Solving Differential Equation via Orthogonal Branciari Metric Spaces

Prakasam, Senthil Kumar; Gnanaprakasam, Arul Joseph; Mani, Gunaseelan; Kumar, Santosh

doi:https://doi.org/10.1155/2023/4943412

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Fixed Point Theory and its Applications in Ordinary and Fractional Differential and Integral Equations

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Research Article | Open Access

Volume 2023 | Article ID 4943412 | https://doi.org/10.1155/2023/4943412

Solving Differential Equation via Orthogonal Branciari Metric Spaces

Senthil Kumar Prakasam,¹Arul Joseph Gnanaprakasam,¹Gunaseelan Mani,²and Santosh Kumar³

Academic Editor: Marija Cvetkovic

Received14 Dec 2022

Revised11 Jan 2023

Accepted31 Mar 2023

Published26 Apr 2023

Abstract

In this paper, we investigate an orthogonal -contraction map concept and prove the fixed-point theorem in an orthogonal complete Branciari metric space (OCBMS). We also provide illustrative examples to support our theorems. We demonstrated the existence of a uniqueness solution to the fourth-order differential equation using a more orthogonal contraction operator in OCBMS as an application of the main results.

1. Introduction

The Branciari metric (BM) concept was introduced by Branciari [1] in the year 2000. The generalization is via the fact that the triangle inequality is replaced by the rectangular inequality for all pairwise distinct points of . Afterwards, many authors studied and elaborated the existence of old fixed-point theorems in the BMS (briefly Branciari metric spaces) [2–7]. The -contraction concept was introduced by Jleli and Samet [8] in 2014. Later, some authors provided a variety of results based on -contraction [9, 10]. Saleh et al. [11] introduced the concept of generalized and -contractions. And also proved fixed-point theorems in CBMS. Eshraghisamani et al. [12] initiated new contractive map and proved fixed-point theorem in BMS.

An orthogonality notion in metric spaces is presented by Gordji et al. in 2017 [13, 14]. Recently, many authors established a variety of fixed-point results in generalized orthogonal metric space (OMS). Nazam et al. [15] demonstrated the concept of -orthogonal interpolation contraction mappings. The notion of metric-like space via a hybird pair of operators was introduced by Ali et al. [16] in 2022. In 2021, Hussain [17] presented another family of fractional symmetric --contractions and builds up some new results for such contraction in the context of -metric space. Mukheimer et al. [18] initiated the concept of orthogonal -contraction mapping and proved fixed-point results in OBMS.

From the above motivation, we prove some fixed-point results in the direction of OBMS. We also give some examples to argue that our results correctly generalize certain results in the literature.

In this article, we present basic definitions and examples in Section 2, prove some fixed-point theorems by orthogonal -contractive mapping in an OCBMS in Section 3, and finally, obtain a unique solution of differential equation using orthogonal contraction operator in Section 4.

2. Preliminaries

Throughout this article, we denote by , , and the nonempty set, the set of positive integers, and the set of positive real numbers, respectively.

The Branciari metric space was introduced by Branciari [1] as follows.

Definition 1. Let and a function s.t (briefly such that) and all
(BM1) iff ;
(BM2);
(BM3)
The pair is called a BMS with Branciari metric .

The following example is on the Branciari metric space (BMS).

Example 1. Let , where and . Define as Then, is a CBMS (briefly complete Branciari metric space). However, we get (1) although and hence, is discontinuous(2)There is nonexistence s.t , and hence, the topology is not a Hausdorff(3) however, there does not exist s.t , and thus, an open ball does not necessitate an open set(4) is not a Cauchy sequence since it converges to both 0 and 2

Now, we give the following concepts, which are used in this paper.

Definition 2. Let be a BMS and be a sequence in and . (1) is convergent to as . We denote this by ;(2) is Cauchy as ;(3) is complete every Cauchy sequence in which converges to some element in .

Eshraghisamani et al. [12] introduced the concept of -contraction as follows.

Definition 3. Let be a BMS. A map is said to be -contraction if there exist and s.t where is the family of all functions which satisfy the following axioms:
is increasing
For each sequence
is continuous.

Using Definition 3, Eshraghisamani et al. [12] proved the following theorem.

Theorem 4. Let be a CBMS and a-contraction function. Then, has a ufp (briefly unique fixed point).

The below example supports Theorem 4.

Example 2. Let be two functions defined as below: where are upper semicontinuous from the right s.t , for all Then, .

In Theorem 4, by replacing the condition (), we get the following remark.

Remark 5. Let be the sequence of s.t and . Then, (1),(2).

In 2017, Gordji et al. [13] introduced the concept of an orthogonal set as follows.

Definition 6. Let and be a binary relation. If holds then is called an orthogonal set.

The following example and Figure 1 are satisfied by Definition 6.

Example 3. Let and define if . It is clear that . Hence, is an orthogonal set.

Example 4. A wheel graph with edge for every , a node connect to each node to every edge of -cycle. Let be the set of all edge of for every . Define if there is a connection from to . Then, is an orthogonal set.

The following orthogonal sequence definition was introduced by Gordji et al. [13] which will be utilized in this paper to prove main results.

Definition 7. Let be an orthogonal set. A sequence is called an orthogonal sequence (shortly, -sequence) if

Again, the concepts of orthogonal continuous also introduced by Gordji et al. [13].

Definition 8. Let be a OMS. Then, a mapping is called orthogonal continuous in if for every -sequence in with as , we have as .

Definition 9. Let be a OBMS. (1), an orthogonal sequence in , converges at a point if (2) are orthogonal sequences in and are said to be orthogonal Cauchy sequence if

Gordji et al. [13] introduced the concept of an orthogonal complete as follows.

Definition 10. Let be a OMS. Then, is called an orthogonal complete, if every orthogonal Cauchy sequence is convergent.

Finally, the following orthogonal-preserving concepts introduced by Gordji et al. [13] is of importance in this paper.

Definition 11. Let be an orthogonal set. A function is called a -preserving if whenever .

Lemma 12. Let be an orthogonal Cauchy sequence in BMS s.t , for some . Then, , for all with .

Eshraghisamani et al. [12] proved fixed-point result on Branciari metric space as follows.

Theorem 13. Let be a complete generalized metric space and a map . Suppose that there exist and function , satisfying the following conditions: (i)For every and nonconstant (ii)For every that , such that then has a ufp.

3. Main Results

Before presenting our main result of this section, we are inspired by the concept of contraction mapping defined by Saleh et al. [11]; we introduce a new concept of an orthogonal -contraction mapping. Then, we prove a fixed-point results in OCBMS.

Definition 14. Let be a OBMS and . Then, is called an orthogonal -contraction w.r.t if s.t. where

Motivated by Theorem 13, we prove the below theorem.

Theorem 15. Let be a OCBMS and is a self-map on . Suppose that and a function hold the axioms: (i) is orthogonal-preserving(ii)For every and nonconstant (iii) with for every that such that then has a ufp.

Proof. Since is orthogonal set, It follows that or . Let If for any , then it is easy to see that is a fixed point of . Consider that for all . Since is -preserving, we have This implies that is an -sequence.
First, we show that . Since satisfies (12), for all , we have Since , we have Thus, is a decreasing sequence; hence, it is convergent and Now, we show that . From (17), we have since ; therefore, . So, by (ii).
On the other hand from (19), we have Then, Thus, Put ; using (22), and condition (iii) of , we get Now, we will show that as . Therefore, , as .

Now, to prove that the sequence is Cauchy, we consider two cases.

Case 1. If then

Case 2. If then

Thus, combining these two cases and using (23), when , we have

Thus, we deduce that is an orthogonal Cauchy sequence.

Completeness of ensures for some .

Now, we want to show that is a fixed point of . From (12), we have

Hence, , and , and therefore, as . Again, by using (ii).

Thus, , and hence, is a fixed point on .

Now, we prove that is unique. Conversely, assume that any two fixed points s.t . From (12), since is preserving, , we have

Now,

This implies that

This is a contradiction. Then has a ufp.

The below example validates the proof of Theorem 15.

Example 5. Let and defined as follow , for all we define the relation and , otherwise.

We observe that

Hence, -preserving, is not a BMS. It is obvious that is a OCBMS.

Let be a map defined by

Now, we define by .

Easily, we can show that satisfies conditions (ii) and (iii) of Theorem 15, satisfies (12), and is fixed point of .

Saleh et al. [11] proved a new contractive maps and their fixed points on BMS as follows:

Theorem 16. Let be a BMS and be an -contraction w.r.t (briefly with respect to) . Then, has a ufp.

In the following theorem, we are going to prove fixed-point theorem on an orthogonal -contraction mapping using continuity hypothesis of .

Theorem 17. Let be a OCBMS with an orthogonal element and a function , orthogonal -contraction w.r.t , the following axioms are satisfy: (i) is orthogonal-preserving.(ii) is with -contraction mapping.Then, has a ufp.

Proof. Since is orthogonal set, It follows that or . Let for all .
If for any , then it is easy to see that is a fixed point of . Consider . Since is -preserving, we have for all . Which implies that is a -sequence.

Using equation (10) and , we have

Consequently, we obtain that where

If , then inequality (41) becomes

This is a contradiction. Hence, we must have , for all . Therefore, inequality (41) becomes which implies from that

Thus, is decreasing sequence and boundary below by 0, so s.t . Suppose that , then from

Taking and , it is clear from (44), (46), and () that , and . Hence, using , we get

This is a contradiction. Therefore, , we have

Now, let us assume that , for some . Then, we have . Using (44), we get

This is a contradiction. To summarize , for all .

Next, to prove is a orthogonal Cauchy sequence in . Now, we consider it as not an orthogonal Cauchy; then, we can find two subsequences , and of s.t is the smallest integer for which

By using a similar argument, we obtain

Now, using (10) and , we have which implies that where

From (48), (51), and Remark 5, we get

Now, let , and , for all . In view of (51), (53), (55), and , we have , for all and . Therefore, using , we obtain which is contradiction. Hence, is orthogonal Cauchy sequence. As is complete, then there exists s.t

Without loss of generality, we consider and , for all . Suppose that , it follows from (10) and that where , which implies that

From Remark 5 and Lemma 12, we have

Let , and , for all ; it follows from (10) and that

This is a contradiction. Therefore, summarize , that is, is a fixed point of . Finally, prove that is ufp.

Consider two different fixed points and in .

Then, , since is an orthogonal-preserving, .

Using (10) and , we deduce that where , which implies that

This is a contradiction. Therefore, has a ufp.

Corollary 18. Let be a OCBMS and . Assume that : (i) is orthogonal-preserving(ii)where , and is nondecreasing and lower semicontinuous s.t . Then, has a ufp.

Proof. Let , for all . From (64), we have for all with , and . Therefore, is orthogonal-preserving.
Now, we define , for all , where is nondecreasing and lower semicontinuous s.t .
From (65), we have Taking and using (66), we have Therefore, all conditions are satisfied in Theorem 17, and hence, has a ufp.

In the following example, validate the proof of Theorem 17.

Example 6. Let , where and Define a map as follows: (1)(2)(3)(4) and(5) if or or

Here, the triangle inequality is not satisfied, so is not a metric on ; we have

It is easy to verify that is a OCBMS. Let be defined as an orthogonality relation on by

Since is not continuous at , and is not continuous, then is neither orthogonal -contraction nor an orthogonal -contraction.

Declare that is an orthogonal -contraction w.r.t , where and , s.t .

Indeed, for , and , we have

Hence, all the hypotheses are satisfied in Theorem 17, and is the ufp of .

4. An Application

The following BVP of a fourth-order differential equation is taken from Saleh et al. [11].

In this section, as an application of Theorem 17, we present the following result which provides an existence and uniqueness solution to the BVP of a fourth-order differential equation through an orthogonal -contraction.

Let is a continuous function. Let represent the space of all continuous functions defined on the interval [0,1]. Define a metric by

It is known that is a complete BMS. Define the green function associated with (72)

Now, we provide the following result regarding the BVP (72) solution.

Theorem 19. Assume that the following axioms are satisfied:
(P1) is orthogonal continuous function
(P2) there exist and s.t, for all , and where is defined by Then, (72) has a unique solution in .

Proof. Define the binary relation on by Observe that is a solution of (72) iff is a solution of the differential equation Then, is an orthogonal-continuous.
Now, we show that is orthogonal-preserving, in (P2), for all with and for all . Then, is an orthogonal-preserving.
Next, we claim that is orthogonal -contraction. We have where . As , for all , we obtain Observe that as . It follows that is an orthogonal -contraction. Therefore, for all , we obtain where , and . Thus, all the axioms of Theorem 17 are fulfilled. Therefore, has a ufp in which is a solution of (72).

5. Conclusion

In this paper, we proved the fixed-point results for orthogonal -contraction map on OCBMS. Furthermore, we presented some examples to strengthen our main results. Also, we provided an application to the BVP of a fourth-order differential equation.

Khalehoghli et al. [19, 20] presented a real generalization of the mentioned Banach’s contraction principle by introducing -metric spaces, where is an arbitrary relation on . We note that in a special case, can be considered as [partially ordered relation], [orthogonal relation], etc. If one can find a suitable replacement for a Banach theorem that may determine the values of fixed points, then many problems can be solved in this -relation. This will provide a structural method for finding a value of a fixed point. It is an interesting open problem to study the fixed-point results on -complete -metric spaces.

Data Availability

This clause is not applicable to this paper.

Additional Points

Rights and Permissions. Open access: this article is distributed under the terms of the Creative Commons Attribution.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Copyright © 2023 Senthil Kumar Prakasam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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