Abstract
The purpose of this work is to introduce new types of contraction mappings in the sense of a multiplicative metric space. Fixed point results for these contraction mappings in multiplicative metric spaces are obtained. Our presented results generalize, extend, and improve results on the topic in the literature. Moreover, our results cannot be directly obtained as a consequence from the corresponding results in metric spaces. We also state some illustrative examples to claim that our results properly generalize some results in the literature. We apply our main results for proving a fixed point theorem involving a cyclic mapping.
MSC:47H09, 47H10.
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1 Introduction and preliminaries
Throughout this paper, we denote by ℕ, , and ℝ the sets of positive integers, positive real numbers, and real numbers, respectively.
The Banach-contraction mapping principle is one of the pivotal results of analysis. It is widely considered as the source of metric fixed point theory. Also its significance lies in its vast applicability in a number of branches of mathematics. This principle is given by the next theorem.
Theorem 1.1 ([1])
Let be a complete metric spaces and be a Banach-contraction mapping, i.e.,
for all , where . Then T has a unique fixed point.
Theorem 1.1 was used to establish the existence of a solution for an integral equation. Since then, because of its simplicity and usefulness, it has become a very famous and popular tool in solving existence problems in many branches of mathematical analysis.
In 1968, Kannan [2, 3] introduced a new type of contraction mapping, which is called the Kannan-contraction type. He also established a fixed point result for such a type.
Let be a complete metric spaces and be a Kannan-contraction mapping, i.e.,
for all , where . Then T has a unique fixed point.
The Kannan fixed point theorem is very important because Subrahmanyam [4] proved that the Kannan theorem characterizes the metric completeness of underlying spaces, that is, a metric space X is complete if and only if every Kannan-contraction mapping on X has a fixed point. There is a large literature dealing with Kannan-contraction mappings and their generalizations, some of which are found in [5, 6], and [7].
A similar contractive condition and fixed point result for this contraction has been introduced by Chatterjea [8].
Theorem 1.3 ([8])
Let be a complete metric spaces and be a Chatterjea-contraction mapping, i.e.,
for all , where . Then T has a unique fixed point.
We see that the conditions (1.1), (1.2), and (1.3) are interesting to study because all conditions are independent.
In 2008, Bashirov et al. [9] defined a new distance, the so-called multiplicative distance, by using the concept of a multiplicative absolute value. After that, by using the idea of a multiplicative distance, Özavşar and Çevikel [10] studied multiplicative metric spaces and some topological properties.
Definition 1.4 ([9])
Let X be a nonempty set. A mapping is said to be a multiplicative metric if it satisfies the following conditions:
-
1.
for all and if and only if ,
-
2.
for all ,
-
3.
for all .
Also, is called a multiplicative metric space.
Example 1.5 ([9])
Let be defined as follows:
where and is defined as follows:
It is easy to see that all conditions of the multiplicative metric are satisfied.
Example 1.6 ([9])
Let be a fixed real number. Then defined by
where , holds for the multiplicative metric conditions.
Remark 1.7 One can extend the multiplicative metric in Example 1.6 to by the following definition:
where .
Definition 1.8 ([10])
Let be a multiplicative metric space, and . We now define a set
which is called a multiplicative open ball of radius ε with center x. Similarly, one can describe a multiplicative closed ball as
Definition 1.9 ([10])
Let be a multiplicative metric space, be a sequence in X, and . If, for every multiplicative open ball , there exists a natural number N such that , then the sequence is said to be multiplicative convergent to x, denoted by as .
Lemma 1.10 ([10])
Let be a multiplicative metric space, be a sequence in X and . Then
Lemma 1.11 ([10])
Let be a multiplicative metric space and be a sequence in X. If the sequence is multiplicative convergent, then the multiplicative limit point is unique.
Definition 1.12 ([10])
Let be a multiplicative metric space and be a sequence in X. The sequence is called a multiplicative Cauchy sequence if, for all , there exists such that for .
Lemma 1.13 ([10])
Let be a multiplicative metric space and be a sequence in X. Then is a multiplicative Cauchy sequence if and only if as .
Definition 1.14 ([10])
Let be a multiplicative metric space. The multiplicative metric spaces X is said to be complete if and only if every Cauchy sequence in X for all converges in X.
Definition 1.15 ([10])
Let be a multiplicative metric space. A point is said to be a multiplicative limit point of if and only if for every . The set of all multiplicative limit points of the set S is denoted by .
Definition 1.16 ([10])
Let be a multiplicative metric space. We call a set multiplicative closed in if S contains all of its multiplicative limit points.
Özavşar and Çevikel also introduced the concepts of Banach-contraction, Kannan-contraction, and Chatterjea-contraction mappings in the sense of multiplicative metric spaces.
Definition 1.17 ([10])
Let be a multiplicative metric space. A self-mapping f is said to be multiplicative Banach-contraction if
for all , where .
Definition 1.18 ([10])
Let be a multiplicative metric space. A self-mapping f is said to be multiplicative Kannan-contraction if
for all , where .
Definition 1.19 ([10])
Let be a multiplicative metric space. A self-mapping f is said to be a multiplicative Chatterjea-contraction if
for all , where .
By using these ideas, they proved some fixed point theorems on complete multiplicative metric spaces. In fact, fixed point results in the framework of multiplicative metric spaces of Özavşar and Çevikel [10] can be directly obtained as a consequence from the corresponding results in metric spaces.
In this paper, we introduce new types of nonlinear mappings, the so-called -Banach-contraction, -Kannan-contraction and -Chatterjea-contraction mappings, in the sense of multiplicative metric spaces and establish the existence of fixed point theorems for such mappings in multiplicative metric spaces by using the concept of cyclic -admissible mappings.
Definition 1.20 ([11])
Let X be a nonempty set, f be a self-mapping on X, and be two mappings. We say that f is a cyclic -admissible mapping if
and
We also give the example of a nonlinear mapping which is not in the range of application of the results of Özavşar and Çevikel [10], but it can be applied to our results. Our fixed point results generalize, extend, and improve results on the topic in the literature. Moreover, our results cannot be directly obtained as a consequence of the corresponding results in metric spaces. Finally, we apply our main results for proving a fixed point theorem involving a cyclic mapping.
2 Fixed point results in multiplicative metric spaces
First of all, we will introduce the concept of a -Banach-contraction mapping in the sense of a multiplicative metric distance.
Definition 2.1 Let be a multiplicative metric space and let be two mappings. The mapping is said to be a multiplicative -Banach-contraction if
for all , where .
Example 2.2 Let and be defined as follows:
for all , where is defined by
Then is a multiplicative metric space.
Define the mappings and as follows:
and
Now we will show that f is a multiplicative -Banach-contraction with .
For and , we get
Similarly, for and , we get
In the other cases, it is easy to see that condition (2.1) holds since . Therefore,
for all , where . Hence, f is a multiplicative -Banach-contraction mapping.
Remark 2.3 In Example 2.2, f is not a multiplicative Banach-contraction mapping. Indeed, for and , we get
for all . Therefore, the class of multiplicative -Banach-contraction mappings is a real wider class of Banach-contraction mappings.
Next, we give some fixed point result for multiplicative -Banach-contraction mappings in complete multiplicative metric spaces.
Theorem 2.4 Let be a complete multiplicative metric space and be a multiplicative -Banach-contraction mapping. Suppose that the following conditions hold:
-
(1)
there exists such that and ;
-
(2)
f is a cyclic -admissible mapping;
-
(3)
one of the following conditions holds:
(3.1) f is continuous;
(3.2) if is a sequence in X such that as and for all , then .
Then f has a fixed point. Furthermore, if and for all fixed point , then f has a unique fixed point.
Proof Starting from a point in condition (1), we get and . We will construct the iterative sequence , where for all . Since f is a cyclic -admissible mapping, we have
and
By a similar method, we get
for all . This implies that
for all . From the -Banach-contractive condition of f, we have
for all . Let such that , then we get
Letting , we get and so the sequence is multiplicative Cauchy. From the completeness of X, there exists such that as .
Now, we assume that f is continuous. Hence, we obtain
Next, we will assume that condition (3.2) holds. Hence . Then we have, for each ,
Letting , we get , that is, . This shows that z is a fixed point of f.
Finally, we show that z is the unique fixed point of f. Assume that y is another fixed point of f. From the hypothesis, we find that and , and hence
This shows that and then . Therefore, z is the unique fixed point of f. This completes the proof. □
Now, we give some illustrative examples to the claim that our results properly generalize the results of Özavşar and Çevikel [10].
Example 2.5 Let and be defined as follows:
for all , where is defined by
It is easy to see that is a complete multiplicative metric space.
Define mappings and as follows:
and
From Remark 2.3, we can see that f is not a multiplicative Banach-contraction mapping. Therefore, the results of Özavşar and Çevikel [10] cannot be used for this case.
Here, we show that by Theorem 2.4 can be guaranteed the existence of a fixed point. From Example 2.2, we find that f is a multiplicative -Banach-contraction with . It is easy to see that there exists such that and . This implies that condition (1) in Theorem 2.4 holds. Also, it is easy to prove that f is a cyclic -admissible mapping. Furthermore, it is easy to see that condition (3.2) in Theorem 2.4 holds.
Therefore, all the conditions of Theorem 2.4 hold and so f has a unique fixed point, .
Corollary 2.6 ([10])
Let be a complete multiplicative metric space and be a multiplicative Banach-contraction mapping. Then f has a unique fixed point. Moreover, for any , the iterative sequence converges to the fixed point.
Proof Setting and for all in Theorem 2.4, we get this result. □
Here we give the concepts of multiplicative -Kannan-contraction and multiplicative -Chatterjea-contraction mappings. Also, we establish the fixed point result for these mappings.
Definition 2.7 Let be a multiplicative metric space and let be two mappings. The mapping is said to be a multiplicative -Kannan-contraction if
for all , where .
Definition 2.8 Let be a multiplicative metric spaces and let be two mappings. The mapping is said to be a multiplicative -Chatterjea-contraction if
for all , where .
Theorem 2.9 Let be a complete multiplicative metric space and be a multiplicative -Kannan-contraction mapping. Suppose that the following conditions hold:
-
(1)
there exists such that and ;
-
(2)
f is a cyclic -admissible;
-
(3)
one of the following conditions holds:
(3.1) f is continuous;
(3.2) if is a sequence in X such that as and for all , then .
Then f has a fixed point. Furthermore, if and for all fixed point , then f has a unique fixed point.
Proof Starting from a point in condition (1), we get and . We will construct the iterative sequence , where for all . Since f is a cyclic -admissible mapping, we have
and
By a similar method, we get
for all . This implies that
for all . From the -Kannan-multiplicative contractive condition of f, we have
for all . Thus we have
for all , where . Let such that , then we get
Letting , we get and so the sequence is multiplicative Cauchy. From the completeness of X, there exists such that as .
Now, we assume that f is continuous. Hence, we obtain
Next, we will assume that condition (3.2) holds. Hence . Then we have
Letting , we get , that is, . This shows that z is a fixed point of f.
Now we show that z is the unique fixed point of f. Assume that y is another fixed point of f. From the hypothesis, we find that and , and hence
This shows that and then . Therefore, z is the unique fixed point of f. This completes the proof. □
Corollary 2.10 ([10])
Let be a complete multiplicative metric space and be a multiplicative Kannan-contraction mapping. Then f has a unique fixed point. Moreover, for any , the iterative sequence converges to the fixed point.
Proof Setting and for all in Theorem 2.9, we get this result. □
Theorem 2.11 Let be a complete multiplicative metric space and be a multiplicative -Chatterjea-contraction mapping. Suppose that the following conditions hold:
-
(1)
there exists such that and ;
-
(2)
f is a cyclic -admissible;
-
(3)
one of the following conditions holds:
(3.1) f is continuous;
(3.2) if is a sequence in X such that as and for all , then .
Then f has a fixed point. Furthermore, if and for all fixed point , then f has a unique fixed point.
Proof Starting from a point in condition (1), we get and . We will construct the iterative sequence , where for all . Since f is a cyclic -admissible mapping, we have
and
By a similar method, we get
for all . This implies that
for all . From the multiplicative -Chatterjea-contraction condition of f, we have
for each . Thus we have
for all , where . Let such that , then we get
Letting , we get and so the sequence is multiplicative Cauchy. From the completeness of X, there exists such that as .
Now, we assume that f is continuous. Hence, we obtain
Next, we will assume that condition (3.2) holds. Hence . Then we have
Letting , we get , that is, . This shows that z is a fixed point of f.
Now we show that z is the unique fixed point of f. Assume that y is another fixed point of f. From the hypothesis, we find that and , and hence
This shows that and then . Therefore, z is the unique fixed point of f. This completes the proof. □
Corollary 2.12 ([10])
Let be a complete multiplicative metric space and be a multiplicative Chatterjea-contraction mapping. Then f has a unique fixed point. Moreover, for any , the iterative sequence converges to the fixed point.
Proof Setting and for all in Theorem 2.11, we get this result. □
3 Some cyclic contractions via cyclic -admissible mapping
In 2003, Kirk et al. [12] introduced the concept of cyclic mappings and cyclic contractions.
Definition 3.1 ([12])
Let A and B be nonempty subsets of a metric space . A mapping is called cyclic if and .
Definition 3.2 ([12])
Let A and B be nonempty subsets of a metric space . A mapping is called a cyclic contraction if there exists such that for all and .
Notice that although a Banach-contraction is continuous, a cyclic contraction need not to be. This is one of the important gains of fixed point results for cyclic mappings. Following [12], a number of fixed point theorems on a cyclic mappings have appeared (see, e.g., [13–21]).
In this section, we apply our main results for proving a fixed point theorems involving a cyclic mapping in multiplicative metric spaces.
Definition 3.3 Let A and B be nonempty subsets of a multiplicative metric space . A mapping is called cyclic if and .
Theorem 3.4 Let A and B be two closed subsets of a complete multiplicative metric space such that and be a cyclic mapping. Assume that
for all and , where . Then f has a unique fixed point in .
Proof Define mappings by
For and , we get
In other cases, we see that the contractive condition (3.2) holds. Therefore, f is a multiplicative -Banach-contraction mapping. It is easy to see that f is a cyclic -admissible mapping. Since , there exists such that and .
Next, we show that condition (3.2) in Theorem 2.4 holds. Let be a sequence in X such that for all and as . Then we have for all . Since B is closed subset of X, we get and then . Now, the conditions (1), (2), and (3.2) of Theorem 2.4 hold. So, f has a unique fixed point in , say z. If , then . Similarly, if , then we have . Therefore . This completes the proof. □
Similarly, we can prove the following theorems.
Theorem 3.5 Let A and B be two closed subsets of a complete multiplicative metric space such that and be a cyclic mapping. Assume that
for all and , where . Then f has a unique fixed point in .
Theorem 3.6 Let A and B be two closed subsets of a complete multiplicative metric space such that and be a cyclic mapping. Assume that
for all and , where . Then f has a unique fixed point in .
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Acknowledgements
The first author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST). The second author gratefully acknowledges the financial support provided by Thammasat University under the TU Research Scholar Contract No. 2/10/2557.
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Yamaod, O., Sintunavarat, W. Some fixed point results for generalized contraction mappings with cyclic -admissible mapping in multiplicative metric spaces. J Inequal Appl 2014, 488 (2014). https://doi.org/10.1186/1029-242X-2014-488
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DOI: https://doi.org/10.1186/1029-242X-2014-488