#### Abstract

In this paper, we introduce the notion of controlled rectangular metric spaces as a generalization of rectangular metric spaces and rectangular -metric spaces. Further, we establish some related fixed point results. Our main results extend many existing ones in the literature. The obtained results are also illustrated with the help of an example. In the last section, we apply our results to a common real-life problem in a general form by getting a solution for the Fredholm integral equation in the setting of controlled rectangular metric spaces.

#### 1. Introduction

The fixed point theory is a growing and exciting field of mathematics with a variety of variant applications in mathematical sciences, proposing newer applications in discrete dynamics and super fractals. The fixed point theory is a fundamental tool to various theoretical and applied fields, such as variational and linear inequalities, the approximation theory, nonlinear analysis, integral and differential equations and inclusions, the dynamic systems theory, mathematics of fractals, mathematical economics (game theory, equilibrium problems, and optimization problems), and mathematical modeling; see [1–3]. In particular, fixed point techniques have been applied in such diverse fields; see [4, 5]. There are particular real-life problems, whose statements are fairly easy to understand, which can be argued using some versions of fixed point theorems; see [6, 7].

The notion of extended -metric spaces was introduced by Kamran et al. [9] as a generalization of metric spaces and -metric spaces [10, 11]. This metric type space has been generalized in several directions (for instance, controlled metric spaces [12], double controlled metric spaces [13], and others [14–19]). In a different perception, Branciari [20] proposed rectangular metric spaces. In the same order, Asim et al. [21] included a control function to initiate the concept of extended rectangular -metric spaces as a generalization of rectangular -metric spaces [22]. In [23], Mlaiki et al. introduced controlled rectangular -metric spaces, which generalize rectangular metric spaces and rectangular -metric spaces.

In this paper, our goal is to introduce the notion of controlled rectangular metric spaces, which is different from controlled rectangular -metric spaces, and generalize rectangular metric spaces as well as rectangular -metric spaces. Further, we prove some fixed point results on such spaces as a generalization of many preexisting results in the literature. Also, we give examples for the justification of our results. In the last, as an application, we give an existence theorem for the Fredholm integral equation in the setting of controlled rectangular metric spaces.

#### 2. Preliminaries

In this section, we collect some basic concepts related to our main results.

*Definition 1 [22]. *A mapping on a nonempty set is called a rectangular -metric space, if there exists a constant such that for all and all distinct different from *and*, the following axioms are satisfied:
(i)(ii)(iii)In this case, the pair is called a rectangular -metric space.

*Definition 2 [9]. *Let be a nonempty set and be a mapping. Then, a mapping is called an extended -metric, if for all , it satisfies the following axioms:
(i)(ii)(iii)The pair is called an extended -metric space.

*Definition 3 [21]. *A mapping on a nonempty set is called an extended rectangular -metric space, if for all and all distinct different from and , the following axioms are satisfied:
(i)(ii)(iii)where is a mapping. In this case, the pair is called an extended rectangular -metric space.

Note that the topology of rectangular metric spaces need not be Hausdorff. For more examples, see the papers of Sarma et al. [24] and Samet [25]. The topological structure of rectangular metric spaces is not compatible with the topology of classic metric spaces; see Example 7 in the paper of Suzuki [26]. In the same direction, extended rectangular -metric spaces cannot be Hausdorff.

*Definition 4 [23]. *A mapping on a nonempty set is called a controlled rectangular -metric space, if for all distinct , the following axioms are satisfied:
(i)(ii)(iii)where is a mapping. In this case, the pair is called a controlled rectangular -metric space.

As a generalization of metric spaces, Mlaiki et al. in [12] introduced the concept of controlled metric spaces as follows.

*Definition 5 [12]. *Let be a nonempty set and . Then, a mapping is called a controlled metric, if for all , it satisfies the following axioms:
(i)(ii)(iii)The pair is called a controlled metric space.

Note that Definition 5 generalizes -metric spaces and is different from Definition 2.

*Example 1 [12]. *Let . Define as
Hence, is a controlled metric space, where is defined as

#### 3. Main Results

In this section, we introduce the notion of controlled rectangular metric spaces. Also, we establish some fixed point results.

*Definition 6. *A mapping on a nonempty set is called a controlled rectangular metric space, if for all and all distinct different from and , the following axioms are satisfied:
(i)(ii)(iii)where is a mapping, In this case, the pair is called a controlled rectangular metric space.

*Remark 7. *(i)Every rectangular metric space and rectangular -metric is a controlled rectangular metric space(ii)Clearly, Definition 6 is different from Definition 4(iii)Every controlled metric space is a controlled rectangular metric space, but its converse is not true in general. See the following example

*Example 2. *Let . Define a mapping by
Then, is a controlled rectangular metric space, where is a mapping defined as
Clearly, is not a controlled metric space if we take and . Then, , , , , and . Here, the triangle inequality is not satisfied:

*Example 3. *Let . Define as
Then, is a controlled rectangular metric space with defined as , for all . However, is not a rectangular metric space; for instance, notice

The concepts of convergence, Cauchyness, and completeness can simply be generalized in terms of controlled rectangular metric spaces.

*Definition 8. *Let be a controlled rectangular metric space. Then,
(i)A sequence in is said to be convergent to , if (ii)A sequence in is called a Cauchy sequence, if (iii) is called a complete controlled rectangular metric space, if every Cauchy sequence in is convergent to some point of

*Definition 9. *Let be a controlled rectangular metric space. Let and . Then,
(i)The open ball is define as(ii)The mapping is called continuous at , if for , there is such that . Thus, if is continuous at , then for any sequence converging to , we have

Note that a rectangular -metric space is not continuous in general, and it is the same for controlled rectangular metric spaces.

Lemma 10. *Let be a controlled rectangular metric space and be a Cauchy sequence in such that , whenever . If for all , then has a unique limit.*

*Proof. *Suppose that a sequence in has two limit points , that is, and . is a Cauchy sequence for , whenever . Hence, from condition (iii) of Definition 6, we have
This implies that
Hence, has a unique limit point in .

*Definition 11. *Let be a controlled rectangular metric space. Then,
(i)For a mapping , we definewhere and . The is called an orbit of .
(ii)A mapping is called -orbitally continuous, if implies , where (iii)A mapping is called -orbitally complete, if every Cauchy sequence in is convergent in

Our main result is similar to the Banach contraction principle in the setting of controlled rectangular metric spaces. Throughout this section, for a mapping and , we consider an orbit .

Theorem 12. *Let be a mapping on a controlled rectangular metric space . Suppose that the following axioms hold:
*(i)*For all , we have**where .
*(ii)*, for any *(iii)* is -orbitally complete*(iv)* is orbitally continuous*(v)*For each , and exist and are finite**Then, has a unique fixed point in .*

*Proof. *Consider an arbitrary point , and define an iterative sequence over as follows:
From equation (12), we have
Recursively, we have
That is,
By taking limit , we get

Next, we show that is a Cauchy sequence. For this, we will take the following two cases.

*Case 1. *Let be odd, that is, , where . Then, from condition (iii) of Definition 6 and equation (16) for , we have
As
therefore, we obtain
Since , the series
converge by the ratio test. Let
Then, equation (22) takes the following form:
By taking limit in equation (26), we get

*Case 2. *Let be even, that is, , where . Then, from condition (iii) of Definition 6 and equation (16) for , we have
As
therefore, we obtain
Since , the series
converge by the ratio test. Let
Then, equation (31) takes the following form:
By taking limit in equation (35), we get

Hence, in both cases, , which shows that is a Cauchy sequence. As is -orbitally complete, so there exists such that . Next, we show that is a fixed point of . As is orbitally continuous, so we have

Since for each , and exist and are finite, so by taking limit and using equation (17), we get

Therefore, . Hence, is a fixed point of . In view of Lemma 10, is the unique fixed point of .

*Example 4. *Let . Define by . Then, is a complete controlled rectangular metric space with defined as . Define a mapping by
Clearly, all the axioms of Theorem 12 are satisfied, and hence, is a fixed point of .

Corollary 13. *Let be a mapping on a complete controlled rectangular metric space . Suppose that the following axioms hold:
*(i)*For all , we have**(ii)**, for any *(iii)* is continuous**Then, has a unique fixed point.*

*Remark 14. *(i)By putting , for all in Theorem 12, we get Theorem 2.1 of George et al. [22](ii)By putting , for all in Theorem 12, we get the following corollary in view of Das and Dey [27]

Corollary 15. *Let be a mapping on a rectangular metric space . Suppose that the following axioms hold:
*(i)*For all , , where *(ii)* is -orbitally complete*(iii)* is orbitally continuous**Then, has a unique fixed point.*

Theorem 16. *Let be a mapping on complete controlled rectangular metric space , which satisfies the following axioms:
*(i)*For all , we have**where .
*(ii)*, for any , where for each *(iii)*For each , , , and exist and are finite**Then, has a unique fixed point in .*

*Proof. *Let us take an arbitrary element and choose and . Then, from equation (41), we obtain
This implies that
where , as . By recursively applying equation (41), we obtain
Thus, by taking the limit in equation (44), we have
Again from equation (41), we have
By using equation (45), we obtain
Now, we will show that is a Cauchy sequence. By following the same procedure as in the proof of Theorem 12 and using equations (45) and (47), we conclude that is a Cauchy sequence. As is complete, so there exists such that
Next, we show that is a fixed point of . From condition (iii) of Definition 6, for any , we have
for each . This implies that