Advertisement

Fixed Point Theory and Applications

, 2010:189684 | Cite as

Some Fixed Point Theorems of Integral Type Contraction in Cone Metric Spaces

  • Farshid Khojasteh
  • Zahra Goodarzi
  • Abdolrahman Razani
Open Access
Research Article

Abstract

We define a new concept of integral with respect to a cone. Moreover, certain fixed point theorems in those spaces are proved. Finally, an extension of Meir-Keeler fixed point in cone metric space is proved.

Keywords

Natural Number Fixed Point Theorem Contractive Condition Normal Constant Normal Cone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

In 2007, Huang and Zhang in [1] introduced cone metric space by substituting an ordered Banach space for the real numbers and proved some fixed point theorems in this space. Many authors study this subject and many fixed point theorems are proved; see [2, 3, 4, 5]. In this paper, the concept of integral in this space is introduced and a fixed point theorem is proved. In order to do this, we recall some definitions, examples, and lemmas from [1, 4] as follows.

Let Open image in new window be a real Banach space. A subset Open image in new window of Open image in new window is called a cone if and only if the following hold:

(i) Open image in new window is closed, nonempty, and Open image in new window ,

(ii) Open image in new window , Open image in new window , and Open image in new window imply that Open image in new window

(iii) Open image in new window and Open image in new window imply that Open image in new window

Given a cone Open image in new window we define a partial ordering Open image in new window with respect to Open image in new window by Open image in new window if and only if Open image in new window We will write Open image in new window to indicate that Open image in new window but Open image in new window , while Open image in new window will stand for Open image in new window int Open image in new window where int  Open image in new window denotes the interior of Open image in new window The cone Open image in new window is called normal if there is a number Open image in new window such that Open image in new window implies Open image in new window for all Open image in new window The least positive number satisfying above is called the normal constant [1].

The cone Open image in new window is called regular if every increasing sequence which is bounded from above is convergent. That is, if Open image in new window is a sequence such that Open image in new window for some Open image in new window , then there is Open image in new window such that Open image in new window . Equivalently, the cone Open image in new window is regular if and only if every decreasing sequence which is bounded from below is convergent [1]. Also every regular cone is normal [4]. In addition, there are some nonnormal cones.

Example 1.1.

Suppose Open image in new window with the norm Open image in new window and consider the cone Open image in new window : Open image in new window . For all Open image in new window , set Open image in new window and Open image in new window Then Open image in new window Open image in new window and Open image in new window Since Open image in new window Open image in new window is not normal constant of Open image in new window Therefore, Open image in new window is non-normal cone.

From now on, we suppose that Open image in new window is a real Banach space, Open image in new window is a cone in Open image in new window with Open image in new window and Open image in new window is partial ordering with respect to Open image in new window . Let Open image in new window be a nonempty set. As it has been defined in [1], a function Open image in new window is called a cone metric on Open image in new window if it satisfies the following conditions:

(i) Open image in new window for all Open image in new window and Open image in new window if and only if Open image in new window

(ii) Open image in new window , for all Open image in new window

(iii) Open image in new window , for all Open image in new window

Then Open image in new window is called a cone metric space.

Example 1.2.

Suppose Open image in new window Open image in new window Open image in new window is a metric space and Open image in new window is defined by Open image in new window Then Open image in new window is a cone metric space and the normal constant of Open image in new window is equal to Open image in new window

Definition 1.3.

Definition 1.4.

Let Open image in new window be a cone metric space and Open image in new window be a sequence in Open image in new window If for any Open image in new window with Open image in new window , there is Open image in new window such that for all Open image in new window Open image in new window then Open image in new window is called a Cauchy sequence in Open image in new window

Definition 1.5.

Let Open image in new window be a cone metric space, if every Cauchy sequence is convergent in Open image in new window then Open image in new window is called a complete cone metric space.

Definition 1.6.

then Open image in new window is called continuous on Open image in new window

The following lemmas are useful for us to prove the main result.

Lemma 1.7.

Let Open image in new window be a cone metric space and Open image in new window a normal cone with normal constant Open image in new window Let Open image in new window be a sequence in Open image in new window Then Open image in new window converges to Open image in new window if and only if

Lemma 1.8.

Let Open image in new window be a cone metric space and Open image in new window a normal cone with normal constant Open image in new window Let Open image in new window be a sequence in Open image in new window . Then Open image in new window is a Cauchy sequence if and only if

Lemma 1.9.

Let Open image in new window be a cone metric space and Open image in new window a sequence in Open image in new window If Open image in new window is convergent, then it is a Cauchy sequence.

Lemma 1.10.

The following example is a cone metric space.

Example 1.11.

Let Open image in new window Open image in new window and Open image in new window Suppose that Open image in new window is defined by Open image in new window where Open image in new window is a constant. Then Open image in new window is a cone metric space.

Theorem 1.12.

Let Open image in new window be a complete cone metric space and Open image in new window a normal cone with normal constant Open image in new window Suppose the mapping Open image in new window satisfies the contractive condition

for all Open image in new window where Open image in new window is a constant. Then Open image in new window has a unique fixed point Open image in new window Also, for all Open image in new window the sequence Open image in new window converges to Open image in new window

2. Certain Integral Type Contraction Mapping in Cone Metric Space

In 2002, Branciari in [6] introduced a general contractive condition of integral type as follows.

Theorem 2.1.

where Open image in new window is nonnegative and Lebesgue-integrable mapping which is summable (i.e., with finite integral) on each compact subset of Open image in new window such that for each Open image in new window , Open image in new window , then Open image in new window has a unique fixed point Open image in new window , such that for each Open image in new window Open image in new window

In this section we define a new concept of integral with respect to a cone and introduce the Branciari's result in cone metric spaces.

Definition 2.2.

Definition 2.3.

The set Open image in new window is called a partition for Open image in new window if and only if the sets Open image in new window are pairwise disjoint and Open image in new window

Definition 2.4.

For each partition Open image in new window of Open image in new window and each increasing function Open image in new window we define cone lower summation and cone upper summation as

respectively.

Definition 2.5.

Suppose that Open image in new window is a normal cone in Open image in new window . Open image in new window is called an integrable function on Open image in new window with respect to cone Open image in new window or to simplicity, Cone integrable function, if and only if for all partition Open image in new window of Open image in new window

where Open image in new window must be unique.

We show the common value Open image in new window by

We denote the set of all cone integrable function Open image in new window by Open image in new window .

Proof.
  1. (1)
     
  1. (2)
     

Definition 2.7.

The function Open image in new window is called subadditive cone integrable function if and only if for all Open image in new window

Example 2.8.

This shows that Open image in new window is an example of subadditive cone integrable function.

Theorem 2.9.

Let Open image in new window be a complete cone metric space and Open image in new window a normal cone. Suppose that Open image in new window is a nonvanishing map and a subadditive cone integrable on each Open image in new window such that for each Open image in new window , Open image in new window . If Open image in new window is a map such that, for all Open image in new window

for some Open image in new window then Open image in new window has a unique fixed point in Open image in new window

Proof.

If Open image in new window then Open image in new window and this is a contradiction, so
We now show that Open image in new window is a Cauchy sequence. Due to this, we show that
By triangle inequality
and by sub-additivity of Open image in new window we get
This means that Open image in new window is a Cauchy sequence and since Open image in new window is a complete cone metric space, thus Open image in new window is convergent to Open image in new window Finally, since

which is a contradiction. Thus Open image in new window has a unique fixed point Open image in new window

Lemma 2.10.

Proof.

Let Open image in new window be a partition of set Open image in new window such that Open image in new window and Open image in new window , then (by Definitions 2.4 and 2.5)

Example 2.11.

respectively, then
In order to obtain inequality (2.29), set Open image in new window and Open image in new window where Open image in new window Hence
It means that
On the other side, Branciari in [6] shows that
Thus inequalities (2.33) and (2.35) imply that
or in other words
Thus by Theorem 2.9, Open image in new window has a fixed point. But, on the other hand,

and this means that Open image in new window does not satisfy in Theorem 1.12.

3. Extension of Meir-Keeler Contraction in Cone Metric Space

In 2006, Suzuki in [7] proved that the integral type contraction (see [6]) is a special case of Meir-Keeler contraction (see [8]). Haghi and Rezapour in [5] extended Meir-Keeler contraction in cone metric space as follows.

Theorem 3.1 (see[5]).

Let Open image in new window be a complete regular cone metric space and Open image in new window has the property (KMC) on Open image in new window that is, for all Open image in new window , there exists Open image in new window such that

for all Open image in new window . Then Open image in new window has a unique fixed point.

An extension of Theorem 3.1 is as follows.

Theorem 3.2.

Let Open image in new window be a complete regular cone metric space and Open image in new window a mapping on Open image in new window . Suppose that there exists a function Open image in new window from Open image in new window into itself satisfying the following:

Open image in new window and Open image in new window for all Open image in new window ,

Open image in new window is nondecreasing and continuous function. Moreover, its inverse is continuous,

for all Open image in new window there exists Open image in new window such that for all Open image in new window

for all Open image in new window

Then Open image in new window has a unique fixed point.

Proof.

Now, we show that Open image in new window is a Cauchy sequence. If this is not, then there is a Open image in new window such that for all natural number Open image in new window there are Open image in new window so that the relation Open image in new window does not hold. Since Open image in new window has continuous inverse thus there exists Open image in new window such that for all natural number Open image in new window there are Open image in new window so that the relation Open image in new window does not hold. For each Open image in new window there exists Open image in new window such that Open image in new window for all Open image in new window with Open image in new window Choose a natural number Open image in new window such that Open image in new window for all Open image in new window Also, take Open image in new window so that the relation Open image in new window does not hold. Then Open image in new window yields
which is a contradiction. Therefore Open image in new window is a Cauchy sequence. Since Open image in new window is a complete cone metric space, there is Open image in new window such that Open image in new window . Since Open image in new window , for all Open image in new window with Open image in new window , thus for each Open image in new window , there is a natural number Open image in new window such that for all Open image in new window , Open image in new window Since Open image in new window thus Open image in new window for all Open image in new window . It means that Open image in new window In the other side, Open image in new window and the limit point is unique in cone metric spaces. Thus Open image in new window has at least one fixed point. Now, if Open image in new window are two distinct fixed points for Open image in new window then

which is a contradiction. Therefore Open image in new window has a unique fixed point.

Remark 3.3.

Open image in new window Set Open image in new window , then Theorem 3.1 is a direct result of Theorem 3.2.

Open image in new window Let Open image in new window be a nonvanishing map and a subadditive cone integrable on each Open image in new window such that for each Open image in new window , Open image in new window If Open image in new window then Open image in new window satisfies all conditions of Theorem 3.2. In other words, Theorem 2.9 is a direct result of Theorem 3.2.

Notes

Acknowledgment

The third author would like to thank the School of Mathematics of the Institute for Research in Fundamental Sciences, Teheran, Iran, for supporting this research (Grant no. 88470119).

References

  1. 1.
    Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications 2007,332(2):1468–1476. 10.1016/j.jmaa.2005.03.087MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. Journal of Mathematical Analysis and Applications 2008,341(1):416–420. 10.1016/j.jmaa.2007.09.070MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ilić D, Rakočević V: Common fixed points for maps on cone metric space. Journal of Mathematical Analysis and Applications 2008,341(2):876–882. 10.1016/j.jmaa.2007.10.065MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Rezapour Sh, Hamlbarani R: Some notes on the paper: "Cone metric spaces and fixed point theorems of contractive mappings". Journal of Mathematical Analysis and Applications 2008,345(2):719–724. 10.1016/j.jmaa.2008.04.049MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Haghi RH, Rezapour Sh: Fixed points of multifunctions on regular cone metric spaces. Expositiones Mathematicae 2010,28(1):71–77. 10.1016/j.exmath.2009.04.001MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Branciari A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. International Journal of Mathematics and Mathematical Sciences 2002,29(9):531–536. 10.1155/S0161171202007524MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Suzuki T: Meir-Keeler contractions of integral type are still Meir-Keeler contractions. International Journal of Mathematics and Mathematical Sciences 2007, 2007:-6.Google Scholar
  8. 8.
    Meir A, Keeler E: A theorem on contraction mappings. Journal of Mathematical Analysis and Applications 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Farshid Khojasteh et al. 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  • Farshid Khojasteh
    • 1
  • Zahra Goodarzi
    • 2
  • Abdolrahman Razani
    • 2
    • 3
  1. 1.Department of Mathematics, Science and Research BranchIslamic Azad UniversityTehranIran
  2. 2.Department of Mathematics, Faculty of ScienceImam Khomeini International UniversityQazvinIran
  3. 3.School of MathematicsInstitute for Research in Fundamental SciencesTehranIran

Personalised recommendations