- Research Article
- Open Access

# A Counterexample on a Theorem by Khojasteh, Goodarzi, and Razani

- IvanD Arandelović
^{1}Email author and - DragoljubJ Kečkić
^{2}

**2010**:470141

https://doi.org/10.1155/2010/470141

© Ivan D. Arandelović and Dragoljub J. Kečkić. 2010

**Received: **27 June 2010

**Accepted: **1 September 2010

**Published: **7 September 2010

## Abstract

In the paper by Khojasteh et al. (2010), the authors tried to generalize Branciari's theorem, introducing the new integral type contraction mappings. In this note we give a counterexample on their main statement (Theorem 2.9). Also we give a comment explaining what the mistake in the proof is, and suggesting what conditions might be appropriate in generalizing fixed point results to cone spaces, where the cone is taken from the infinite dimensional space.

## Keywords

- Fixed Point Theorem
- Normal Cone
- Cauchy Sequence
- Type Contraction
- Fixed Point Result

## 1. Introduction

In the paper [1], Branciari proved the following fixed point theorem with integral-type contraction condition.

Theorem 1.1.

where is nonnegative measurable mapping, having finite integral on each compact subset of such that for each , . Then has a unique fixed point , such that for each , .

There are many generalizations of fixed point results to the so-called cone metric spaces, introduced by several Russian authors in mid-20th. These spaces are re-introduced by Huang and Zhang [2]. In the same paper, the notion of convergent and Cauchy sequences are given.

Definition 1.2.

Let be a Banach space. By we denote the zero element of . A subset of is called a cone if

(1) is closed, nonempty, and ;

Given a cone , we define partial ordering on with respect to by if and only if . We will write to indicate that and , whereas will stand for (interior of ).

We say that is a solid cone if and only if .

for each . The least positive satisfying (1.2) is called the normal constant of .

Definition 1.3.

Let be a nonempty set. Suppose that a mapping satisfies:

(1) for all and if and only if ;

Then, is called a cone metric on , and is called a cone metric space.

Definition 1.4.

Let be a solid cone metric space, let , and let be a sequence in . Then

(1) converges to if for every there exists a positive integer such that for all . We denote this by or ;

(2) is a cone Cauchy sequences if for every there exists a positive integer such that for all ;

(3) is a complete cone metric space if every Cauchy sequence is convergent.

In the paper [3] Khojasteh et al. tried to generalize Branciari fixed point result to the cone metric spaces. They introduce the concept of integration along the interval as a limit of Cauchy sums.

Definition 1.5 (see [3]).

Using this concept, they stated the following statement (Theorem in [3]).

Theorem 1.6 (see [3]).

for some , then has a unique fixed point in .

However, the last statement is not true. This will be proved in the next section.

## 2. Constructing the Counterexample

It is obvious that is a normal solid cone with normal constant equals to .

Proposition 2.1.

- (a)
and (b) obvious.

- (c)

Proposition 2.2.

(a) is integrable on every segment and .

(b) is a nonvanishing subadditive function such that for all there holds .

- (a)

Proposition 2.3.

Let the space be defined by (2.1) and (2.3). Let be given by for , and , otherwise, and let .

The space together with the mappings and satisfies all assumptions of Theorem 1.6. On the other hand, has no fixed point.

Proof.

which completes the proof of the first statement.

On the other hand, has no fixed point. Namely, if we suppose that is a fixed point for , it means that for all , and moreover for all , and also for all , by induction. By continuity of , it follows that implying !

## 3. A Comment

and the existence of the total ordering on . However, in infinite dimensional case, such conclusion invokes continuity of the function inverse to . Even for the linear mappings this is not always true, but only under additional assumption that initial mapping is bijective. This asserts the well known Banach open mapping theorem. In the absence of some generalization of the open mapping theorem to nonlinear case, it is necessary to include continuity of the inverse function in the assumptions, as it was done in [4].

## Authors’ Affiliations

## References

- 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 - 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 - Khojasteh F, Goodarzi Z, Razani A:
**Some fixed point theorems of integral type contraction in cone metric spaces.***Fixed Point Theory and Applications*2010, Article ID 189684**2010:**-13.Google Scholar - Sabetghadam F, Masiha HP:
**Common fixed points for generalized -pair mappings on cone metric spaces.***Fixed Point Theory and Applications*2010, Article ID 718340**2010:**-8.Google Scholar

## Copyright

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.