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

*Fixed Point Theory and Applications*
**volume 2011**, Article number: 484717 (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.

Let be a mapping satisfying the inequality

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

Several papers have been written on the Gregus fixed point theorem. For example, see [2–6]. We can combine the Gregus condition by the following inequality, where is a mapping on metric space :

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.

Let be a closed convex subset of a complete metrizable topological vector space and a mapping that satisfies

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.

Let endowed with the Euclidean metric and . Let defined by . Let and such that . Then for all such that , we have that

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.

## References

- 1.
Greguš M Jr.:

**A fixed point theorem in Banach space.***Unione Matematica Italiana. Bollettino. A*1980,**17**(1):193–198. - 2.
Ćirić LjB:

**On a generalization of a Greguš fixed point theorem.***Czechoslovak Mathematical Journal*2000,**50**(3):449–458. 10.1023/A:1022870007274 - 3.
Fisher B, Sessa S:

**On a fixed point theorem of Greguš.***International Journal of Mathematics and Mathematical Sciences*1986,**9**(1):23–28. 10.1155/S0161171286000030 - 4.
Jungck G:

**On a fixed point theorem of Fisher and Sessa.***International Journal of Mathematics and Mathematical Sciences*1990,**13**(3):497–500. 10.1155/S0161171290000710 - 5.
Mukherjee RN, Verma V:

**A note on a fixed point theorem of Greguš.***Mathematica Japonica*1988,**33**(5):745–749. - 6.
Murthy PP, Cho YJ, Fisher B:

**Common fixed points of Greguš type mappings.***Glasnik Matematički. Serija III*1995,**30**(2):335–341. - 7.
Olaleru JO, Akewe H:

**An extension of Gregus fixed point theorem.***Fixed Point Theory and Applications*2007,**2007:**-8.

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**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### Cite this article

Moradi, S. A Counterexample to "An Extension of Gregus Fixed Point Theorem".
*Fixed Point Theory Appl* **2011, **484717 (2011). https://doi.org/10.1155/2011/484717

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### Keywords

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