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A Counterexample on a Theorem by Khojasteh, Goodarzi, and Razani
Fixed Point Theory and Applications volume 2010, Article number: 470141 (2010)
Abstract
In the paper by Khojasteh et al. (2010), the authors tried to generalize Branciari's theorem, introducing the new integral type contraction mappings. In this note we give a counterexample on their main statement (Theorem 2.9). Also we give a comment explaining what the mistake in the proof is, and suggesting what conditions might be appropriate in generalizing fixed point results to cone spaces, where the cone is taken from the infinite dimensional space.
1. Introduction
In the paper [1], Branciari proved the following fixed point theorem with integral-type contraction condition.
Theorem 1.1.
Let be a complete metric space,
, and
is a mapping such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ1_HTML.gif)
where is nonnegative measurable mapping, having finite integral on each compact subset of
such that for each
,
. Then
has a unique fixed point
, such that for each
,
.
There are many generalizations of fixed point results to the so-called cone metric spaces, introduced by several Russian authors in mid-20th. These spaces are re-introduced by Huang and Zhang [2]. In the same paper, the notion of convergent and Cauchy sequences are given.
Definition 1.2.
Let be a Banach space. By
we denote the zero element of
. A subset
of
is called a cone if
(1) is closed, nonempty, and
;
(2),
, and
imply
;
(3).
Given a cone , we define partial ordering
on
with respect to
by
if and only if
. We will write
to indicate that
and
, whereas
will stand for
(interior of
).
We say that is a solid cone if and only if
.
Let be a solid cone in
and let
be the corresponding partial ordering on
with respect to
. We say that
is a normal cone if and only if there exists a real number
such that
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ2_HTML.gif)
for each . The least positive
satisfying (1.2) is called the normal constant of
.
Definition 1.3.
Let be a nonempty set. Suppose that a mapping
satisfies:
(1) for all
and
if and only if
;
(2) for all
;
(3) for all
.
Then, is called a cone metric on
, and
is called a cone metric space.
Definition 1.4.
Let be a solid cone metric space, let
, and let
be a sequence in
. Then
(1) converges to
if for every
there exists a positive integer
such that for all
. We denote this by
or
;
(2) is a cone Cauchy sequences if for every
there exists a positive integer
such that for all
;
(3) is a complete cone metric space if every Cauchy sequence is convergent.
In the paper [3] Khojasteh et al. tried to generalize Branciari fixed point result to the cone metric spaces. They introduce the concept of integration along the interval as a limit of Cauchy sums.
Definition 1.5 (see [3]).
Let be a normal solid cone, and let
. We say that
is integrable on
if and only if the following sums:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ3_HTML.gif)
converge to the same element of , where
form a partition of
. Clearly,
stands for
. This element is denoted by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ4_HTML.gif)
We say that is subadditive if and only if for any
there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ5_HTML.gif)
Using this concept, they stated the following statement (Theorem in [3]).
Theorem 1.6 (see [3]).
Let be a complete cone metric space and let
be a normal cone. Suppose that
is a nonvanishing map which is subadditive cone integrable on each
and such that for each
,
. If
is a map such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ6_HTML.gif)
for some , then
has a unique fixed point in
.
However, the last statement is not true. This will be proved in the next section.
2. Constructing the Counterexample
Consider the Banach space
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ7_HTML.gif)
with the norm , and the cone
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ8_HTML.gif)
It is obvious that is a normal solid cone with normal constant equals to
.
Consider the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ9_HTML.gif)
and the mapping given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ10_HTML.gif)
Proposition 2.1.
(a)
-
(b)
is a cone metric space.
-
(c)
A sequence
is convergent (in
) to
if and only if
. Also
is Cauchy sequence (in
) if and only if
is a Cauchy sequence with respect to norm in
.
-
(d)
is a complete cone metric space.
Proof.
-
(a)
and (b) obvious.
-
(c)
Let
(in
), and let
. Then, the function
, and we have that for all
and for all
there holds
(2.5)
Let , and let
. Then there exists
such that for all
,
. Also for all
there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ12_HTML.gif)
Hence, implying
.
The similar argument proves the second part of the statement concerning Cauchy sequences.
-
(d)
The set
can be represented as
, where
is the bounded linear functional given by
. Therefore,
is a closed subset of
and hence complete in the norm. By part (c) of this proposition, it implies that
is a complete cone metric space.
Let denote the function identically equal to
. Consider the mapping
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ13_HTML.gif)
for and
.
Proposition 2.2.
(a) is integrable on every segment
and
.
(b) is a nonvanishing subadditive function such that for all
there holds
.
Proof.
-
(a)
The integrability of
on
follows immediately from its continuity. Further, let
be a partition of
. Then
for some partition
of
, and we have
(2.8)
and similarly .
-
(b)
Follows from the part (a).
Proposition 2.3.
Let the space be defined by (2.1) and (2.3). Let
be given by
for
, and
, otherwise, and let
.
The space together with the mappings
and
satisfies all assumptions of Theorem 1.6. On the other hand,
has no fixed point.
Proof.
We only have to check the inequality (1.6). Note that for all and all
we have
. Also, note that
is a linear mapping, and
. Therefore
. Thus (1.6) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ15_HTML.gif)
Taking into account Proposition 2.2, part (a), we have for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ16_HTML.gif)
Putting , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ17_HTML.gif)
which completes the proof of the first statement.
On the other hand, has no fixed point. Namely, if we suppose that
is a fixed point for
, it means that
for all
, and moreover
for all
, and also for all
, by induction. By continuity of
, it follows that
implying
!
3. A Comment
The mistake in the proof of Theorem 1.6 given in [3] is in the following. The authors from conclude that
also, which is unjustifiable. The original Branciari's proof [1] deals with one-dimensional integral, and such conclusion is valid due to the implication
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F470141/MediaObjects/13663_2010_Article_1288_Equ18_HTML.gif)
and the existence of the total ordering on . However, in infinite dimensional case, such conclusion invokes continuity of the function inverse to
. Even for the linear mappings this is not always true, but only under additional assumption that initial mapping is bijective. This asserts the well known Banach open mapping theorem. In the absence of some generalization of the open mapping theorem to nonlinear case, it is necessary to include continuity of the inverse function in the assumptions, as it was done in [4].
References
Branciari A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. International Journal of Mathematics and Mathematical Sciences 2002,29(9):531–536. 10.1155/S0161171202007524
Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications 2007,332(2):1468–1476. 10.1016/j.jmaa.2005.03.087
Khojasteh F, Goodarzi Z, Razani A: Some fixed point theorems of integral type contraction in cone metric spaces. Fixed Point Theory and Applications 2010, Article ID 189684 2010:-13.
Sabetghadam F, Masiha HP: Common fixed points for generalized -pair mappings on cone metric spaces. Fixed Point Theory and Applications 2010, Article ID 718340 2010:-8.
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Arandelović, I., Kečkić, D. A Counterexample on a Theorem by Khojasteh, Goodarzi, and Razani. Fixed Point Theory Appl 2010, 470141 (2010). https://doi.org/10.1155/2010/470141
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DOI: https://doi.org/10.1155/2010/470141