Fixed Point Theory and Applications

, 2011:484717

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

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## 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.

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, 3, 4, 5, 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

(iii),

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. 1.
Greguš M Jr.: A fixed point theorem in Banach space. Unione Matematica Italiana. Bollettino. A 1980,17(1):193–198.
2. 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. 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. 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. 5.
Mukherjee RN, Verma V: A note on a fixed point theorem of Greguš. Mathematica Japonica 1988,33(5):745–749.
6. 6.
Murthy PP, Cho YJ, Fisher B: Common fixed points of Greguš type mappings. Glasnik Matematički. Serija III 1995,30(2):335–341.
7. 7.
Olaleru JO, Akewe H: An extension of Gregus fixed point theorem. Fixed Point Theory and Applications 2007, 2007:-8.

## Copyright information

© Sirous Moradi. 2011

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.

## Authors and Affiliations

1. 1.Department of MathematicsFaculty of Science, Arak UniversityArakIran