Abstract
The main aim of this paper is to study and establish some new fixed point theorems for contractive maps that satisfied Mizoguchi-Takahashi’s condition in the setting of bicomplex-valued metric spaces. These new results improve and generalize the Banach contraction principle and some well-known results in the literature. Finally, as applications of our results, we give the existence and uniqueness of the solution of a nonlinear integral equation.
1. Introduction
Fixed point theorems concern maps of a set into itself that, under certain conditions, admit a fixed point, that is, a point such that . The knowledge of the existence of fixed points has relevant applications in many branches of analysis and topology. In 2011, Azam et al. [1] introduced the so-called complex-valued metric spaces and proved the existence of fixed points under some contraction conditions. Very recently, Choi et al. [2] introduced the notion of bicomplex-valued metric space which is a generalization of the complex-valued metric space and established sufficient conditions for the existence of common fixed points of a pair of mappings satisfying a contractive condition.
The following new fixed point theorem is one of the main results of this paper. It can be considered a bicomplex-valued metric version of the Banach contraction principle and will generalize and improve some well-known results in the literature. Finally, we strengthen our results by giving an application to find an analytical solution for a nonlinear integral equation. The results thus derived in this study are generally extended and generalized in character. Moreover, the results presented here may be very useful in the study of engineering problems like boundary value problems (see, e. g., [3–7]) in future works.
2. Preliminaries
In this section, we recall some definitions and terminologies, used to prove the main results.
2.1. Bicomplex Numbers
The set of bicomplex numbers denoted by is the first setting in an infinite sequence of multicomplex sets which are generalizations of the set of complex numbers . Here, we recall the set of bicomplex numbers (see, for example, [8–10]):
Since each element in can be written as or , we can also express as where , , and are independent imaginary units such that . The product of and defines a hyperbolic unit such that . The product of units is commutative and is defined as
With the addition and multiplication of two bicomplex numbers defined in the obvious way, the set makes up a commutative ring. Three important subsets of can be specified as
Each of the set is isomorphic to the field of complex numbers, while is the set of the so-called hyperbolic numbers.
2.2. Conjugation and Moduli
Three kinds of conjugation , and can be defined on bicomplex numbers. With the bicomplex number specified as in (2) and the bar denoting complex conjugation in , we define
It is easy to check that each conjugation has the following properties:
Here, and
With each kind of conjugation, one can define a specific bicomplex modulus as
It can be shown that , where or .
A norm of a bicomplex number denoted by is defined by which, upon choosing , gives
For any two bicomplex numbers , one can easily verify that , where is a nonnegative real number. Further, for any two bicomplex numbers, holds.
Next, we recall some necessary definitions and lemmas.
First, define a partial order on as follows:
Let and . Define a partial order relation on as follows:
if and only if and . We find that if any one of the following properties holds: where is a partial order relation on . So that
We write if and , i.e., one of , , and is satisfied and we write if only is satisfied.
Given the definition of the complex metric space introduced in [1], we extend the definition of bicomplex analysis as follows.
Definition 1. Letbe a nonempty set. A functionalis called a bicomplex-valued metric onif forthe following conditions are satisfied:
Then, is called a bicomplex-valued metric space.
Some known examples of bicomplex-valued metric, which show a bicomplex-valued metric space, are the following.
Example 2. The set of real numbers together with the functionalfor all where is the usual real modulus. One can easily check that , are bicomplex-valued metric spaces.
Definition 3. Letbe a sequence in a bicomplex-valued metric space. The sequenceis said to converge toif and only if for any, there existsdepending onsuch thatfor all. It is denoted byasor.
Definition 4. A sequencein a bicomplex-valued metric spaceis said to be a Cauchy sequence if and only if for any, there existsdepending onsuch thatfor all
Definition 5. A bicomplex-valued metric spaceis said to be complete if and only if every Cauchy sequence inconverges in.
Lemma 6 (see [2]). Letbe a bicomplex-valued metric space andbe a sequence in. Then,converges toif and only ifas.
Lemma 7 (see [2]). Letbe a bicomplex-valued metric space andbe a sequence insuch that. Then, for any, .
3. Statement of Results
Definition 8 (see [11, 12]). A functionis said to be a MT-function if it satisfies Mizoguchi-Takahashi’s condition, i.e.,for all.
Clearly, if is a nondecreasing function, then is a MT-function.
Notice that is a MT-function if and only if for each there exist and such that for all , for more detail, (see [4], Remark 2.5 (iii)).
An example which is not an MT-function is given hereunder. Let be defined by
Since , is not a MT-function.
Recently, Du [13] first proved the following characterizations of MT-functions which are quite useful for proving our main results.
Theorem 9. [13] Letbe a function. Then, the following statements are equivalent.(a) is a MT-function(b)For each , there exist and such that for all (c)For each , there exist and such that for all (d)For each , there exist and such that for all (e)For each , there exist and such that for all (f)For any nonincreasing sequence in , we have (g) is a function of contractive factor, that is, for any strictly decreasing sequence in , we have
Theorem 10. Letbe a complete bicomplex-valued metric space and letbe a mapping on. Suppose that there exists a MT-functionsuch thatThen, has a unique fixed point on .
Notice that need not be continuous.
Proof. Let be given. Define the sequence by
For each , by (15), we have which implies
Let for . Then, by (18), we have
So, we know that is a strictly decreasing sequence in . Applying (g) of Theorem 9, we obtain
That is,
Let
Then, . For each , by (18) again, we have
For any with , we get
Since , . Hence, by the last inequality, we obtain that is, as Therefore, is a Cauchy sequence in the complete bicomplex-valued metric space. Then, there exists such that
Next, we prove that . Assume that . For each , by (15), we have which deduces
Since as , from Lemma 6 and Lemma 7 and by taking the limit from both sides of (29), we get
Since , then which is a contradiction. Therefore, , that is, is a fixed point of .
Finally, we want to show the uniqueness of the fixed point of . Let be a fixed point of and . By (15), we obtain which implies then
Since , we have which deduces and hence which is a contradiction. Therefore , that is, has a unique fixed point.
Example 11. Letandbe defined by, whereis the usual real modulus. One can easily check thatis a complete bicomplex-valued metric onandfor all.
Define and by ,
Then, is a MT-function.
For all we have then, satisfies (15).
So all the hypotheses of Theorem 10 are fulfilled. It is therefore possible to apply Theorem 10 to get the fact that has a unique fixed point on (Here, is the unique fixed point of ).
The following fixed point theorem established in metric space is immediate from Theorem 10.
Theorem 12. Letbe a complete metric space and letbe a mapping on. Suppose that there exists a MT-functionsuch thatThen, has a unique fixed point on .
Since any nondecreasing function or any nonincreasing function is an MT-function, by applying Theorem 10, we have the following results.
Corollary 13. Letbe a complete bicomplex-valued metric space and letbe a mapping on. Suppose that there exists a nondecreasing (or nonincreasing) functionsuch thatThen, has a unique fixed point on .
Corollary 14. Letbe a complete metric space and letbe a mapping on. Suppose that there exists a nondecreasing (nonincreasing) functionsuch thatThen, has a unique fixed point on .
Corollary 15 (bicomplex-valued metric version of Banach contraction principle). Letbe a bicomplex-valued complete metric space and letbe a contraction mapping on, that is,for all , where . Then, has a unique fixed point on .
The aim of the next work is to generalize our results by finding a common fixed point for functions defined on a bicomplex-valued complete metric space.
Theorem 16. Letbe a complete bicomplex-valued metric space andbemappings on. Suppose that there exists a MT-functionsuch thatfor all and , then, there exists a unique common fixed point such that
Proof. For any Putting and , then let and . Inductively, choose a sequence in so that and for all .
It follows from the property of the function that if is odd and for , then
Similarly, if is even, we also obtain that which implies that, for all
Arguing like in the proof of Theorem 16, we prove that is a Cauchy sequence in the complete bicomplex-valued metric space. Then, there exists such that
Next, we prove that . Assume that . For each , by (42), we have then then letting and by Lemma 6 and Lemma 7, we have
We obtain a contradiction. Hence, . As then
Therefore, , i.e., . If there exists another point such that , then using an argument similar to the above, we get i.e.,
We obtain a contradiction, hence .
By the same way, if there exists a unique point such that . We will show that . Suppose that , we have then which is a contradiction. Then, so .
The proof is completed.
4. Application
In this section, we give existence theorems for the Volterra-type integral equation. For this purpose, let be the space of all continuous realvalued functions on . Note that is a complete bicomplex-valued metric space by considering , then .
Consider the Volterra-type integral equation as where is the unknown solution, and .
The kernel of the integral equation is defined on .
Theorem 17. Assume that the following conditions are satisfied: (i) is continuous function(ii)there exists such thatfor all and .
Then, the nonlinear Volterra Equation (56) has a unique solution.
Proof. Define a mapping as follows:
Let . By the simple calculation and applying assumption (), for each , we can write
Then,
It follows that satisfies all the conditions of Theorem 10 where for all . Hence, has a unique fixed point. This yields that there exists a unique solution of the nonlinear Volterra integral equation.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All the authors contributed equally and significantly to writing this article. All the authors read and approved the final manuscript.
Acknowledgments
The third author extend his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant (R.G.P.1/184/41). The first author wish to acknowledge the approval and the support of this research study from the National Natural Science Foundation of China (11601525) and Natural Science Foundation of Hunan Province (2020JJ40644).