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.
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.
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:
The following lemmas are useful for us to prove the main result.
The following example is a cone metric space.
2. Certain Integral Type Contraction Mapping in Cone Metric Space
In 2002, Branciari in  introduced a general contractive condition of integral type as follows.
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.
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).
An extension of Theorem 3.1 is as follows.
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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