Some Fixed Point Theorems of Integral Type Contraction in Cone Metric Spaces
© Farshid Khojasteh et al. 2010
Received: 1 October 2009
Accepted: 11 February 2010
Published: 23 February 2010
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.
In 2007, Huang and Zhang in  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 [2–5]. 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 be a real Banach space. A subset of is called a cone if and only if the following hold:
(i) is closed, nonempty, and ,
(ii) , , and imply that
(iii) and imply that
Given a cone we define a partial ordering with respect to by if and only if We will write to indicate that but , while will stand for int where int denotes the interior of The cone is called normal if there is a number such that implies for all The least positive number satisfying above is called the normal constant .
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 . Equivalently, the cone is regular if and only if every decreasing sequence which is bounded from below is convergent . Also every regular cone is normal . In addition, there are some nonnormal cones.
Suppose with the norm and consider the cone : . For all , set and Then and Since is not normal constant of Therefore, is non-normal cone.
From now on, we suppose that is a real Banach space, is a cone in with and is partial ordering with respect to . Let be a nonempty set. As it has been defined in , a function is called a cone metric on if it satisfies the following conditions:
(i) for all and if and only if
(ii) , for all
(iii) , for all
Then is called a cone metric space.
Suppose is a metric space and is defined by Then is a cone metric space and the normal constant of is equal to
Let be a cone metric space and be a sequence in If for any with , there is such that for all then is called a Cauchy sequence in
Let be a cone metric space, if every Cauchy sequence is convergent in then is called a complete cone metric space.
then is called continuous on
The following lemmas are useful for us to prove the main result.
Let be a cone metric space and a sequence in If is convergent, then it is a Cauchy sequence.
The following example is a cone metric space.
Let and Suppose that is defined by where is a constant. Then is a cone metric space.
for all where is a constant. Then has a unique fixed point Also, for all the sequence converges to
2. Certain Integral Type Contraction Mapping in Cone Metric Space
In 2002, Branciari in  introduced a general contractive condition of integral type as follows.
where is nonnegative and Lebesgue-integrable mapping which is summable (i.e., with finite integral) on each compact subset of such that for each , , then has a unique fixed point , such that for each
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.
The set is called a partition for if and only if the sets are pairwise disjoint and
where must be unique.
We show the common value by
We denote the set of all cone integrable function by .
If , then for (2) for and .
- (1)Suppose that and are partitions for and respectively. That is,(2.6)
- (2)Suppose is an partition for , that is(2.9)
This shows that is an example of subadditive cone integrable function.
for some then has a unique fixed point in
which is a contradiction. Thus has a unique fixed point
and this means that does not satisfy in Theorem 1.12.
3. Extension of Meir-Keeler Contraction in Cone Metric Space
In 2006, Suzuki in  proved that the integral type contraction (see ) is a special case of Meir-Keeler contraction (see ). Haghi and Rezapour in  extended Meir-Keeler contraction in cone metric space as follows.
Theorem 3.1 (see).
for all . Then has a unique fixed point.
An extension of Theorem 3.1 is as follows.
Let be a complete regular cone metric space and a mapping on . Suppose that there exists a function from into itself satisfying the following:
and for all ,
is nondecreasing and continuous function. Moreover, its inverse is continuous,
for all there exists such that for all
Then has a unique fixed point.
which is a contradiction. Therefore has a unique fixed point.
Set , then Theorem 3.1 is a direct result of Theorem 3.2.
Let be a nonvanishing map and a subadditive cone integrable on each such that for each , If then satisfies all conditions of Theorem 3.2. In other words, Theorem 2.9 is a direct result of Theorem 3.2.
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).
- 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.2005.03.087MathSciNetView ArticleMATHGoogle Scholar
- 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.2007.09.070MathSciNetView ArticleMATHGoogle Scholar
- 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.2007.10.065MathSciNetView ArticleMATHGoogle Scholar
- 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.2008.04.049MathSciNetView ArticleMATHGoogle Scholar
- Haghi RH, Rezapour Sh: Fixed points of multifunctions on regular cone metric spaces. Expositiones Mathematicae 2010,28(1):71–77. 10.1016/j.exmath.2009.04.001MathSciNetView ArticleMATHGoogle Scholar
- 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/S0161171202007524MathSciNetView ArticleMATHGoogle Scholar
- 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
- Meir A, Keeler E: A theorem on contraction mappings. Journal of Mathematical Analysis and Applications 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6MathSciNetView ArticleMATHGoogle Scholar
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