- Research Article
- Open Access
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
The Erratum to this article has been published in Fixed Point Theory and Applications 2011 2011:346059
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.
- Natural Number
- Fixed Point Theorem
- Contractive Condition
- Normal Constant
- Normal Cone
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
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 .
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.
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).
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