# 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.

## 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.

Definition 1.2.

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

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

## Authors’ Affiliations

## References

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## Copyright

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