Research Article  Open Access
Wardowski Type Contractions and the FixedCircle Problem on Metric Spaces
Abstract
In this paper, we present new fixedcircle theorems for selfmappings on an metric space using some Wardowski type contractions, contractive, and weakly contractive selfmappings. The common property in all of the obtained theorems for Wardowski type contractions is that the selfmapping fixes both the circle and the disc with the center and the radius .
1. Introduction
Fixedpoint theory has many applications in different fields; see [1–10]. Recently, using Wardowski’s technique, some new fixedpoint theorems on metric spaces [11] and some new fixedcircle theorems on metric spaces [12, 13] have been obtained. Our aim in this paper is to obtain various fixedcircle 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 fixedcircle theorems. By means of this notion, we define new types of an contraction such as HardyRogers type contraction and Reich type contraction and present some fixedcircle results on metric spaces. Also, we give an illustrative example of a selfmapping 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 selfmappings on metric spaces. In Section 5, we give an application of fixedcircle results obtained by Wardowski technique to integral type contractive selfmappings.
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 [16–18] 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.

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 selfmapping. If for every then the circle is called the fixed circle of .
3. Contraction and HardyRogers 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 selfmapping 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 selfmapping 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 fixedcircle theorem.
Theorem 11. Let be an metric space, be an contractive selfmapping 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 selfmapping 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 selfmapping be defined asfor all with . Then it can be easily seen that is not an contractive selfmapping. Indeed, if for , then, using Lemma 3 and the contractive property, we get which is a contradiction since . Hence is not an contractive selfmapping. But fixes every circle where .
Related to the number of the elements of the set , the number of the fixed circles of an contractive selfmapping 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 selfmapping as for all . Then the selfmapping is an contractive selfmapping 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 HardyRogers type contraction.
Definition 15. Let be an metric space and be a selfmapping on . If there exist , , and such that for all the following holdswhere then the selfmapping is called a HardyRogers type contraction on .
Proposition 16. Let be an metric space. If a selfmapping on is a HardyRogers 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 HardyRogers type contraction condition can be changed as followswhere
Now using the HardyRogers type contraction condition, we prove the following fixedcircle theorem.
Theorem 18. Let be an metric space, be a HardyRogers type contractive selfmapping 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 HardyRogers 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 HardyRogers type contractive selfmapping 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 selfmapping on . If there exist , , and such that for all the following holdswhere then the selfmapping is called a Reich type contraction on .
Proposition 21. Let be an metric space. If a selfmapping 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 selfmapping 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 selfmapping on . If there exist , , and such that for all the following holds where then the selfmapping is called a Chatterjea type contraction on .
Proposition 25. Let be an metric space. If a selfmapping 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 selfmapping 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 selfmapping as for all . Then the selfmapping is an contractive selfmapping with , and . Indeed, we obtain for , and Then we have Also we obtain Therefore, the selfmapping fixes the circle and the disc .
Also the selfmapping is a HardyRogers type contractive selfmapping resp., a Reich type contractive selfmapping and a Chatterjea type contractive selfmapping on with , (resp., , and .
4. Contractive and Weakly Contractive SelfMappings on Metric Spaces
First, in this section we present this wellknown 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 selfmapping in an metric space.
Definition 30. Let be a selfmapping on an metric space We say that is contractive selfmapping if there exist and such that for all we have