Fixed Point Theory and Applications

, 2011:484717 | Cite as

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 Open image in new window be a Banach space and Open image in new window be a closed convex subset of Open image in new window . In 1980 Greguš [1] proved the following results.

Theorem 1.1.

Let Open image in new window be a mapping satisfying the inequality

for all Open image in new window , where Open image in new window , and Open image in new window . Then Open image in new window 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 Open image in new window is a mapping on metric space Open image in new window :

for all Open image in new window , where Open image in new window , and Open image in new window .

Definition 1.2.

Let Open image in new window be a topological vector space on Open image in new window . The mapping Open image in new window is said to be an Open image in new window such that for all Open image in new window

(i) Open image in new window ,

(ii) Open image in new window Open image in new window Open image in new window ,

(iii) Open image in new window ,

(iv) Open image in new window for all Open image in new window with Open image in new window ,

(v)if Open image in new window and Open image in new window , then Open image in new window .

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

Theorem 1.3.

Let Open image in new window be a closed convex subset of a complete metrizable topological vector space Open image in new window and Open image in new window a mapping that satisfies

for all Open image in new window , where Open image in new window is an Open image in new window on Open image in new window , Open image in new window , and Open image in new window . Then Open image in new window 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, Open image in new window or Open image in new window .

If Open image in new window , then Open image in new window , and hence inequality (2.1) is true. If Open image in new window , then Open image in new window , and so Open image in new window , and hence inequality (2.1) is true. So condition (1.3) holds for Open image in new window and Open image in new window , but Open image in new window 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.MathSciNetMATHGoogle Scholar
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    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/S0161171290000710MathSciNetCrossRefMATHGoogle Scholar
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    Mukherjee RN, Verma V: A note on a fixed point theorem of Greguš. Mathematica Japonica 1988,33(5):745–749.MathSciNetMATHGoogle Scholar
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    Murthy PP, Cho YJ, Fisher B: Common fixed points of Greguš type mappings. Glasnik Matematički. Serija III 1995,30(2):335–341.MathSciNetMATHGoogle Scholar
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    Olaleru JO, Akewe H: An extension of Gregus fixed point theorem. Fixed Point Theory and Applications 2007, 2007:-8.MathSciNetCrossRefMATHGoogle Scholar

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

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