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# On Fractional Differential Inclusion for an Epidemic Model via L-Fuzzy Fixed Point Results

1
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

2
Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria

* Corresponding Author: Mohammed Shehu Shagari. Email:

(This article belongs to the Special Issue: Computational Aspects of Nonlinear Operator and Fixed Point Theory with Applications)

*Computer Modeling in Engineering & Sciences* **2023**, *137*(2), 1937-1956. https://doi.org/10.32604/cmes.2023.028239

**Received** 06 December 2022; **Accepted** 30 January 2023; **Issue published** 26 June 2023

## Abstract

The real world is filled with uncertainty, vagueness, and imprecision. The concepts we meet in everyday life are vague rather than precise. In real-world situations, if a model requires that conclusions drawn from it have some bearings on reality, then two major problems immediately arise, viz. real situations are not usually crisp and deterministic; complete descriptions of real systems often require more comprehensive data than human beings could recognize simultaneously, process and understand. Conventional mathematical tools which require all inferences to be exact, are not always efficient to handle imprecisions in a wide variety of practical situations. Following the latter development, a lot of attention has been paid to examining novel L-fuzzy analogues of conventional functional equations and their various applications. In this paper, new coincidence point results for single-valued mappings and an L-fuzzy set-valued map in metric spaces are proposed. Regarding novelty and generality, the obtained invariant point notions are compared with some well-known related concepts via non-trivial examples. It is observed that our principal results subsume and refine some important ones in the corresponding domains. As an application, one of our results is utilized to discuss more general existence conditions for realizing the solutions of a non-integer order inclusion model for COVID-19.## Keywords

From the beginning of the universe, man has been exerting great efforts in understanding nature and then coming up with a good connection between life and what it requires. This struggle is broken down into three phases, namely, understanding the surrounding ambient, acknowledgment of creativity, and preparing for the future. In these strives, a lot of challenges like linguistic interpretation, characterization of inter-connected phenomena into suitable categories, application of non-liberal ideas, vagueness in data analysis, and more than a handful of others, affect the precision of results. These problems, common with everyday activities can be avoided by using the concepts of fuzzy sets because they are more flexible than crisp sets. Numerous fields of mathematics, the social sciences, and engineering have undergone enormous upheavals since Zadeh [1] introduced fuzzy sets. The fundamental ideas of fuzzy sets have been refined and used in several contexts. In 1981, Heilpern [2] proved an invariant point theorem for fuzzy contraction mappings, which is a fuzzy analogue of invariant point theorems due to Nadler [3] and Banach [4]. Following [2], a number of authors have studied the existence of invariant points of fuzzy set-valued maps; for example, see [5–9]. Initiated by Goguen [10], L-fuzzy sets are a particularly intriguing development of the fuzzy set notion that substitutes a complete distributive lattice for the range set’s interval [0,1].

A recent study by Rashid et al. [11] introduced the idea of L-fuzzy mappings (Lmap) and examined a pair of Lmaps that are

On the other hand, fractional differential inclusions arise in different problems in mathematical physics, bio-mathematics, control theory, critical point theory for non-smooth energy functionals, differential variational inequalities, fuzzy set arithmetic, traffic theory, and so on. Usually, the first most investigated problem in the study of differential inclusion is the criteria for the existence of its solutions. In this context, several authors have applied different invariant point techniques and topological methods to establish the existence results of differential inclusions in abstract spaces. In the current literature, we can find much work on fractional-order models coming-up with different measures for curbing the novel corona virus (COVID-19). Lately, Rahman et al. [13] investigated a fractional non-integer order fuzzy dynamical system and established an epidemic model for COVID-19. Their proposed model is examined for solvability, using an invariant point method. On close developments, we can cite [14–18].

Following the existing findings, we notice that L-fuzzy invariant point results using the characterizations of

The rest of the paper is structured as follows: Section 2 contains the fundamentals needed to establish our principal proposal. Results and discussion are presented in Section 3. Section 4 is concerned with the application of one of the proposed notions in the fractional differential inclusion model for COVID-19. The summary and conclusion of our obtained key notions are presented in Section 5.

We collate, herein, a few fundamentals which are useful to our principal results. These basic concepts are picked from [1–3,8,9,19,20]. Throughout this paper, the sets

Let

where

Definition 2.1. [21] The mapping

Definition 2.2. [22] The mapping

Recently, Monairah et al. [23] came up with variants of Definitions 2.1 and 2.2 in the fashion given hereunder.

Definition 2.3 [23]. The mapping

Definition 2.4. [23]

The following two lemmas are essential to the discussion of our principal findings:

Lemma 2.1. [23] Let

Lemma 2.2. [23] Let

(i)

(ii) For each

(iii) For each

(iv) For each

(v) For each

(vi) For any sequence

(vii)

Definition 2.5. [24] A relation

(i) reflexive;

(ii) antisymmetric;

(iii) transitive.

A set L together with a partial ordering

Definition 2.6. [24] Let L be a nonempty set and (L,

Definition 2.7. [24] A pset

(i) a lattice, if

(ii) a complete lattice, if

(iii) distributive lattice if

A pset L is called a complete lattice if for every doubleton

Definition 2.8. [10] An L-fuzzy set (L-fset) ∇ on a nonempty set

Denote the class of all L-fuzzy sets on a nonempty set

Definition 2.9. [10] The

Definition 2.10. [11,12] Let

Consistent with Rashid et al. [11,12], let

Definition 2.11. [19,25] Let

We represent the set of all L-fuzzy invariant points of

We begin this section by introducing the idea of coincidence point results for an Lmap and single-valued mappings.

Theorem 3.1. Let

(

(

(

for all

Then,

Proof. By

Further, for every

Let

Given that

From (4), we claim that there exists

Assume that this claim is not true, that is,

that is,

Let

Given that

Hence, it if follows from (8) that

For any

Thus,

Employing the defining property of

We propose that

a contradiction for all

Definition 3.1. [26] A nonempty subset ∇ of

We denote the class of all bounded proximal subsets of

In what follows, we study another coincidence point result in connection with

Theorem 3.2.Let

(

(

for all

Then,

Proof. Let

Since

Hence, Theorem 3.1 can be applied to find

We provide the following example to support the hypotheses of Theorem 3.1.

Example 3.1. Let

Consider an Lmap

Then, for every

Further, define the mappings

Then, we observe that the following hold:

(

(

and

To prove that

(i)

(iii)

(iv)

(vi)

This shows that

Define

Since

Now, we claim that

for all

To verify (11), we check the possibilities:

Case 1. If x = 1, m = 2 and

Case 2. For x = 1, m > 2 and

Case 3. For x = 2, m > 2 and

Case 4. For x > 2, m > x and

Thus, following cases (1)−(4), we have demonstrated that (11) is valid. Hence, all the assumptions of Theorem 3.1 are agreed with. Whence,

Now, notice that if we take L = [0,1] and w = 1,

for all

Similarly, let

for all

Also, we note that

for all m > 2. Thus, all the Mizoguchi-Takahashi type results are not valid here.

Fig. 1 represents the Lattice in Example 3.1.

Consider a nonempty subset A of

Corollary 3.1. Let

(i) for every

(ii)

(iii) there exists

Then,

Proof. Define the mappings

The next observation follows from Corollary 3.1.

Corollary 3.2. Let

(i) for every

(ii)

(iii) there exist

Then,

Corollary 3.3.Let

(i) for every

(ii)

(iii) there exists

Then,

Proof. Take

Corollary 3.4. Let

for all

Then,

Proof. Take

Corollary 3.5. Let

for all

Proof. First, note that

4 An Application to Fractional Differential Inclusion for an Epidemic Model

Of recent, Ahmed et al. [14] examined the importance of lock-down in managing the escalation of COVID-19, using the following non-integer order epidemic model:

where the total population under study, P(t) is partitioned into four units, viz. a susceptible population that is free from lock-down W(t), a susceptible population that is not free from lock-down

where

Consequently, the model (14) takes the form:

with the conditions:

where

It is a fact that in general, differential equations are not efficient tools to analyze non-statistical uncertainties, since the derivative of a solution to any differential equation automatically enjoys all the regularity properties of the concerned mapping and of the solution itself. This hereditary property is not found under the setting of differential inclusions. With this information, we extend problem (14) to its set-valued version given as

where

Definition 4.1. [33] Let

where

Definition 4.2. [33] Let

Lemma 4.1. [33] Let

In particular, if

Given Lemma 4.1, the integral reformulation of problem (17) which is equivalent to the model (14) is given by

Let

where

and

Definition 4.3. [34] Let

For each

Let

Then, the set of all selections of V can be regarded as the set of all selections of an Lmap

Definition 4.4. A function

and

Definition 4.5. [34] A set-valued mapping

Definition 4.6. An Lmap

Let

Theorem 4.1. Assume that the following conditions are satisfied:

(

(

for almost all

Then, the differential inclusion (19) has at least one solution in

Proof. We start by resolving (19) into an L-fuzzy invariant point problem. Accordingly, let

Then, define an Lmap

By setting

It is clear that the L-fuzzy invariant points of

Case I.

Since V has compact values, it follows that

Hence,

Case 2. Next, we prove that (13) is satisfied. Recall that for every

By

Define

Then, from (22) and (23), we obtain

Whence,

Taking supremum over all of

for all

This article established new coincidence point results for single-valued mappings and an Lmap (Theorems 3.1 and 3.2) by using a modified version of an

The results of this paper, examined in metric space, are indeed fundamental. It follows that an ample amount of future work can be highlighted. Accordingly, the underlying space can be taken to other generalized, pseudo or quasi-metric spaces, such as b-MS, metric-like space, and fuzzy MS. On the flip side, the involved Lmap can be extended to some hybrid set-valued maps, such as fuzzy soft set-valued maps, intuitionistic maps, L-fuzzy soft set-valued maps, and so on. As a result of these suggested modifications, the contractive inequalities obtained herein will be modified. The latter possible variants will pave the way for better applications.

Acknowledgement: The authors gratefully acknowledge the technical and financial support provided by the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia.

Funding Statement: The Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia has funded this project under Grant Number (G: 220-247-1443).

Conflicts of Interest: The authors declare that they have no conflicts to report regarding the present study.

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*Computer Modeling in Engineering & Sciences*,

*137*

*(2)*, 1937-1956. https://doi.org/10.32604/cmes.2023.028239

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*Comput. Model. Eng. Sci.*, vol. 137, no. 2, pp. 1937-1956. 2023. https://doi.org/10.32604/cmes.2023.028239

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