Abstract
We prove common fixed point theorem for coincidentally commuting nonself mappings satisfying generalized contraction condition of Ćirić type in cone metric space. Our results generalize and extend all the recent results related to non-self mappings in the setting of cone metric space.
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1. Introduction
Recently, Huang and Zhang [1] introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. The category of cone metric spaces is larger than metric spaces and there are different types of cones. Subsequently, many authors like Abbas and Jungck [2], Abbas and Rhoades [3], Ilić and Rakočević [4], Raja andVaezpour[5] have generalized the results of Huang and Zhang [1] and studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces. However, authors like Janković et al. [6], Jungck et al. [7], Kadelburg et al. [8, 9], Radenović and Rhoades [10], Rezapour and Hamlbarani [11] studied the existence of common fixed points of a pair of self and nonself mappings satisfying a contractive type condition in the situation in which the cone does not need be normal.
The study of fixed point theorems for nonself mappings in metrically convex metric spaces was initiated by Assad and Kirk [12]. Utilizing the induction method of Assad and Kirk [12], many authors like Assad [13], Ćirić [14], Hadžić [15], Hadžić and Gajić [16], Imdad and Kumar [17], Rhoades [18, 19] have obtained common fixed point in metrically convex spaces. Recently, Ćirić and Ume [20] defined a wide class of multivalued nonself mappings which satisfy a generalized contraction condition and proved a fixed point theorem which generalize the results of Itoh [21] and Khan [22].
Very recently, Radenović and Rhoades [10] extended the fixed point theorem of Imdad and Kumar [17] for a pair of nonself mappings to nonnormal cone metric spaces. Janković et al. [6] proved new common fixed point results for a pair of nonself mappings defined on a closed subset of metrically convex cone metric space which is not necessarily normal by adapting Assad-Kirk's method.
The aim of this paper is to prove common fixed point theorems for coincidentally commuting nonself mappings satisfying a generalized contraction condition of Ćirić type in the setting of cone metric spaces. Our results generalize mainly results of Ćirić and Ume [20] and all the recent results related to nonself mappings in the setting of cone metric space.
2. Definitions and Preliminaries
We recall some basic definitions and preliminaries that will be needed in the sequel.
Definition 2.1 (see [1]).
Let be a real Banach space. A subset
of
is called a Cone if and only if
(1) is nonempty, closed and
;
(2),
and
;
(3).
For a given cone , a partial ordering is defined as
on
with respect to
by
, if and only if
. It is denoted as
to indicate that
but
, while
will stand for
, where
denotes the interior of
.
The cone is called normal, if there is a number
such that for all
,
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ1_HTML.gif)
The least positive number satisfying (2.1) is called the normal constant of
. It is clear that
. There are nonnormal cones also.
The definition of a cone metric space given by Huang and Zhang [1] is as follows.
Definition 2.2 (see [1]).
Let be a nonempty set. Suppose that
is a real Banach space,
is a cone with
and
is a partial ordering with respect to
.
If the mapping satisfies the following:
(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.
Example 2.3 (see [1]).
Let ,
and
such that
, where
is a constant. Then
is a cone metric space.
Definition 2.4 (see [1]).
Let be a cone metric space and
a sequence in
. Then, one has the following.
(1) converges to
, if for every
with
, there is
such that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ2_HTML.gif)
It is denoted by or
,
.
(2)If for any , there is a number
such that for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ3_HTML.gif)
then is called a Cauchy sequence in
.
(3) is a complete cone metric space, if every Cauchy sequence in
is convergent.
(4)A self mapping is said to be continuous at a point
, if
implies that
for every
in
.
The following two lemmas of Huang and Zhang [1] will be required in the sequel.
Lemma 2.5 (see [1]).
Let be a cone metric space and
a normal cone with normal constant
. A sequence
in
converges to
if and only if
as
.
Lemma 2.6 (see [1]).
Let be a cone metric space and
a normal cone with normal constant
. A sequence
in
is a Cauchy sequence if and only if
as
.
The following Corollary of Rezapour [23] will be needed in the sequel.
Corollary 2.7 (see [23]).
Let , the real Banach space.
(i)If and
, then
.
(ii)If and
, then
.
(iii)If for each
, then
.
The following remarks of Radenović and Rhoades [10] will be needed in the sequel.
Remark 2.8 (see [10]).
If ,
and
, then there exists
such that for all
, it follows that
.
Remark 2.9 (see [10]).
If and
, then
, where
is a sequence and
is a given point in
.
Remark 2.10 (see [10]).
If and
,
, then
for each cone
.
Remark 2.11 (see [10]).
If is a real Banach space with a cone
and if
, where
and
, then
.
3. Main Results
In the following, we suppose that is a Banach space,
is a cone in
with
and
is partial ordering with respect to
.
Theorem 3.1.
Let be a complete cone metric space and
a nonempty closed subset of
such that for each
and
there exists a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ4_HTML.gif)
Suppose that are two nonself mappings satisfying for all
with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ5_HTML.gif)
and are nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ6_HTML.gif)
Also assume that
(i);
(ii);
(iii) is closed in
;
Then there exists a coincidence point of and
in
. Moreover, if
and
are coincidentally commuting, then
and
have a unique common fixed point in
.
Proof.
Two sequences and
are constructed in the following way. Let
. As
, by (i) there exists a point
such that
. Since
, from (ii) it follows that
. Let
be such that
. Since
, there exists
such that
.
If , 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%2F408086/MediaObjects/13663_2009_Article_1278_Equ7_HTML.gif)
Since , there exists a point
such that
and thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ8_HTML.gif)
Assume that .
Thus repeating the arguments, two sequences and
are obtained such that
(i);
(ii);
(iii) whenever
, then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ9_HTML.gif)
Next, we claim that is a Cauchy sequence in
. The following are derived. Let us denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ10_HTML.gif)
Obviously, two consecutive terms cannot lie in . Note that, if
, then
and
belong to
. Now, three cases are distinguished.
Case 1.
If , then
. Now from (3.2),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ11_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ12_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ13_HTML.gif)
Now foursubcasesarise.
Subcase 1.1.
If and
, then (3.8) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ14_HTML.gif)
Subcase 1.2.
If and
, then (3.8) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ15_HTML.gif)
Subcase 1.3.
If and
, then (3.8) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ16_HTML.gif)
Subcase 1.4.
If and
, then (3.8) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ17_HTML.gif)
Combining allSubcases1.1, 1.2, 1.3, and 1.4, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ18_HTML.gif)
where . Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ19_HTML.gif)
Case 2.
If , then
. Now,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ20_HTML.gif)
Proceeding as in Case 1,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ21_HTML.gif)
Case 3.
If , then
,
and
. Now,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ22_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ23_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ24_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ25_HTML.gif)
Again foursubcasesarise.
Subcase 3.1.
If , then (3.20) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ26_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ27_HTML.gif)
where using Case 2,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ28_HTML.gif)
Then (3.24) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ29_HTML.gif)
Subcase 3.2.
If , then (3.20) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ30_HTML.gif)
Proceeding as in Subcase 3.1, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ31_HTML.gif)
Subcase 3.3.
If , then (3.20) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ32_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ33_HTML.gif)
where using Case 2,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ34_HTML.gif)
Then (3.30) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ35_HTML.gif)
Subcase 3.4.
If , then (3.20) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ36_HTML.gif)
Proceeding as in Subcase 3.1, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ37_HTML.gif)
Combining all fourSubcases3.1, 3.2, 3.3, and 3.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ38_HTML.gif)
where by (3.3). Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ39_HTML.gif)
Now, combining main Cases 1, 2, and 3, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ40_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ41_HTML.gif)
Following the procedure of Assad and Kirk [12], it can be easily shown by induction that for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ42_HTML.gif)
By triangle inequality, for , it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ43_HTML.gif)
From Remark 2.9, , which implies by Definition 2.4(2) that
is a Cauchy sequence in
which is a closed subset of the complete cone metric space and hence is complete. Then there exists a point
such that
as
. Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ44_HTML.gif)
Since , there exists a point
such that
. By the construction of
, it was seen that there exists a subsequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ45_HTML.gif)
We will prove that . Consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ46_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ47_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ48_HTML.gif)
Now again four cases arise.
Case 1.
If , then (3.43) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ49_HTML.gif)
Case 2.
If , then (3.43) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ50_HTML.gif)
Case 3.
If , then (3.43) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ51_HTML.gif)
Case 4.
If , then (3.43) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ52_HTML.gif)
Combining Cases 1, 2, 3, and 4, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ53_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ54_HTML.gif)
where .
Let be given with
. From
as
and Definition 2.4(1),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ55_HTML.gif)
From as
and by Definition 2.4 (1),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ56_HTML.gif)
From the definition of convergence in cone metric space and by (3.52) and (3.53), inequality (3.43) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ57_HTML.gif)
Therefore, for each
. Then by (iii) of Corollary 2.7, we have
, that is,
which implies that
is the coincidence point of
and
.
Since and
are coincidentally commuting,
for
which implies
. Consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ58_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ59_HTML.gif)
Thus and
. Two cases arise.
Case 1.
If and
, then (3.55) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ60_HTML.gif)
Case 2.
If and
, then (3.55) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ61_HTML.gif)
Combining Cases 1 and 2, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ62_HTML.gif)
Since by (3.3), it follows from Remark 2.11 that
which implies that
. Thus
.
Uniqueness: if is another common fixed point of
and
in
, then
. Now by (3.2), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ63_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ64_HTML.gif)
Thus and
. Two cases arise.
Case 1.
If and
, then (3.60) becomes
. Since
, by Remark 2.11 we have
which implies that
is the unique common fixed point of
and
.
Case 2.
If and
, then (3.60) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ65_HTML.gif)
Since , by Remark 2.11 we have
which implies
is the unique common fixed point of
and
. Hence
is the unique common fixed point of
and
in
.
The following example illustrates Theorem 3.1.
Example 3.2.
Let ,
,
,
and
. Define two nonself mappings
as
and
for all
.
Now let us see that conditions (i)–(iii) in Theorem 3.1 are satisfied.
It may be seen that and
. Then
and
. Also,
as
. Moreover
is closed in
.
Next, we shall see that inequality (3.2) is satisfied by taking and
. It is easy to see that
.
Now, LHS of inequality (3.2) is . Taking
and
, it follows that
.
Next, RHS of inequality (3.2) is , where
,
and
. Then RHS of inequality (3.2) is
if
and
. Thus LHS of inequality (3.2)
RHS of inequality (3.2). Similarly, LHS of inequality (3.2)
RHS of inequality (3.2) for all possible cases of
and
. Thus all the conditions of Theorem 3.1 are satisfied. Hence "0" is the unique common fixed point of
and
in
.
Corollary 3.3.
Let be a complete cone metric space and
a nonempty closed subset of
such that for each
and
there exists a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ66_HTML.gif)
Suppose that is a nonself mapping satisfying for all
with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F408086/MediaObjects/13663_2009_Article_1278_Equ67_HTML.gif)
and are nonnegative real numbers such that
. Also assume that
. Then there exists a unique fixed point of
in
.
Proof.
The proof of this corollary follows by taking , the identity mapping of
in Theorem 3.1.
Remark 3.4.
Our results generalize the results of Radenović and Rhoades [10] and Janković et al. [6] and extend the results of Ćirić and Ume [20] to cone metric space for single valued mappings.
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The authors would like to thank the referees for their valuable suggestions which lead to the improvement of the presentation of the paper.
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Sumitra, R., Rhymend Uthariaraj, V., Hemavathy, R. et al. Common Fixed Point Theorem for Non-Self Mappings Satisfying Generalized Ćirić Type Contraction Condition in Cone Metric Space. Fixed Point Theory Appl 2010, 408086 (2010). https://doi.org/10.1155/2010/408086
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DOI: https://doi.org/10.1155/2010/408086