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  • Letter to the Editor
  • Open Access

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

Fixed Point Theory and Applications20112011:484717

https://doi.org/10.1155/2011/484717

  • Received: 29 November 2010
  • Accepted: 21 February 2011
  • Published:

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
(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.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Arak University, Arak, 38156-8-8349, Iran

References

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