Research Article  Open Access
Erdal Karapınar, Hassen Aydi, Andreea Fulga, "On Hybrid Wardowski Contractions", Journal of Mathematics, vol. 2020, Article ID 1632526, 8 pages, 2020. https://doi.org/10.1155/2020/1632526
On Hybrid Wardowski Contractions
Abstract
The goal of this work is to introduce the concept of hybrid Wardowski contractions. We also prove related fixedpoint results. Moreover, some illustrated examples are given.
1. Introduction
Let represent the collection of functions so that(i) is strictly increasing(ii) for each sequence in , iff (iii) there is so that
Definition 1 (see [1]). A mapping is called a Wardowski contraction if there exist and such that for all ,
Example 1 (see [1]). The functions defined by(1)(2)(3)(4)belong to .
Wardowski [1] introduced a new proper generalization of Banach contraction. For other related papers in the literature, see [2–10]. The main result of Wardowski is as follows.
Theorem 1 (see [1]). Let be a complete metric space, and let be an contraction. Then, has a unique fixed point, say , in and for any point , the sequence converges to .
Theorem 2 (see [11]). Let be a complete metric space and be a given mapping such thatfor all , where , , are nonnegative real numbers such that . Then, admits a unique fixed point in .
In the paper [12], the concept of interpolative Hardy–Rogerstype contractions was introduced.
Definition 2. (see [12]). On a metric space , a selfmapping is an interpolative Hardy–Rogerstype contraction if there exist and with , such thatfor all , where .
Theorem 3 (see [12]). Let be a complete metric space and be an interpolative Hardy–Rogerstype contraction. Then, has a fixed point in .
The interpolation concept was used in other new papers related to fixedpoint theory. For example, see [13–17]. In this paper, we consider new contractive type selfmappings, named as hybrid Wardowski contractions. Our fixedpoint results will be supported by concrete examples.
2. Main Results
Let be a metric space and be a selfmapping on this space. For and , such that , we define the following expression:
On the other hand, let represent the set of functions such that(i) is strictly increasing(ii) there exists such that , for every
Definition 3. A mapping is called a hybrid Wardowski contraction, if there is such thatIn particular, if inequality (5) holds for , we say the mapping is a 0hybrid Wardowski contraction.
Theorem 4. A hybrid Wardowski contraction selfmapping on a complete metric space admits exactly one fixed point in .
Proof. Taking an arbitrary point , we consider the sequence defined by the relation , . According to this construction, it is easy to see that if there is so that , turns into a fixed point of . We shall presume that for all ,On account of (4), for and , we have thatDenoting by , we haveand from (5), it follows thatwhich gives usIf , then the above inequality becomeswhich is a contradiction. Consequently, and then there exists such thatSupposing that , we have and by , we obtainwhich is a contradiction. Therefore,In order to prove that is a Cauchy sequence in , we suppose that there exist and the sequences of positive integers, with such thatfor any .
Thus, we haveWhen , using (14) and (15), it followsBy using the triangle inequality, we haveSo,Moreover, sincewe haveSo, the inequalityoccurs for all , and using (5), there exists such thatwhereMoreover, since the function is increasing, we haveAnd letting ,That is a contradiction, so and then, . Consequently, the sequence is Cauchy and by completeness of , it converges to some point .
There exists a subsequence such that for all ; then,On the contrary, if there is a natural number such that for all , applying (5), for and , we havewhereWe suppose that . Inasmuch asLetting in inequality (29), we find thatwhich contradicts . Therefore, .
We claim now that admits only one fixed point. If there exists another point , , such that , then and we havewhich is a contradiction.
Example 2. Let be endowed with the standard metric . Let the mapping be defined by . Take , , , , , and . Then, we have the following: For , For and ,Thus, all assumptions of Theorem 4 hold, and has a unique fixed point. On the other hand, for and , we haveThus, it is not a Wardowski contraction, since for every function and
Theorem 5. A 0hybrid Wardowski contraction selfmapping on a complete metric space admits a fixed point in provided that for each sequence in , iff .
Proof. Following the same reasoning from the proof of the previous theorem, we can assume that for all ,On account of (4), for and , we have thatUsing the same notation, , and taking into account , by (5), we haveWe can remark that the case , is not possible since the above inequality becomesa contradiction. Therefore, for all , and then, there exists such thatWe claim that . Indeed, if we suppose that , taking the limit as in (40), we havewhich contradicts We conclude thatLet and now; we haveAnd taking into account (44),Therefore, and sincewe obtain that and so . Thus, is a Cauchy sequence on a complete metric space and there exists such that . Of course, it easy to see that, for and , we haveIf we suppose that there is a subsequence such that , then we havewhich means that is a fixed point of . Therefore, we can assume that for every , and by (5), we obtainLetting and taking into account the previous considerations, we have and then . Consequently, is a fixed point of .
Example 3. Let be a set endowed with the metric (Table 1).
And the mapping is defined as .
First, we remark that Theorem 1 is not satisfied, since for and ,Hence, for any and , we can writeChoosing , , , and , for and , we have

3. Consequences
Considering in Theorem 5 and , we obtain Theorem 2. Considering in Theorem 5 and , we obtain Theorem 3. Considering in Theorem 4, , and , we obtain Theorem 3.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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Copyright © 2020 Erdal Karapınar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.