Journal of Applied Mathematics

Volumeย 2011, Article IDย 127521, 14 pages

http://dx.doi.org/10.1155/2011/127521

## 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 -sequence with respect to the stronger Meir-Keeler cone-type mapping and the asymptotic -sequence with respect to the weaker Meir-Keeler cone-type mapping and prove some common fixed point theorems for these two asymptotic sequences in cone metric spaces with regular cone . Our results generalize some recent results.

#### 1. Introduction and Preliminaries

Let be a metric space, a subset of , and a map. We say is contractive if there exists such that for all , The well-known Banachโs fixed point theorem asserts that if , is contractive and is complete, then has a unique fixed point in . 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 Banachโs theorem.

(Kannan [2]) There exists such that for all ,

(Chatterjea [3]) There exists such that for all ,

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 be a complete metric space and a map. Suppose there exists a function satisfying , for all and is right upper semicontinuous such that
**
Then, has a unique fixed point in .*

Later, Meir-Keeler [5], using a result of Chu and Diaz [6], extended Boyd-Wongโs result to mappings satisfying the following more general condition: 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 be a complete metric space and let be a Meir-Keeler contraction, that is, for every , there exists such that implies for all . Then, has a unique fixed point.*

Subsequently, some authors worked on this notion of Meir-Keeler contraction (e.g., [7โ10]).

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., [13โ27]).

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 be a real Banach space and a nonempty subset of . , where denotes the zero element of , is called a cone if and only if(i) is closed,(ii), , ,(iii) and .

For given a cone , we can define a partial ordering with respect to by or if and only if for all . The real Banach space equipped with the partial ordered induced by is denoted by . We shall write to indicate that but , while will stand for , where denotes the interior of .

Proposition 1.4 (see [28]). *Suppose is a cone in a real Bancah space . Then,*(i)*If and , then .*(ii)*If and , then .*(iii)*If and , then .*(iv)*If and for each , then .*

Proposition 1.5 (see [29]). *Suppose , , and . Then, there exists such that for all .*

The cone is called normal if there exists a real number such that for all ,

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

The cone is called regular if every increasing sequence which is bounded from above is convergent, that is, if is a sequence such that for some , then there is such that as . Equivalently, the cone 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 be a nonempty set, a real Banach space, and a cone in . Suppose the mapping satisfies(i), for all ,(ii) if and only if ,(iii), for all ,(iv), for all .Then, is called a cone metric on , and is called a cone metric space.

*Definition 1.7 (see [11]). *Let be a cone metric space, and let be a sequence in and . If for every with there is such that
then is said to be convergent and converges to .

*Definition 1.8 (see [11]). *Let be a cone metric space, and let be a sequence in . We say that is a Cauchy sequence if for any with , there is such that

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

*Remark 1.10 (see [11]). *If is a normal cone, then converges to if and only if as . Further, in the case is a Cauchy sequence if and only if as .

In this paper, we introduce the notions of the asymptotic -sequence with respect to the stronger Meir-Keeler cone-type mapping and the asymptotic -sequence with respect to the weaker Meir-Keeler cone-type mapping and prove some common fixed point theorems for these two asymptotic sequences in cone metric spaces with regular cone .

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

In 1973, Geraghty [30] introduced the following generalization of Banachโs contraction principle.

Theorem 2.1 (see [30]). *Let be a complete metric space, and let denote the class of the functions which satisfy the condition
**
Let be a mapping satisfying
**
where . Then, has a unique fixed point .*

In this section, we first introduce the notions of the stronger Meir-Keeler cone-type mapping and the asymptotic -sequence with respect to this stronger Meir-Keeler cone-type mapping , and we next prove some common fixed point theorems for the asymptotic -sequence in cone metric spaces.

*Definition 2.2. *Let be a cone metric space with cone , and let
Then, the function is called a stronger Meir-Keeler cone-type mapping, if for each with there exists such that for with there exists such that .

*Example 2.3. *Let , a normal cone, , and let be the Euclidean metric. Define by where , , then is a stronger Meir-Keeler cone-type mapping.

*Example 2.4. *Let , a normal cone, , and let be the Euclidean metric. Define by for , then is a stronger Meir-Keeler cone-type mapping.

*Definition 2.5. *Let be a cone metric space with a cone , a stronger Meir-Keeler cone-type mapping, and let
be a sequence of mappings. Suppose that there exists such that the sequence satisfy that
Then, we call an asymptotic -sequence with respect to this stronger Meir-Keeler cone-type mapping .

*Example 2.6. *Let and a normal cone in . Let
and we define the mapping by
Let the asymptotic -sequence of mappings, , be
and let be
Then, is a stronger Meir-Keeler cone-type mapping and for , and let be an asymptotic -sequence with respect to this stronger Meir-Keeler cone-type mapping .

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

Theorem 2.7. *Let be a complete cone metric space, a regular cone in , and let be a stronger Meir-Keeler cone-type mapping. Suppose
**
is an asymptotic -sequence with respect to this stronger Meir-Keeler cone-type mapping . Then, has a unique common fixed point in .*

*Proof. *Since is an asymptotic -sequence with respect to this stronger Meir-Keeler cone-type mapping , there exists such that

Given and we define the sequence recursively as follows:
Hence, for each , we have

Thus, the sequence is descreasing. Regularity of guarantees that the mentioned sequence is convergent. Let . Then, there exists such that for all

For each , since is a stronger Meir-Keeler type mapping, for these and we have that for with , there exists such that . Thus, by (*), we can deduce
and it follows that for each
So,

We now claim that for . For with , we have
and hence , since . So is a Cauchy sequence. Since is a complete cone metric space, there exists such that .

We next prove that is a unique periodic point of , for all . Since for all ,
we have . This implies that . So, is a periodic point of , for all .

Let be another periodic point of , for all . Then,
Then, .

Since , we have that is also a periodic point of , for all . Therefore, , for all , that is, is a unique common fixed point of .

*Example 2.8. *It is easy to get that is a unique common fixed point of the asymptotic -sequence of Example 2.6.

If the stronger Meir-Keeler cone-type mapping for some , then we are easy to get the following corollaries.

Corollary 2.9. *Let be a complete cone metric space, a regular cone of a real Banach space , and let . Suppose the sequence of mappings
**
satisfy that for some ,
**
Then, has a unique common fixed point in .*

Corollary 2.10 (see [11]). *Let be a complete cone metric space, a regular cone of a real Banach space , and let . Suppose the mapping satisfies that for some ,
**
Then, has a unique fixed point in .*

*Definition 2.11. *Let be a cone metric space with a cone , and let
be stronger Meir-Keeler cone-type mappings with
Suppose the sequence , satisfy that for some ,
Then, we call a generalized asymptotic -sequence with respect to the stronger Meir-Keeler cone-type mappings .

*Example 2.12. *Let and a normal cone in . Let
and we define the mapping by
Let , be
and let be
Then, be stronger Meir-Keeler cone-type mappings with
and for , let be a generalized asymptotic -sequence with respect to the stronger Meir-Keeler cone-type mappings .

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

Theorem 2.13. *Let be a complete cone metric space, a regular cone of a real Banach space , let
**
be stronger Meir-Keeler cone-type mappings with
**
and let
**
be a generalized asymptotic -sequence with respect to the stronger Meir-Keeler cone-type mappings . Then, has a unique common fixed point in .*

*Example 2.14. *It is easy to get that is a unique common fixed point of the generalized -sequence of Example 2.12.

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

In this section, we first introduce the notions of the weaker Meir-Keeler cone-type mapping and the asymptotic -sequence with respect to this weaker Meir-Keeler cone-type mapping , and we next prove some common fixed point theorems for the asymptotic -sequence in cone metric spaces.

*Definition 3.1. *Let be a cone metric space with cone , and let
Then, the function is called a weaker Meir-Keeler cone-type mapping, if for each with there exists such that for with there exists such that .

*Example 3.2. *Let , a normal cone, , and let be the Euclidean metric. Define by for , then is a weaker Meir-Keeler cone-type mapping.

*Definition 3.3. *Let be a cone metric space with a cone , be a weaker Meir-Keeler cone-type mapping, and let
be a sequence of mappings. Suppose that there exists such that the sequence satisfy that
Then, we call an asymptotic -sequence with respect to this weaker Meir-Keeler cone-type mapping .

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

Theorem 3.4. *Let be a complete cone metric space, a regular cone in , and let be a weaker Meir-Keeler cone-type mapping, and also satisfies the following conditions:*(i)*; for all ,*(ii)*for , if , then ,*(iii)* is decreasing.**Suppose that
**
is an asymptotic -sequence with respect to this weaker Meir-Keeler cone-type mapping . Then, has a unique common fixed point in .*

*Proof. *Since is an asymptotic -sequence with respect to this weaker Meir-Keeler cone-type mapping , there exists such that

Given and we define the sequence recursively as follows:
Hence, for each , we have

Since is decreasing. Regularity of guarantees that the mentioned sequence is convergent. Let , . We claim that . On the contrary, assume that . Then, by the definition of the weaker Meir-Keeler cone-type mapping, there exists such that for with there exists such that . Since , there exists such that , for all . Thus, we conclude that . So, we get a contradiction. So, , and so .

Next, we let , and we claim that the following result holds:
We will prove (3.7) by contradiction. Suppose that (3.7) is false. Then, there exists some such that for all , there are with satisfying:(1) is even and is odd,(2),(3) is the smallest even number such that the conditions (1), (2) hold.By (2), we have , and
Letting . Then, by the condition (ii) of this weaker Meir-Keeler cone-type mapping , we have
a contradiction. So, is a Cauchy sequence. Since is a complete cone metric space, there exists such that .

We next prove that is a unique periodic point of , for all . Since for all ,
we have . This implies that . So, is a periodic point of , for all .

Let be another periodic point of , for all . Then,
Then, .

Since , we have that is also a periodic point of , for all . Therefore, , for all , that is, is a unique common fixed point of .

#### Acknowledgment

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

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