Common Fixed Point Theorems on Tricomplex Valued Metric Space

Abstract

In this paper, we introduce the notion of tricomplex valued metric space and prove some common fixed point theorems. The presented results generalize and expand some of the literature well-known results. We also explore some applications of our key results.

1. Introduction

Fixed point theory plays an important role in applications of many branches of mathematics. There has been a number of generalizations of the usual notion of a metric space (see [1–8] and the references therein). Serge [9] made a pioneering attempt in the development of special algebras. He conceptualized commutative generalization of complex numbers as bicomplex numbers, tricomplex numbers, and so on as elements of an infinite set of algebra. Subsequently during the 1930s, other researchers also contributed in this area [10–12]. However, unfortunately the next fifty years failed to witness any advancement in this field. Afterward, Price [13] developed the bicomplex algebra and function theory. Recently renewed interest in this subject finds some significant applications in different fields of mathematical sciences as well as other branches of science and technology. Also, one can see the attempts in [14]. An impressive body of work has been developed by a number of researchers. Among them, an important work on elementary functions of bicomplex numbers has been done by Luna-Elizarrarás et al. [15]. Choi et al. [16] proved some common fixed point theorems in connection with two weakly compatible mappings in bicomplex valued metric spaces. Jebril et al. [17] proved some common fixed point theorems under rational contractions for a pair of mappings in bicomplex valued metric spaces. In 2021, Beg et al. [18] proved the following fixed point theorems on bicomplex valued metric spaces.

Theorem 1. Let be a complete bicomplex valued metric space with degenerated and for all and such thatfor all , where are nonnegative real numbers with . Then, have a unique common fixed point.

In this paper, inspired by Theorem 1, we prove some common fixed point theorems on tricomplex metric space with applications.

2. Preliminaries

Throughout this paper, we denote the set of real, complex, bicomplex, and tricomplex numbers, respectively, as , , , and . Price [13] defined the bicomplex number as follows:where , and independent units are such that and ; we denote the set of bicomplex numbers as follows:i.e.,where and . Price [13] defined the tricomplex number as follows:where and independent units are such that , and ; we denote the set of tricomplex numbers as follows:i.e.,where and . If and be any two tricomplex numbers, then the sum is and the product is .

There are four idempotent elements in ; they are out of which and are nontrivial such that and . Every tricomplex number can be uniquely be expressed as the combination of and , namely,

This representation of is known as the idempotent representation of tricomplex number, and the complex coefficients and are known as idempotent components of the bicomplex number .

An element is said to be invertible if there exists another element in such that and is said to be inverse (multiplicative) of . Consequently, is said to be the inverse (multiplicative) of . An element which has an inverse in is said to be the nonsingular element of and an element which does not have an inverse in is said to be the singular element of .

An element is nonsingular if and only if and singular if and only if .

The inverse of is defined as

The norm of is a positive real valued function and is defined bywhere .

The linear space with respect to defined norm is a norm linear space; also is complete; therefore, is the Banach space. If , then holds instead of ; therefore, is not the Banach algebra. The partial order relation on is defined as follows: let be the set of tricomplex numbers and and , then if and only if and , i.e., if one of the following conditions is fulfilled:(a), (b), (c), (d),

In particular, we can write if and , i.e., one of (b), (c), and (d) is fulfilled and we will write if only (d) is fulfilled.

For any two tricomplex numbers , we can verify the following:(1)(2)(3), where is a nonnegative real number(4) and the equality holds only when at least one of and is nonsingular(5) if is a nonsingular(6) if is a nonsingular

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

Definition 1. Let be a nonempty set and such that(A1), for all and if and only if (A2) for all (A3) for all Then, is called the tricomplex valued metric on and is called the tricomplex valued metric space.

Example 1. Let and be defined by . Then, is a tricomplex valued metric space.

Definition 2. Let be a tricomplex valued metric space. A sequence in is said to be a convergent and converges to if for every , there exists such that , for all , and it is denoted by .

Lemma 1. Let be a tricomplex valued metric space. A sequence converges to iff .

Proof. Let be a convergent sequence and converges to a point , and let be any real number. SupposeThen, , and for this , there exists such that for all . Therefore,Hence, .
Conversely, let . Then, for each , there exists a real number such that for all ,Then, for this , there exists such thatTherefore,Hence, converges to a point .

Definition 3. Let be a tricomplex valued metric space. A sequence in is said to be a Cauchy sequence in if for any , there exists such that for all and .

Definition 4. Let be a tricomplex valued metric space. Let be any sequence in . Then, if every Cauchy sequence in is convergent in , then is said to be a complete tricomplex valued metric space.

Lemma 2. Let be a tricomplex valued metric space and be a sequence in . Then, is a Cauchy sequence in iff .

Proof. Let is a Cauchy sequence in . Let be any real number. SupposeThen, , and for this , there exists such that , for all . Therefore,And, this implies thatConversely, let . Then, for each , there exists a real number such that for all ,Then, for this , there exists a natural number such thatTherefore,Hence, is a Cauchy sequence.

Definition 5. Let and be self mappings of nonvoid set . A point is called a common fixed point of and if .

3. Main Result

In this section, we prove common fixed point theorem in a tricomplex valued metric space using rational type contraction condition.

Theorem 2. If and are self mapping defined on a complete tricomplex valued metric space such thatfor all where are nonnegative reals with , then and have a unique common fixed point.

Proof. Let be an arbitrary point in and define , . Then,Since implies , therefore,which implies thatSince , therefore,so thatAlso,Since implies , therefore,which implies thatAs , therefore,Putting , we have (for all )Therefore, for any , we havewhich implies thatIn view of Lemma 2, the sequence is Cauchy. Since is complete, there exists some such that as . On the contrary, let so that , thenAlso, for all , we haveAs , we getTherefore, . Similarly, we can derive that . Let (in ) be another common fixed point of and , i.e., . Then,which implies thatSince , therefore,Therefore, (as ).

Remark 1. (see [13]). We have

Example 2. Considering , define a mapping by , for all , where is the usual real modulus. Then, is a complete tricomplex valued metric space. Every real number is a tricomplex number but every tricomplex number is not necessarily a real number. Therefore, is not a metric space. Now, we consider a self mapping defined byfor all . Then,Thus, all the hypothesis of Theorem 2 are fulfilled with and . Hence, and have a unique common fixed point.

Example 3. Considering , define a mapping by , for all . Then, is a complete tricomplex valued metric space. Every real number is a tricomplex number but every tricomplex number is not necessarily a real number. Therefore, is not a metric space. Now, we consider a self mappings defined byfor all . Then,Every real number is a tricomplex number but every tricomplex number is not necessarily a real number. Therefore, we cannot find common fixed point for such mappings on metric space. Thus, all the hypothesis of Theorem 2 are fulfilled with and . Hence, and have a unique common fixed point.
By setting in Theorem 2, one deduces the following.

Corollary 1. If is a self mapping defined on a complete tricomplex valued metric space such thatfor all , where are nonnegative reals with , then has a unique fixed point.

Theorem 3. Let be a complete tricomplex valued metric space and the mappings such thatfor all , where and and are nonnegative reals with . Then, , have a unique common fixed point.

Proof. Let be an arbitrary point in . Define and , Now, we distinguish two cases. First, if (for ) and , thenSince and , therefore,which implies thatSince , therefore,so thatAlso,Since and , we getwhich implies thatSince , therefore,Now, with , we have (for all )So, for any , we haveand henceforthOn using Lemma 2, we conclude that is a Cauchy sequence. Since is a complete, then there exists such that as . Now, we assert that ; otherwise, and we havewhich implies thatTherefore, , i.e., . Similarly, we can derive that . Assume that in is an another common fixed point of and . Then,Since , thenHence, .
Second, we consider that the case (for any ) implies . Now, if , then . Thus, we have , so there exist and such that . Similarly, we can derive that there exist and such that . As (due to definition), it implies , so that which in turn yields that . Similarly, one can also have . As , it implies ; therefore, , is a common fixed point of and . Assume that in is an another common fixed point of and . Then,Since , thenThis implies that .
If implies that , then also proof can be completed on the preceding lines.

Example 4. Considering , define a mapping by , for all , where is the usual real modulus. Then, is a complete tricomplex valued metric space. Now, we consider a self mapping defined byLet , and , then . Let , thenTherefore,Hence, the conditions of Corollary 1 are fulfilled. Therefore, 0 is the unique fixed point of .

Example 5. Let , , for all . Define bywhere is a closed path in containing a zero. Then, then we prove that (, ) is a complete tricomplex valued metric space. We haveNow, we define the mapping byUsing the Cauchy integral formula when the mappings and are analytic, we getClearly,for all , where are nonnegative reals with . Therefore, all the conditions of Theorem 2 are fulfilled, then the mappings and have a unique common fixed point in .

Definition 6. (see [19]). Two self mappings and of a metric space are said to be commuting iff for all in .

Example 6. Considering , define a mapping by , for all , where is the usual real modulus. Then, is a complete tricomplex valued metric space. Now, we consider a self mappings defined byThen,So, and are commute. Now,Therefore,However,Thus, is not a contraction. Hence, and have a unique common fixed point.
We can prove that common fixed point theorems on tricomplex valued metric space under commuting mappings.

Definition 7. (see [19]). Two self mappings and of a metric space are said to be weakly compatible if the mappings commute at their coincidence points, i.e., for some implies .
We can prove common fixed point theorems on tricomplex valued metric space under weakly compatible mappings.

4. Applications

In this section, we give applications using Theorem 2 and Corollary 1.

Let be a set of all real continuous functions on equipped with metric for all and , where is the usual real modulus. Then, is a complete tricomplex valued metric space. Now, we consider the system of nonlinear Fredholm integral equation as follows:where . Assume that and are continuous, where is a given function in . We define a partial order in as iff .

Theorem 4. Suppose that is a complete tricomplex valued metric space equipped with metric for all , and be continuous operator on defined by

If there exists such that for all with and satisfying the following inequality:then the integral operators defined by (85) and (86) have a unique common solution.

Proof. ConsiderTherefore,Hence, all the hypothesis of Theorem 2 are fulfilled with and , and so the integral operators and defined by (85) and (86) have a unique common solution.

Theorem 5. Let be a complete tricomplex valued metric space with the metricwhere . Ifthen the linear systemof linear equations with unknowns has a unique solution.

Proof. Define bywhere , , andNow,Hence, all the conditions of Corollary 1 are satisfied with , and and so the linear system of equation has a unique solution.

5. Conclusion and Future Work

In this paper, we introduce the notion of tricomplex valued metric space and proved some common fixed point theorems on tricomplex valued metric space. An illustrative example and applications on tricomplex valued metric space is given. In 2013, Patel et al. [20] proved common fixed points for a pair of maps on metric space. It is an interesting open problem to study the tricomplex valued metric space instead of metric space and obtain common fixed points for a pair of maps on tricomplex valued metric space. In 2013, Rouzkard et al. [21] proved existence and uniqueness theorems on ordered metric spaces via generalized distances. It is an interesting open problem to study the ordered tricomplex valued metric space instead of ordered metric space and obtain fixed point theorems on ordered tricomplex valued metric space. Recently, Altun et al. [22, 23] proved the best proximity point theorems on complete metric space. It is an interesting open problem to study the best proximity theorems on tricomplex valued metric space instead of best proximity point theorems on complete metric space.

Data Availability

No data were used to support this work.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.