Abstract
In this paper, we update the well-known fixed point theorem of Kannan using the interpolation notion in the realm of quasi-metric spaces. We consider some asymmetric versions. We also present some illustrative examples in support of the obtained results.
1. Introduction and Preliminaries
In 2018, Karapinar [1] published a new type of contractions obtained from the definition of the Kannan contraction by interpolation as follows.
Theorem 1 (see [11, Theorem 2.2]). Let be a complete metric space and be an interpolative Kannan type contraction, i.e., a self-map such that there exist so thatfor all with . Then, has a unique fixed point in .
This theorem has been generalized in 2019 by Gaba et al. [2] where they introduced the concept of -interpolative Kannan contractions as follows.
Definition 1. Let be a metric space and be a self-map. We shall call a relaxed -interpolative Kannan contraction, if there exist with such that
The interpolative method has been successfully applied to a diverse range of contractions (see [3–8]).
In 1982, Reilly et al.[9] obtained a quasi-metric version of the celebrated Banach contraction principle. Since then, and especially in the last decade, several authors have contributed to the development of the fixed point theory in the framework of quasi-metric spaces (see [10, 11]). It is in the same spirit that we provide here a quasi-metric version of the interpolative Kannan fixed point theorem.
Our basic reference for quasi-metric spaces is [12].
Definition 2. (see [13]). Let be a function where is a nonempty set. This function is called a quasi-pseudometric (respectively, -quasi-metric) on if and (respectively, and ) hold, where
The condition is known as the -condition. Furthermore, given a quasi-pseudometric on , there is a natural function , defined by for all . It is named as the conjugate of . For a -quasi-metric on , the distance function defined by for all is a metric on .
The classical example of a -quasi-metric is the truncated difference.
Example 1. (truncated difference). Set . Given asUnder these conditions, forms a -quasi-metric. Further, the pair becomes a -quasi-metric space.
Each quasi-metric on induces a topology on which has as a base the family of open balls where .
If satisfies the separation axiom (resp. ) on , we say that is a (resp. a Hausdorff) quasi-metric space. Note that a -quasi-metric space is if and only if for each , the condition implies .
A quasi-metric space is called bicomplete if the metric space is complete.
Definition 3. (convergence in quasi-pseudometric spaces, see [12]).
Let be a quasi-pseudometric space. We say that the sequence -converges to , or left-converges to , ifWe denote this by . More precisely, converges to with respect to .
In a similar manner, a sequence -converges to or right-converges to with respect to , ifWe denote this by . A sequence in the setting of a quasi-pseudometric space -converges to in the case that the sequence converges to from left and right, that is,Moreover, it is denoted as (or, , if there is no confusion).
Remark 1. From the definition of -convergence, we have
As demonstrated in [7, Example 1.7.], the reverse implication does not hold in general.
Definition 4. (compare [13]).(1)A sequence in a quasi-pseudometric is called left-Cauchy if for every , there exists such that(2)Similarly, we define right-Cauchyness. A sequence in a quasi-metric space is called right -Cauchy if it is a left -Cauchy sequence in the quasi-metric space .(3)The quasi-metric space is called left (right) -sequentially complete if every left (right) -Cauchy sequence converges with respect to the topology .(4) is called -sequentially complete if every Cauchy sequence in the metric space converges with respect to the topology .
Remark 2. One can easily be convinced that both bicompleteness and left (right) -sequential completeness imply -sequential completeness. However, the rest of implications does not hold in general. For more details, the reader can consult [12].
2. Revisiting the Interpolative Kannan Mappings
Let us recall that an interpolative Kannan contraction on a metric space is a self-mapping such that there exist for whichfor all with .
Before proceeding to the main results of this paper, we would like to correct an inaccuracy that appears in the proof of Theorem 2.2 in the paper by Karapinar [1].
Theorem 2. (see [11, Theorem 2.2]).
Let be a complete metric space and be an interpolative Kannan type contraction, i.e., a self-map such that there exist so thatfor all with . Then, has a unique fixed point in .
Proof. Following the proof presented by Karapinar, let , and construct the sequence by for all positive integers . Taking and in (10), we derive thatwhich yieldsIt is not difficult to see that inequality (10) is asymmetric, and to fill in this gap, we also consider the case of and in (10). So taking and in (10), we havewhich yieldsSince , it is obvious that ; hence, it is routine to check that is a Cauchy sequence which converges to the unique fixed point of .
This gap was revealed in Example 2.3 of [1]. Indeed, according to [11, Example 2.3], let be a set endowed with a metric such that , . Define the self-mapping on by and . The author claimed that for and , the self-mapping forms an interpolative Kannan type contraction. However, considering the pair , note that . We have
That is, inequality (10) fails for .
However, considering the pair , note that . We have
This suggests a modification of the so-called interpolative Kannan contraction in the following way.
Definition 5. Let be a metric space. A self-mapping is called a generalized interpolative Kannan contraction if there exist for whichfor all with and .
Remark 3. Observe then that for in (17), a generalized interpolative Kannan contraction is actually just an interpolative Kannan contraction in the sense and spirit of Karapinar.
Using this definition, we get the following.
Theorem 3 (compare [11, Theorem 2.2]). Let be a complete metric space and be a generalized interpolative Kannan contraction. Then, has a unique fixed point in .
Proof. The proof is merely a copy of that of [11, Theorem 2.2.] and needs not be repeated. One just has to observe that the right hand side of inequality (17) is symmetric in . This has actually been captured in the proof of Theorem 2. Moreover, [11, Example 2.3.] satisfies the hypothesis of Theorem 3.
3. Interpolative Kannan Mappings and Fixed Points in Quasi-Metric Spaces
The discussion we made in the previous section suggests, in a natural way, the following notions.
Definition 6. Let be a quasi-metric space.
A -interpolative Kannan contraction on is a mapping such that there exist for whichfor all with and .
A -interpolative Kannan contraction on is a mapping such that there exist for whichfor all with .
Then, we have the following easy, but useful, consequence of the interpolative Kannan contraction for metric spaces.
Proposition 1. Let be a -quasi-metric space. If is a -interpolative Kannan contraction or a -interpolative Kannan contraction on , then the following conditions hold:(1) is a generalized interpolative Kannan contraction on the metric space .(2)For any , is a Cauchy sequence in the metric space .
Proof. For (1), suppose that is a -interpolative Kannan contraction on . So there exist and for whichfor all with and . Similarly, given , we also havefor all with and .
Thus, given ,Then, is a generalized interpolative Kannan contraction on the metric space .
(2) Since, by (1), is a generalized interpolative Kannan contraction on the metric space , it follows from the modified proof of classical interpolative Kannan contraction principle that is a Cauchy sequence in the metric space .
By using the preceding proposition, three quasi-metric versions of the generalized interpolative Kannan principle are easily deduced.
Theorem 4. Every -(resp. every -) interpolative Kannan contraction on a bicomplete -quasi-metric space on has a unique fixed point.
Proof. Let be a or a interpolative Kannan contraction. Since is a complete metric space and, by Proposition 1 (1), is a generalized interpolative Kannan on , we deduce a modified proof of classical interpolative Kannan contraction principle that has a unique fixed point.
Theorem 5. Every -interpolative Kannan contraction on a Hausdorff -sequentially complete -quasi-metric space has a unique fixed point.
Proof. Let be a -interpolative Kannan contraction on the Hausdorff -sequentially complete -quasi- metric space . Fix an . By Proposition 1 (2), the sequence is a Cauchy sequence in the metric space . Hence, there is such that converges to with respect to , i.e., as . Since is a -interpolative Kannan contraction, there exist for whichConsequently, as . From Hausdorffness of , we deduce that . Finally, suppose that is a fixed point of . From (10), we haveand thus since the space is . This concludes the proof.
Corollary 1. Every -interpolative Kannan contraction on a -quasi-metric space such that is Hausdorff and -sequentially complete has a unique fixed point.
Proof. Let be a -interpolative Kannan contraction on . Put ; then, is a -interpolative Kannan contraction on a Hausdorff -sequentially complete -quasi-metric space . From Theorem 5, we deduce that has a unique fixed point.
Theorem 6. Every -interpolative Kannan contraction on a -sequentially complete -quasi-metric space has a unique fixed point.
Proof. Let be a -interpolative Kannan contraction on the -sequentially complete -quasi-metric space . Fix . As in the proof of Theorem 5 (see Proposition 1), is a Cauchy sequence in the metric space . Hence, there is such that converges to with respect to , i.e., as . Since is a -interpolative Kannan contraction, there exist for whichfor all . Consequently, as . From the triangle inequality,We deduce . Therefore, because is a -quasi-metric space, which is . Finally, suppose that is a fixed point of . Then,and thus . This concludes the proof.
Corollary 2. Every -contraction on a -quasi-metric space which is such that is -sequentially complete has a unique fixed point.
Proof. Let be a -interpolative Kannan contraction on . Put . Then, is a -interpolative Kannan contraction on the -sequentially complete -quasi-metric space . From Theorem 6, we deduce that has a unique fixed point.
Remark 4. Note that Theorems 5 and 6 remain valid if “-sequentially complete” is replaced with “left -sequentially complete” or “right K-sequentially complete.”
We conclude the paper with an example which shows that Theorem 5 cannot be generalized to -sequentially complete -quasi-metric spaces.
Example 2. Let and let be the -quasi-metric on given by for all , and otherwise. Clearly, is both left and right -sequentially complete, so it is -sequentially complete. Now, define as for all . Of course, has no fixed point. However, it is a -interpolative Kannan contraction with and since for each with , one has
In light of the above example, a question is in order.
Open Question 1. Can Theorem 6 be generalized to -sequentially complete -quasi-metric spaces? It is our belief that the answer is “No”; however, so far, we have failed to provide a counter-example.
4. Conclusion
As mentioned in the introduction, the interesting concept of -interpolative Kannan contractions was introduced in [2]. So, the authors plan, in a different manuscript, to discuss asymmetric versions of such a concept by first outlining the asymmetric nature of this notion, even for a classical interpolative Kannan contraction in a metric space.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Disclosure
Our proofs are inspired by the recent work of Daǧ et al. [14]. The presentation of the manuscript is as proceedings.
Conflicts of Interest
The authors declare that they have no conflicts of interest concerning the publication of this article.
Authors’ Contributions
All authors contributed equally and significantly in writing this article.
Acknowledgments
The first author (Y.U.G.) wishes to acknowledge that this work was carried out with the aid of a grant from the Carnegie Corporation provided through the African Institute for Mathematical Sciences.