ be a real Banach space and ๐‘ƒ a nonempty subset of ๐ธ. ๐‘ƒโ‰ {๐œƒ}, where ๐œƒ denotes the zero element of ๐ธ, is called a cone if and only if(i)๐‘ƒ is closed,(ii)๐‘Ž,๐‘โˆˆโ„, ๐‘Ž,๐‘โ‰ฅ0, ๐‘ฅ,๐‘ฆโˆˆ๐‘ƒโ‡’๐‘Ž๐‘ฅ+๐‘๐‘ฆโˆˆ๐‘ƒ,(iii)๐‘ฅโˆˆ๐‘ƒ and โˆ’๐‘ฅโˆˆ๐‘ƒโ‡’๐‘ฅ=๐œƒ.

For given a cone ๐‘ƒโŠ‚๐ธ, we can define a partial ordering with respect to ๐‘ƒ by ๐‘ฅโ‰ผ๐‘ฆ or ๐‘ฅโ‰ฝ๐‘ฆ if and only if ๐‘ฆโˆ’๐‘ฅโˆˆ๐‘ƒ for all ๐‘ฅ,๐‘ฆโˆˆ๐ธ. The real Banach space ๐ธ equipped with the partial ordered induced by ๐‘ƒ is denoted by (๐ธ,โ‰ผ). We shall write ๐‘ฅโ‰บ๐‘ฆ to indicate that ๐‘ฅโ‰ผ๐‘ฆ but ๐‘ฅโ‰ ๐‘ฆ, while ๐‘ฅโ‰ผ๐‘ฆ will stand for ๐‘ฆโˆ’๐‘ฅโˆˆint(๐‘ƒ), where int(๐‘ƒ) denotes the interior of ๐‘ƒ.

Proposition 1.4 (see [28]). Suppose ๐‘ƒ is a cone in a real Bancah space ๐ธ. Then,(i)If ๐‘’โ‰ผ๐‘“ and ๐‘“โ‰ช๐‘”, then ๐‘’โ‰ช๐‘”.(ii)If ๐‘’โ‰ช๐‘“ and ๐‘“โ‰ผ๐‘”, then ๐‘’โ‰ช๐‘”.(iii)If ๐‘’โ‰ช๐‘“ and ๐‘“โ‰ช๐‘”, then ๐‘’โ‰ช๐‘”.(iv)If ๐‘Žโˆˆ๐‘ƒ and ๐‘Žโ‰ผ๐‘’ for each ๐‘’โˆˆint(๐‘ƒ), then ๐‘Ž=๐œƒ.

Proposition 1.5 (see [29]). Suppose ๐‘’โˆˆint(๐‘ƒ), ๐œƒโ‰ผ๐‘Ž๐‘›, and ๐‘Ž๐‘›โ†’๐œƒ. Then, there exists ๐‘›0โˆˆโ„• such that ๐‘Ž๐‘›โ‰ช๐‘’ for all ๐‘›โ‰ฅ๐‘›0.

The cone ๐‘ƒ is called normal if there exists a real number ๐พ>0 such that for all ๐‘ฅ,๐‘ฆโˆˆ๐ธ,๐œƒโ‰ผ๐‘ฅโ‰ผ๐‘ฆโŸนโ€–๐‘ฅโ€–โ‰ค๐พโ€–๐‘ฆโ€–.(1.6)

The least positive number ๐พ satisfying above is called the normal constant of ๐‘ƒ.

The cone ๐‘ƒ is called regular if every increasing sequence which is bounded from above is convergent, that is, if {๐‘ฅ๐‘›} is a sequence such that๐‘ฅ1โ‰ผ๐‘ฅ2โ‰ผโ‹ฏโ‰ผ๐‘ฅ๐‘›โ‰ผโ‹ฏโ‰ผ๐‘ฆ,(1.7) for some ๐‘ฆโˆˆ๐ธ, then there is ๐‘ฅโˆˆ๐ธ such that โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโ€–โ†’0 as ๐‘›โ†’โˆž. Equivalently, the cone ๐‘ƒ 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 ๐‘‹ be a nonempty set, ๐ธ a real Banach space, and ๐‘ƒ a cone in ๐ธ. Suppose the mapping ๐‘‘โˆถ๐‘‹ร—๐‘‹โ†’(๐ธ,โ‰ผ) satisfies(i)๐œƒโ‰ผ๐‘‘(๐‘ฅ,๐‘ฆ), for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹,(ii)๐‘‘(๐‘ฅ,๐‘ฆ)=๐œƒ if and only if ๐‘ฅ=๐‘ฆ,(iii)๐‘‘(๐‘ฅ,๐‘ฆ)=๐‘‘(๐‘ฆ,๐‘ฅ), for all ๐‘ฅ,๐‘ฆโˆˆ๐‘‹,(iv)๐‘‘(๐‘ฅ,๐‘ฆ)+๐‘‘(๐‘ฆ,๐‘ง)โ‰ฝ๐‘‘(๐‘ฅ,๐‘ง), for all ๐‘ฅ,๐‘ฆ,๐‘งโˆˆ๐‘‹.Then, ๐‘‘ is called a cone metric on ๐‘‹, and (๐‘‹,๐‘‘) is called a cone metric space.

Definition 1.7 (see [11]). Let (๐‘‹,๐‘‘) be a cone metric space, and let {๐‘ฅ๐‘›} be a sequence in ๐‘‹ and ๐‘ฅโˆˆ๐‘‹. If for every ๐‘โˆˆ๐ธ with ๐œƒโ‰ช๐‘ there is ๐‘›0โˆˆโ„• such that ๐‘‘๎€ท๐‘ฅ๐‘›๎€ธ,๐‘ฅโ‰ช๐‘,โˆ€๐‘›>๐‘›0,(1.8) then {๐‘ฅ๐‘›} is said to be convergent and {๐‘ฅ๐‘›} converges to ๐‘ฅ.

Definition 1.8 (see [11]). Let (๐‘‹,๐‘‘) be a cone metric space, and let {๐‘ฅ๐‘›} be a sequence in ๐‘‹. We say that {๐‘ฅ๐‘›} is a Cauchy sequence if for any ๐‘โˆˆ๐ธ with ๐œƒโ‰ช๐‘, there is ๐‘›0โˆˆโ„• such that ๐‘‘๎€ท๐‘ฅ๐‘›,๐‘ฅ๐‘š๎€ธโ‰ช๐‘,โˆ€๐‘›,๐‘š>๐‘›0.(1.9)

Definition 1.9 (see [11]). Let (๐‘‹,๐‘‘) be a cone metric space. If every Cauchy sequence is convergent in ๐‘‹, then ๐‘‹ is called a complete cone metric space.

Remark 1.10 (see [11]). If ๐‘ƒ is a normal cone, then {๐‘ฅ๐‘›} converges to ๐‘ฅ if and only if ๐‘‘(๐‘ฅ๐‘›,๐‘ฅ)โ†’๐œƒ as ๐‘›โ†’โˆž. Further, in the case {๐‘ฅ๐‘›} is a Cauchy sequence if and only if ๐‘‘(๐‘ฅ๐‘›,๐‘ฅ๐‘š)โ†’๐œƒ as ๐‘š,๐‘›โ†’โˆž.

In this paper, we introduce the notions of the asymptotic ๐’ฎโ„ณ๐’ฆ-sequence with respect to the stronger Meir-Keeler cone-type mapping ๐œ‰โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’[0,1) and the asymptotic ๐’ฒโ„ณ๐’ฆ-sequence with respect to the weaker Meir-Keeler cone-type mapping ๐œ™โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’int(๐‘ƒ)โˆช{๐œƒ} and prove some common fixed point theorems for these two asymptotic sequences in cone metric spaces with regular cone ๐‘ƒ.

2. Common Fixed Point Theorems for the Asymptotic ๐’ฎโ„ณ๐’ฆ-Sequences

In 1973, Geraghty [30] introduced the following generalization of Banachโ€™s contraction principle.

Theorem 2.1 (see [30]). Let (๐‘‹,๐‘‘) be a complete metric space, and let ๐‘† denote the class of the functions ๐›ฝโˆถ[0,โˆž)โ†’[0,1) which satisfy the condition ๐›ฝ๎€ท๐‘ก๐‘›๎€ธโŸถ1โŸน๐‘ก๐‘›โŸถ0.(2.1) Let ๐‘“โˆถ๐‘‹โ†’๐‘‹ be a mapping satisfying ๐‘‘(๐‘“๐‘ฅ,๐‘“๐‘ฆ)โ‰ค๐›ฝ(๐‘‘(๐‘ฅ,๐‘ฆ))โ‹…๐‘‘(๐‘ฅ,๐‘ฆ),for๐‘ฅ,๐‘ฆโˆˆ๐‘‹,(2.2) where ๐›ฝโˆˆ๐‘†. Then, ๐‘“ has a unique fixed point ๐‘งโˆˆ๐‘‹.

In this section, we first introduce the notions of the stronger Meir-Keeler cone-type mapping ๐œ‰โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’[0,1) and the asymptotic ๐’ฎโ„ณ๐’ฆ-sequence with respect to this stronger Meir-Keeler cone-type mapping ๐œ‰, and we next prove some common fixed point theorems for the asymptotic ๐’ฎโ„ณ๐’ฆ-sequence in cone metric spaces.

Definition 2.2. Let (๐‘‹,๐‘‘) be a cone metric space with cone ๐‘ƒ, and let [๐œ‰โˆถint(๐‘ƒ)โˆช{๐œƒ}โŸถ0,1).(2.3) Then, the function ๐œ‰ is called a stronger Meir-Keeler cone-type mapping, if for each ๐œ‚โˆˆint(๐‘ƒ) with ๐œ‚โ‰ซ๐œƒ there exists ๐›ฟโ‰ซ๐œƒ such that for ๐‘ฅ,๐‘ฆโˆˆ๐‘‹ with ๐œ‚โ‰ผ๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ช๐›ฟ+๐œ‚ there exists ๐›พ๐œ‚โˆˆ[0,1) such that ๐œ‰(๐‘‘(๐‘ฅ,๐‘ฆ))<๐›พ๐œ‚.

Example 2.3. Let ๐ธ=โ„, ๐‘ƒ={๐‘ฅโˆˆ๐ธโˆถ๐‘ฅโ‰ฝ๐œƒ} a normal cone, ๐‘‹=[0,โˆž), and let ๐‘‘โˆถ๐‘‹ร—๐‘‹โ†’๐ธ be the Euclidean metric. Define ๐œ‰โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’[0,1) by ๐œ‰(๐‘‘(๐‘ฅ,๐‘ฆ))=๐›พ where ๐›พโˆˆ[0,1), ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, then ๐œ‰ is a stronger Meir-Keeler cone-type mapping.

Example 2.4. Let ๐ธ=โ„, ๐‘ƒ={๐‘ฅโˆˆ๐ธโˆถ๐‘ฅโ‰ฝ๐œƒ} a normal cone, ๐‘‹=[0,โˆž), and let ๐‘‘โˆถ๐‘‹ร—๐‘‹โ†’๐ธ be the Euclidean metric. Define ๐œ‰โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’[0,1) by ๐œ‰(๐‘‘(๐‘ฅ,๐‘ฆ))=โ€–๐‘‘(๐‘ฅ,๐‘ฆ)โ€–/(โ€–๐‘‘(๐‘ฅ,๐‘ฆ)โ€–+1) for ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, then ๐œ‰ is a stronger Meir-Keeler cone-type mapping.

Definition 2.5. Let (๐‘‹,๐‘‘) be a cone metric space with a cone ๐‘ƒ, ๐œ‰โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’[0,1) a stronger Meir-Keeler cone-type mapping, and let ๎€ฝ๐‘“๐‘›๎€พ๐‘›โˆˆโ„•,๐‘“๐‘›โˆถ๐‘‹โŸถ๐‘‹(2.4) be a sequence of mappings. Suppose that there exists ๐›ผโˆˆโ„• such that the sequence {๐‘“๐‘›}๐‘›โˆˆโ„• satisfy that ๐‘‘๎€ท๐‘“๐›ผ๐‘–๐‘ฅ,๐‘“๐›ผ๐‘—๐‘ฆ๎€ธโ‰ผ๐œ‰(๐‘‘(๐‘ฅ,๐‘ฆ))โ‹…๐‘‘(๐‘ฅ,๐‘ฆ),โˆ€๐‘ฅ,๐‘ฆโˆˆ๐‘‹,and๐‘–,๐‘—โˆˆโ„•.(2.5) Then, we call {๐‘“๐‘›}๐‘›โˆˆโ„• an asymptotic ๐’ฎโ„ณ๐’ฆ-sequence with respect to this stronger Meir-Keeler cone-type mapping ๐œ‰.

Example 2.6. Let ๐ธ=โ„2 and ๐‘ƒ={(๐‘ฅ,๐‘ฆ)โˆˆโ„2|๐‘ฅ,๐‘ฆโ‰ฝ๐œƒ} a normal cone in ๐ธ. Let ๎€ฝ๐‘‹=(๐‘ฅ,0)โˆˆโ„2๎€พโˆช๎€ฝโˆฃ๐‘ฅโ‰ฅ0(0,๐‘ฆ)โˆˆโ„2๎€พ,โˆฃ๐‘ฆโ‰ฅ0(2.6) and we define the mapping ๐‘‘โˆถ๐‘‹ร—๐‘‹โ†’๐ธ by ๎‚€9๐‘‘((๐‘ฅ,0),(๐‘ฆ,0))=5||||,||||๎‚,๎‚€||||,3๐‘ฅโˆ’๐‘ฆ๐‘ฅโˆ’๐‘ฆ๐‘‘((0,๐‘ฅ),(0,๐‘ฆ))=๐‘ฅโˆ’๐‘ฆ5||||๎‚,๎‚€9๐‘ฅโˆ’๐‘ฆ๐‘‘((๐‘ฅ,0),(0,๐‘ฆ))=๐‘‘((0,๐‘ฆ),(๐‘ฅ,0))=53๐‘ฅ+๐‘ฆ,๐‘ฅ+5๐‘ฆ๎‚.(2.7) Let the asymptotic ๐’ฎโ„ณ๐’ฆ-sequence of mappings, {๐‘“๐‘›}๐‘›โˆˆโ„•, ๐‘“๐‘›โˆถ๐‘‹โ†’๐‘‹ be ๐‘“๐‘›(๐‘ฅ,0)=(0,3๐‘›๐‘“๐‘ฅ),๐‘›๎‚ต1(0,๐‘ฆ)=3๐‘›+1๎‚ถ,๐‘ฆ,0(2.8) and let ๐œ‰โˆถint(๐‘ƒ)โ†’[0,1) be โŽงโŽชโŽจโŽชโŽฉ1๐œ‰((๐‘ฅ,๐‘ฆ))=31โ€–๐‘‘(๐‘ฅ,๐‘ฆ)โ€–,if๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ผ2,โ€–๐‘‘(๐‘ฅ,๐‘ฆ)โ€–1โ€–๐‘‘(๐‘ฅ,๐‘ฆ)โ€–+1,if๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ซ2.(2.9) Then, ๐œ‰ is a stronger Meir-Keeler cone-type mapping and for ๐›ผ=2, and let {๐‘“๐‘›}๐‘›โˆˆโ„• be an asymptotic ๐’ฎโ„ณ๐’ฆ-sequence with respect to this stronger Meir-Keeler cone-type mapping ๐œ‰.

Now, we will prove the following common fixed point theorem of the asymptotic ๐’ฎโ„ณ๐’ฆ-sequence with respect to this stronger Meir-Keeler cone-type mapping for cone metric spaces with regular cone.

Theorem 2.7. Let (๐‘‹,๐‘‘) be a complete cone metric space, ๐‘ƒ a regular cone in ๐ธ, and let ๐œ‰โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’[0,1) be a stronger Meir-Keeler cone-type mapping. Suppose ๎€ฝ๐‘“๐‘›๎€พ๐‘›โˆˆโ„•,๐‘“๐‘›โˆถ๐‘‹โŸถ๐‘‹(2.10) is an asymptotic ๐’ฎโ„ณ๐’ฆ-sequence with respect to this stronger Meir-Keeler cone-type mapping ๐œ‰. Then, {๐‘“๐‘›}๐‘›โˆˆโ„• has a unique common fixed point in ๐‘‹.

Proof. Since {๐‘“๐‘›}๐‘›โˆˆโ„• is an asymptotic ๐’ฎโ„ณ๐’ฆ-sequence with respect to this stronger Meir-Keeler cone-type mapping ๐œ‰, there exists ๐›ผโˆˆโ„• such that ๐‘‘๎€ท๐‘“๐›ผ๐‘–๐‘ฅ,๐‘“๐›ผ๐‘—๐‘ฆ๎€ธโ‰ผ๐œ‰(๐‘‘(๐‘ฅ,๐‘ฆ))โ‹…๐‘‘(๐‘ฅ,๐‘ฆ),โˆ€๐‘ฅ,๐‘ฆโˆˆ๐‘‹,and๐‘–,๐‘—โˆˆโ„•.(2.11)
Given ๐‘ฅ0โˆˆ๐‘‹ and we define the sequence {๐‘ฅ๐‘›} recursively as follows: ๐‘ฅ๐‘›=๐‘“๐›ผ๐‘›๐‘ฅ๐‘›โˆ’1,โˆ€๐‘›โˆˆโ„•.(2.12) Hence, for each ๐‘›โˆˆโ„•, we have ๐‘‘๎€ท๐‘ฅ๐‘›,๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘“=๐‘‘๐›ผ๐‘›๐‘ฅ๐‘›โˆ’1,๐‘“๐›ผ๐‘›+1๐‘ฅ๐‘›๎€ธ๎€ท๐‘‘๎€ท๐‘ฅโ‰ผ๐œ‰๐‘›โˆ’1,๐‘ฅ๐‘›๎€ท๐‘ฅ๎€ธ๎€ธโ‹…๐‘‘๐‘›โˆ’1,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโ‰ช๐‘‘๐‘›โˆ’1,๐‘ฅ๐‘›๎€ธ.(โˆ—)
Thus, the sequence {๐‘‘(๐‘ฅ๐‘›,๐‘ฅ๐‘›+1)} is descreasing. Regularity of ๐‘ƒ guarantees that the mentioned sequence is convergent. Let lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐‘›,๐‘ฅ๐‘›+1)=๐œ‚โ‰ฅ0. Then, there exists ๐œ…0โˆˆโ„• such that for all ๐‘›โ‰ฅ๐œ…0๎€ท๐‘ฅ๐œ‚โ‰ผ๐‘‘๐‘›,๐‘ฅ๐‘›+1๎€ธโ‰ช๐œ‚+๐›ฟ.(2.13)
For each ๐‘›โˆˆโ„•, since ๐œ‰ is a stronger Meir-Keeler type mapping, for these ๐œ‚ and ๐›ฟโ‰ซ0 we have that for ๐‘ฅ๐œ…0+๐‘›,๐‘ฅ๐œ…0+๐‘›+1โˆˆ๐‘‹ with ๐œ‚โ‰ผ๐‘‘(๐‘ฅ๐œ…0+๐‘›,๐‘ฅ๐œ…0+๐‘›+1)โ‰ช๐›ฟ+๐œ‚, there exists ๐›พ๐œ‚โˆˆ[0,1) such that ๐œ‰(๐‘‘(๐‘ฅ๐œ…0+๐‘›,๐‘ฅ๐œ…0+๐‘›+1))<๐›พ๐œ‚. Thus, by (*), we can deduce ๐‘‘๎€ท๐‘ฅ๐œ…0+๐‘›,๐‘ฅ๐œ…0+๐‘›+1๎€ธ๎€ท๐‘‘๎€ท๐‘ฅ=๐œ‰๐œ…0+๐‘›โˆ’1,๐‘ฅ๐œ…0+๐‘›๎€ท๐‘ฅ๎€ธ๎€ธโ‹…๐‘‘๐œ…0+๐‘›โˆ’1,๐‘ฅ๐œ…0+๐‘›๎€ธโ‰ช๐›พ๐œ‚๎€ท๐‘ฅโ‹…๐‘‘๐œ…0+๐‘›โˆ’1,๐‘ฅ๐œ…0+๐‘›๎€ธ,(2.14) and it follows that for each ๐‘›โˆˆโ„•๐‘‘๎€ท๐‘ฅ๐œ…0+๐‘›,๐‘ฅ๐œ…0+๐‘›+1๎€ธโ‰ช๐›พ๐œ‚๎€ท๐‘ฅโ‹…๐‘‘๐œ…0+๐‘›โˆ’1,๐‘ฅ๐œ…0+๐‘›๎€ธโ‰ชโ‹ฏโ‰ช๐›พ๐‘›๐œ‚๎€ท๐‘ฅโ‹…๐‘‘๐œ…0+1,๐‘ฅ๐œ…0+2๎€ธ.(2.15) So, lim๐‘›โ†’โˆž๐‘‘๎€ท๐‘ฅ๐œ…0+๐‘›,๐‘ฅ๐œ…0+๐‘›+1๎€ธ=๐œƒ,since๐›พ๐œ‚<1.(2.16)
We now claim that lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐œ…0+๐‘›,๐‘ฅ๐œ…0+๐‘š)=๐œƒ for ๐‘š>๐‘›. For ๐‘š,๐‘›โˆˆโ„• with ๐‘š>๐‘›, we have ๐‘‘๎€ท๐‘ฅ๐œ…0+๐‘›,๐‘ฅ๐œ…0+๐‘š๎€ธโ‰ผ๐‘šโˆ’1๎“๐‘–=๐‘›๐‘‘๎€ท๐‘ฅ๐œ…0+๐‘–,๐‘ฅ๐œ…0+๐‘–+1๎€ธโ‰บ๐›พ๐œ‚๐‘šโˆ’11โˆ’๐›พ๐œ‚๐‘‘๎€ท๐‘ฅ๐œ…0+1,๐‘ฅ๐œ…0+2๎€ธ,(2.17) and hence ๐‘‘(๐‘ฅ๐‘›,๐‘ฅ๐‘š)โ†’๐œƒ, since 0<๐›พ๐œ‚<1. So {๐‘ฅ๐‘›} is a Cauchy sequence. Since (๐‘‹,๐‘‘) is a complete cone metric space, there exists ๐œˆโˆˆ๐‘‹ such that lim๐‘›โ†’โˆž๐‘ฅ๐‘›=๐œˆ.
We next prove that ๐œˆ is a unique periodic point of ๐‘“๐‘—, for all ๐‘—โˆˆโ„•. Since for all ๐‘—โˆˆโ„•, ๐‘‘๎€ท๐œˆ,๐‘“๐›ผ๐‘—๐œˆ๎€ธ๎€ท=๐‘‘๐œˆ,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅ+๐‘‘๐‘›,๐‘“๐›ผ๐‘—๐œˆ๎€ธ๎€ท=๐‘‘๐œˆ,๐‘ฅ๐‘›๎€ธ๎€ท๐‘“+๐‘‘๐›ผ๐‘›๐‘ฅ๐‘›โˆ’1,๐‘“๐›ผ๐‘—๐œˆ๎€ธ๎€ท=๐‘‘๐œˆ,๐‘ฅ๐‘›๎€ธ๎€ท๐‘‘๎€ท๐‘ฅ+๐œ‰๐‘›โˆ’1๎€ท๐‘ฅ,๐œˆ๎€ธ๎€ธโ‹…๐‘‘๐‘›โˆ’1๎€ธ๎€ท,๐œˆโ‰ช๐‘‘๐œˆ,๐‘ฅ๐‘›๎€ธ+๐›พ๐œ‚๎€ท๐‘ฅโ‹…๐‘‘๐‘›โˆ’1๎€ธ,,๐œˆ(2.18) we have ๐‘‘(๐œˆ,๐‘“๐›ผ๐‘—๐œˆ)โ†’๐œƒ. This implies that ๐œˆ=๐‘“๐›ผ๐‘—๐œˆ. So, ๐œˆ is a periodic point of ๐‘“๐‘—, for all ๐‘—โˆˆโ„•.
Let ๐œ‡ be another periodic point of ๐‘“๐‘–, for all ๐‘–โˆˆโ„•. Then, ๐‘‘๎€ท๐‘“(๐œ‡,๐œˆ)=๐‘‘๐›ผ๐‘–๐œ‡,๐‘“๐›ผ๐‘—๐œˆ๎€ธโ‰ผ๐œ‰(๐‘‘(๐œ‡,๐œˆ))โ‹…๐‘‘(๐œ‡,๐œˆ)โ‰ช๐›พ๐œ‚๐‘‘(๐œ‡,๐œˆ).(2.19) Then, ๐œ‡=๐œˆ.
Since ๐‘“๐‘–๐œˆ=๐‘“๐‘–(๐‘“๐›ผ๐‘–๐œˆ)=๐‘“๐›ผ๐‘—(๐‘“๐‘–๐œˆ), we have that ๐‘“๐‘–๐œˆ is also a periodic point of ๐‘“๐‘–, for all ๐‘—โˆˆโ„•. Therefore, ๐œˆ=๐‘“๐‘–๐œˆ, for all ๐‘—โˆˆโ„•, that is, ๐œˆ is a unique common fixed point of {๐‘“๐‘›}๐‘›โˆˆโ„•.

Example 2.8. It is easy to get that (0,0) is a unique common fixed point of the asymptotic ๐’ฎโ„ณ๐’ฆ-sequence {๐‘“๐‘›}๐‘›โˆˆโ„• of Example 2.6.

If the stronger Meir-Keeler cone-type mapping ๐œ‰(๐‘ก)=๐‘ for some ๐‘โˆˆ[0,1), then we are easy to get the following corollaries.

Corollary 2.9. Let (๐‘‹,๐‘‘) be a complete cone metric space, ๐‘ƒ a regular cone of a real Banach space ๐ธ, and let ๐‘โˆˆ[0,1). Suppose the sequence of mappings ๎€ฝ๐‘“๐‘›๎€พ๐‘›โˆˆโ„•,๐‘“๐‘›โˆถ๐‘‹โŸถ๐‘‹(2.20) satisfy that for some ๐›ผโˆˆโ„•, ๐‘‘๎€ท๐‘“๐›ผ๐‘–๐‘ฅ,๐‘“๐›ผ๐‘—๐‘ฆ๎€ธโ‰ผ๐‘โ‹…๐‘‘(๐‘ฅ,๐‘ฆ),โˆ€๐‘ฅ,๐‘ฆโˆˆ๐‘‹,and๐‘–,๐‘—โˆˆโ„•.(2.21) Then, {๐‘“๐‘›}๐‘›โˆˆโ„• has a unique common fixed point in ๐‘‹.

Corollary 2.10 (see [11]). Let (๐‘‹,๐‘‘) be a complete cone metric space, ๐‘ƒ a regular cone of a real Banach space ๐ธ, and let ๐‘โˆˆ[0,1). Suppose the mapping ๐‘“โˆถ๐‘‹โ†’๐‘‹ satisfies that for some ๐›ผโˆˆโ„•, ๐‘‘(๐‘“๐›ผ๐‘ฅ,๐‘“๐›ผ๐‘ฆ)โ‰ผ๐‘โ‹…๐‘‘(๐‘ฅ,๐‘ฆ),โˆ€๐‘ฅ,๐‘ฆโˆˆ๐‘‹.(2.22) Then, ๐‘“ has a unique fixed point in ๐‘‹.

Definition 2.11. Let (๐‘‹,๐‘‘) be a cone metric space with a cone ๐‘ƒ, and let ๐œ‰,๐œ‰๐‘–,๐‘—[โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’0,1),โˆ€๐‘–,๐‘—โˆˆโ„•(2.23) be stronger Meir-Keeler cone-type mappings with sup๐‘–,๐‘—โˆˆโ„•๐œ‰๐‘–,๐‘—(๐‘ก)โ‰ค๐œ‰(๐‘ก)โˆ€๐‘กโˆˆ๐‘ƒ.(2.24) Suppose the sequence {๐‘“๐‘›}๐‘›โˆˆโ„•, ๐‘“๐‘›โˆถ๐‘‹โ†’๐‘‹ satisfy that for some ๐›ผโˆˆโ„•, ๐‘‘๎€ท๐‘“๐›ผ๐‘–๐‘ฅ,๐‘“๐›ผ๐‘—๐‘ฆ๎€ธโ‰ผ๐œ‰๐‘–,๐‘—(๐‘‘(๐‘ฅ,๐‘ฆ))โ‹…๐‘‘(๐‘ฅ,๐‘ฆ),โˆ€๐‘ฅ,๐‘ฆโˆˆ๐‘‹,๐‘–,๐‘—โˆˆโ„•.(2.25) Then, we call {๐‘“๐‘›}๐‘›โˆˆโ„• a generalized asymptotic ๐’ฎโ„ณ๐’ฆ-sequence with respect to the stronger Meir-Keeler cone-type mappings {๐œ‰๐‘–,๐‘—}๐‘–,๐‘—โˆˆโ„•.

Example 2.12. Let ๐ธ=โ„2 and ๐‘ƒ={(๐‘ฅ,๐‘ฆ)โˆˆโ„2|๐‘ฅ,๐‘ฆโ‰ฝ๐œƒ} a normal cone in ๐ธ. Let ๎€ฝ๐‘‹=(๐‘ฅ,0)โˆˆโ„2๎€พโˆช๎€ฝโˆฃ๐‘ฅโ‰ฅ0(0,๐‘ฆ)โˆˆโ„2๎€พ,โˆฃ๐‘ฆโ‰ฅ0(2.26) and we define the mapping ๐‘‘โˆถ๐‘‹ร—๐‘‹โ†’๐ธ by ๎‚€9๐‘‘((๐‘ฅ,0),(๐‘ฆ,0))=5||||,||||๎‚,๎‚€||||,3๐‘ฅโˆ’๐‘ฆ๐‘ฅโˆ’๐‘ฆ๐‘‘((0,๐‘ฅ),(0,๐‘ฆ))=๐‘ฅโˆ’๐‘ฆ5||||๎‚,๎‚€9๐‘ฅโˆ’๐‘ฆ๐‘‘((๐‘ฅ,0),(0,๐‘ฆ))=๐‘‘((0,๐‘ฆ),(๐‘ฅ,0))=53๐‘ฅ+๐‘ฆ,๐‘ฅ+5๐‘ฆ๎‚.(2.27) Let {๐‘“๐‘›}๐‘›โˆˆโ„•, ๐‘“๐‘›โˆถ๐‘‹โ†’๐‘‹ be ๐‘“๐‘›(๐‘ฅ,0)=(0,2๐‘›๐‘“๐‘ฅ),๐‘›๎‚ต1(0,๐‘ฆ)=2๐‘›+1๎‚ถ,๐‘ฆ,0(2.28) and let ๐œ‰๐‘–,๐‘—,๐œ‰โˆถ๐‘ƒโ†’[0,1) be ๐œ‰๐‘–,๐‘—โŽงโŽชโŽจโŽชโŽฉ1(๐‘ก)=21,if๐‘กโ‰ผ1,2+14โ€–๐‘กโ€–๐‘–+๐‘—๐œ‰โŽงโŽชโŽจโŽชโŽฉ3,if๐‘กโ‰ซ1,(๐‘ก)=4,if๐‘กโ‰ผ3,โ€–๐‘กโ€–โ€–๐‘กโ€–+1,if๐‘กโ‰ซ3.(2.29) Then, {๐œ‰๐‘–,๐‘—}๐‘–,๐‘—โˆˆโ„• be stronger Meir-Keeler cone-type mappings with sup๐‘–,๐‘—โˆˆโ„•๐œ‰๐‘–,๐‘—(๐‘ก)โ‰ผ๐œ‰(๐‘ก)โˆ€๐‘กโˆˆ๐‘ƒ,(2.30) and for ๐›ผ=2, let {๐‘“๐‘›}๐‘›โˆˆโ„• be a generalized asymptotic ๐’ฎโ„ณ๐’ฆ-sequence with respect to the stronger Meir-Keeler cone-type mappings {๐œ‰๐‘–,๐‘—}๐‘–,๐‘—โˆˆโ„•.

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

Theorem 2.13. Let (๐‘‹,๐‘‘) be a complete cone metric space, ๐‘ƒ a regular cone of a real Banach space ๐ธ, let ๐œ‰,๐œ‰๐‘–,๐‘—[โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’0,1),โˆ€๐‘–,๐‘—โˆˆโ„•(2.31) be stronger Meir-Keeler cone-type mappings with sup๐‘–,๐‘—โˆˆโ„•๐œ‰๐‘–,๐‘—(๐‘ก)โ‰ผ๐œ‰(๐‘ก)โˆ€๐‘กโˆˆ๐‘ƒ,(2.32) and let ๎€ฝ๐‘“๐‘›๎€พ๐‘›โˆˆโ„•,๐‘“๐‘›โˆถ๐‘‹โŸถ๐‘‹(2.33) be a generalized asymptotic ๐’ฎโ„ณ๐’ฆ-sequence with respect to the stronger Meir-Keeler cone-type mappings {๐œ‰๐‘–,๐‘—}๐‘–,๐‘—โˆˆโ„•. Then, {๐‘“๐‘›}๐‘›โˆˆโ„• has a unique common fixed point in ๐‘‹.

Example 2.14. It is easy to get that (0,0) is a unique common fixed point of the generalized ๐’ฎโ„ณ๐’ฆ-sequence {๐‘“๐‘›}๐‘›โˆˆโ„• of Example 2.12.

3. Common Fixed Point Theorems for the Asymptotic ๐’ฒโ„ณ๐’ฆ-Sequences

In this section, we first introduce the notions of the weaker Meir-Keeler cone-type mapping ๐œ™โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’int(๐‘ƒ)โˆช{๐œƒ} and the asymptotic ๐’ฒโ„ณ๐’ฆ-sequence with respect to this weaker Meir-Keeler cone-type mapping ๐œ™, and we next prove some common fixed point theorems for the asymptotic ๐’ฒโ„ณ๐’ฆ-sequence in cone metric spaces.

Definition 3.1. Let (๐‘‹,๐‘‘) be a cone metric space with cone ๐‘ƒ, and let ๐œ™โˆถint(๐‘ƒ)โˆช{๐œƒ}โŸถint(๐‘ƒ)โˆช{๐œƒ}.(3.1) Then, the function ๐œ™ is called a weaker Meir-Keeler cone-type mapping, if for each ๐œ‚โˆˆint(๐‘ƒ) with ๐œ‚โ‰ซ๐œƒ there exists ๐›ฟโ‰ซ๐œƒ such that for ๐‘ฅ,๐‘ฆโˆˆ๐‘‹ with ๐œ‚โ‰ผ๐‘‘(๐‘ฅ,๐‘ฆ)โ‰ช๐›ฟ+๐œ‚ there exists ๐‘›0โˆˆโ„• such that ๐œ™๐‘›0(๐‘‘(๐‘ฅ,๐‘ฆ))โ‰ช๐œ‚.

Example 3.2. Let ๐ธ=โ„, ๐‘ƒ={๐‘ฅโˆˆ๐ธโˆถ๐‘ฅโ‰ฝ๐œƒ} a normal cone, ๐‘‹=[0,โˆž), and let ๐‘‘โˆถ๐‘‹ร—๐‘‹โ†’๐ธ be the Euclidean metric. Define ๐œ™โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’int(๐‘ƒ)โˆช{๐œƒ} by ๐œ™(๐‘‘(๐‘ฅ,๐‘ฆ))=(1/3)๐‘‘(๐‘ฅ,๐‘ฆ) for ๐‘ฅ,๐‘ฆโˆˆ๐‘‹, then ๐œ™ is a weaker Meir-Keeler cone-type mapping.

Definition 3.3. Let (๐‘‹,๐‘‘) be a cone metric space with a cone ๐‘ƒ, ๐œ™โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’int(๐‘ƒ)โˆช{๐œƒ} be a weaker Meir-Keeler cone-type mapping, and let ๎€ฝ๐‘“๐‘›๎€พ๐‘›โˆˆโ„•,๐‘“๐‘›โˆถ๐‘‹โ†’๐‘‹(3.2) be a sequence of mappings. Suppose that there exists ๐›ผโˆˆโ„• such that the sequence {๐‘“๐‘›}๐‘›โˆˆโ„• satisfy that ๐‘‘๎€ท๐‘“๐›ผ๐‘–๐‘ฅ,๐‘“๐›ผ๐‘—๐‘ฆ๎€ธโ‰ผ๐œ™(๐‘‘(๐‘ฅ,๐‘ฆ)),โˆ€๐‘ฅ,๐‘ฆโˆˆ๐‘‹,๐‘–,๐‘—โˆˆโ„•.(3.3) Then, we call {๐‘“๐‘›}๐‘›โˆˆโ„• an asymptotic ๐’ฒโ„ณ๐’ฆ-sequence with respect to this weaker Meir-Keeler cone-type mapping ๐œ‰.

Now, we will prove the following common fixed point theorem of the asymptotic ๐’ฒโ„ณ๐’ฆ-sequence with respect to this weaker Meir-Keeler cone-type mapping for cone metric spaces with regular cone.

Theorem 3.4. Let (๐‘‹,๐‘‘) be a complete cone metric space, ๐‘ƒ a regular cone in ๐ธ, and let ๐œ™โˆถint(๐‘ƒ)โˆช{๐œƒ}โ†’int(๐‘ƒ)โˆช{๐œƒ} be a weaker Meir-Keeler cone-type mapping, and ๐œ™ also satisfies the following conditions:(i)๐œ™(๐œƒ)=๐œƒ; ๐œ™(๐‘ก)โ‰ช๐‘ก for all ๐‘กโ‰ซ๐œƒ,(ii)for ๐‘ก๐‘›โˆˆint(๐‘ƒ)โˆช{๐œƒ}, if lim๐‘›โ†’โˆž๐‘ก๐‘›=๐›พโ‰ซ๐œƒ, then lim๐‘›โ†’โˆž๐œ™(๐‘ก๐‘›)โ‰ช๐›พ,(iii){๐œ™๐‘›(๐‘ก)}๐‘›โˆˆโ„• is decreasing.Suppose that ๎€ฝ๐‘“๐‘›๎€พ๐‘›โˆˆโ„•,๐‘“๐‘›โˆถ๐‘‹โŸถ๐‘‹(3.4) is an asymptotic ๐’ฒโ„ณ๐’ฆ-sequence with respect to this weaker Meir-Keeler cone-type mapping ๐œ™. Then, {๐‘“๐‘›}๐‘›โˆˆโ„• has a unique common fixed point in ๐‘‹.

Proof. Since {๐‘“๐‘›}๐‘›โˆˆโ„• is an asymptotic ๐’ฒโ„ณ๐’ฆ-sequence with respect to this weaker Meir-Keeler cone-type mapping ๐œ‰, there exists ๐›ผโˆˆโ„• such that ๐‘‘๎€ท๐‘“๐›ผ๐‘–๐‘ฅ,๐‘“๐›ผ๐‘—๐‘ฆ๎€ธโ‰ผ๐œ™(๐‘‘(๐‘ฅ,๐‘ฆ)),โˆ€๐‘ฅ,๐‘ฆโˆˆ๐‘‹,๐‘–,๐‘—โˆˆโ„•.(3.5)
Given ๐‘ฅ0โˆˆ๐‘‹ and we define the sequence {๐‘ฅ๐‘›} recursively as follows: ๐‘ฅ๐‘›=๐‘“๐›ผ๐‘›๐‘ฅ๐‘›โˆ’1,โˆ€๐‘›โˆˆโ„•.(3.6) Hence, for each ๐‘›โˆˆโ„•, we have ๐‘‘๎€ท๐‘ฅ๐‘›,๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘“=๐‘‘๐›ผ๐‘›๐‘ฅ๐‘›โˆ’1,๐‘“๐›ผ๐‘›+1๐‘ฅ๐‘›๎€ธ๎€ท๐‘‘๎€ท๐‘ฅโ‰ผ๐œ™๐‘›โˆ’1,๐‘ฅ๐‘›๎€ท๐‘‘๎€ท๐‘“๎€ธ๎€ธ=๐œ™๐›ผ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’2,๐‘“๐›ผ๐‘›๐‘ฅ๐‘›โˆ’1๎€ธ๎€ธโ‰ผ๐œ™2๎€ท๐‘‘๎€ท๐‘ฅ๐‘›โˆ’2,๐‘ฅ๐‘›โˆ’1๎€ธ๎€ธโ‰ผโ‹ฏโ‰ผ๐œ™๐‘›๐‘‘๎€ท๐‘ฅ0,๐‘ฅ1๎€ธ.(3.7)
Since {๐œ™๐‘›(๐‘‘(๐‘ฅ0,๐‘ฅ1))}๐‘›โˆˆโ„• is decreasing. Regularity of ๐‘ƒ guarantees that the mentioned sequence is convergent. Let lim๐‘›โ†’โˆž๐œ™๐‘›(๐‘‘(๐‘ฅ0,๐‘ฅ1))=๐œ‚, ๐œ‚โ‰ฅ๐œƒ. We claim that ๐œ‚=๐œƒ. On the contrary, assume that ๐œƒโ‰ช๐œ‚. Then, by the definition of the weaker Meir-Keeler cone-type mapping, there exists ๐›ฟโ‰ซ0 such that for ๐‘ฅ0,๐‘ฅ1โˆˆ๐‘‹ with ๐œ‚โ‰ผ๐‘‘(๐‘ฅ0,๐‘ฅ1)โ‰ช๐›ฟ+๐œ‚ there exists ๐‘›0โˆˆโ„• such that ๐œ™๐‘›0(๐‘‘(๐‘ฅ0,๐‘ฅ1))โ‰ช๐œ‚. Since lim๐‘›โ†’โˆž๐œ™๐‘›(๐‘‘(๐‘ฅ,๐‘“๐‘ฅ))=๐œ‚, there exists ๐‘š0โˆˆโ„• such that ๐œ‚โ‰ผ๐œ™๐‘š๐‘‘(๐‘ฅ0,๐‘ฅ1)โ‰ช๐›ฟ+๐œ‚, for all ๐‘šโ‰ฅ๐‘š0. Thus, we conclude that ๐œ™๐‘š0+๐‘›0(๐‘‘(๐‘ฅ0,๐‘ฅ1))โ‰ช๐œ‚. So, we get a contradiction. So, lim๐‘›โ†’โˆž๐œ™๐‘›(๐‘‘(๐‘ฅ0,๐‘ฅ1))=๐œƒ, and so lim๐‘›โ†’โˆž๐‘‘(๐‘ฅ๐‘›,๐‘ฅ๐‘›+1)=๐œƒ.
Next, we let ๐‘๐‘š=๐‘‘(๐‘ฅ๐‘š,๐‘ฅ๐‘š+1), and we claim that the following result holds: foreach๐œ€โ‰ซ๐œƒ,thereis๐‘›0(๐œ€)โˆˆโ„•suchthatforall๐‘š,๐‘›โ‰ฅ๐‘›0๐‘‘๎€ท๐‘ฅ(๐œ€),๐‘š,๐‘ฅ๐‘š+1๎€ธโ‰ช๐œ€.(โˆ—โˆ—) We will prove (3.7) by contradiction. Suppose that (3.7) is false. Then, there exists some ๐œ€โ‰ซ๐œƒ such that for all ๐‘˜โˆˆโ„•, there are ๐‘š๐‘˜,๐‘›๐‘˜โˆˆโ„• with ๐‘š๐‘˜>๐‘›๐‘˜โ‰ฅ๐‘˜ satisfying:(1)๐‘š๐‘˜ is even and ๐‘›๐‘˜ is odd,(2)๐‘‘(๐‘ฅ๐‘š๐‘˜,๐‘ฅ๐‘›๐‘˜)โ‰ฝ๐œ€,(3)๐‘š๐‘˜ is the smallest even number such that the conditions (1), (2) hold.By (2), we have lim๐‘˜โ†’โˆž๐‘‘(๐‘ฅ๐‘š๐‘˜,๐‘ฅ๐‘›๐‘˜)=๐œ€, and ๎€ท๐‘ฅ๐œ€โ‰ผ๐‘‘๐‘š๐‘˜,๐‘ฅ๐‘›๐‘˜๎€ธ๎€ท๐‘ฅโ‰ผ๐‘‘๐‘š๐‘˜,๐‘ฅ๐‘š๐‘˜+1๎€ธ๎€ท๐‘ฅ+๐‘‘๐‘š๐‘˜+1,๐‘ฅ๐‘›๐‘˜+1๎€ธ๎€ท๐‘ฅ+๐‘‘๐‘›๐‘˜+1,๐‘ฅ๐‘›๐‘˜๎€ธ๎€ท๐‘ฅโ‰ผ๐‘‘๐‘š๐‘˜,๐‘ฅ๐‘š๐‘˜+1๎€ธ๎€ท๐‘‘๎€ท๐‘ฅ+๐œ™๐‘š๐‘˜,๐‘ฅ๐‘›๐‘˜๎€ท๐‘ฅ๎€ธ๎€ธ+๐‘‘๐‘›๐‘˜+1,๐‘ฅ๐‘›๐‘˜๎€ธ.(3.8) Letting ๐‘˜โ†’โˆž. Then, by the condition (ii) of this weaker Meir-Keeler cone-type mapping ๐œ™, we have ๐œ€โ‰ผ๐œƒ+lim๐‘˜โ†’โˆž๐œ™๎€ท๐‘‘๎€ท๐‘ฅ๐‘š๐‘˜,๐‘ฅ๐‘›๐‘˜๎€ธ๎€ธ+๐œƒโ‰ช๐œ€,(3.9) a contradiction. So, {๐‘ฅ๐‘›} is a Cauchy sequence. Since (๐‘‹,๐‘‘) is a complete cone metric space, there exists ๐œˆโˆˆ๐‘‹ such that lim๐‘›โ†’โˆž๐‘ฅ๐‘›=๐œˆ.
We next prove that ๐œˆ is a unique periodic point of ๐‘“๐‘—, for all ๐‘—โˆˆโ„•. Since for all ๐‘—โˆˆโ„•, ๐‘‘๎€ท๐œˆ,๐‘“๐›ผ๐‘—๐œˆ๎€ธ๎€ท=๐‘‘๐œˆ,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅ+๐‘‘๐‘›,๐‘“๐›ผ๐‘—๐œˆ๎€ธ๎€ท=๐‘‘๐œˆ,๐‘ฅ๐‘›๎€ธ๎€ท๐‘“+๐‘‘๐›ผ๐‘›๐‘ฅ๐‘›โˆ’1,๐‘“๐›ผ๐‘—๐œˆ๎€ธ๎€ท=๐‘‘๐œˆ,๐‘ฅ๐‘›๎€ธ๎€ท๐‘‘๎€ท๐‘ฅ+๐œ™๐‘›โˆ’1๎€ท,๐œˆ๎€ธ๎€ธโ‰ช๐‘‘๐œˆ,๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅ+๐‘‘๐‘›โˆ’1๎€ธ,,๐œˆ(3.10) we have ๐‘‘(๐œˆ,๐‘“๐›ผ๐‘—๐œˆ)โ†’๐œƒ. This implies that ๐œˆ=๐‘“๐›ผ๐‘—๐œˆ. So, ๐œˆ is a periodic point of ๐‘“๐‘—, for all ๐‘—โˆˆโ„•.
Let ๐œ‡ be another periodic point of ๐‘“๐‘–, for all ๐‘–โˆˆโ„•. Then, ๐‘‘๎€ท๐‘“(๐œ‡,๐œˆ)=๐‘‘๐›ผ๐‘–๐œ‡,๐‘“๐›ผ๐‘—๐œˆ๎€ธโ‰ผ๐œ™(๐‘‘(๐œ‡,๐œˆ))โ‰ช๐‘‘(๐œ‡,๐œˆ).(3.11) Then, ๐œ‡=๐œˆ.
Since ๐‘“๐‘–๐œˆ=๐‘“๐‘–(๐‘“๐›ผ๐‘–๐œˆ)=๐‘“๐›ผ๐‘—(๐‘“๐‘–๐œˆ), we have that ๐‘“๐‘–๐œˆ is also a periodic point of ๐‘“๐‘–, for all ๐‘—โˆˆโ„•. Therefore, ๐œˆ=๐‘“๐‘–๐œˆ, for all ๐‘—โˆˆโ„•, that is, ๐œˆ is a unique common fixed point of {๐‘“๐‘›}๐‘›โˆˆโ„•.

Acknowledgment

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

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