Abstract
In this paper, we extend the result of Wardowski (Fixed Point Theory Appl. 2012:94, 2012) by applying some weaker conditions on the self map of a complete metric space and on the mapping F, concerning the contractions defined by Wardowski. With these weaker conditions, we prove a fixed point result for F-Suzuki contractions which generalizes the result of Wardowski.
MSC:74H10, 54H25.
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1 Introduction and preliminaries
Throughout this article, we denote by ℝ the set of all real numbers, by the set of all positive real numbers, and by ℕ the set of all natural numbers.
In 1922, Polish mathematician Banach [1] proved a very important result regarding a contraction mapping, known as the Banach contraction principle. It is one of the fundamental results in fixed point theory. Due to its importance and simplicity, several authors have obtained many interesting extensions and generalizations of the Banach contraction principle (see [2–9] and references therein). Subsequently, in 1962, M Edelstein proved the following version of the Banach contraction principle.
Theorem 1.1 [10]
Let (X, d) be a compact metric space and let be a self-mapping. Assume that holds for all with . Then T has a unique fixed point in X.
In 2008, Suzuki [2] proved generalized versions of Edelstein’s results in compact metric space as follows.
Theorem 1.2 [2]
Let (X, d) be a compact metric space and let be a self-mapping. Assume that for all with ,
Then T has a unique fixed point in X.
In 2012, Wardowski [11] introduce a new type of contractions called F-contraction and prove a new fixed point theorem concerning F-contractions. In this way, Wardowski [11] generalized the Banach contraction principle in a different manner from the well-known results from the literature. Wardowski defined the F-contraction as follows.
Definition 1.3 Let be a metric space. A mapping is said to be an F-contraction if there exists such that
where is a mapping satisfying the following conditions:
-
(F1)
F is strictly increasing, i.e. for all such that , ;
-
(F2)
For each sequence of positive numbers, if and only if ;
-
(F3)
There exists such that .
We denote by ℱ, the set of all functions satisfying the conditions (F1)-(F3). For examples of the function F the reader is referred to [12] and [11].
Remark 1.4 From (F1) and (1) it is easy to conclude that every F-contraction is necessarily continuous.
Wardowski [11] stated a modified version of the Banach contraction principle as follows.
Theorem 1.5 [11]
Let be a complete metric space and let be an F-contraction. Then T has a unique fixed point and for every the sequence converges to .
Very recently, Secelean [12] proved the following lemma.
Lemma 1.6 [12]
Let be an increasing mapping and be a sequence of positive real numbers. Then the following assertions hold:
-
(a)
if , then ;
-
(b)
if , and , then .
By proving Lemma 1.6, Secelean showed that the condition (F2) in Definition 1.3 can be replaced by an equivalent but a more simple condition,
(F2′)
or, also, by
(F2″) there exists a sequence of positive real numbers such that .
Remark 1.7 Define by , then . Note that with the F-contraction reduces to a Banach contraction. Therefore, the Banach contractions are a particular case of F-contractions. Meanwhile there exist F-contractions which are not Banach contractions (see [11, 12]).
In this paper, we use the following condition instead of the condition (F3) in Definition 1.3:
(F3′) F is continuous on .
We denote by the set of all functions satisfying the conditions (F1), (F2′), and (F3′).
Example 1.8 Let , , , . Then .
Remark 1.9 Note that the conditions (F3) and (F3′) are independent of each other. Indeed, for , satisfies the conditions (F1) and (F2) but it does not satisfy (F3), while it satisfies the condition (F3′). Therefore, . Again, for , , , where denotes the integral part of α, satisfies the conditions (F1) and (F2) but it does not satisfy (F3′), while it satisfies the condition (F3) for any . Therefore, . Also, if we take , then and . Therefore, .
In view of Remark 1.9, it is meaningful to consider the result of Wardowski [11] with the mappings instead . Also, we define the F-Suzuki contraction as follows and we give a new version of Theorem 1.5.
Definition 1.10 Let be a metric space. A mapping is said to be an F-Suzuki contraction if there exists such that for all with
where .
2 Main results
Theorem 2.1 Let T be a self-mapping of a complete metric space X into itself. Suppose and there exists such that
Then T has a unique fixed point and for every the sequence converges to .
Proof Choose and define a sequence by
If there exists such that , the proof is complete. So, we assume that
For any we have
i.e.,
Repeating this process, we get
From (4), we obtain , which together with (F2′) and Lemma 1.6 gives , i.e.,
Now, we claim that is a Cauchy sequence. Arguing by contradiction, we assume that there exist and sequences and of natural numbers such that
So, we have
It follows from (5) and the above inequality that
On the other hand, from (5) there exists , such that
Next, we claim that
Arguing by contradiction, there exists such that
It follows from (6), (8), and (10) that
This contradiction establishes the relation (9). Therefore, it follows from (9) and the assumption of the theorem that
From (F3′), (7), and (11), we get . This contradiction shows that is a Cauchy sequence. By completeness of , converges to some point x in X. Finally, the continuity of T yields
Now, let us to show that T has at most one fixed point. Indeed, if be two distinct fixed points of T, that is, . Therefore,
then we get
which is a contradiction. Therefore, the fixed point is unique. □
Theorem 2.2 Let be a complete metric space and be an F-Suzuki contraction. Then T has a unique fixed point and for every the sequence converges to .
Proof Choose and define a sequence by
If there exists such that , the proof is complete. So, we assume that
Therefore,
For any we have
i.e.,
Repeating this process, we get
From (14), we obtain , which together with (F2′) and Lemma 1.6 gives
Now, we claim that is a Cauchy sequence. Arguing by contradiction, we assume that there exist and sequences and of natural numbers such that
So, we have
It follows from (15) and the above inequality that
From (15) and (17), we can choose a positive integer such that
So, from the assumption of the theorem, we get
It follows from (12) that
From (F3′), (15), and (18), we get . This contradiction shows that is a Cauchy sequence. By completeness of , converges to some point in X. Therefore,
Now, we claim that
Again, assume that there exists such that
Therefore,
which implies that
It follows from (21) and (22) that
Since , by the assumption of the theorem, we get
Since , this implies that
So, from (F1), we get
It follows from (21), (23), and (24) that
This is a contradiction. Hence, (20) holds. So, from (20), for every , either
or
holds. In the first case, from (19), (F2′), and Lemma 1.6, we obtain
It follows from (F2′) and Lemma 1.6 that . Therefore,
Also, in the second case, from (19), (F2′), and Lemma 1.6, we obtain
It follows from (F2′) and Lemma 1.6 that . Therefore,
Hence, is a fixed point of T. Now let us show that T has at most one fixed point. Indeed, if are two distinct fixed points of T, that is, , then . So, we have and from the assumption of the theorem, we obtain
which is a contradiction. Thus, the fixed point is unique. □
Example 2.3 Consider the sequence as follows:
Let and . Then is complete metric space. Define the mapping by and for every . Since
T is not a Banach contraction and a Suzuki contraction. On the other hand taking , we obtain the result that T is an F-contraction with . To see this, let us consider the following calculation. First observe that
For , we have
Since and , we have
So, from (25), we get
For , similar to , we have
For , we have
Since , we have
We know that . Therefore
So from (26), we get
Therefore for all . Hence T is an F-contraction and .
For , , , and in the above example, we compare the rate of convergence of the Banach contraction (-contraction) and F-contractions for , , and in Table 1.
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Acknowledgements
The second author was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (Under NUR Project ‘Theoretical and Computational fixed points for Optimization problems’ No. 57000621).
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Piri, H., Kumam, P. Some fixed point theorems concerning F-contraction in complete metric spaces. Fixed Point Theory Appl 2014, 210 (2014). https://doi.org/10.1186/1687-1812-2014-210
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DOI: https://doi.org/10.1186/1687-1812-2014-210