Abstract
In this paper, some common fixed point theorems have been established for two Banach pairs of mappings with -contraction defined on a complex valued metric space satisfying contractive condition involving product. Some consequences which are associated with properties and are also obtained.
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Introduction
Fixed point theory which has important applications such as game theory, military, economics, statistics and medicine is one of the most famous theory in mathematics. It is well known that the Banach’s contraction principle is a fundamental result in fixed point theory. There are many generalizations of this principle. A new generalization of contraction mapping has been introduced and called -contraction mappings on metric spaces which are related with another function by Beiranvand [2]. In 2009, Morales and Rojas [7, 8] have extended -contraction mappings to cone metric spaces by proving fixed point theorems for -Kannan, -Chatterjea, -Zamfirescu, -weakly contraction mappings. Besides, Sintunavarat and Kumam [15, 16] use different types of -weak contractions to generalize some contractions which are existing in the literature. Subrahmanyam has initiated the concept of Banach operator of type in complete metric spaces. Afterwards, Chen and Li [4] have introduced the notion of Banach operator pairs as a new class of non-commuting maps and have proved various best approximation results using some common fixed point theorems for -nonexpansive mappings.
Many authors have generalized and established the notion of a metric spaces in the recent past such as rectangular metric spaces, semi-metric spaces, quasi-metric spaces, quasi-semi metric spaces, pseudo-metric spaces, 2-metric spaces, D-metric spaces, G-metric spaces, K-metric spaces, cone metric spaces and etc. Recently, Azam et al. [1]. have introduced the complex valued metric spaces which is a generalization of the metric space and also have obtained some fixed point results for a pair of mappings for contraction condition satisfying rational expressions which are not meaningful in cone metric spaces. Therefore, many results of analysis can not be generalized to cone metric spaces. Later, one can study the progresses of a host of results of analysis involving divisions in the framework of complex valued metric spaces. There exist various paper on complex valued metric spaces such as [1, 3, 9, 12, 18] and [13, 14].
The purpose of this paper is to prove common fixed point theorems for two Banach pairs of mappings satisfying contractive condition including product in complex valued metric space using - contraction. Also, we obtain some consequences related with and properties which are defined by Jeong and Rhoades [6].
Basic facts and definitions
Let be the set of complex numbers and . Define a partial order on as follows:
It follows that
if one of the following conditions is satisfied:
In particular, we write if and one of (i), (ii) and (iii) is satisfied and we write if only (iii) is satisfied. Note that
Definition
[18] Let and the ’max’ function for the partial order relation is defined on by:
-
i.
;
-
ii.
;
-
iii.
.
Using the previous definition, we have the following lemma:
Lemma 2.1
[18] Let and the partial order relation is defined on . Then, the following statements are obvious:
-
i.
If ;
-
ii.
If ;
-
iii.
If and so on.
Now, we give the definition of complex valued metric space which has been introduced by Azam et al. [1].
Definition
Let be a nonempty set. Suppose that the mapping satisfies:
- .:
-
, for all and if and only if ;
- .:
-
for all ;
- .:
-
, for all .
Then, is called a complex valued metric on and is called a complex valued metric space.
Let be a complex valued metric space and be a sequence in and . We say that the sequence converges to if for every , with there is such that for all , We denote this by , or as . The sequence is Cauchy sequence if for every , with there is such that for all , The metric space is a complete complex valued metric space, if every Cauchy sequence is convergent.
Now, we give some lemmas which we require to prove the main results.
Lemma 2.2
[1] Let be a complex valued metric space and let be a sequence in . Then, converges to if and only if as .
Lemma 2.3
[1] Let be a complex valued metric space and let be a sequence in . Then, is Cauchy sequence if and only if as .
Here, some essential notions are given about -contraction.
Definition
[2] Let be a metric space and be two functions. A mapping is said to be -contraction if there exists such that
for all .
Example 1
[10] Let be with the usual metric. Let define two mappings as
It is clear that, is not a contraction but it is contraction since,
Definition
[2] Let be a metric space. Then,
-
i.
A mapping is said to be sequentially convergent, if the sequence in is convergent whenever is convergent.
-
ii.
is said to be sub-sequentially convergent, if the sequence has a convergent subsequence whenever is convergent.
Definition
[17] Let be a self-mapping of a normed space . Then, is called a Banach operator of type if
for some and for all .
Definition
[4] Let and be self-mappings of a nonempty subset of a normed linear space . Then, is a Banach operator pair, if any one of the following conditions is satisfied:
-
i.
,
-
ii.
for each ,
-
iii.
for each ,
-
iv.
for some .
Now, we give an example which illustrates Banach operator pairs.
Example 2
Let be the set of complex numbers and define by
where and for all Then, is complex valued metric space. Suppose that , and be self-mappings of as follows:
Note that , then we have the following conditions:
-
i.
,
-
ii.
for ,
-
iii.
where ,
-
iv.
for ,
Hence, is a Banach pair. And also is said to be a Banach pair with . Moreover, the unique common fixed point of , and is in .
However, we give the following definitions which are and properties.
Definition
[6] If a map satisfies for each , then it is said to have property , where is the set of fixed points of the mapping . If for each , then we say that and have property .
It is obvious that, if is a map which has a fixed point , then is also fixed point of for every natural number . However, the converse is false and the related examples were given in [6].
Some fixed point and periodic point results
Theorem 3.1
Let , and be continuous self-mappings of a complete complex valued metric space . Assume that the mapping is an injective and sub-sequentially convergent. If the mappings , and satisfy
for all where with , then
-
i.
and have a unique common fixed point,
-
ii.
if and are Banach pairs, then , and have a unique common fixed point in ,
-
iii.
and have property .
Proof
(i)Take as an arbitrary element and define the sequences
for all Then,
using (1) and triangle inequality
which implies that
and
We deduce that
where . Continuing in this way, we have
for all .
So for any , we obtain
and so,
This implies that is a Cauchy sequence. Since is complete, there exists such that . Since is sub-sequentially convergent, has a convergent subsequence such that . As is continuous,
By the uniqueness of the limit, . Since and are continuous, and Again since is continuous,
Now, we prove . Assume that is even, then
By (1), for all , we have
Also, for every , we can write
Letting , we have
which amounts to say that
Because , , i.e., . As is injective, . Thus, is the fixed point of .
And to show we suppose that is odd, then
Now, using and triangle inequality, we get
Since , we obtain . Hence, As is injective, , i.e., is the fixed point of , too.
Now, we demonstrate that and have a unique common fixed point. For this, assume that is an another common fixed point of and .
Thus, we have Since , consequently provides that We know that is injective, is the unique common fixed point of and .
(ii) Since we assume that and are Banach pairs; and commute at the fixed point of and , respectively. This implies that for . So, which gives that is another fixed point of . It is true for , too. By the uniqueness of fixed point of , . Hence, , is unique common fixed point of , and in .
(iii) By (i), and have a common fixed point in . Define . Then, by
From (i), we know that
Let . Then,
which implies that
and
and then
Then, the following step is obviously recognized
where . Continuing in this way, we have
for all Since as , we get and . By the injectivity of , we get . Using , we also obtain and consequently and have the property
Using the same technique as in Theorem 3.1, the following two theorems can be proved.
Theorem 3.2
Let be continuous self-mappings on a complete complex valued metric also be injective and sub-sequentially convergent mapping satisfying the following inequality;
for all and with . Then, the following the statements hold:
-
i.
and have a unique common fixed point
-
ii.
if and are Banach pairs, then , and have a unique common fixed point in .
-
iii.
and have property .
Theorem 3.3
If , and an injective and sub-sequentially convergent mapping are self-mappings defined on a complete complex valued metric space satisfying
for all and . Then, the following hold:
-
i.
and have a unique common fixed point,
-
ii.
if and are Banach pairs, then , and have a unique common fixed point in .
-
iii.
and have property .
Theorem 3.4
Let , and be continuous self-mappings of a complete complex valued metric space . Assume that is an injective and sub-sequentially convergent mapping. If the mappings , and satisfy
for all and Then,
-
i.
and have a unique common fixed point,
-
ii.
if and are Banach pairs, then , and have a unique common fixed point in .
-
ii.
and have property .
Proof
Choose as an arbitrary element and define the sequences
for each Then, using (4) and triangle inequality
then we get
Case 1:Let be maximum, then
which is a contradiction.
Case 2: Suppose that be maximum, then it must be
and using the triangle inequality, we have
which is a contradiction too. Then, we have to investigate other case.
Case 3: Assume that is maximum, then
Thus, for all , we get
Since as , we can write for all where
So for any , because , using same procedure as in Theorem 3.1, we get is a Cauchy sequence. Since is complete, there exists such that . Since is sub-sequentially convergent, has a convergent subsequence such that . As is continuous,
By the uniqueness of the limit, . Since and are continuous, and Again since is continuous,
Hence, the existence and uniqueness of common fixed point of and is proved using the same method in Theorem 3.1.
(ii) Now, we show that , and have the unique common fixed point. we assume that and are Banach pairs; and commute at the fixed point of and , respectively. This implies that for . So, which gives that is another fixed point of . It is true for , too. By the uniqueness of fixed point of , . Hence, , is unique common fixed point of , and in .
(iii) and have the property , and is based on an argument similar to the process used in Theorem 3.1.
Corollary 3.5
Let be a complete complex valued metric space and . If there exists an injective and sub-sequentially mapping such that for all and with :
then
-
i.
has a unique fixed point,
-
ii.
if is a Banach pair, then , have a unique common fixed point in .
-
iii.
has property .
Proof
By taking in Theorem 3.2, (i) and (ii) can be obtained. Now, we prove that has property . Let . Then,
which implies that
Then,
where . Since as , we get and . By the injectivity of , , has property .
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The authors are grateful to the reviewers for their careful reviews and useful comments.
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Öztürk, M., Kaplan, N. Common fixed points of -contraction mappings in complex valued metric spaces. Math Sci 8, 129 (2014). https://doi.org/10.1007/s40096-014-0129-2
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DOI: https://doi.org/10.1007/s40096-014-0129-2