[Retracted] Intuitionistic Fuzzy Fixed Point Theorems in Complex-Valued <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.20973 12.533" width="8.20973pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g></svg>-</span>Metric Spaces with Applications to Fractional Differential Equations

Tabassum, Rehana; Shagari, Mohammed Shehu; Azam, Akbar; Mohamed, OM Kalthum S. K.; Bakery, Awad A.

doi:https://doi.org/10.1155/2022/2261199

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Volume 2022 | Article ID 2261199 | https://doi.org/10.1155/2022/2261199

[Retracted] Intuitionistic Fuzzy Fixed Point Theorems in Complex-Valued -Metric Spaces with Applications to Fractional Differential Equations

Rehana Tabassum,¹Mohammed Shehu Shagari,²Akbar Azam,³OM Kalthum S. K. Mohamed,^4,5and Awad A. Bakery^4,6

Academic Editor: Muhammad Gulzar

Received17 Apr 2022

Revised08 May 2022

Accepted09 Jun 2022

Published21 Jun 2022

Abstract

Among various efforts in advancing fuzzy mathematics, a lot of attentions have been paid to examine novel intuitionistic fuzzy analogues of the classical fixed point results. Along this direction, the idea of intuitionistic fuzzy mapping (IFM) is used in this paper to establish some fixed point (FP) results in complex-valued -metric spaces. Moreover, from application perspective, one of our results is rendered to provide an existence condition for a solution of Caputo-type fractional differential equations. A few nontrivial illustrations are also furnished to authenticate and indicate the usability of the presented results.

1. Introduction

Many FP results of contraction type mappings are specifically beneficial to find the existence and uniqueness of solution to different mathematical problems. In this regard, the Banach contraction principle [1] has become a very powerful source in various fields of applied mathematical analysis. Afterwards, several researchers improved and refined this result using different mappings in various generalized metric spaces (MS). For instance, Nadler refined the Banach FP result for multivalued mappings (MVM). Bakhtin [2] initiated the notion of -MS as a refinement of classical MS, and Czerwik [3] proved the contraction mapping principle in -MS. Therefore, a large amount of research has been brought up to obtain FPs of several mappings in -MS.

It is a familiar fact that fixed point results concerning rational contractions cannot be improved or even meaningless in cone metric spaces. To circumvent this problem, Azam et al. [4] have given a brilliant idea of complex-valued MS and improved the Banach contraction principle for a pair of mappings obeying the rational inequality in the bodywork of complex-valued MS. In sequel, Sintunavarat and Kumam [5] presented some common FP theorems using the control functions in contractive condition instead of constants. Subsequently, Ahmad et al. [6] have reported some new FP results for multivalued mappings obeying the greatest lower bound property in the context of complex-valued MS. Later, there has been much progress in the study of complex-valued MS by many authors [6–9]. In continuation to this, Rao et al. [10] brought in a new view of complex-valued -MS and proved some common FP results in complex-valued -MS. Meanwhile, Mukheimer [11] refined the results of Azam et al. [4] and Bhatt et al. [9].

It is well-known that most of the problems in existing nature have several uncertainties, and to handle these uncertainties, there are various theories including fuzzy set [12], the soft set [13], the fuzzy soft set [14], and intuitionistic fuzzy set (IF-set) [15]. However, many physical problems with vague information can be tackled more precisely by means of IF-set approach. Keeping this in view, some FP results for MVM are refined to IFMs given by Shen et al. [16]. However, research on IFMs to establish the fixed point results is relatively recent, and few work has been done for IFMs on various spaces (e.g., see [17–22]).

Stirred by the above-mentioned investigations, we present some FP results for IFMs in the bodywork of set of an IF-set [23] in complete complex-valued -MS. Further, some nontrivial examples and an application are given for the reliability of our main results.

2. Preliminaries

We launch here some basic definitions and results which will be useful in what comes hereafter. Let be the set of complex numbers and . We depict a partial order and on as follows: (i) if and only if and (ii) if and only if and

Definition 1 (see [10]). Let be a nonempty set and be a real number. A function is termed a complex-valued -MS, if for all , the following conditions hold: (i) and if and only if (ii)(iii)Then, is termed a complex-valued -metric on and the pair is termed a complex-valued -MS.

Remark 2. Let be a complex-valued -MS. If then is a complex-valued MS. If and , then is a MS.

Example 1. Let and a mapping is given by for all . Then, is a complex-valued -MS with .

Definition 3 (see [10]). Let be a complex-valued -MS. A point is an interior point of a set whenever we can find :

A point is said to be a limit point of a set whenever for every

is termed an open set if each element of is an interior point of .

Definition 4 (see [10]). Let be a sequence in and . If for every with there is : for all , then is said to be a convergent sequence which converges to , and we denote this by If for every with there is : for all , and hence is said to be a Cauchy sequence in A complex-valued -MS is termed a complete space if every Cauchy sequence is convergent in .

Lemma 5 (see [10]). Let be a complex-valued -MS and be a sequence in . Then, converges to if and only if as .

Lemma 6 (see [10]). Let be a complex-valued -MS and be a sequence in . Then, is a Cauchy sequence if and only if as .
Let be the collection of all nonempty closed and bounded subsets of We denote for and for and For we have

Definition 7. Let be a complex-valued -MS and be a MVM. Define for and . Moreover, for and

Definition 8. Let be a complex-valued -MS. A nonempty subset of is said to be bounded from below if we can find some : for all .

Definition 9. Let be a complex-valued -MS. A MVM is said to be bounded from below if for each we can find :

Definition 10. Let be a complex-valued -MS. A MVM is said to have a lower bound property on if for any , the MVM given by is termed bounded from below. This means for , there is an element : for all , where is a lower bound of associated with .

Definition 11. Let be a complex-valued -MS. A MVM has the greatest lower bound (g.l.b) property on if the g.l.b of exists in for all . We denote by the g.l.b of , i.e.,

Definition 12 (see [15]). Let be a universal set. An IF-set in is an object of the form , where and denote the membership and nonmembership values of in obeying for every .

Definition 13 (see [15]). Let be an IF-set. Then, the set of is a crisp set depicted by and is given by

Definition 14 (see [24]). A mapping is termed a triangular norm (-norm), if the following conditions are obeyed: (i) for all (ii) for all (iii)If and then (iv) for all Minimum -norm depicted by is given by for all .

Definition 15 (see [24]). Fuzzy negation is a decreasing map such that If is continuous and strictly nonincreasing, then it is termed strict. Fuzzy negations with for all are termed strong fuzzy negations. The example of fuzzy negation is a standard negation given by for all

Definition 16 (see [23]). Let be an IF-set of and and be a triangular norm and a fuzzy negation, respectively. Then, set of is a crisp set depicted by and is given by

Remark 17. If we take and then set is reduced into original idea of a cut set by Atanassov [15].

Definition 18 (see [16]). Let be an arbitrary set and be a MS. A mapping is termed an IFM if is a mapping from into (class of all intuitionistic fuzzy subsets of ).

3. Main Results

In this section, first we present few definitions which will be useful in the proof of our main ideas and then establish illustrations to validate their hypotheses.

Definition 19. A point is said to be an intuitionistic fuzzy FP of an IFM if we can find :

Definition 20. Let be a complex-valued -MS and be an intuitionistic fuzzy map. Suppose that for each , we can find : We can depict for

Definition 21. Let be a complex-valued -metric space and be an intuitionistic fuzzy map. Suppose that for each , we can find : An IFM is said to have the greatest lower bound (g.l.b) property on if the exists in for all We denote the g.l.b of by and is given by

Theorem 22. Let be a complete complex-valued -MS and be a pair of IFMs obeying the g.l.b property. Assume that for each , we can find : If for all , where and are nonnegative real numbers with , where . Then, and have a common FP in .

Proof. Suppose that is an arbitrary and fixed element of ; then by assumption , so we can take and therefore from (16), It follows that As then we obtain Therefore, Thus, we can find some : It yields Using the g.l.b property of and we get This implies Inductively, we can develop a sequence in : for where .
Now for and using the triangular inequality of Therefore, since and as .
By Lemma 6, is a Cauchy sequence in , which is complete; so we can find some : .
However, from (16), we have This implies that Since , Therefore, This implies that we can find some : It yields Next, However, since as .
By Lemma 5, we have . Since is closed, so . Similarly, it follows that . Thus, and have a common FP in .
By setting in Theorem 22, we have the following result.

Corollary 23. Let be a complete complex-valued -MS and be a pair of IFMs obeying the g.l.b property. Assume that for each , we can find : If for all where and are nonnegative real numbers with , where . Then, and have a common FP in .

By letting in Theorem 22, we have the following corollary.

Corollary 24. Let be a complete complex-valued -MS and be an IFM obeying the g.l.b property. Assume that for each , we can find : If for all where and are nonnegative real numbers with , where . Then, has a FP in .

Corollary 25. Let be a complete complex-valued -MS and be a pair of MVM obeying the g.l.b property for all , and are nonnegative real numbers with , where . Then, and have a common FP in .

Example 2. Let , where , for and .
Then, is a complete complex-valued -MS with . Assume that and a pair of IFMs are given by

If and then we have

Thus, the contractive condition of Theorem 22 becomes trivial when .

Assume that without loss of generality, and and then, we have

Thus, clearly for and any value of and , we have

Hence, all the conditions of Theorem 22 are obeyed to obtain a common FP of and .

Theorem 26. Let be a complete complex-valued -MS and be a pair of IFMs obeying the g.l.b property. Assume that for each , we can find : If for all , where and are nonnegative real numbers with and for any . Thus, we can find in :

Proof. Let be an arbitrary but fixed element in . Then by hypothesis, . So we can find : . From (47), it is easy to see that This implies Therefore, . From (46), we have From here, by following the remaining steps in the proof of Theorem 26, we obtain From (51), we have Using the triangular inequality of It follows that . From (46), we have By repetition of the above steps and using the fact that is a complex-valued -MS, we can generate a sequence in : Inductively, we can construct a sequence in : Now, for with , using triangle inequality and the iterative scheme (58), we have Consequently, Hence, is a Cauchy sequence in . Since is a closed subspace of a complete MS , therefore, we can find : as . Now, to show that , from (46), we have This implies By definition, we can find some : Using the g.l.b property of and we get From triangle inequality, it follows that Therefore, Applying in (66), we have . This implies as . Since is closed, therefore . Similarly, one can show that .
Hence, By setting in Theorem 26, we obtain the following result as a corollary.

Corollary 27. Let be a complete complex-valued -MS and be an IFM obeying the g.l.b property. Assume that for each , we can find : If for all where and are nonnegative real numbers with and for any . Thus, we can find in :

Corollary 28. Let be a complete complex-valued -MS and be a pair of MVM obeying the g.l.b property for all , and are nonnegative real numbers with , and for any . Thus, we can find in :

Theorem 29. Let be a complete complex-valued -MS and be an IFM obeying the g.l.b property. Assume that for each , we can find : If for all where and are nonnegative real numbers with Then, has a FP in .

Proof. Suppose that is an arbitrary and fixed element of , then by assumption, , so we can take ; therefore from Theorem 26, This implies Thus, for all
Since , then we obtain It yields Therefore, we can find some : This implies Using the g.l.b property of we have However, This implies where .
Inductively, we develop a sequence in : Now for and using the triangle inequality of , we have However, Letting therefore, .
By Lemma 6, is a Cauchy sequence in which is complete, so we can find some : . Next, we show that . From Theorem 26, we have This implies Since , then we obtain It yields So, we can find : Therefore, Using the g.l.b property, we have By triangle inequality, Thus, This implies As we get By Lemma 5, we have as . Since is closed, therefore, has a FP.

Corollary 30. Let be a complete complex-valued -MS and be a MVM obeying the g.l.b property such that for all , and are nonnegative real numbers with Then, has a FP in .

4. Applications to Fractional Mixed Volterra-Fredholm Integrodifferential Equations with Integral Boundary Conditions

In this section, first some definitions and results are given from the existing literature and then Corollary 24 of Theorem 22 is applied to study the existence of a solution of fractional mixed Volterra-Fredholm integrodifferential equation with integral boundary conditions.

Definition 31 [25, 26]. Let be a function given on ; then, the Caputo fractional order derivative of is given by where and denotes the integer part of .

Definition 32 [25]. Let be a function given almost everywhere on , for ; we depict provided that the Lebesgue integral exists.

Lemma 33 [27]. Let be a continuous function, be the Caputo fractional derivative, and ; then, the solution of fractional mixed Volterra-Fredholm integrodifferential equation with boundary conditions, is given by

Theorem 34. Let be a complete complex-valued -metric space endowed with the metric given by and be the space of all continuous functions from into . Suppose that the function and be chosen: the following conditions are obeyed:
(C1) For each and , we have , where (C2) We can find a constant : for all (C3) We can find some constants : for all (C4) We can find continuous functions : (C5) We can find a continuous function and a continuous nondecreasing function : where the function satisfies , provided is continuous
(C6) We can find a multivalued function and some constants with and : where Then, the Caputo integrodifferential equations (102) and (103) have a solution.

Proof. By Lemma 33, the equivalent integral reformulation of problem (100)–(101) is given by By (C1), for each , we have , where By (C2), for all and each we have By (C2)-(C5), we have Let and . Then, where .
Choose : . Then by (C6), we have Let be any two arbitrary mappings and be the given multivalued mapping. Consider an intuitionistic fuzzy mapping as follows: By letting and , we obtain Therefore, the inequality (117) can be written as Multiplying inequality (120) by , we get By definition, inequality (121) implies Hence, the conclusion of Theorem 34 follows the application of Corollary 24.

Example 3. Consider the following fractional integrodifferential equation: with integral boundary conditions

Here,

However,

Notice that by Lemma 33, Condition (C2) of Theorem 34 can be verified by direct calculation.

Let and be given by

Then, is a complete complex-valued -MS. Suppose that is given by

Assume without loss of generality that for , , and . Thus, we have

Clearly, for any value of and and we have

Hence, all the conditions of Theorem 34 are obeyed. We conclude that the fractional mixed Volterra-Fredholm integrodifferential equation has a solution in .

5. Conclusion

A number of practical and theoretic problems in economics, engineering, management sciences, and medical science and a substantial number of other fields involve vagueness and the complexity of modeling data possessing nonstatistical uncertainties. Prototypal mathematical techniques are not usually successful because the imprecisions in these domains may be of various kinds. Several innovative models such as fuzzy set theory, rough set theory, intuitionistic fuzzy set theory, and other related mathematical tools have been established to manipulate data with incomplete information. On the other hand, fractional differential equations (FDEs) have instigated, in recent years, considerable interests due to their enormous applications in both physical sciences, chemical process in engineering, and social sciences. Different methods for solving FDEs and investigating the existence of their solutions were proposed by many authors; some of these techniques are series methods, the iteration method, the operational calculus, and so on. In this direction, one may see the textbooks [25, 26, 28] for both introductory and detail analysis.

Motivated by the above developments, in this article, we have used the ideas of complex-valued -MS to establish sufficient conditions for the existence of common FPs of a pair of IFMs obeying some contractive conditions involving rational inequalities. In Theorem 22, we presented a Banach type results involving rational expression, while the Banach type locally contractive condition is discussed in Corollary 24. On the other hand, Theorem 26 is a slight improvement of Definition 19. Moreover, in continuation of the role of the classical Banach contraction theorem in the study of existence of solutions of nonlinear integrodifferential equations, we included the application section. In this regard, Corollary 24 is applied to provide some conditions for the existence of a solution of mixed Volterra-Fredholm integrodifferential equation with integral boundary conditions. In each case of the above-mentioned key results, examples are supplied to support or authenticate their usability. While the current paper is theoretical, at least a note justifying its study is in order. To start with, in several MS, e.g., the cone MS, FP results involving rational expressions cannot be refined, whereas, in complex-valued MS, this is feasible. Thus, by incorporating the notion of complex-valued IFMs into the existing concepts, our remarkable results will be beneficial and significant in the modeling and solution of optimization problems in mathematical analysis.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

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

Acknowledgments

This work was funded by the University of Jeddah, Saudi Arabia, under grant No. UJ-21-DR-93. The authors, therefore, acknowledge with thanks the University technical and financial support.

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Copyright © 2022 Rehana Tabassum et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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