Unique Fixed Point Results and Its Applications in Complex-Valued Fuzzy -Metric Spaces

Abstract

The goal of this paper is to extend the concept of complex-valued fuzzy metric space to complex-valued fuzzy -metric spaces and to discuss various existence results for fixed points to ensure their existence and uniqueness. To demonstrate the viability of the proposed strategies, a nontrivial example is used. Finally, applications to integral equations and initial value problems in mechanical engineering are discussed to demonstrate the superiority of the obtained results.

1. Introduction and Preliminaries

Fixed point theory combines topology, geometry, and analysis in an amazing way. Fixed point theory has emerged as a powerful tool in the study of nonlinear analysis in recent years. In fixed point theory and many other mathematical subjects, multiple separate objects are considered. As a result, mathematics is not only about numbers and shapes but also about prepositions, fluid flows, vector connections, and chemical interactions, among other things. Many researchers investigated the significance of various features of symmetry and demonstrated how they might be applied to many types of mathematical problems [1, 2]. There are several generalizations of the concept of metric spaces in the literature. Azam et al. developed the idea of complex-valued metric space and discovered that the Banach contraction principle may be applied to complex-valued metric spaces [3]. They studied its applications to complex integral equations. After that, fixed point theorems have been studied by many authors in complex-valued metric spaces [4–8].

The concept of -metric spaces has been introduced by Bakhtin and Czerwik [9, 10]. Later on, many authors studied fixed point theorems for single and multivalued mappings in -metric spaces for instance [11, 12]. In [13], the author generalized the concept of -metric spaces by introducing the setting of complex-valued -metric spaces. Many other researchers worked on complex-valued -metric, and they extended generalized fixed point theorems in the sense of complex-valued -metric spaces (see [14, 15] and the references therein).

The concept of fuzzy sets was given by Zadeh [2] and opened the door of new direction in mathematical research. Pao-Ming and Ying-Ming established the notion of fuzzy metric spaces [16]. Afterwards, George and Veeramani improved the settings of fuzzy metric spaces [17]. Heilpern introduced the concept of fuzzy mapping and obtained fixed point results for fuzzy mappings [18]. Heilpern’s work was further extended by many authors, for instance, see [19–21]. Shukla et al. worked on the neighborhood structure of fuzzy fixed point [22]. Several other researchers worked on fuzzy metric spaces and obtained the generalizations of related results [23, 24].

George and Veeramani generalized the concept of fuzzy metric to the context of complex-valued fuzzy metric and obtained the complex-valued fuzzy version of Banach contraction mapping result in different forms [17]. Also, they obtain some related fixed point results with valid examples.

In this paper, we introduce the setting of complex-valued fuzzy -metric spaces to generalize the setting of complex-valued -metric space and establish the complex-valued fuzzy version of the Banach contraction principle. We also provide examples to back up our findings. The paper concludes with an application to integral and differential equation.

All over the manuscript we have symbolized the set of complex numbers by . We mark some shortcut representation used in this manuscript, as -norm for a complex-valued continuous triangular norm, CF -metric for complex-valued fuzzy -metric, and s.t. for such that.

Let The elements are denoted by and respectively. The set . Clearly for iff Let the unit closed complex interval be symbolized by and the open unit complex interval by .

Definition 1 (see [17]). Define an ordered relation on by if and only if . The relations and indicate that and , respectively.
Let . If there exists such that it the lower bound of , that is, and for every lower bound of , then is called the greatest lower bound of

Definition 2 (see [25]). Let be a nonempty set. A complex fuzzy set is characterized by a mapping such that domain is and the range in the closed unit complex interval

Definition 3 (see [17]). A binary equation is said to be complex-valued -norm if the following conditions hold: (1)(2) whenever (3)(4)for all

Some fundamental examples of a -norm are as follows: (1), for all (2), for all (3), for all

Definition 4 (see [17]). Let be a complex-valued fuzzy metric space. A sequence in is known as a Cauchy sequence if The complex-valued fuzzy metric space is complete if every Cauchy sequence is convergent in .

Definition 5 (see [17]). A sequence is monotonic with respect to if either or

Lemma 6 (see [17]). Let be a complex-valued fuzzy metric space. If and then

Lemma 7 (see [17]). Let be complex-valued fuzzy metric space. A sequence in converges to iff holds

Remark 8 (see [17]). Let then: (a)If the sequence is monotonic with respect to and there exist with , then there exists such that (b)Although the partial ordering is not a linear order on , the pair is a lattice(c)If and there exists with , then and both exist

Remark 9 (see [17]). Let , then (a)If and , then (b)If and , then (c)If and , then

Definition 10 (see [15]). Let be a nonempty set and let be a given real number. A function is called a complex-valued -metric on if, for all , the following conditions are satisfied: (i)(ii) if and only if (iii)(iv)The pair is called a complex-valued -metric space.

Example 1 (see [15]). Let . Define the mapping by for all . Then, is complex-valued -metric space with .

Definition 11 (see [17]). Let be a nonempty set, a continuous complex-valued -norm, and a complex fuzzy set on satisfying conditions: (1)(2) for every if and only if (3)(4)(5) is continuous for all and Then, the triplet is said to be a complex-valued fuzzy metric space, and is called a complex-valued fuzzy metric on . The functions denote the degree of nearness and the degree of nonnearness between and with respect to the complex parameter , respectively.

Example 2 (see [17]). Let . Define by for all Define complex fuzzy set as for each Then, is complex-valued fuzzy metric spaces.

2. Fixed Point Results in Complex-Valued Fuzzy -Metric Spaces

We start this section with the following definition.

Definition 12. is said a complex-valued fuzzy -metric space if is an arbitrary set, is a -norm, and is a fuzzy set on meeting the points below for all and provided a number : (1)(2) for every if and only if (3)(4)(5) is continuous for all and Then, the triplet is said to be a complex-valued fuzzy metric space, and is called a complex-valued fuzzy metric on .

Example 3. Let be a complex-valued fuzzy metric defined by such that be a real number. Then, is CF -matric space with

Proof. (1), (2), (3), and (5) are obvious. Here, we prove (4). For an arbitrary integer , we have Since is an increasing function for , one can write Thus, we have

Remark 13. CF -metric is the generalization of complex-valued fuzzy metric space. It is obvious from example that is every CF -metric is complex-valued fuzzy metric for Similarly, some important results like Lemmas 6 and 7 and definitions of convergence and Cauchy presented in Section 1 can also be defined in the same manner in CF -metric space as mentioned in complex-valued fuzzy metric space.

Theorem 14. Let be a complete CF -metric space and let be mapping enjoying the following condition: for all and Then, has a unique fixed point , for all

Proof. Let Define a sequence in by If for some . Then clearly, has a fixed point. Suppose for all . To show that is a Cauchy sequence, let define Since by Remark 8, the exists. For using (6), we get which implies Therefore, by definition, we get Thus, is monotonic in . Using Remark 8 and from (11), there exists , with From inequality (9), we have for all and so for every , which yields from (12) Since and applying Remark 9, we must obtained Thus, Hence, Therefore, from (16), we have that is a Cauchy sequence. From the completeness of and Lemma 7, we get that there exists such that Now for and , it yields from (6) that that is Now, for any , Taking and using (17), (19), and Remark 9, we get that for all ; that is,
Now, we have to show the uniqueness of fixed point of On contrary, suppose be another fixed point of Then, there exists such that than from (6) we have which is a contradiction. Therefore, we must obtain for all Hence,

Corollary 15. Let be a complete CF -metric space and let be mapping enjoying the following condition: for all and . Then, has a unique fixed point , for all

Proof. By the use of Theorem 14, has a fixed point as observes all conditions. But implies that is another fixed point of By uniqueness of fixed point, we have As fixed point of is also a fixed point of Thus, has a unique fixed point.

Corollary 16. Let be a complete CF -metric space and let be mapping enjoying the following condition: for all and . Then, has a unique fixed point , for all

Example 4. Let and -norm be defined by for all Define as Then, is a CF -metric space. Define as

Then, we have the following cases.

Case 1. If then

Case 2. If and then and .

Case 3. If and then and .

Case 4. If and then and .

Case 5. If and then and

Case 6. If and then and .

Case 7. If and then and .

Case 8. If and then and .

The above-mentioned cases observe all conditions of Theorem 14 with Thus, the fuzzy contractive mapping has a unique fixed point, which is

Theorem 17. Let be a complete CF -metric space with for Let be mapping enjoying the following conditions: (i)There exists and such that for all (ii)There exists such that for all Then, has a unique fixed point in

Proof. It is enough to proof that is complete and for all Let be a Cauchy sequence in Since is complete thus by the use of Lemma 7, there exists such that for all Now for all Since for every also . By using the properties of -norm and Remark 9, we obtain Taking and using Remark 9, we get Therefore,
For every , it yields from (26) that is Thus, for all we get Taking and using Remark 9, we have Therefore,

Theorem 18. Let be a complete CF -metric space such that for any sequence with , we get for all Let be a mapping observing that for all , where . Then, has a unique fixed point in

Proof. Let Define a sequence in by If for some . Then clearly, has a fixed point. Suppose for all . To show that is a Cauchy sequence, let define Since by Remark 8, the exists. For by the use of (??) and Lemma 6, we get which yields Therefore, by definition, we obtain Hence, is monotonic in , and by the use of Remark 8 and (39), there exists such that For , once again from (34), we have for all and , we have As using (40) and assumption, we get From (40) and (43) Thus, is a Cauchy sequence in Since is complete, by Lemma 7, there exists such that For (34) yields that Taking and by (45) and Remark 9, we have ; that is,
Now to investigate the uniqueness of fixed point, let on contrary that be any other fixed point of So there exist with ; then, (34) yields Continuing this way, we obtain Using , it follows that which is contradiction. Thus, that is,

Example 5. Let and -norm be defined by for all Define as Then, is a CF -metric space. Define as For , we obtain that for all values of we have , and for only , we have . Thus, all conditions of Theorem 18 are satisfied so, is a unique fixed point of .

Example 6. Let and for every let Let define by For every Similarly Note that

Thus, all conditions of Corollary 15 are satisfied for and , so has a fixed point which is a solution of the integral equation or the differential equation

3. Application

Integral equations have plenty applications in many scientific fields. It is a ripely rising field in abstract theory. One of its significant approach in the study of integral equations is to apply fixed point results to the function defined by the right-hand side of the equation or to develop homotopy methods, which are highly considered in fixed point theory to find the approximate solution. In this section, firstly, we study application of our main Theorem 14 the existence of unique solution to Fredholm integral equation.

Theorem 19. Let be the spaces of continuous real valued functions defined on interval , where . The Fredholm integral equation is Let and be a CF -metric defined as follows: If there exists with where holds. Then, (58) has a unique solution in

Proof. Let define as Then For all , we have so, which implies that Therefore, Since all conditions of Theorem 14 are satisfied, thus (58) has a unique solution in

Next, we study the application of Theorem 18, in mechanical engineering, since the system of auto mobile suspension is an achievable application for the system of spring mass in the field of engineering. We are going to study the motion of an auto mobile spring when its motion is upon a craggy and cleft road, where the forcing term is the craggy road and bumps noticed provide the absorbing. Tension, gravity, and earth quick are the possible external forces acting on the system. We express spring mass by and the external force acting on it by . Then the following initial value problem represents the damped motion of the spring mass system under the action of external force where express the damping constant and is a continuous mapping. Clearly, the problem (68) is equivalent to the following integral equation

where represents the corresponding Green’s function and defined as where is a constant ratio. Consider the set of real valued functions . For , consider CF -mertic space defined by for all WE have to show that problem (68) has a solution iff there exists in , a solution of the integral equation (69).

Theorem 20. Consider problem (68), suppose the following conditions are satisfied: (i)(ii)Then, the integral equation (69) has a unique solution in

Proof. Let define an operator Now, this yields that Consequently, we get Thus, by Theorem 18, we obtained the existence of unique solution to integral equation (69).

4. Conclusion

In this article, we presented the generalization of CF -metric space and successfully obtained the generalization of Banach contraction principle to the new established setting herein. In support of our obtained results, we have constructed some examples, and with the help of derived result, we guaranteed the existence of unique solution to integral equation, which makes it possible for more integral equations to be verified in such conditions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Acknowledgments

The last author N. Mlaiki would like to thank Prince Sultan University for paying APC and for the support through the TAS Research Lab.