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In this manuscript, we introduce the concept of complex-valued triple controlled metric spaces as an extension of rectangular metric type spaces. To validate our hypotheses and to show the usability of the Banach and Kannan fixed point results discussed herein, we present an application on Fredholm-type integral equations and an application on higher degree polynomial equations.

Since the breakthrough of Banach [

In this manuscript, following the path of the work done in [

In what follows, owing to Azam et al. [

Let

(

Following [

Let

Then, the pair

As an extension of complex-valued extended

Let

The authors in [

Let

Highly motivated by the abovementioned concepts, we now present the definition of complex-valued triple controlled metric spaces.

Let

Throughout the rest of this paper, we will denote a complex-valued triple controlled metric space by (CV-TCMS). Next, we present the topology of (CV-TCMSs).

Let

We say that a sequence

We say that a sequence

We say that

Let

Note that a CV rectangular metric space is a CV-TCMS. The converse is not true. Next, we present an example that confirms this statement.

Let

Note that

In this paper, we prove the Banach and Kannan fixed point results in the setting of CV-TCMSs. Two related applications are also investigated.

Let

Then,

First, we have

Now, let

Let

From now on, we consider the following case.

Assume that for all natural numbers

If

Now, given that

Since

Let

Thus, by Subcase

Now, similar to Subcase

Now, if there exists

Therefore, in view of the assumptions in the theorem, as

In closing, assume there exist two fixed points of

Let

Then,

First of all, note that for all

Consequently,

Since

Therefore,

Also, for all

By (

By the argument of the proof of Theorem

As

At the limit

Hence,

Consider the set

Note that

Assume that for all

Then, the above integral equation has a unique solution.

Let

Now, we have

Thus,

Therefore, all the hypotheses of Theorem

The following is an application on higher degree polynomial equations.

For any natural number

It is not difficult to see that if

Notice that, since

Hence,

Moreover, it is easy to see that for all

Note that all the conditions of Theorem

Finally, we would like to leave the following questions.

Let

Under what conditions does

Let

Under what conditions does

Data sharing is not applicable to this article as no data set were generated or analyzed during the current study.

The authors declare no conflict of interest.

All the authors have equally contributed to the final manuscript.

The authors would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.