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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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F484717/MediaObjects/13663_2010_Article_737_Equ1_HTML.gif)
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
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F484717/MediaObjects/13663_2010_Article_737_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F484717/MediaObjects/13663_2010_Article_737_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F484717/MediaObjects/13663_2010_Article_737_Equ4_HTML.gif)
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
Greguš M Jr.: A fixed point theorem in Banach space. Unione Matematica Italiana. Bollettino. A 1980,17(1):193–198.
Ćirić LjB: On a generalization of a Greguš fixed point theorem. Czechoslovak Mathematical Journal 2000,50(3):449–458. 10.1023/A:1022870007274
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
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
Mukherjee RN, Verma V: A note on a fixed point theorem of Greguš. Mathematica Japonica 1988,33(5):745–749.
Murthy PP, Cho YJ, Fisher B: Common fixed points of Greguš type mappings. Glasnik Matematički. Serija III 1995,30(2):335–341.
Olaleru JO, Akewe H: An extension of Gregus fixed point theorem. Fixed Point Theory and Applications 2007, 2007:-8.
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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
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DOI: https://doi.org/10.1155/2011/484717