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Common Fixed Point Theorems for Four Mappings on Cone Metric Type Space
Fixed Point Theory and Applications volume 2011, Article number: 589725 (2011)
Abstract
In this paper we consider the so called a cone metric type space, which is a generalization of a cone metric space. We prove some common fixed point theorems for four mappings in those spaces. Obtained results extend and generalize well-known comparable results in the literature. All results are proved in the settings of a solid cone, without the assumption of continuity of mappings.
1. Introduction
Replacing the real numbers, as the codomain of a metric, by an ordered Banach space we obtain a generalization of metric space. Such a generalized space, called a cone metric space, was introduced by Huang and Zhang in [1]. They described the convergence in cone metric space, introduced their completeness, and proved some fixed point theorems for contractive mappings on cone metric space. Cones and ordered normed spaces have some applications in optimization theory (see [2]). The initial work of Huang and Zhang [1] inspired many authors to prove fixed point theorems, as well as common fixed point theorems for two or more mappings on cone metric space, for example, [3–14].
In this paper we consider the so-called a cone metric type space, which is a generalization of a cone metric space and prove some common fixed point theorems for four mappings in those spaces. Obtained results are generalization of theorems proved in [13]. For some special choices of mappings we obtain theorems which generalize results from [1, 8, 15]. All results are proved in the settings of a solid cone, without the assumption of continuity of mappings.
The paper is organized as follows. In Section 2 we repeat some definitions and well-known results which will be needed in the sequel. In Section 3 we prove common fixed point theorems. Also, we presented some corollaries which show that our results are generalization of some existing results in the literature.
2. Definitions and Notation
Let 
be a real Banach space and 
a subset of 
. By 
we denote zero element of 
and by 
the interior of 
. The subset 
is called a cone if and only if
(i)
is closed, nonempty and 
;
(ii)
, 
, and 
imply 
;
(iii)
.
For a given cone 
, a partial ordering 
with respect to 
is introduced in the following way: 
if and only if 
. One writes 
to indicate that 
, but 
. If 
, one writes 
.
If 
, the cone 
is called solid.
In the sequel we always suppose that 
is a real Banach space, 
is a solid cone in 
, and 
is partial ordering with respect to 
.
Analogously with definition of metric type space, given in [16], we consider cone metric type space.
Definition 2.1.
Let 
be a nonempty set and 
a real Banach space with cone 
. A vector-valued function 
is said to be a cone metric type function on 
with constant 
if the following conditions are satisfied:
()
for all 
and 
if and only if 
;
()
for all 
;
()
for all 
.
The pair 
is called a cone metric type space (in brief CMTS).
Remark 2.2.
For 
in Definition 2.1 we obtain a cone metric space introduced in [1].
Definition 2.3.
Let 
be a CMTS and 
a sequence in 
.
()
converges to 
if for every 
with 
there exists 
such that 
for all 
. We write 
, or 
, 
.
()If for every 
with 
there exists 
such that 
for all 
, then 
is called a Cauchy sequence in 
.
If every Cauchy sequence is convergent in 
, then 
is called a complete CMTS.
Example 2.4.
Let 
be orthonormal basis of 
with inner product 
. Let 
, and define
where 
represents class of element 
with respect to equivalence relation of functions equal almost everywhere. We choose 
and
We show that 
is a solid cone. Let 
, 
, with property 
. Since scalar product is continuous, we get 
, 
. Clearly, it must be 
, 
, and, hence, 
, that is, 
is closed. It is obvious that 
, and for 
, and all 
, we have 
, 
. Finally, if 
we have 
and 
, 
, and it follows that 
, 
, and, since 
is complete, we get 
. Let us choose 
. It is obvious that 
, since if not, for every 
there exists 
such that 
. If we choose 
, we conclude that it must be 
, hence 
, which is contradiction.
Finally, define 
by
Then it is obvious that 
is CMTS with 
. Let 
, 
, 
be functions such that 
, 
, 
, and 
, 
, with 
give 
, 
, and 
, which proves 
, but 
.
The following properties are well known in the case of a cone metric space, and it is easy to see that they hold also in the case of a CMTS.
Lemma 2.5.
Let 
be a CMTS over-ordered real Banach space 
with a cone 
. The following properties hold 
.
()If 
and 
, then 
.
()If 
for all 
, then 
.
()If 
, where 
and 
, then 
.
()Let 
in 
and let 
. Then there exists positive integer 
such that 
for each 
.
Definition 2.6 (see [17]).
Let 
be mappings of a set 
. If 
for some 
, then 
is called a coincidence point of 
and 
, and 
is called a point of coincidence of 
and 
.
Definition 2.7 (see [17]).
Let 
and 
be self-mappings of set 
and 
. The pair 
is called weakly compatible if mappings 
and 
commute at all their coincidence points, that is, if 
for all 
.
Lemma 2.8 (see [5]).
Let 
and 
be weakly compatible self-mappings of a set 
. If 
and 
have a unique point of coincidence 
, then 
is the unique common fixed point of 
and 
.
3. Main Results
Theorem 3.1.
Let 
be a CMTS with constant 
and 
a solid cone. Suppose that self-mappings 
are such that 
, 
and that for some constant 
for all 
there exists
such that the following inequality
holds. If one of 
, 
, 
, or 
is complete subspace of 
, then 
and 
have a unique point of coincidence in 
. Moreover, if 
and 
are weakly compatible pairs, then 
, 
, 
, and 
have a unique common fixed point.
Proof.
Let us choose 
arbitrary. Since 
, there exists 
such that 
. Since 
, there exists 
such that 
. We continue in this manner. In general, 
is chosen such that 
, and 
is chosen such that 
.
First we prove that
where 
, which will lead us to the conclusion that 
is a Cauchy sequence, since 
(it is easy to see that 
). To prove this, it is necessary to consider the cases of an odd integer 
and of an even 
.
For 
, 
, we have 
, and from (3.2) there exists
such that 
. Thus we have the following three cases:
(i)
;
(ii)
, which, because of property 
, implies 
;
(iii)
, that is, by using 
,
which implies 
.
Thus, inequality (3.3) holds in this case.
For 
, 
, we have
where
Thus we have the following three cases:
(i)
;
(ii)
, which implies 
;
(iii)
, which implies 
.
So, inequality (3.3) is satisfied in this case, too.
Therefore, (3.3) is satisfied for all 
, and by iterating we get
Since 
, for 
we have
Now, by 
and 
, it follows that for every 
there exists positive integer 
such that 
for every 
, so 
is a Cauchy sequence.
Let us suppose that 
is complete subspace of 
. Completeness of 
implies existence of 
such that 
. Then, we have
that is, for any 
, for sufficiently large 
we have 
. Since 
, there exists 
such that 
. Let us prove that 
. From 
and (3.2), we have
where
Therefore we have the following four cases:
(i)
, as 
;
(ii)
, as 
;
(iii)
, that is,
(iv)
, that is, because of 
,
which implies
since from 
and 
we have 
, and therefore 
.
Therefore, 
for each 
. So, by 
we have 
, that is, 
, 
is a coincidence point, and 
is a point of coincidence of 
and 
.
Since 
, there exists 
such that 
. Let us prove that 
. From 
and (3.2), we have
where
Therefore we have the following four cases:
(i)
;
(ii)
;
(iii)
;
(iv)
.
By the same arguments as above, we conclude that 
, that is, 
. So, 
is a point of coincidence of 
and 
, too.
Now we prove that 
is unique point of coincidence of pairs 
and 
. Suppose that there exists 
which is also a point of coincidence of these four mappings, that is, 
. From (3.2),
where
Using 
we get 
, that is, 
. Therefore, 
is the unique point of coincidence of pairs 
and 
. If these pairs are weakly compatible, then 
is the unique common fixed point of 
, 
, 
, and 
, by Lemma 2.8.
Similarly, we can prove the statement in the cases when 
, 
, or 
is complete.
We give one simple, but illustrative, example.
Example 3.2.
Let 
, 
, and 
. Let us define 
for all 
. Then 
is a CMTS, but it is not a cone metric space since the triangle inequality is not satisfied. Starting with Minkowski inequality (see [18]) for 
, by using the inequality of arithmetic and geometric means, we get
Here, 
.
Let us define four mappings 
as follows:
where 
, 
, 
, and 
. Since 
we have trivially 
and 
. Also, 
is a complete space. Further, 
, that is, there exists 
such that (3.2) is satisfied.
According to Theorem 3.1, 
and 
have a unique point of coincidence in 
, that is, there exists unique 
and there exist 
such that 
. It is easy to see that 
, 
, and 
.
If 
is weakly compatible pair, we have 
, which implies 
, that is, 
. Similarly, if 
is weakly compatible pair, we have 
, which implies 
, that is, 
. Then 
, and 
is the unique common fixed point of these four mappings.
The following two theorems can be proved in the same way as Theorem 3.1, so we omit the proofs.
Theorem 3.3.
Let 
be a CMTS with constant 
and 
a solid cone. Suppose that self-mappings 
are such that 
, 
and that for some constant 
for all 
there exists
such that the following inequality
holds. If one of 
, 
, 
, or 
is complete subspace of 
, then 
and 
have a unique point of coincidence in 
. Moreover, if 
and 
are weakly compatible pairs, then 
, 
, 
, and 
have a unique common fixed point.
Theorem 3.4.
Let 
be a CMTS with constant 
and 
a solid cone. Suppose that self-mappings 
are such that 
, 
and that for some constant 
for all 
there exists
such that the following inequality
holds. If one of 
, 
, 
, or 
is complete subspace of 
, then 
and 
have a unique point of coincidence in 
. Moreover, if 
and 
are weakly compatible pairs, then 
, 
, 
, and 
have a unique common fixed point.
Theorems 3.1 and 3.4 are generalizations of [13, Theorem 2.2]. As a matter of fact, for 
, from Theorems 3.1 and 3.4, we get [13, Theorem 2.2].
If we choose 
and 
, from Theorems 3.1, 3.3, and 3.4 we get the following results for two mappings on CMTS.
Corollary 3.5.
Let 
be a CMTS with constant 
and 
a solid cone. Suppose that self-mappings 
are such that 
and that for some constant 
for all 
there exists
such that the following inequality
holds. If 
or 
is complete subspace of 
, then 
and 
have a unique point of coincidence in 
. Moreover, if 
is a weakly compatible pair, then 
and 
have a unique common fixed point.
Corollary 3.6.
Let 
be a CMTS with constant 
and 
a solid cone. Suppose that self-mappings 
are such that 
and that for some constant 
for all 
there exists
such that the following inequality
holds. If 
or 
is complete subspace of 
, then 
and 
have a unique point of coincidence in 
. Moreover, if 
is a weakly compatible pair, then 
and 
have a unique common fixed point.
Corollary 3.7.
Let 
be a CMTS with constant 
and 
a solid cone. Suppose that self-mappings 
are such that 
and that for some constant 
for all 
there exists
such that the following inequality
holds. If 
or 
is complete subspace of 
, then 
and 
have a unique point of coincidence in 
. Moreover, if 
is a weakly compatible pair, then 
and 
have a unique common fixed point.
Theorem 3.8.
Let 
be a CMTS with constant 
and 
a solid cone. Suppose that self-mappings 
are such that 
, 
and that there exist nonnegative constants 
, 
, satisfying
such that for all 
inequality
holds. If one of 
, 
, 
, or 
is complete subspace of 
, then 
and 
have a unique point of coincidence in 
. Moreover, if 
and 
are weakly compatible pairs, then 
, 
, 
, and 
have a unique common fixed point.
Proof.
We define sequences 
and 
as in the proof of Theorem 3.1. First we prove that
where
which implies that 
is a Cauchy sequence, since, because of (3.32), it is easy to check that 
. To prove this, it is necessary to consider the cases of an odd and of an even integer 
.
For 
, 
, we have 
, and from (3.33) we have
that is,
Therefore,
that is, inequality (3.34) holds in this case.
Similarly, for 
, 
, we have 
, and from (3.33) we have
that is,
Thus,
and inequality (3.34) holds in this case, too.
By the same arguments as in Theorem 3.1 we conclude that 
is a Cauchy sequence.
Let us suppose that 
is complete subspace of 
. Completeness of 
implies existence of 
such that 
. Then, we have
that is, for any 
, for sufficiently large 
we have 
. Since 
, there exists 
such that 
. Let us prove that 
. From 
and (3.33), we have
The sequence 
converges to 
, so for each 
there exists 
such that for every 
because of (3.32). Now, by 
it follows that 
, that is, 
. So, we have 
, that is, 
is a coincidence point, and 
is a point of coincidence of mappings 
and 
.
Since 
, there exists 
such that 
. Let us prove that 
, too. From 
and (3.33), we have
and by the same arguments as above, we conclude that 
, that is, 
. Thus, 
is a point of coincidence of mappings 
and 
, too.
Suppose that there exists 
which is also a point of coincidence of these four mappings, that is, 
. From (3.33) we have
and (because of 
) it follows that 
. Therefore, 
is the unique point of coincidence of pairs 
and 
, and we have 
. If 
and 
are weakly compatible pairs, then 
is the unique common fixed point of 
, 
, 
, and 
, by Lemma 2.8.
The proofs for the cases in which 
, 
, or 
is complete are similar.
Theorem 3.8 is a generalization of [13, Theorem 2.8]. Choosing 
from Theorem 3.8 we get the following corollary.
Corollary 3.9.
Let 
be cone metric space and 
a solid cone. Suppose that self-mappings 
are such that 
, 
and that there exist nonnegative constants 
, 
, satisfying 
, such that for all 
inequality
holds. If one of 
, 
, 
, or 
is complete subspace of 
, then 
and 
have a unique point of coincidence in 
. Moreover, if 
and 
are weakly compatible pairs, then 
, 
, 
, and 
have a unique common fixed point.
If we choose 
and 
, from Theorem 3.8, we get the following result for two mappings on CMTS.
Corollary 3.10.
Let 
be a CMTS with constant 
and 
a solid cone. Suppose that self-mappings 
are such that 
and that there exist nonnegative constants 
, 
, satisfying
such that for all 
inequality
holds. If one of 
or 
is complete subspace of 
, then 
and 
have a unique point of coincidence in 
. Moreover, if 
is a weakly compatible pair, then 
and 
have a unique common fixed point.
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Acknowledgments
The authors are indebted to the referees for their valuable suggestions, which have contributed to improve the presentation of the paper. The first two authors were supported in part by the Serbian Ministry of Science and Technological Developments (Grant no. 174015).
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Cvetković, A., Stanić, M., Dimitrijević, S. et al. Common Fixed Point Theorems for Four Mappings on Cone Metric Type Space. Fixed Point Theory Appl 2011, 589725 (2011). https://doi.org/10.1155/2011/589725
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DOI: https://doi.org/10.1155/2011/589725