Abstract
In this paper, we introduce a generalization of rectangular -metric spaces, by changing the rectangular inequality as follows: , for all distinct We prove some fixed point theorems, and we use our results to present a nice application in the last section of this paper.
1. Introduction
It will not be an exaggeration if we say that Banach [1] in 1922 introduced in some way a new area in mathematics, which is called fixed point theory, and that is due to the fact that he proved the existence and uniqueness of a fixed point for self-contractive mappings in metric spaces. Since 1922, mathematicians around the world start to generalize his result either by changing the type of contractions or by generalizing the type of metric spaces (see [2–19]). The question here is what is the point of all these generalizations? Well, in fact, the answer to that is quite simple and that is the larger the class of functions or metrics, the more fields that results can be applied to, such as computer sciences and engineering.
In this paper, and inspired by the work done in [20–27], we introduce the notion of controlled rectangular -metric spaces as a generalization of the rectangular metric spaces and rectangular -metric spaces. In the second section, we present some preliminaries; in the third section, we prove our main result; in the fourth section, we present an application of our result to polynomial equations; and in the closing section, we give a conclusion with some open questions.
2. Preliminaries
The concept of rectangular metric spaces was introduced by Branciari in [28] as follows.
Definition 1 [28] (rectangular (or Branciari) metric spaces). Let be a nonempty set. A mapping is called a rectangular metric on if for any and all distinct points ; it satisfies the following conditions:
() if and only if
()
()
In this case, the pair is called a rectangular metric space.
In [29], George et al. introduced the concept of -rectangular metric spaces as follows.
Definition 2 [29] (rectangular -metric spaces). Let be a nonempty set. A mapping is called a rectangular -metric on if there exists a constant such that for any and all distinct points ; it satisfies the following conditions:
() if and only if
()
()
In this case, the pair is called a rectangular metric space.
As a generalization of rectangular -metric spaces, Abdeljawad et al. in [30] introduced the concept of extended Branciari -distance spaces as follows.
Definition 3 [30]. For a nonempty set and a mapping , we say that a function is called an extended Branciari -distance if it satisfies (i) if and only if (ii)(iii)for all and all distinct . The couple of the symbols denotes an extended Branciari -distance space (shortly, -metric space).
Now, we present the definition of controlled rectangular -metric spaces.
Definition 4. Let be a nonempty set, a function , and We say that is a controlled rectangular -metric space if all distinct ; we have (1) if and only if (2)(3)
Next, we present the topology of controlled rectangular -metric spaces.
Definition 5. Let be a controlled rectangular -metric space. (1)A sequence is called -convergent in a controlled rectangular -metric space , if there exists such that (2)A sequence is called -Cauchy if and only if exists and is finite(3)A controlled rectangular -metric space is called -complete if for every -Cauchy sequence in , there exists , such that (4)Let define an open ball in a controlled rectangular -metric space by
Notice that rectangular metric spaces and rectangular -metric spaces are controlled rectangular -metric spaces, but the converse is not always true. In the following example, we present a controlled rectangular -metric space which is not a rectangular metric space.
Example 1. Let , where and be the set of positive integers. We define by where is a constant bigger than Now, define by It is not difficult to check that is a controlled rectangular -metric space. However, is not a rectangular metric space; for instance, notice that
3. Main Results
Theorem 6. Letbe a controlled rectangular-metric space anda self-mapping onIf there exists, such thatand, thenhas a unique fixed point in
Proof. Let and define the sequence as follows: . Now, by the hypothesis of the theorem, we have Note that if we take the limit of the above inequality as , we deduce that as . Now, consider . Thus, for all , we have two cases.
Case 1. Let for some integers . So, if for , we have . Choose and . Then, ; that is, is a periodic point of . Thus, Since , we get , so ; that is, is a fixed point of .
Case 2. Suppose that for all integers . Let be two natural numbers; to show that is a -Cauchy sequence, we need to consider two subcases:
Subcase 1. Assume that By property (R_3) of the controlled rectangular -metric spaces, we have Thus, Therefore, Now, using the fact that , the above inequalities imply the following: Since , we deduce Note that the series converges by the ratio test, which implies that converges as .
Subcase 2. . First of all, note that which leads us to conclude that as . Similar to Subcase 1, we have Hence, Since , we deduce By using the ratio test, it is not difficult to see that the series converges. Hence, converges as and go toward Thus, by Subcases 1 and 2, we deduce that the sequence is a -Cauchy sequence. Since is a -complete extended rectangular -metric space, we deduce that converges to some We claim that is a fixed point of Note that there exists an integer such that . Due to Case 2, for all . Similarly, for all . Hence, we are in Case 1, so is a fixed point of .
Also, there exists an integer such that . Again, necessarily, and for all . Thus, . Therefore, we may assume that for all , we have
Now, taking the limit as , we deduce that ; that is, and is a fixed point of as desired.
Finally, to show uniqueness assume, there exist two fixed points of say and such that By the contractive property of , we have which leads us to a contradiction. Thus, has a unique fixed point as required.
Theorem 7. Letbe a complete extended rectangular-metric space anda self-mapping onsatisfying the following condition; for all, there existssuch thatAlso, ifand for all, we havethen, has a unique fixed point in
Proof. Let and define the sequence as follows: First of all, note that for all , we have Since , one can easily deduce that So, let . Hence, Therefore, Also, for all , we have Thus, by using the fact that as , we deduce that Now, similar to the proof of Cases 1 and 2 of Theorem 6, we deduce that the sequence is a -Cauchy sequence. Since is a -complete extended rectangular -metric space, we conclude that converges to some Using the argument in the proof of Theorem 6, we may assume that for all , we have . Thus, Taking the limit of the above inequalities, we get Thus, which implies that , and hence, is a fixed point of Finally, to show uniqueness, assume there exist two fixed points of say and such that By the contractive property of , we have which leads us to a contradiction. Thus, has a unique fixed point as required.
4. Application
In closing, we present the following application for our results.
Theorem 8. For any natural number, the equationhas a unique real solution.
Proof. First of all, note that if , Equation (3.1) does not have a solution. So, let , and for all , let and It is not difficult to see that is a -complete controlled rectangular -metric space. Now, let Notice that since , we can deduce that Thus, Hence, On the other hand, notice that for all , we have Thus, Finally, note that satisfies all the hypothesis of Theorem 6. Therefore, has a unique fixed point in , which implies that Equation (3.1) has a unique real solution as desired.
Example 2. The following equation has a unique real solution.
Proof. The proof is a direct consequence of Theorem 6, by taking
5. Conclusion
In closing, we would like to bring to the readers’ attention to the following open questions:
Question 1. Let be a controlled rectangular -metric space and a self-mapping on Also, assume that for all distinct , there exists such that What are the other hypotheses we should add so that has a unique fixed point in the whole space ?
Question 2. Let be a controlled rectangular -metric space and a self-mapping on Also, assume that for all distinct , there exists such that What are the other hypotheses we should add so that has a unique fixed point in the whole space ?
Data Availability
Data availability is not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All three authors contributed equally to this manuscript.
Acknowledgments
The first and third authors would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RG-DES-2017-01-17.