Abstract
In this article, we present an orthogonal -contraction mapping concepts and prove a fixed point theorem on orthogonal complete Branciari metric spaces. As an application, we apply our major results to solving integral equations.
1. Introduction
The general metric concept was introduced by Branciari [1] in 2000 and which is known as the Branciari metric. Later, many authors were interested to the Branciari metric space for extending the results of Branciari-metric spaces (see [2–7]). The -contraction concept was introduced by Jleli and Samet [8] in 2014. It is based on some fixed point results [9, 10]. An orthogonality concept in metric spaces was introduced by Gordji et al. [11, 12]. Several authors proved the fixed point results in the generalized orthogonal metric space of Branciari metric spaces (BMS) [13–17]. The -contraction concept was introduced by Cho [17] in 2018. In this article, we present the new concepts of -contractive orthogonal mapping and prove fixed point theorems in an orthogonal complete Branciari metric space (OCBMS). We also give an example to our current results for using the integral equation solved, respectively.
2. Preliminaries
The basic definitions and results are required in the next section as follows.
Definition 1 (see [1]). Let be a non-empty set and a mapping such that for all and all
(BM1)
(BM2)
(BM3)
The metric is called a Branciari metric, and the pair is called a BMS.
Definition 2 (see [1]). Let be a BMS. A self-map is called -contraction if there exist and such that :
where is the family of all functions which satisfy the following conditions:
() is increasing
() For each sequence
() is continuous.
Remark 3. We know that every -contraction mapping is continuous.
The following notes are subsequently adopted: (1) is the class of all functions which satisfy
Definition 4 (see [17]). Let be a BMS. A mapping is called -contraction with respect to if there exists such that (for all ):
where is the class of all functions in which the following conditions are satisfied ():
()
() for all
() If and are two sequence in with such that then
Example 1 (see [17]). Let be two functions defined by (a), for all where (b), for all where is a lower semicontinuous and increasing function with Then,
Cho [17] proved the following theorem.
Theorem 5 (see [17]). Let be a complete BMS and an -contraction mapping. Then, has a unique fixed point.
Remark 6. Let are sequences of such that and . Then,
Lemma 7 (see [7]). Let be a Cauchy sequence in a BMS such that , for some . Then, , for all In particular, diverge to if
Definition 8 (see [11]). Let and be a binary relation. If satisfies the following condition: then it is called an orthogonal set. We denote this -set by .
Example 2. Let and define if there exists such that . It is easy to see that for all . Hence is an -set.
Example 3 (see [11]). A wheel graph is a graph (see, for example, Figure 1) with vertices for each , a single vertex connect to all vertex to all vertices of an -cycle. Let be the set of all vertices of for each . Define if there is a connection from to . Then, is an -set.
Definition 9 (see [11]). Let be an -set. A sequence is called an orthogonal sequence (shortly, -sequence) if
Definition 10 (see [11]). The triplet is called an orthogonal metric space if is an -set and is a metric space.
Definition 11 (see [11]). Let be an orthogonal metric space. Then, a mapping is said to be orthogonally continuous in if for each -sequence in with as , we have as . Also, is said to be -continuous on if is -continuous in each .
Definition 12 (see [11]). Let be an orthogonal metric space. Then, is said to be an orthogonally complete, if every Cauchy -sequence is convergent.
Definition 13 (see [11]). Let be an -set. A mapping is said to be -preserving if whenever . Also, is said to be weakly -preserving if or whenever for all .
3. Major Results
In this section, we present the generalized orthogonal -contraction notion.
Definition 14. Let be an -set and a mapping such that for all and all
(OBM1) if and only if
(OBM2)
(OBM3) for all .
The metric is an orthogonal Branciari metric (shortly OBM), and the pair is an orthogonal BMS (shortly OCBMS).
Definition 15. Let be a OCBMS and . Then, is said to be generalized orthogonal -contraction with respect to if there exist such that
Theorem 16. Let be a complete OCBMS with an orthogonal element and a self-mapping . Suppose that there exist and such that the following conditions hold: (i) is -preserving;(ii) is generalized orthogonal -contraction mapping;(iii) is -continuous.Then, has a unique fixed point.
Proof. Since is an -set, It follows that or . Let for all . If for any , then it is clear that is a fixed point of . Now, we consider for all . Since is -preserving, we have for all . This implies is an -sequence. Using contractive Condition (6) and , we have which is implies that Hence, inequality (11) becomes (in view of ()) that Therefore, the sequence is non-increasing and bounded below by 0. Then, such that We can claim that then Setting and In view of (11),(13), and (), we have and , for all . Therefore, applying the condition , we deduce which is a contradiction, and therefore Now, we consider , for some . Then, also . Using (11), we get which is a contradiction. Hence, we conclude that .
Next, we show that is a Cauchy sequence in . On the contrary, assume that it is not Cauchy, then there exists an for which we can find two subsequences and of such that , for all and
Suppose that is the least integer exceeding satisfying inequality (17). Then,
Using (17), (18), and the triangular inequality, we get
As ,
Employing the triangular inequality once again, we get
On letting and using (15) as well as (20) we get
Now, using (6) and , we obtain
Consequently, we deduce that
Let and . Then, in view of Remark 6 and (24), we have and So, on using , we obtain which is a reductio ad absurdum. Therefore, must be a Cauchy sequence in . Since is a complete, then there exists such that , then,
As is continuous, then we get that (due to (26))
Using Lemma 7, we have that is, is a fixed point of . On the contrary, assume that there are two fixed points such that . From (6), since is preserving, we have
This is implies that which is a reductio ad absurdum. Then, has a unique fixed point.
Example 4. Let , where and Define the binary relation on by if . Define a mapping defined by , for all .
It is easy to see that is an orthogonal complete BMS. Let be defined as for all . Clearly, is an orthogonal preserving and orthogonal continuous. Observe that is an -contraction with respect to , where and such that .
Let ; then,
Hence, all the hypotheses of Theorem 16 are satisfied, and is the unique fixed point of .
4. Applications
As an application of Theorem 16, we find an existence and uniqueness of the solution of the following integral equation:
Let be the set of real continuous functions defined on and the mapping defined by
Obviously, is a solution of integral Equation (32) iff is a fixed point of .
Theorem 17. Suppose that
(R1) The mappings , and are continuous functions;
(R2) there exist and such that
(R3) .
Then, the integral Equation (32) has a unique solution in .
Proof. Define the orthogonality relation on by Define given by for all . It is easy to see that is complete orthogonal BMS. For each with and , we have Accordingly, and so . Then, is -preserving. Let with . Suppose that . For each , we have Thus, which implies that for each . We consider -simulation mapping , where and such that . Then, Hence, all the conditions of Theorem 16 are fulfilled. Therefore, the integral equation has a unique solution.
5. Conclusion
In this article, we proved the fixed point theorems for orthogonal -contraction mapping on orthogonal complete BMS.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All authors equally contributed to this work. All authors read and approved the final manuscript.
Acknowledgments
The authors would like to thank the anonymous referees for their valuable comments and suggestions, which led to a considerable improvement of the article. The first author, Aiman Mukheimer, would like to thank Prince Sultan University for paying APC and for the support through the TAS research lab. The work was supported by the Higher Education Commission of Pakistan too, and the last author, Imran Abbas Baloch, would like to thank HEC of Pakistan.