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Common Fixed Point Theorem for Four Non-Self Mappings in Cone Metric Spaces
Fixed Point Theory and Applications volume 2010, Article number: 983802 (2010)
Abstract
We extend a common fixed point theorem of Radenovic and Rhoades for four non-self-mappings in cone metric spaces.
1. Introduction and Preliminaries
Recently, Huang and Zhang [1] generalized the concept of a metric space, replacing the set of real numbers by ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. Subsequently, the study of fixed point theorems in such spaces is followed by some other mathematicians; see [2–8]. The aim of this paper is to prove a common fixed point theorem for four non-self-mappings on cone metric spaces in which the cone need not be normal. This result generalizes the result of Radenović and Rhoades [5].
Consistent with Huang and Zhang [1], the following definitions and results will be needed in the sequel.
Let be a real Banach space. A subset
of
is called a cone if and only if
(a) is closed, nonempty and
;
(b),
,
implies
;
(c).
Given a cone , we define a partial ordering
with respect to
by
if and only if
. A cone
is called normal if there is a number
such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ1_HTML.gif)
The least positive number satisfying the above inequality is called the normal constant of , while
stands for
(interior of
).
Definition 1.1 (see [1]).
Let be a nonempty set. Suppose that the mapping
satisfies
(d1) for all
and
if and only if
;
(d2) for all
;
(d3) for all
.
Then is called a cone metric on
, and
is called a cone metric space.
The concept of a cone metric space is more general than that of a metric space.
Definition 1.2 (see [1]).
Let be a cone metric space. One says that
is
-
(e)
a Cauchy sequence if for every
with
, there is an
such that for all
,
;
-
(f)
a Convergent sequence if for every
with
, there is an
such that for all
,
for some fixed
.
A cone metric space is said to be complete if every Cauchy sequence in
is convergent in
. It is known that
converges to
if and only if
as
. It is a Cauchy sequence if and only if
.
Remark 1.3 (see [9]).
Let be an ordered Banach (normed) space. Then
is an interior point of
if and only if
is a neighborhood of
.
Corollary 1.4 (see [10]).
-
(1)
If
and
, then
.Indeed,
implies
.
(2)If and
, then
.Indeed,
implies
.
-
(3)
If
for each
, then
.
If ,
, and
, then there exists an
such that for all
we have
.
If is a real Banach space with cone
and if
where
and
, then
.
We find it convenient to introduce the following definition.
Definition 1.7 (see [5]).
Let be a complete cone metric space and
a nonempty closed subset of
, and
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ2_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ3_HTML.gif)
for all ,
,
, then
is called a generalized
-contractive mapping of
into
.
Definition 1.8 (see [2]).
Let and
be self-maps on a set
(i.e.,
). If
for some
in
, then
is called a coincidence point of
and
, and
is called a point of coincidence of
and
. Self-maps
and
are said to be weakly compatible if they commute at their coincidence point; that is, if
for some
, then
.
2. Main Result
The following theorem is Radenović and Rhoades [5] generalization of Imdad and Kumar's [12] result in cone metric spaces.
Theorem 2.1.
Let be a complete cone metric space and
a nonempty closed subset of
such that for each
and
there exists a point
(the boundary of
) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ4_HTML.gif)
Suppose that are such that
is a generalized
-contractive mapping of
into
, and
(i),
(ii),
(iii) is closed in
.
Then the pair has a coincidence point. Moreover, if pair
is weakly compatible, then
and
have a unique common fixed point.
The purpose of this paper is to extend the above theorem for four non-self-mappings in cone metric spaces. We begin with the following definition.
Definition 2.2.
Let be a complete cone metric space and
a nonempty closed subset of
, and
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ5_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ6_HTML.gif)
for all ,
,
, then
is called a generalized
-contractive mappings pair of
into
.
Notice that by setting and
in (2.2), one deduces the slightly generalized form of (1.3).
We state and prove our main result as follows.
Theorem 2.3.
Let be a complete cone metric space and
a nonempty closed subset of
such that for each
and
there exists a point
(the boundary of
) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ7_HTML.gif)
Suppose that are such that
is a generalized
-contractive mappings pair of
into
, and
(I),
(II),
(III) and
(or
and
) are closed in
.Then
(IV) has a point of coincidence,
(V) has a point of coincidence.
Moreover, if and
are weakly compatible pairs, then
,
,
, and
have a unique common fixed point.
Proof.
Firstly, we proceed to construct two sequences and
in the following way.
Let be arbitrary. Then (due to
) there exists a point
such that
. Since
, one concludes that
. Thus, there exists
such that
. Since
there exists a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ8_HTML.gif)
Suppose that . Then
which implies that there exists a point
such that
. Otherwise, if
, then there exists a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ9_HTML.gif)
Since there exists a point
with
, so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ10_HTML.gif)
Let be such that
. Thus, repeating the foregoing arguments, one obtains two sequences
and
such that
(a),
,
(b) or
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ11_HTML.gif)
(c) or
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ12_HTML.gif)
We denote that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ13_HTML.gif)
Note that , as if
, then
, and one infers that
which implies that
. Hence
. Similarly, one can argue that
.
Now, we distinguish the following three cases.
Case 1.
If , then from (2.2)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ14_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ15_HTML.gif)
Clearly, there are infinite many such that at least one of the following four cases holds:
-
(1)
(2.13)
-
(2)
(2.14)
-
(3)
(2.15)
-
(4)
(2.16)
which implies , that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ20_HTML.gif)
From (1), (2), (3), and () it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ21_HTML.gif)
Similarly, if , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ22_HTML.gif)
If , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ23_HTML.gif)
Case 2.
If , then
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ24_HTML.gif)
which in turn yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ25_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ26_HTML.gif)
Now, proceeding as in Case 1, we have that (2.18) holds.
If , then
. We show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ27_HTML.gif)
Using (2.21), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ28_HTML.gif)
By noting that , one can conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ29_HTML.gif)
in view of Case 1.
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ30_HTML.gif)
and we proved (2.24).
Case 3.
If , then
. We show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ31_HTML.gif)
Since , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ32_HTML.gif)
From this, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ33_HTML.gif)
By noting that , one can conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ34_HTML.gif)
in view of Case 1.
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ35_HTML.gif)
and we proved (2.28).
Similarly, if , then
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ36_HTML.gif)
From this, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ37_HTML.gif)
By noting that , one can conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ38_HTML.gif)
in view of Case 1.
Thus, in all Cases 1–3, there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ39_HTML.gif)
and there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ40_HTML.gif)
Following the procedure of Assad and Kirk [13], it can easily be shown by induction that, for , there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ41_HTML.gif)
From (2.38) and by the triangle inequality, for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ42_HTML.gif)
From Remark 1.5 and Corollary 1.4(1), .
Thus, the sequence is a Cauchy sequence. Then, as noted in [14], there exists at least one subsequence
or
which is contained in
or
, respectively, and finds its limit
Furthermore, subsequences
and
both converge to
as
is a closed subset of complete cone metric space
. We assume that there exists a subsequence
for each
, then
. Since
as well as
are closed in
, and
is Cauchy in
, it converges to a point
. Let
, then
. Similarly,
a subsequence of Cauchy sequence
also converges to
as
is closed. Using (2.2), one can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ43_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ44_HTML.gif)
Let . Clearly at least one of the following four cases holds for infinitely many
:
-
(1)
(2.42)
-
(2)
(2.43)
-
(3)
(2.44)
-
(4)
(2.45)
In all cases we obtain for each
. Using Corollary 1.4(3) it follows that
or
. Thus,
, that is,
is a coincidence point of
,
.
Further, since Cauchy sequence converges to
and
,
, there exists
such that
. Again using (2.2), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ49_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ50_HTML.gif)
Hence, we get the following cases:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ51_HTML.gif)
Since , using Remark 1.6 and Corollary 1.4(3), it follows that
; therefore,
, that is,
is a coincidence point of
.
In case and
are closed in
,
or
. The analogous arguments establish (IV) and (V). If we assume that there exists a subsequence
with
as well
being closed in
, then noting that
is a Cauchy sequence in
, foregoing arguments establish (IV) and (V).
Suppose now that and
are weakly compatible pairs, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ52_HTML.gif)
Then, from (2.2),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ53_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ54_HTML.gif)
Hence, we get the following cases:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ55_HTML.gif)
Since , using Remark 1.6 and Corollary 1.4(3), it follows that
. Thus,
.
Similarly, we can prove that . Therefore
, that is,
is a common fixed point of
,
,
, and
.
Uniqueness of the common fixed point follows easily from (2.2).
The following example shows that in general ,
,
, and
satisfying the hypotheses of Theorem 2.3 need not have a common coincidence justifying two separate conclusions (IV) and (V).
Example 2.4.
Let ,
,
,
, and
defined by
, where
is a fixed function, for example,
. Then
is a complete cone metric space with a nonnormal cone having the nonempty interior. Define
,
,
, and
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ56_HTML.gif)
Since . Clearly, for each
and
there exists a point
such that
. Further,
,
,
, and
,
,
, and
are closed in
.
Also,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ57_HTML.gif)
Moreover, for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ58_HTML.gif)
that is, (2.2) is satisfied with .
Evidently, and
. Notice that two separate coincidence points are not common fixed points as
and
, which shows necessity of weakly compatible property in Theorem 2.3.
Next, we furnish an illustrate example in support of our result. In doing so, we are essentially inspired by Imdad and Kumar [12].
Example 2.5.
Let ,
,
, and
defined by
, where
is a fixed function, for example,
. Then
is a complete cone metric space with a nonnormal cone having the nonempty interior. Define
,
,
, and
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ59_HTML.gif)
Since . Clearly, for each
and
there exists a point
such that
. Further,
,
, and
.
Also,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ60_HTML.gif)
Moreover, if and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ61_HTML.gif)
Next, if , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ62_HTML.gif)
Finally, if , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ63_HTML.gif)
Therefore, condition (2.2) is satisfied if we choose . Moreover
is a point of coincidence as
as well as
whereas both the pairs
and
are weakly compatible as
and
. Also,
,
,
, and
are closed in
. Thus, all the conditions of Theorem 2.3 are satisfied and
is the unique common fixed point of
,
,
, and
. One may note that
is also a point of coincidence for both the pairs
and
.
Remark 2.6.
-
(1)
Setting
and
in Theorem 2.3, one deduces Theorem 2.1 due to [5].
-
(2)
Setting
and
in Theorem 2.3, we obtain the following result.
Corollary 2.7.
Let be a complete cone metric space and
a nonempty closed subset of
such that for each
and
there exists a point
(the boundary of
) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ64_HTML.gif)
Suppose that satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ65_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F983802/MediaObjects/13663_2009_Article_1376_Equ66_HTML.gif)
for all ,
,
, and
has the additional property that for each
,
,
has a unique fixed point.
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Acknowledgments
The authors would like to express their sincere appreciation to the referees for their very helpful suggestions and many kind comments. This project was supported by the National Natural Science Foundation of China (10461007 and 10761007) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (2008GZS0076 and 2009GZS0019).
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Huang, X., Zhu, C. & Wen, X. Common Fixed Point Theorem for Four Non-Self Mappings in Cone Metric Spaces. Fixed Point Theory Appl 2010, 983802 (2010). https://doi.org/10.1155/2010/983802
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DOI: https://doi.org/10.1155/2010/983802