- Letter to the Editor
- Open Access

# A Counterexample to "An Extension of Gregus Fixed Point Theorem"

- Sirous Moradi
^{1}Email author

**2011**:484717

https://doi.org/10.1155/2011/484717

© Sirous Moradi. 2011

**Received:**29 November 2010**Accepted:**21 February 2011**Published:**14 March 2011

## Abstract

In the paper by Olaleru and Akewe (2007), the authors tried to generalize Gregus fixed point theorem. In this paper we give a counterexample on their main statement.

## Keywords

- Banach Space
- Vector Space
- Point Theorem
- Differential Geometry
- Convex Subset

## 1. Introduction

Let be a Banach space and be a closed convex subset of . In 1980 Greguš [1] proved the following results.

Theorem 1.1.

for all , where , and . Then has a unique fixed point.

for all , where , and .

Definition 1.2.

Let be a topological vector space on . The mapping is said to be an such that for all

(i) ,

(ii) ,

(iii) ,

(iv) for all with ,

(v)if and , then .

In 2007, Olaleru and Akewe [7] considered the existence of fixed point of when is defined on a closed convex subset of a complete metrizable topological vector space and satisfies condition (1.2) and extended the Gregus fixed point.

Theorem 1.3.

for all , where is an on , , and . Then has a unique fixed point.

Here, we give an example to show that the above mentioned theorem is not correct.

## 2. Counterexample

Example 2.1.

We have two cases, or .

If , then , and hence inequality (2.1) is true. If , then , and so , and hence inequality (2.1) is true. So condition (1.3) holds for and , but has not fixed point.

## Authors’ Affiliations

## References

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## 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.