#### Abstract

Complex Urysohn integral equations and complex Fredholm integral equation of the second kind have intensified the attention of appreciable researchers to their solution due to their comprehensive applications. This study is devoted to the existence and uniqueness of solution to these integral equations in the setting of complete complex double controlled metric spaces via fixed point theory. For this motive, a fixed point result together with a numerical example for the convergence behavior of operator to the fixed point is analyzed in the context of the quoted metric spaces.

#### 1. Introduction and Preliminaries

A self-mapping , where , is called a contraction if , such thatwhere is a metric space. Furthermore, if the metric space is complete and satisfies the inequality (1), then in such case, possesses a unique fixed point. Inequality (1) is often used to define contractivity, which implies the continuity of and is known as the Banach contraction principle. The generalization of Banach contraction principle stated by means of contraction mapping is one of the active areas of research for the practitioners of nonlinear phenomenon. Among these generalizations, weakly contractive mappings are one of the eminent results from scholars, which can be found in the articles [1, 2]. Moreover, Alqahtani et al. [3] have proved a result which is a generalization of Das and Gupta [4] and Jaggi [5], in which the rational type contractions were considered. Those considered rational type contractions are themselves, in fact, generalizations of Banach contraction principle. For more studies, refer to [611].

Apart from the generalization of contractive conditions, researchers have dedicated plentiful time to the solution of diversified amount of equations via the fixed point method. Such generalizations include solving linear system of equations, differential equations, fractional differential and integral equations, etc. Abdeljawad et al. generalized the double controlled metric spaces [12] to a complex double controlled metric spaces [13], in which a contraction theorem was presented. Furthermore, the solutions to complex Riemann–Liouville fractional operator, Atangana–Baleanu fractional integral operator, and nonlinear telegraph equation were obtained via the fixed point method. For further study, we encourage readers to study [14].

Integral equations play a key role in radioactive heat transfer problems, electricity and magnetism, kinetic theory of gases, mathematical economics, mathematical problems of radioactive equilibrium, quantum mechanics, fluid mechanics, optimization, optimal controlled systems, communication theory, radiation, population genetics, potential theory, queuing theory, geophysics, medicine, the particle transport problems of astrophysics, nuclear reactor theory, renewal theory, acoustics, fluid mechanics, hereditary phenomena in physics and biology, steady-state heat conduction, fracture mechanics, and continuum mechanics. Therefore, researchers have dedicated a lot of attention towards solving these equations (for instance, see [1519] and references therein). Urysohn integral equations occupy comparable priority in the diversified area of research in multiplex subjects like mathematics, biological sciences, physical sciences, and so on. The Urysohn integral equations are defined aswhere is a continuous function on the interval , and are kernels defined on the complex region, and .

The solution of Urysohn integral equations has been handled via distinctive approaches such as [20, 21] and references therein. One of the approaches is the fixed point approach (several articles can be found on such topic, for instance, see [22, 23] and references therein).

Likewise, Fredholm integral equation is one of the significant integral equations and is given bywhere is a complex continuous function on the given interval and is a complex parameter, the kernel is a known continuous separable kernel function on a complex plane, and . It is always assumed that functions and are complex-valued. In addition, the sets and are linearly independent on the interval , and are the real limits of the integral. Similarly, the applications of Fredholm integral equation can be found in the aforementioned fields of the research, which highlight the dominance of integral (3) in the area of research. Therefore, the authors conceive to find the solution of (3) via variant strategies. Similarly, the solution of Fredholm integral equation is managed in distinctive methods. An illustration of some of those methods has been analyzed in [2426]. One of the methods to solve integral (3) is the -spline wavelet method by Maleknejad and Nosrati Sahlan [27]. Similarly, He proposed a variational iteration method [2830] to solve integral (3). The Adomian decomposition method and various numerical methods have also been applied to obtain the solution to (3). Fixed point theory is an alternative supporting approach to solve (3). Numerous researchers have followed this method. For further study, one can see [3133].

As mentioned earlier, there are diverse methodologies to solve an integral equation. The fixed point technique is among the various other techniques. In the current study, we have extended the result proven in [3] to common fixed point weak contraction in the setting of complete complex double controlled metric type spaces with a numerical experiment. Moreover, we apply our proven result to obtain sufficient conditions for the existence and uniqueness of complex-valued Urysohn integral equations and Fredholm integral equation.

Theorem 1 (see [3]). Let be a complete extended -metric space and be a continuous self-mapping such that for all ,where and

Consider that for each and , , where for all . Then, has a fixed point say . In addition, for , we have .

In [13], Panda et al. proposed a more generalized structure of double controlled metric spaces, i.e., complex double controlled metric spaces, where a couple of fixed point results are proved. In addition, the solutions to a couple of fractional operators and telegraph equation have been investigated via the fixed point technique. To achieve the main objectives of this work, we need to revisit some basic concepts from literature.

Definition 1. Consider the set of complex numbers and suppose , are complex numbers. A partial order on can be defined asTherefore, it can be inferred that , if one of the given property holds.(1) and .(2) and .(3) and .(4) and .The notation can be listed as , if , along with property (4) to be satisfied by considering the equivalent fashion, i.e.,(1).(2) and .(3).(4)If and .

Definition 2. Let be a nonempty set and be a distance mapping, where is a complex-valued set. Moreover, consider the functions . A complex-valued double controlled metric can be defined as(1); .(2).(3).. The pair is called a complex-valued double controlled metric space.

Example 1. Consider a nonempty set , and we define distances as follows (Table 1).
Lets define the functions, by (Table 2).
From the above data, it can be analyzed thatwhich shows that is not a complex controlled metric space and hence nor a complex extended -metric space. On the other hand,Similarly, all the remaining cases can be verified, which shows that is a complex double controlled metric space.

Definition 3. (see [13]). Let be a complex double controlled metric type space and let be a sequence and .
For any , there exists such that ,(1)A sequence is said to be convergent to if ; then, as . Mathematically, it can be expressed as .(2)A sequence , is said to be Cauchy if for , .(3)The metric space is said to be complete if every Cauchy sequence is convergent in .

#### 2. Fixed Point Result

In this section, a fixed point result is established which generalizes the result in [3] to a common weak contraction in the sense of complete complex double controlled metric space.

Theorem 2. Let be a complete complex double controlled metric space. Furthermore, let be two self-operators. Now for and , such that for all distinct , we havewhere

Moreover, for and for any ,

Additionally, for each , suppose that

Then, have a common fixed point.

Proof. Let be an arbitrary point. Suppose the sequence satisfies the conditions of the theorem and is defined asConsidertogether withfrom which it is concluded thatNowFrom (17), we get that .
In order to proceed, consider the following cases from (16).

Case 1. If , using (9) implies ; as and , we get , which is an obvious contradiction as .

. If , thenIn order to conclude, consider the following subcases.
: if , which followswhile from (18), we haveBy simple calculation from (20), we getwhich is a contradiction to inequality (19).
(Case 2)ii: if , thenMoreover, from (18), it follows thatFrom inequality (23), it can be deduced thatwhich is a contradiction to (22). Consequently, Case 2 cannot be true.

. If , then using (9), we haveSince and , then from (25),Since , letting , it is obvious thatFurther, we prove that the given sequence is a Cauchy sequence. Let be integers such that , and we haveHere is used. DefineHence,The ratio test together with (10) implies that the limit of real number sequence exists, and so is a Cauchy sequence. In fact, the ratio test is applied to the term and using (30), we havewhich follows that the sequence is a Cauchy sequence. Now, it is known that the double controlled metric space is a complete space. So, the sequence converges to some point say . We claim that is the common fixed point for . Using triangle inequality, it follows thatNow it is claimed that . Using triangle inequality, continuity of , and (9), it follows thatFrom (27) and (32) and letting , we get which implies . Similarly, it can be checked that . Thus, have a common fixed point say . Now we show that the given point is unique; for this, consider , such that and . From (4), it follows thatIt can be deduced that . Similarly, the uniqueness of fixed point for can also be checked. Therefore, have a unique common fixed point.
This completes the proof.
If in Theorem 2, . Then, we get the following corollary.

Corollary 1. Let be a complete complex double controlled metric space. Furthermore, suppose that are two self-operators, such that distinct ,where and . With the additional assumption for and any ,

Moreover, for each , suppose that

Then, have a common unique fixed point.

Corollary 2. Let be a complete complex double controlled metric space. Furthermore, suppose that are two self-operators, such that distinct , we havewhere such that each for such that . With the additional assumption for and for any ,

Moreover, for each , suppose that

Then, have a common unique fixed point.

If and , then Corollary 1 reduces to the following corollary.

Corollary 3. Let be a complete complex double controlled metric space. Furthermore, suppose that is a self-operator, such that

With the additional assumption for and for any ,

Moreover, for each , suppose that

Then, has a unique fixed point.

Example 2. Let and define a distance function by , where and . Consider the functions by and , respectively. Then, it is easy to verify that is a complete complex double controlled type metric space. Now consider the operators by , and . Then, the left hand side of the inequality (9) implies thatNow to calculate right hand side of the inequality (9), considerWith simple calculations, it can be analyzed that and . Then, for and , inequality (9) is satisfied. Furthermore for each , conditions (11) and (12) are also satisfied, respectively. Hence, by Theorem 2, and possess a common unique fixed point, i.e., 0. For visualizing the co-relation of left and right hand side of the inequality (9), consider Figure 1.
The real and imaginary parts can be separately visualized in Figures 2 and 3, respectively.
It can be seen that Figures 2 and 3 provide the same visuals. This is because of the equal values of real and complex parts as can be observed in Table 3. Furthermore, the numerical comparison of some values of left hand side and right hand side of inequality (9) is given in Table 3.
To visualize the convergence behavior of and graphically, consider Figures 4 and 5, respectively.
For checking the convergence behavior of and numerically, consider Tables 4 and 5, respectively.

#### 3. Existence of a Unique Solution of Complex Urysohn Integral Equations

Let be the set of all continuous functions from the closed and bounded interval to . Consider a distance function defined bywhere . Furthermore, consider the functions defined by and , respectively. Then, is a complete complex double controlled metric space.

In this section, we propose the existence and uniqueness of solution via the fixed point method for the complex-valued Urysohn integral equations.where and are continuous complex-valued kernels on the given interval and is continuous and complex-valued on the interval .

Consider the operators defined by

Urysohn integral (47) would have a unique common solution if the following conditions:hold, where , , and are defined in (9). Consider

If , then

It can be concluded that inequality (9) is satisfied. Additionally, for any , conditions (11) and (12) are also satisfied. Consequently, all hypotheses of Theorem 2 are satisfied. Therefore, system (47) possesses a common unique solution.

#### 4. Existence of a Unique Solution of Complex Fredholm Integral Equation

Let be the set of all continuous functions from the closed and bounded interval to . Consider a distance function defined bywhere . Furthermore, consider the functions , defined by and , respectively. Then, is a complete complex double controlled metric space.

In this section, we inspect the existence and uniqueness of solution via the fixed point method for complex Fredholm integral equation.where each of the component of (53) is defined in Section 1.

Consider the operator defined by

The Fredholm integral equation would have a unique solution if the following condition:holds.

Considerwhich implies

Consequently, all hypotheses of the Corollary 3 are satisfied. Therefore, integral (53) has a unique solution.

#### 5. Conclusion

In this study, we have proved an extended rational weak contraction for common fixed point in the setting of complex double controlled type metric spaces. Our results are validated by a numerical example and existence and uniqueness for complex Urysohn integral equations and complex Fredholm integral equation. The existence and uniqueness of a solution for certain type of equation hold remarkable applications towards applied sciences, like applied mathematics, physics, biology, engineering, and so on. It helps the researchers to know about the nature, specific conditions involved, exact range, and domain of a solution to any under discussion problem and thus saves a lot of efforts, time, and energy. To contribute to the worldly problems, we have also considered complex Urysohn and Fredholm integral equations for existence and uniqueness of a solution. The Urysohn and Fredholm integral equations have remarkable applications towards geophysics, mechanics, quantum mechanics, population genetics, fluid mechanics, etc.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.