#### Abstract

In this manuscript, owing to the concept of orthogonal coupled contraction mappings type I and II, we prove coupled fixed point theorem in orthogonal metric spaces. In order to strengthen our main results, a suitable example is presented. Moreover, the results we obtained supplement and improve previous research findings. A fruitful application is also supplied to endorse our outcomes.

#### 1. Introduction

One of the simple and widely applicable result in nonlinear analysis is Banach contraction principle and it is prolonged in in the last century. One ordinary approach to bolster the Banach contraction principle is to restore the metric by weird generalized metric spaces. In 2017, Eshaghi Gordji et al. [1] established the idea of orthogonality and offered a framework to enlarge the results. In the same year, Eshaghi Gordji and Habibi [2] extended this work and proved some fixed point theorem in generalized -metric spaces. Later, in 2019, Gordji and Habibi [3] demonstrated fixed point theorems in -connected -metric spaces. By applying altering distance functions, Gungor and Turkoglu [4], in 2019, established fixed point results on -metric spaces. Using orthogonal -contraction mappings, Sawangsup et al. [5] proved some fixed point theorem on -complete metric spaces. In 2021, Beg et al. [6] proved fixed point theorems on -complete -metric spaces. In 2021, Arul et al. [7] proved fixed point theorems on -metric spaces. The concept of -triangular -admissibility introduced by Arul et al. [8] proved fixed point theorems on -metric spaces in 2022.

In 1987, on the other extreme, Gue and Lakshmikantham [9] prompted the conception of a coupled fixed point in partially ordered metric spaces. Following that, Bhaskar and Lakshmikantham [10], in 2006, demonstrated presence of coupled fixed point theorems by utilizing the mixed monotone property. The same coupled fixed point theorems on complete cone metric spaces was exposed by Sabetghadam et al. [11], in 2009. Afterwards, Guneseelan et al. [12] analyzed these results on complex partial -metric space. Motivated by the above work, here, we prove coupled fixed point theorems on -complete metric spaces.

#### 2. Preliminaries

In 2021, Gunaseelan et al. [13] proved the following theorem.

Theorem 1. *Let be a complete complex partial metric space and the mapping such that
where and are nonnegative constants with . Then, there exists a unique coupled fixed point of .*

Now, let us recall some basic concepts, which will be used in the sequel.

*Definition 2 (see [1]). *Let and be a binary relation such that or
then, it is called an orthogonal set (briefly -set). We denote this -set by .

*Definition 3 (see [1]). *Let be an -set. A sequence is called an orthogonal sequence (briefly, -sequence) if or

*Definition 4 (see [1]). *A triplet is called an orthogonal metric space (briefly -metric space) if is an -set and is a metric space.

*Definition 5 (see [1]). *Let be an -set. A mapping is said to be -preserving (briefly ) if whenever and . Also, is said to be weakly -preserving (briefly ) if or whenever and .

*Definition 6 (see). *Let be a nonempty set and be a mapping. A point is said to be a coupled fixed point of if and .

#### 3. Main Result

This section presents the new results motivated by Theorem 1 and an -set, we introduce new -coupled contraction mappings of type and .

*Definition 7. *Let be an -metric space. A function is called an -coupled contraction mapping of type (briefly -) on if for all with and ,
where and are nonnegative constants with .

*Definition 8. *Let be an -metric space. A function is called an -coupled contraction mapping of type (briefly -) on if for all with and ,
where and are nonnegative constants with .

Theorem 9. *Let be an -complete metric space with an orthogonal element and be a mapping such that
*(i)* is *(ii)* is -**Then, there exists a unique coupled fixed point of .*

*Proof. *By the definition of an -set, we can find satisfying or
and we can find satisfying or

It follows that or and or . Let

If , for any , then is a coupled fixed point of . Suppose that or , . Then, or

for all . Since is , we have

. Therefore, and are -sequences. Since is -,

Set

Then, we have

Since , we get

If , then

Hence, and , which implies that is a coupled fixed point of . Let . For each ,

Similarly, we can derive that

Thus, which implies that and are Cauchy -sequences. By the definition of an -complete, we can find satisfying and . Now,

By choice of and , we have

Since is -, we get

As , we get

Therefore, .

Similarly, we can prove that . Assume that is another coupled fixed point of satisfying . Then, and . Since is , we get

Since is an -, we get

Thus, we have

Since , we obtain

Therefore, and , which is a absurdity. So, has a unique coupled fixed point.

*Example 1. *Let and for all . Define a relation on by
Then, is an -complete metric space. Define a mapping by . Let and . Then, and . Now,

It follows that

Therefore, is . Then, for all , , we get

Therefore, all the hypotheses of Theorem 9 are fulfilled with . Hence, has a unique coupled fixed point .

Theorem 10. *Let be an -complete metric space with an orthogonal element and be a mapping such that
*(i)* is *(ii)* is -**Then, there exists a unique coupled fixed point of .*

*Proof. *By the definition of an -set, we can find satisfying
and we can find satisfying

It follows that or and or . Let

If , for any , then, is a coupled fixed point of . Suppose that or , . Then for all . Since is , we have

. Therefore and are -sequences. Since is -, we derive that

Similarly, we can derive that

Set

Then, we have

Since , then , we get

I*f*, then

Hence, and , which implies that is a coupled fixed point of . Let . For each ,

Similarly, we can derive that

Thus, which implies that and are Cauchy -sequences. By the definition of an -complete, we can find satisfying and . Now,

By choice of and , we have

Since is -, we get

As , we get

Therefore, .

Similarly, we can prove that . Assume that is another coupled fixed point of satisfying . Then, and . Since is , we get

Since is an -, we get and

Thus, we have

Since , we obtain

Therefore, and , which is a absurdity. So, has a unique coupled fixed point.

*Example 2. *Let and for all . Define a relation on by
Then, is an -complete metric space. Define a mapping by . Let and . Then and . Now

It follows that

Therefore, is . Then, for all , , we get

Therefore, all the hypotheses of Theorem 10 are fulfilled with . Hence, has a unique coupled fixed point .

#### 4. Supportive Application

Let and . Consider the integral equations: where (a) and are continuous,(b) is continuous and measurable at , ;(c), and , .

Theorem 11. *Assume that the conditions â€“ hold. Suppose that
for each and . Then, equation (60) has a unique solution in .*

*Proof. *Let . Define a relation on by
Define a mapping by
for all . Thus, is a -complete metric space. Define by

For each with and , we have

It follows that and so . Then, is . Let with and . Suppose that . For every , we have which implies that

Therefore, all the hypotheses of Theorem 9 are fulfilled and hence equation (60) has a unique solution.

#### 5. Open problem

In this article, we proved coupled fixed point theorems for orthogonal coupled contraction mappings of type and in -complete metric spaces. An illustrative example is presented to strengthen our obtained main results. Agarwal and Karapinar [14] proved coupled fixed point theorems in -metric spaces in 2013. Here, the intriguing open problem is to investigate the coupled fixed point theorems on orthogonal -metric spaces instead of coupled fixed point theorems on -metric space. In 2014, RoldĂˇn et al. [15] introduced multidimensional fixed point theorems. It is an intriguing open problem to investigate the orthogonal multidimensional fixed point theorems instead of orthogonal coupled fixed point theorem. In 2016, RoldĂˇn, Sintunavarat [17] proved common coupled fixed point theorems in fuzzy metric spaces using the CLRg property. It is an intriguing open problem to investigate the orthogonal common coupled fixed point theorems instead of orthogonal coupled fixed point theorem.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no competing interests.

#### Authorsâ€™ Contributions

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