Abstract
In this article, we present a generalization of the double controlled metric like spaces, called quasi double controlled metric like spaces, by assuming that the symmetric condition is not necessary satisfied. Moreover, the self distance is not necessary zero.
Keywords:
fixed point; double controlled metric type spaces; double controlled metric like spaces; quasi double controlled metric like spaces MSC:
47H10; 54H25
1. Introduction
Extension of metric spaces has been the focus of many researchers in mathematics, the reason behind this focus is that the more general the metric is the more fields can be applied on. All started by generalizing metric spaces to partial metric spaces and to b-metric spaces. For example in partial metric spaces the author assumed that the self distance may be not zero. In b-metric spaces the author add a constant bigger than 1 in the triangle inequality. In the next paragraph we present some more details.
Bakhtin in [1], initiated a generalization of metric spaces called b-metric spaces. Lately, several generalizations of the b-metric spaces were initiated such as extended b-metric spaces by Kamran et al. [2]. In 2018, Mlaiki et al. [3], introduced the concept of controlled metric type spaces which will be denoted in the sequel by . Few months later, Abdeljawad et al. in [4], initiated a more general metric type so called double controlled metric type spaces denoted . In 2020, Mlaiki in [5], introduced a generalization of so called double controlled metric like spaces denoted , where he assumed that the self distance is not necessary zero. Another type of extension of metric spaces, is where we assume that we do not necessary have the symmetry condition which called quasi metric spaces for more details we refer the reader to the following references [6,7,8,9,10,11,12,13]. In this work, we introduce the concept of double controlled quasi metric like spaces denoted , where we assume that there is no symmetric condition in the . Since in this work we are introducing a new type of metric spaces, we are going to prove the existence of a fixed point for mapping that satisfying the Banach contraction principle, and nonlinear contraction principle. Also, we present an application in polynomial equations.
2. Preliminaries
The first extension of b-metric spaces was initiated by Kamran et al. in [2].
Definition 1
([2]). Consider the set and . If the function satisfies the following conditions for all ,
();
();
()
then, the pair is called an extended b-metric spaces.
Later, Mlaiki et al. [3] introduced the an interesting extension to the space proposed by Kamran et al. in [2].
Definition 2
([3]). Given a nonempty set and . The function is a controlled metric type if
();
();
(),
for all . The pair is called a controlled metric type space.
In the same year, a generalization of was initiated in [4] as follows.
Definition 3
([4]). Consider the set and be non-comparable functions. The function if we have
() if and only if ;
();
(),
for all then the pair is called a double controlled metric type space.
In 2020, as an extension of , was introduced by assuming that the self distance is not necessary zero see [5].
Definition 4
([5]). Consider the set and be non-comparable functions. if the function satisfies the following conditions;
();
();
(),
for all the pair is called a double controlled metric like space.
Now, we introduce the notion of as follows.
Definition 5.
Consider the set and be non-comparable functions. if the function satisfies the following conditions;
();
(),
for all the pair is called a double controlled quasi metric like space.
Note that in Definition 5, we do not have the symmetry condition. Next, we present the topology of .
Definition 6.
Let be a , and . The upper closed ball of radius r centered and the lower closed ball of radius r centered are defined by,
and
respectively. Now, we define the notions of a circle and a disc on a quasi-metric space as follows: Let and . The circle and the disc are
and
Definition 7.
- Let a , a sequence is convergent to some b in if and only if
- A sequence is a left Cauchy sequence if and only if for all , we have exists and finite.
- A sequence is a right Cauchy sequence if and only if for all , we have exists and finite.
- A sequence is a Cauchy sequence if and only if is left and right Cauchy.
- We say that is left complete, right complete and dual complete if and only if each left-Cauchy, right Cauchy and Cauchy sequence in is convergent respectively.
Example 1.
Let be defined by
Consider such that
where for , and and are in . Also, let and
Note that, is a with control functions
It is not difficult to see that is a but it is not a
3. Main Results
Now, we prove the Banach contraction principle in
Theorem 1.
Let be a dual complete defined by the functions . Let be a mapping such that
for all , where . For , take . Suppose that
Also, assume that for every , we have
Then admits a unique fixed point.
Proof.
Let in be the sequence that satisfies the conditions of our theorem. Fom (1), we obtain
Similarly, we can show that;
Let such that Thus,
Note that, we are using the fact that . Let
Hence, we have
Now, note that
Thus, by the ratio test we deduce that exists and then the real sequence is -Cauchy. Finally, if we take the limit in the inequality (6) as we deduce that
that is, the sequence is right Cauchy in Similarly we can show by the use of (5) that the sequence is left Cauchy in Therefore, we deduce that is a Cauchy sequence in Since is dual complete, we can conclude that the sequence converges to some that is
Now, We prove that Note that,
Taking the limit as we deduce that
Using again the triangle inequality and (1),
Now, we illustrate Theorem 1 by the following example.
Example 2.
Let For all we have:
Now, let and It is easy to see that is a dual complete . Now, let , defined by
Thus,
Hence,
Note that it is not difficult to see that satisfies all the hypothesis of Theorem 1. Therefore, has a unique fixed point in
Definition 8.
Let . For some , let be the orbit of . We say that the function is -orbitally lower semi-continuous at if for such that , we have .
Corollary 1.
Let be a dual complete defined by the functions . Let Let and such that
Take . Suppose that
Then Also, is -orbitally lower semi-continuous at ϑ.
In the next theorem, we study the nonlinear case, but first we remind the reader of the following set of comparison functions.
Definition 9
([14]). Define Φ to be the set of functions that satisfies the following properties
- Φ is monotone increasing;
- for all ;
- ;
- Φ is continuous;
- converges to 0 ;
- converges
Next, we present the following lemma.
Lemma 1
([14]). 1. and ⟹;
2. and ⟹;
3. and ⟹.
Theorem 2.
Let be a dual complete defined by the functions . Consider the map and assume that there exists such that
for all Moreover, assume that for each , we have
where . If the and are continuous, then has a unique fixed point with for each .
Proof.
Let and be as in the hypothesis of the theorem. Suppose that there exists , such that , then clearly is the fixed point. Hence, we may assume that for each n. From the condition (12), we have
where clearly . If for some n, we accept that , then by (14) and the fact that for all we have we deduce that
which is a contradiction. Thus, for all we obtain . From which, it follows that . Then, we obtain by induction,
By using the properties of , we obtain that
Similarly, we can show that
Now, using the same technique in the proof of Theorem 1, we can easily deduce that
By condition (13) and by using the ratio test we deduce that is an -Cauchy sequence. The fact that is a dual complete implies that as such that . Hence, we can easily deduce that . Finally, assume that and are two fixed points of such that . From the assumption (12), we have
which leads to a contradiction. Hence, Similarly, we can show that Therefore as desired. □
4. Application
In closing, we present the following application for our results.
Theorem 3.
Let a natural number the equation
has a unique real solution in
Proof.
Before anything else, note that if Equation (17), does not have a solution. So, let and for all let and Now, let and It’s clear that is a dual complete . Now, let
Notice that, since we can deduce that Thus,
Hence,
Finally, note that it is not difficult to see that satisfies all the hypothesis of Theorem 1. Therefore, has a unique fixed point in which implies that Equation (17) has a unique real solution as desired. □
5. Future Work
In this section, we propose to endowed the by a graph in order to obtain the triplet . Indeed, the concept of fixed point theory with a graph has been widely studied. We refer the reader to the paper of Jachymski [15] for more details on this problem. Inspired by several works in this field, we can introduce the concept of G-contraction where G is the graph associated to and next we present a conjecture related to the existence and uniqueness of fixed point for such contractions in a with a graph. First, in Figure 1, we present an example to illustrate this approach.
Let a with graph where the set , and defined on the figure and to be defined by
and
Now, define the self mapping on as follows;
It is easy to see that has a unique fixed point. Let be a on a set . Let be the diagonal of . A graph G is defined by the pair where V is a set of vertices coinciding with and E is the set of its edges with . From now on, assume that G has no parallel edges.
Definition 10.
Let t and g be two vertices in a graph G. A path in G from t to g of length q () is a sequence of distinct vertices so that , and for .
The graph G may be converted to a weighted graph by assigning to each edge the distance given by the between its vertices.
Notation: Let .
Definition 11.
Let be a complete endowed with a graph G. The mapping is said to be a -contraction if
- there is a function so thatwhere ϕ is a nondecreasing function and converges to 0 for each .
Definition 12.
The mapping is called orbitally G-continuous if for all and any positive sequence ,
Conjecture 1.
Let be a complete with a graph G. Let be a -contraction which is orbitally G-continuous. Suppose the following property : for any in , if and , then there is a subsequence with , holds. Further, suppose that, for each ,
Also, assume that for every , we have
Then the restriction of to possesses a fixed point.
6. Conclusions
We introduced the concept of double controlled quasi metric like spaces, we proved the existence and uniqueness of a fixed point for self mapping in such spaces that satisfies the Banach contraction principle. Also, we proved the same result for mapping that satisfies nonlinear type of contraction. We presented an application of our result to polynomial equations. In the last section we provided the reader with an idea about future work on endowed with a graph.
Author Contributions
Conceptualization, S.H., A.K.S. and N.M.; Methodology, S.H., A.K.S., N.M. and D.R.; Supervision, N.M.; Validation, S.H., A.K.S., N.M. and D.R.; Writing—original draft preparation, S.H., A.K.S., N.M. and D.R.; Writing—review and editing, S.H., A.K.S., N.M. and D.R. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Acknowledgments
The first and the third authors would like to thank Prince Sultan University for paying the publication fees for this work through TAS LAB.
Conflicts of Interest
The authors declare no conflict of interest.
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