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Journal of Mathematics
Volume 2018, Article ID 9127486, 9 pages
https://doi.org/10.1155/2018/9127486
Research Article

Wardowski Type Contractions and the Fixed-Circle Problem on -Metric Spaces

1Department of Mathematics and General Sciences, Prince Sultan University Riyadh 11586, Saudi Arabia
2Balikesir University, Department of Mathematics, 10145 Balikesir, Turkey

Correspondence should be addressed to Nabil Mlaiki; moc.liamg@2102ikialmn

Received 18 February 2018; Accepted 5 September 2018; Published 10 October 2018

Academic Editor: Ming-Sheng Liu

Copyright © 2018 Nabil Mlaiki 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.

Abstract

In this paper, we present new fixed-circle theorems for self-mappings on an -metric space using some Wardowski type contractions, -contractive, and weakly -contractive self-mappings. The common property in all of the obtained theorems for Wardowski type contractions is that the self-mapping fixes both the circle and the disc with the center and the radius .

1. Introduction

Fixed-point theory has many applications in different fields; see [110]. Recently, using Wardowski’s technique, some new fixed-point theorems on -metric spaces [11] and some new fixed-circle theorems on metric spaces [12, 13] have been obtained. Our aim in this paper is to obtain various fixed-circle results using this technique. In Section 2, we recall some necessary background on -metric spaces and give new examples. In Section 3, we introduce the notion of an -contraction to obtain fixed-circle theorems. By means of this notion, we define new types of an -contraction such as Hardy-Rogers type -contraction and Reich type -contraction and present some fixed-circle results on -metric spaces. Also, we give an illustrative example of a self-mapping satisfying all of the conditions of the obtained theorems. In Section 4, we prove the existence along with the conditions that give us uniqueness of a fixed circle for -contractive and weakly -contractive self-mappings on -metric spaces. In Section 5, we give an application of fixed-circle results obtained by Wardowski technique to integral type contractive self-mappings.

2. Preliminaries

In this section, we recall some necessary notions, relations, and results about -metric spaces.

Definition 1 (see [14]). Let be a nonempty set and be a function satisfying the following conditions for all if and only if

Then is called an -metric on and the pair is called an -metric space.

Example 2 (see [15]). Let (or ) and the function be defined as for all (or ). Then the function is an -metric on (or ). This -metric is called the usual -metric on (or ).

Lemma 3 (see [14]). Let be an -metric space and . Then we have

The relationships between a metric and an -metric were studied in different papers (see [1618] for more details). In [17], a formula of an -metric space which is generated by a metric was investigated as follows.

Let be a metric space. Then the function defined byfor all , is an -metric on . The -metric is called the -metric generated by [18]. We note that there exists an -metric which is not generated by any metric as seen in the following example.

Example 4. Let be a nonempty set, the function be any metric on , and the function be defined by for all . Then the function is an -metric and is an -metric space. Indeed,

for , we have

using Table 1, we can easily see that the condition is satisfied.

Table 1

Also the -metric is not generated by any metric . Conversely, suppose that there exists a metric such that for all . Then we get and Therefore, we obtain which is a contradiction. Consequently, is not generated by any metric

In [19] and [14], a circle and a disc are defined on an -metric space as follows, respectively:and

We give an example.

Example 5. Let be a nonempty set, the function be any metric on , and the -metric space be defined as Example 4. Let us consider the circle according to the -metricThen we have the following cases
Case 1. If then .
Case 2. If then
Case 3. If then , where .

Definition 6 (see [19]). Let be an -metric space, be a circle, and be a self-mapping. If for every then the circle is called the fixed circle of .

3. -Contraction and Hardy-Rogers Type -Contraction on -Metric Spaces

At first, we recall the definition of the following family of functions which was introduced by Wardowski in [20].

Definition 7 (see [20]). Let be the family of all functions such that
is strictly increasing
for each sequence in the following holds: if and only if
there exists such that .

The following is an example of some functions that satisfies conditions , , and of Definition 7.

Example 8 (see [20]). defined by .
defined by .
defined by .
defined by .
Note that these four functions satisfy conditions , , and of Definition 7.

Now we introduce the following new contraction type using this family of functions.

Definition 9. Let be an -metric space. A self-mapping on is said to be an -contraction if there exist , , and such that for all the following holds: Now, we present the following proposition.

Proposition 10. Let be an -metric space. If a self-mapping on is an -contraction with , then we have .

Proof. Assume that . From the definition of an -contraction, we getInequality (14) contradicts with the definition of since and . Therefore, it should be .

Using this new type contraction, we give the following fixed-circle theorem.

Theorem 11. Let be an -metric space, be an -contractive self-mapping with , and . Then is a fixed circle of . especially fixes every circle where .

Proof. Let . If , by the definition of we have . Hence, using the -contractive property and the fact that is increasing, we obtainwhich also lead to a contradiction. Therefore, and that is . Consequently, is a fixed circle of .
Now we show that also fixes any circle with . Let and assume that . By the -contractive property, we haveSince is increasing, then we findBut , which leads us to a contradiction. Thus, and . Hence, is a fixed circle of .

Remark 12. Notice that, in Theorem 11, the -contractive self-mapping fixes the disc with the center and the radius . Therefore, the center of any fixed circle is also fixed by .

In the following example, we see that the converse statement of Theorem 11 is not always true.

Example 13. Let be an -metric space, be any point, and the self-mapping be defined asfor all with . Then it can be easily seen that is not an -contractive self-mapping. Indeed, if for , then, using Lemma 3 and the -contractive property, we get which is a contradiction since . Hence is not an -contractive self-mapping. But fixes every circle where .

Related to the number of the elements of the set , the number of the fixed circles of an -contractive self-mapping can be infinite as seen in the following example.

Example 14. Let , the metric be defined as for all , and the -metric be defined as in Example 4. Let us define the self-mapping as for all . Then the self-mapping is an -contractive self-mapping with , , and . Indeed, we get Using Theorem 11, we have Therefore, fixes the circle and the disc . Evidently, the number of the fixed circles of is infinite.

In the following definition, we introduce the notion of a Hardy-Rogers type -contraction.

Definition 15. Let be an -metric space and be a self-mapping on . If there exist , , and such that for all the following holdswhere then the self-mapping is called a Hardy-Rogers type -contraction on .

Proposition 16. Let be an -metric space. If a self-mapping on is a Hardy-Rogers type -contraction with then we have .

Proof. Suppose that . Using the hypothesis, we obtain which is a contradiction since . Therefore, we get .

Remark 17. Using Proposition 16, a Hardy-Rogers type -contraction condition can be changed as followswhere

Now using the Hardy-Rogers type -contraction condition, we prove the following fixed-circle theorem.

Theorem 18. Let be an -metric space, be a Hardy-Rogers type -contractive self-mapping with , and be defined as in Theorem 11. If , then is a fixed circle of . especially fixes every circle where .

Proof. Let and . Using the Hardy-Rogers type -contraction property, Proposition 16, Lemma 3, and the fact that is increasing, we get which is a contradiction. Hence and so . Consequently, is a fixed circle of . By the similar arguments used in the proof of Theorem 11, also fixes any circle where .

Corollary 19. (1) Let be an -metric space, be a Hardy-Rogers type -contractive self-mapping with , and be defined as in Theorem 11. If for all then fixes the disc .
(2) If we consider and in Definition 15, then we obtain the concept of an -contractive mapping.

In Definition 15, if we get then we have the following definition.

Definition 20. Let be an -metric space and be a self-mapping on . If there exist , , and such that for all the following holdswhere then the self-mapping is called a Reich type -contraction on .

Proposition 21. Let be an -metric space. If a self-mapping on is a Reich type -contraction with then we get .

Proof. The proof follows easily since .

Remark 22. Using Proposition 21, a Reich type -contraction condition can be changed as followswhere

Theorem 23. Let be an -metric space, be a Reich type -contractive self-mapping with , and be defined as in Theorem 11. Then is a fixed circle of . Also, fixes every circle where . In other words, fixes the disc .

Proof. The proof follows easily since

In Definition 15, if we get and , then we have the following definition.

Definition 24. Let be an -metric space and be a self-mapping on . If there exist , , and such that for all the following holds where then the self-mapping is called a Chatterjea type -contraction on .

Proposition 25. Let be an -metric space. If a self-mapping on is a Chatterjea type -contraction with then we get .

Proof. The proof follows easily.

Theorem 26. Let be an -metric space, be a Chatterjea type -contractive self-mapping with , and be defined as in Theorem 11. If for all then is a fixed circle of . Also, fixes every circle where . In other words, fixes the disc .

Proof. The proof follows easily by the similar arguments used in the proofs of Theorems 11 and 18.

Now we give the following illustrative example.

Example 27. Let be the set of all complex numbers. Consider the set where is any complex number with and the metric is defined as in [18] such that for all . Let us define the self-mapping as for all . Then the self-mapping is an -contractive self-mapping with , and . Indeed, we obtain for , and Then we have Also we obtain Therefore, the self-mapping fixes the circle and the disc .

Also the self-mapping is a Hardy-Rogers type -contractive self-mapping resp., a Reich type -contractive self-mapping and a Chatterjea type -contractive self-mapping on with , (resp., , and .

4. -Contractive and Weakly -Contractive Self-Mappings on -Metric Spaces

First, in this section we present this well-known interesting class of functions.

Definition 28. Denote by the family of nondecreasing functions where is the -th iterate of

Lemma 29. For every function the following holds: if is nondecreasing, then, for each , implies that

Now, we define the -contractive self-mapping in an -metric space.

Definition 30. Let be a self-mapping on an -metric space We say that is -contractive self-mapping if there exist and such that for all we have

Theorem 31. Let be a -contractive self-mapping with on an -metric space , and consider the circle Thus, for every , either fixes or maps to the interior of Moreover, if for every we have , then is a unique fixed circle of in

Proof. If , then since is -contractive we have If , then we are in the case where maps to the interior of If , then by using the fact that is a nondecreasing function we have Now, if , then the above inequality implies that which leads to a contradiction. Hence, in this case we must have Thus, and that is
Therefore, either fixes or maps to the interior of as required.
To prove the second part of our theorem, we may assume that , for all Now, we only need to show that if there exists where , then , and that will prove the uniqueness. So, first let , and that is , and also let be an arbitrary fixed point of (i.e., ) we have two cases.
Case 1. If then by using the fact that is a nondecreasing function we have Now, if then the above inequality implies that which leads to a contradiction. Hence, in this case we must have and that is
Case 2. If then once again by using the fact that is a nondecreasing function we have which leads us to a contradiction.
Therefore, is the unique fixed circle of in as desired.

Next, we give the definition of a weakly -contractive self-mapping.

Definition 32. Let be a self-mapping on an -metric space We say that is a weakly -contractive self-mapping with if there exist and such that for all we have

Theorem 33. Let be a weakly -contractive self-mapping with on an -metric space and consider the circle Thus, for every either fixes or maps to the interior of Moreover, if for every , we have , then is a unique fixed circle of in

Proof. If , then since is weakly -contractive we have If , then we are in the case where maps to the interior of If , then by using the fact that is a nondecreasing function we have Now, if , then the above inequality implies that which leads to a contradiction. Hence, in this case we must have Thus, and that is
Therefore, either fixes or maps to the interior of as required.
To prove the second part of our theorem, we may assume that , for all . Now, we only need to show that if there exists , where , then , and that will prove the uniqueness. So, first let , and that is , and also let be an arbitrary fixed point (i.e., ) we have two cases.
Case 1. If then by using the fact that is a nondecreasing function we have Now, if , then the above inequality implies that which leads to a contradiction. Hence, in this case we must have and that is
Case 2. If then once again by using the fact that is a nondecreasing function we have which leads us to a contradiction.
Therefore, is the unique fixed circle of in as desired.

5. An Application to Integral Type Contractive Self-Mappings

We assume that is a Lebesgue-integrable mapping which is summable (that is, with finite integral) on each compact subset of , nonnegative, and such that, for each ,

Now we give the following definition.

Definition 34. Let be an -metric space and be defined as in (59). A self-mapping on is said to be an integral type -contraction if there exist , , and such that for all the following holds:

Proposition 35. Let be an -metric space and be defined as in (59). If a self-mapping on is an integral type -contraction with then we get .

Proof. Suppose that . From the definition of an integral type -contraction, we haveInequality (61) contradicts with the definition of since and . Hence, it should be .

Using this new definition, we get the following fixed-circle result.

Theorem 36. Let be an -metric space, be defined as in (59), be an integral type -contraction with , and be defined as in Theorem 11. Then is a fixed circle of .

Proof. Let . Assume that . Then, by the definition of , we get Using the fact that is increasing property, we have andFrom inequality (64) and the definition of integral type -contractivity, we obtain which is a contradiction. Therefore, we find . Consequently, is a fixed circle of .

Remark 37. An integral type -contractive self-mapping fixes also the disc .
If we set the function in Theorem 36 as for all , then we get Theorem 11.
By the similar argument used in Definition 34, the notions of an integral Hardy-Rogers type -contractive self-mapping, an integral Reich type -contractive self-mapping, an integral Chatterjea type -contractive self-mapping, and obtained corresponding fixed-circle theorems can be defined.

Finally, we give the following example.

Example 38. Let be the -metric space with the usual -metric and the function be defined by for all . Let us define the self-mapping as for all . The self-mapping is an integral type -contractive self-mapping with , , and . Indeed, we get for . Then we have Also we obtain Consequently, fixes the circle and the disc .

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) Group no. RG-DES-2017-01-17.

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