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

## 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

(1.1)

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

Several papers have been written on the Gregus fixed point theorem. For example, see [26]. We can combine the Gregus condition by the following inequality, where is a mapping on metric space :

(1.2)

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

(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.

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

(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.

## 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.

Authors

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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