Analysis of Fractional Differential Inclusion Models for COVID-19 via Fixed Point Results in Metric Space

Alansari, Monairah; Shagari, Mohammed Shehu

doi:https://doi.org/10.1155/2022/8311587

Journal of Function Spaces

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Abstract Introduction and Preliminaries Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Unique and Non-Unique Fixed Points and their Applications 2022

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Research Article | Open Access

Volume 2022 | Article ID 8311587 | https://doi.org/10.1155/2022/8311587

Analysis of Fractional Differential Inclusion Models for COVID-19 via Fixed Point Results in Metric Space

Monairah Alansari¹and Mohammed Shehu Shagari²

Academic Editor: Santosh Kumar

Received08 Apr 2022

Revised17 Jun 2022

Accepted28 Jun 2022

Published16 Jul 2022

Abstract

We examine in this paper some new problems on coincidence point and fixed point theorems for multivalued mappings in metric space. By applying the characterizations of a modified -function, under the name -function, a few novel fixed point results different from the existing fixed point theorems are launched. It is well-known that differential equation of either integer or fractional order is not sufficient to capture ambiguity, since the derivative of a solution to any differential equation inherits all the regularity properties of the mapping involved and of the solution itself. This does not hold in the case of differential inclusions. In particular, fractional-order differential inclusion models are more suitable for describing epidemics. Thus, as a generalization of a newly launched existence result for fractional-order model for COVID-19, using Banach and Shauder fixed point theorems, we investigate solvability criteria of a novel Caputo-type fractional-order differential inclusion model for COVID-19 by applying a standard fixed point theorem of multivalued contraction. Stability analysis of the proposed model in the framework of Ulam-Hyers is also discussed. Nontrivial comparative illustrations are constructed to show that our ideas herein complement, unify and, extend a significant number of existing results in the corresponding literature.

1. Introduction and Preliminaries

Numerous challenges in practical world defined by nonlinear functional equations can be simplified by reconfiguring them to their equivalent fixed point problems. Fixed point theory yields relevant tools for solving problems emanating in various arms of sciences. The fixed point theorem, commonly named as the Banach fixed point theorem (see [1]), came up in clear form in Banach thesis in 1922, where it was availed to study the existence of a solution to an integral equation. Since then, because of its importance, it has gained a number of refinements by many authors. In some modifications of the principle, the inequality is weakened, see, for example [2, 3], and in others, the topology of the ambient space is relaxed, see [4–7] and the references therein. Along the lane, three prominent improvements of the Banach fixed point theorem was presented by Ciric [2], Reich [8], and Rus [9].

Nadler [10] launched a multivalued improvement of the Banach contraction mapping principle. Nadler’s contraction mapping principle opened up the concept of metric fixed point theory of multivalued contraction in nonlinear analysis. In line with [10], a number of refinements of fixed point theorems of multivalued contractions have been presented, famously, by Berinde-Berinde [11], Du [12, 13], Mizoguchi and Takahashi [14], Pathak [15], and Reich [16, 17], to cite a few. Fixed point theorems for multivalued mappings are highly advantageous in optimal control theory and have been commonly used to solve several problems in economics, game theory, biomathematics, qualitative physics, viability theory, and many more.

Differential inclusions are found to be of great usefulness in studying dynamical systems and stochastic processes. A few examples include sweeping process, granular systems, nonlinear dynamics of wheeled vehicles, and control problems. In particular, fractional differential inclusions arise in several problems in mathematical physics, biomathematics, control theory, critical point theory for non-smooth energy functionals, differential variational inequalities, fuzzy set arithmetic, traffic theory, etc. Usually, the first most concerned problem in the study of differential inclusion is the conditions for existence of its solutions. In this direction, several authors have applied different fixed point approaches and topological methods to obtain existence results of differential inclusions in abstract spaces. In the current literature, we can find many works on fractional-order models proposing different measures for curbing the novel corona virus (COVID-19) (see, for example, Ali et al. [18], Yu et al. [19], Xu et al. [20], Shaikh et al. [21], and the references therein). Recently, Ahmed et al. [22] constructed a Caputo-type fractional-order model and studied the significance and effect of the lockdown in curbing COVID-19. They ([22]) investigated the existence and uniqueness of solutions of the fractional-order corona virus model by applying the Banach and Schauder fixed point theorems. One of the pioneer results of fixed point theory using fractional-order model was presented by Boccaletti et al. [23]. For some recent results and applications of fraction calculus, we refer [24–26].

Following the above developments, we consider in this paper some problems on coincidence point and fixed point theorems for multivalued mappings. By applying the characterizations of -function, a few new fixed point results different from the fixed point theorems due to Berinde-Berinde [11], Du [13], Mizoguchi-Takahashi [14], Nadler [10], Reich [17], and Rus [27] are launched. It is a common knowledge that differential equation of either integer or fractional order is not sufficient to capture ambiguity, since the derivative of a solution to the differential equation inherits the regularity properties of the mapping and of the function . This is no longer the case with differential inclusions. In particular, fractional-order differential inclusions models are more suitable for describing epidemics (see, e.g., [28]). Differential inclusions are not only models for handling dynamic processes but also provide powerful analytic tools to prove existence theorems such as in control theory, to derive sufficient conditions of optimality, play a significant role in the theory of control conditions under uncertainty. Thus, as a generalization of the existence theorem presented by Ahmed et al. [22], in the sequel, we investigate solvability conditions of a new Caputo-type fractional differential inclusions model for COVID-19 by applying a fixed point theorem of multivalued contraction. Stability analysis of the proposed model in the context of Ulam-Hyers is also obtained. Our results herein complement, unify, and extend the above-mentioned articles and a few others in the comparable literature. A few nontrivial comparative illustrations are constructed to indicate that our obtained ideas properly advanced corresponding results in the literature.

In what follows, we recall some preliminary concepts that are useful to our main results. Throughout this paper, the set , and represent the set of real numbers, nonnegative real numbers, and the set of natural numbers, respectively. Let be a metric space. Denote by , , and , the family of nonempty subsets of , the collection of all nonempty closed and bounded subsets of , and the class of all nonempty compact subsets of , respectively. For , the mapping is given by where is named the Hausdorff-Pompeiu metric induced by the metric . For example, if we consider the set of real numbers endowed with the standard metric, then for any two closed intervals and , we have .

Let be point-valued mappings and be a multivalued mapping. A point in is a coincidence point of and if . If is the identity mapping on , then is named a fixed point of . We denote the set of fixed points of and the set of coincidence point of and by and , respectively.

Let be a real-valued function. For , we recall that

Definition 1. (see [12]). is named an -function if it obeys the Mizoguchi-Takahashi’s condition, that is, , for each .

Remark 2. (see [12]). (i)If is given as , then is an -function(ii)If the function is either increasing or decreasing, then is an -function

Definition 3. is named a -function if it obeys the condition: For each , we can find such that

Definition 4. (see [12]). A function is named a function of contractive factor, if for any strictly decreasing sequence in , we have .

Definition 5. A function is named a function of -contractive factor, if for any sequence in from and after some fixed terms, it is strictly nonincreasing and , for some .
The following example recognizes the existence of -function and function of -contractive factor.

Example 6.
Let be a sequence in given by Define by

Then, it is clear that is a -function, is a strictly decreasing sequence from and after the eight term and for some . Whence, is also a function of -contractive factor. An example which is not a -function is provided hereunder.

Example 7.
Let be given by Since , then is not a -function.

Remark 8. (i)Note that if for all and for some , then becomes an -function, provided is a -function(ii)If we define as for all and , then is a -function

The following Lemma is in consistent with [16, Lemma 18].

Lemma 9.
Let be a -function. Then given by is also a -function for each and some .

Proof. Obviously, and . Let be fixed. Since is a -function, we can find and such that for all . Assume that . Then, for all . Thus, is a -function.

The following result due to Nadler [26] is the first metric fixed point theorem for multivalued contractions.

Theorem 10. (see [10]). Let be a complete metric space and be a multivalued -contraction, that is, we can find such that for all . Then, .

In 2007, Berinde-Berinde [11] presented the following notable fixed point Theorem.

Theorem 11. (see [11]). Let be a complete metric space, be a multivalued mapping, and be an -function. Assume that we can find such that for all with . Then, .

Observe that if we take in Theorem 11, we realize the Mizoguchi-Takahashi fixed point theorem [14] which partially answered the problem posed in Reich [8].

Theorem 12. (see [8]). Let be a complete metric space, be a multivalued mapping, and be an -function. Suppose that for all with . Then, .

In [8], Reich raised the question whether Theorem 12 is also valid when is replaced with . In 1989, Mizoguch-Takahashi [14] responded to this puzzle in affirmative via the following result.

Theorem 13. (see [14]). Let be a complete metric space, be a multivalued mapping, and be an -function. Suppose that for all . Then, .

Let be a nonempty subset of and be a mapping. We recall that the set is -invariant if . Not long ago, Du [13] obtained the following important fixed point and coincidence point result.

Theorem 14. (see [13]). Let be a complete metric space, be a multivalued mapping, be a continuous point-valued mapping, and be an -function. Assume that the following conditions hold:
is -invariant for each ;
we can find a function such that for all . Then, .

Notice that Mizoguchi-Takahashi fixed point theorem (13) is an extension of Nadler’s fixed point theorem (10), but its original proof is not friendly. Alternative proof presented in [29] is also difficult.

Definition 15. (see [9]). Let be a metric space. A single-valued mapping is named:
Rus contraction if we can find with such that for all , Ciric-Reich-Rus contraction if we can find with such that for all ,

In [9], it was proved that every Rus and Ciric-Reich-Rus contraction has a unique fixed point. These results have been extended to multivalued mappings in the following manner.

Theorem 16. (see [27]). Let be a complete metric space and be a multivalued mapping. Assume that we can find with such that for all : Then, .

Theorem 17. (see [17]). Let be a complete metric space and be a multivalued mapping. Assume that we can find with such that for all : Then, .

For more variants of fixed point results of multivalued contractions, the interested reader may consult [30–33] and the references therein.

2. Main Results

In line with the characterizations of -function, we begin this section by launching a few characterizations of -function in Lemma 18. Its proof is a slight adaption of [17, Theorem 2.1].

Lemma 18.
Let . Then, the following statements are equivalent: (i) is a -function(ii)For each , we can find and such that for all (iii)For each , we can find and such that for all (iv)For each , we can find and such that for all (v)For each , we can find and such that for all (vi)For any sequence in , from and after some fixed term, it is nonincreasing and (vii) is a function of -contractive factor, that is, for any sequence in , from and after some fixed term, it is strictly decreasing and

The following existence theorem for coincidence point and fixed point is one of the main results of this paper.

Theorem 19.
Let be a complete metric space, be a multivalued mapping, be continuous point-valued mappings, and be a -function. Suppose that the following conditions are obeyed:
for each , ;
we can find three mappings such that for all , where with .

Then, .

Proof. By , we note that for each , for all . So for each , it follows from that for all ,

Further, for each , . Whence, for each , (16) gives

Let and choose . If , then , that is, and the proof is finished. Otherwise, if , then consider a function given by . By Lemma 9, is a -function and for all . From (2.2), it follows that

Since , then we can find such that . Thus, (18) can be written as

From (19), we claim that we can find such that

Assume that this claim is not true, that is, . Then, we get that is, , contradicting (19). Now, if , then and so . Otherwise, we can find such that

Let for each . Proceeding on similar steps as above, we can construct a sequence in with for each and

Given that is a -function, then by Lemma 18:

Whence,

Take , then . Since for all , then by (23), is a strictly decreasing sequence of positive real numbers. Therefore, for each , we have

Whence, it follows from (26) that

For any with , by (27), we get

Thus, . This proves that is a Cauchy sequence in . The completeness of implies that we can find such that as . Since for each , it follows from condition that for each ,

Using the continuity of the functions and , we have

We claim that . Assume contrary so that . Since the function is continuous, then from condition , we realize a contradiction. Whence, . Since is closed, we have . By condition , . Consequently, .

The following example shows the generality of our Theorem 19 over Theorems 10, 11, 17, and 16 due to Nadler, Berinde-Berinde, Reich, and Rus, respectively.

Example 20.
Let and for all . Let be a multivalued mapping and be mappings given by and , the identity mapping on . Define the function by for all and some . Also, define the mappings by for all . Then, we realize the following: (i)for each , ;(ii);(iii) and are continuous

Clearly, for all and some . Whence, is a -function. Furthermore, it is a routine to verify that condition holds for all .

Now, notice that the mapping does not obey the hypotheses of Theorem 10 due to Nadler. To see this, let and , then for all . Moreover, to see that Theorem 11 due to Berinde-Berinde fails in this instance, let and for all . Then, for all ,

Moreover, to see that Theorems 17 and 16 of Reich and Rus are also not applicable to this example, again take and . Then, by setting and in Theorems 1.17 and 1.16, respectively, we have

A slight modification of Example A of Du [13] provided below shows the generality of our Theorem 19 over Mizoguch-Takahash’s [14] and Du’s [13] fixed point theorems.

Example 21.
Let be the Banach space of all bounded real sequences endowed with the uniform norm , and let be the canonical basis of . Let be a sequence of positive real numbers obeying and for all (for example, take and ). It follows that is convergent. Set for all , and let be a bounded and complete subset of . Then, is a complete metric space and if .

Let be a multivalued mapping and be three mappings, respectively, given by

Then, we notice that the following results hold:

To show that and are continuous, it is suffices to prove that and are nonexpansive. So we consider the following six possibilities: (i)(ii)(iii) for any (iv) for any (v) for any (vi) for any and

Consequently, is nonexpansive, and, since , then and are continuous.

Next, define the function by

Also, define the mappings by

Then, we observe that for all and some . It follows that is a -function. Moreover, we claim that for all and with , where is the Hausdorff metric induced by the norm .

To see (40), we consider the following cases:

Case 1. For and , we have

Case 2. For and , we have

Case 3. For and , we have

Case 4. For and , we have

Case 5. For and , we have

Case 6. For and , we have

Therefore, from Cases (1)–(6), we have shown that Condition (40) is obeyed. Consequently, all the assertions of Theorem 19 are obeyed. It follows that .

Now, observe that if we take the sequence as earlier given, that is, , where for all and let for all , then is an -function, provided is a -function. Thus, (a)for and any , we have

Whence, Mizoguch-Takahashi’s Theorem 13 does not hold in this case. (b)Let the function be given by and and be as given in the above Example. Then, for and with , the above Case 3 becomes

Case : that is, Case 3 also hold. On the other hand, notice that that is, the main result of Du [17, Theorem 19] is not applicable here.

3. Consequences

In this section, we deduce some significant consequences of Theorem 19.

Corollary 2.
Let be a complete metric space, be a multivalued mapping, be a continuous point-valued mapping, and be a -function. Suppose that (i) is -invariant (i.e. ) for each (ii)we can find a mapping such that for all and with .

Then, .

Proof. Take as for all in Theorem 19.

The following result is a direct consequence of Corollary 2.

Corollary 23.
Let be a complete metric space, be a multivalued mapping, be a continuous point-valued mapping, and be a -function. Suppose that (i) is -invariant (i.e., ) for each (ii)we can find and a mapping such that for all and with .

Then, .

Corollary 24.
Let be a complete metric space, be a multivalued mapping, be a continuous point-valued mapping, and be a -function. Suppose that (i) is -invariant (i.e. ) for each (ii)we can find such thatfor all and with .

Then, .

Proof. Define as for all in Corollary 23.

By applying Corollary 2, we deduce a generalized version of the primitive Ciric-Reich-Rus fixed point theorem for multivalued mapping as follows.

Corollary 25.
Let be a complete metric space, be a multivalued mapping, and be a -function. Suppose that we can find a mapping such that for all and with .

Then, .

Proof. Take , the identity mapping on in Corollary 2.

Remark 26. (i)If we take , where , is a -function, and set , then Corollary 25 reduces to Theorem 13 due to Mizoguchi-Takahashi [14].(ii)If is a monotonic increasing function such that for each and , then by setting , where and , Corollary 24 generalizes [14, Corollary 2.2]. Also, Corollary 24 includes Theorem 1.2 in [29] as a special case, by extending the range of from the family of bounded proximal subsets of to .(iii)If we take and for all and , where not all of and are identically zeros, then Corollary 25 reduces to Theorem 1.10(iv)If we put , where , is a -function, take , the identity mapping on , and set , then Corollary 24 reduces to Theorem 11 due to Berinde-Berinde [11].(v)If we define the multivalued mapping as for all , where is a single-valued mapping on , then all the results presented herein can be reduced to their single-valued counterparts(vi)It is clear that more consequences of our main result can be deduced, but we skip them due to the length of the paper

4. Applications to Caputo-Type Fractional Differential Inclusions Model for COVID-19

Very recently, Ahmed et al. [22] investigated the significance of lockdown in curbing the spread of COVID-19 via the following fractional-order epidemic model: where the total population under study, is divided into four components, namely susceptible population that are not under lockdown , susceptible population that are under lock-down , infective population that are not under lockdown , infective population that are under lock-down , and cumulative density of the lockdown program . For the meaning of the rest parameters and numerical simulations of (55), we refer the reader to [22]. The above model (55) is simplified as follows: where

Consequently, the model (55) takes the form: with the condition: where denotes the transpose operation.

In this section, we extend problem (55) to its multivalued analogue given by where is a multivalued mapping ( is the power set of ). We launch existence criteria for solutions of the inclusion problem (60) for which the right hand side is nonconvex with the aid of standard fixed point theorem for multivalued contraction mapping. First, we outline some preliminary concepts of fractional calculus and multivalued analysis as follows.

Definition 27. (see [34]). Let and . Then, the Riemann-Liouville fractional integral order for a function is given as where is the gamma function given by .

Definition 28. (see [34]). Let , and . Then, the Caputo fractional derivative of order for a function is given as

Lemma 29. (see [34]). Let , and . Then, In particular, if , then .

In view of Lemma 29, the integral reformulation of problem 16 which is equivalent to the model 13 is given by

Let denotes the Banach space of all continuous functions from to equipped with the norm given by where and .

Definition 30.
Let be a nonempty set. A single-valued mapping is named a selection of a multivalued mapping , if for each .
For each , we define the set of all selections of a multi-valued mapping by

Definition 31. A function is a solution of problem (60) if there is a function with on such that and .

Definition 32. A multivalued mapping with nonempty compact convex values is said to be measurable, if for every , the function is measurable.
The following is the main result of this section.

Theorem 33. Assume that the following conditions are obeyed:
(N₁) is such that is measurable for each
(N₂) We can find a continuous function such that for all , for almost all and for almost all .

Then, the differential inclusion problem (60) has at least one solution on , provided that , where .

Proof. First, we convert the differential inclusions (60) into a fixed point problem. For this, let and consider the multivalued mapping given by

Clearly, the fixed points of are solutions of problem (60). Now, we prove that obeys all the conditions of Theorem 10 under the following cases.

Case 1. is nonempty and closed for every . Since the multi-valued mapping is measurable, by the measurable selection theorem (see, e.g. [35], Theorem III. 6), it admits a measurable selection . Furthermore, by condition , we get , that is, and hence is integrably bounded. Thus, is nonempty. Now, we show that is closed for each . Let be such that in . Then, and we can find such that for each , Since has compact values, we pass onto a subsequence to obtain that converges to . Therefore, and for each , we have Thus, .

Case 2. Next, we prove that we can find such that for each . Let and . Then, we can find such that for each , By , . Whence, we can find such that Define by Since the multivalued mapping is measurable (see ([35], Proposition III.4)), we can find a function which is a measurable selection of . Thus, , and for each , we have . For each , take Then, from (73) and (76), we realize Therefore, . On similar steps, interchanging the roles of and , we have Note that if we take and for all and , then (54) coincides with (78). Whence, Corollary 25 can be applied to conclude that the mapping has at least one fixed point in which corresponds to the solutions of Problem 4.6.

Example 34. Consider the Caputo-type fractional differential inclusion problem given by where the multivalued mapping is given as Obviously, the mapping is measurable for each . In this case, we can take for all , and thus, for almost all . Note that for each , we have

Moreover, . Whence, . Consequently, by Theorem 38, Problem (68) has at least one solution on

5. Stability Results

Investigated as a type of data dependence, the concept of Ulam stability was initiated by Ulam [36] and developed by Hyers [37], Rassias [38], and later on by many authors. In this section, we study an Ulam-Hyers type stability of the proposed fractional-order model 4.6. In [22], the stability result of the model 4.4 has been obtained in the framework of single-valued mappings. But, it is a known fact that multivalued mappings often have more fixed points than their corresponding single-valued mappings. Whence, the set of fixed points of set-valued mappings becomes more interesting for the study of stability. First, we give some needed definitions as follows.

Let and consider the following inequality:

Definition 35. The proposed problem (60) is Ulam-Hyers stable if we can find a real number such that for every and for each solution of the inequality (82), we can find a solution of problem (60) and two functions with and on such that for almost all , where .

Remark 36. A function is a solution of the inequality (82) if and only if we can find a continuous function and with on such that the following properties hold: (i)(ii)

Lemma 37. Suppose that obeys the inequality (82), then we can find a function with on such that

Proof. From of Remark 36, we have , and by Lemma 29, we get Therefore, from of Remark 36, we realize

Now, we present the main result of this section as follows.

Theorem 38. Assume that the following conditions are obeyed: (i)the multivalued mapping is measurable for each (ii)for all , we can find , on and a continuous function such that for almost all , (iii), where .Then the fractional-order inclusion model (60) is Ulam-Hyers stable.

Proof.
Let , where obeys (82) and is a solution of problem (60). Then, we can find two functions with and on such that for every , Lemma 37 can be applied to have that is, , where . Consequently, the proposed problem (60) is Ulam-Hyers stable.

6. Conclusions

A new coincidence and fixed point theorem of multivalued mapping defined on a complete metric space has been presented in this work by using the characterizations of a modified -function, named -function. It has been noted herein that our result is a generalization of the fixed point theorems due Berinde-Berinde [11], Du [13], Mizoguchi-Takahashi [14], Nadler [10], Reich [17], Rus [27], and a few others in the corresponding literature. Though the conjecture raised by Reich [17] has now been proven valid in an almost complete form in [11, 13, 14], however, our main result (Theorem 19) provided a more general affirmative response to this problem. Moreover, from application perspective, we launched an existence theorem for nonlinear fractional-order differential inclusions model for COVID-19 via a standard fixed point theorem of multivalued mapping. Ulam-Hyers stability analysis of the considered model was also discussed. It is interesting to note that more useful analysis and results may be obtained if the metric on the ground set in this context is either quasi or pseudo metric. For better management of uncertainty, and since every fixed point theorem of contractive multivalued mapping has its fuzzy set-valued analogue, the result of this paper can as well be discussed in the framework of fuzzy fixed point theory and related hybrid models of fuzzy mathematics. Furthermore, in order to obtain effective measures for curbing Covid-19, other than observing the significance of lockdown, numerical simulations and better analytic tools of the proposed fractional-order differential inclusions model are another future directions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

Conceptualization was made by M. Alansari. Methodology was made by M. S. Shagari. Formal analysis was made by M. S. Shagari. Review and editing was made by M. Alansari. Funding acquisition was made by M. Alansari. Writing, review, and editing was made by M. S. Shagari. In addition, all authors have read and approved the final manuscript for submission and possible publication.

Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under the grant no. G: 234-247-1443. The author, therefore, acknowledges with thanks DSR for technical and financial support.

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Copyright © 2022 Monairah Alansari and Mohammed Shehu Shagari. 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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