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Computer Modeling in Engineering & Sciences
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DOI: 10.32604/cmes.2023.028239

ARTICLE

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

Maha Noorwali1 and Mohammed Shehu Shagari2,*

1Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
2Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria
*Corresponding Author: Mohammed Shehu Shagari. Email: shagaris@ymail.com
Received: 06 December 2022; Accepted: 30 January 2023; Published: 06 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: Hausdorff metric; L-fuzzy set; L-fuzzy set-valued map; Caputo fractional differential inclusion; COVID-19

1  Introduction

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 βFL-admissible in order to prove a common invariant point theorem. Rashid et al. [12] established the ideas of DτL and μL∞ distances for L-fuzzy sets and generalized the existing invariant point theorems for fuzzy and multi-valued mappings as an improvement of the notion of Hausdorff distance and μ∞-metric for fuzzy sets.

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 ℳ𝒯-function and 𝒟-function are yet to be adequately examined. As a result of the latter observation, it becomes clear that applications of such ideas in the areas of analyzing solvability conditions of epidemic models are undoubtedly missing in the literature. Hence, by utilizing the recently established auxiliary functions, under the name 𝒟-function, this paper introduces new contractive inequalities for Lmaps and then examines criteria for the existence of L-fuzzy invariant points for such mappings. In line with the awareness that non-integer order differential inclusions are more suitable for analyzing the situation with non-statistical uncertainties, general existence conditions for obtaining the solutions of a new Caputo non-integer order inclusion model for COVID-19 are studied, availing one of the proposed L-fuzzy contractions. Since non-classical analysis is a known better tool for understanding and managing epidemic models, the idea developed herein is hoped to cause a spike in the use of fuzzy analysis to study various physical phenomena. Within the setting of differential equations and invariant point theory with metrics, the idea proposed in this work unifies and extends a few important findings in the corresponding literature.

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.

2  Preliminaries

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 R, ϝ+ and N, represent the sets of real numbers, nonnegative real numbers and the set of natural numbers, respectively.

Let (℧,μ) be a metric space (MS). Denote by CB(℧), the collection of all nonempty closed and bounded subsets of ℧. For A,B∈CB(℧), the mapping Υ:CB(℧)×CB(℧)⟶ϝ+ defined as

Υ(A,B)=max{supς∈Bμ(ς,A),supω∈Aμ(ω,B)},

where μ(ς,A)=infω∈Aμ(ς,ω), is recognized as the Hausdorff metric.

Definition 2.1. [21] The mapping ρℳ𝒯:(0,∞)⟶[0,1) is called an ℳ𝒯-function if it satisfies the Mizoguchi-Takahashi’s condition.

Definition 2.2. [22] The mapping ρ:ϝ+⟶[0,1) is termed a function of contractive factor, if for any nonincreasing sequence {ςx}x≥1 in ϝ+, we have 0≤supx∈Nρ(ςx)<1.

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 ρ:ϝ+⟶[0,1k) is termed a 𝒟-function if it satisfies the condition: for every t∈ϝ+, there exists k∈(1,∞): lim supr⟶t+ρ(r)<1k.

Definition 2.4. [23] ρ:ϝ+⟶[0,1k) is called a function of 1k-contractive factor, if for any sequence {ςx}x≥1 in ϝ+ from and after some fixed terms, it is nonincreasing and 0≤supx∈Nρ(ςx)<1k, for some k∈(1,∞).

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

Lemma 2.1. [23] Let ρ:ϝ+⟶[0,1k) be a 𝒟-function. Then, ρ:ϝ+⟶[0,1k) defined as ρ(t)=ρ(t)+1k2 is also a 𝒟-function for every t∈ϝ+ and some k∈(1,∞).

Lemma 2.2. [23] Let ρ:ϝ+⟶[0,1k),  k∈(1,∞). Then, the statements hereunder are equivalent:

(i)   ρ is a 𝒟-function.

(ii)   For each t∈ϝ+, there exist σt(1)∈[0,1k) and δt(1)>0 such that ρ(s)≤σt(1) for all s∈(t,t+δt(1)).

(iii)   For each t∈ϝ+, there exist σt(2)∈[0,1k) and δt(2)>0 such that ρ(s)≤σt(2) for all s∈[t,t+δt(2)].

(iv)   For each t∈ϝ+, there exist σt(3)∈[0,1k) and δt(3)>0 such that ρ(s)≤σt(3) for all s∈(t,t+δt(3)].

(v)   For each t∈ϝ+, there exist σt(4)∈[0,1k) and δt(4)>0 such that ρ(s)≤σt(4) for all s∈[t,t+δt(4)].

(vi)   For any sequence {ςx}x≥1 in ϝ+, from and after certain term, it is decreasing and 0≤supx∈Nρ(ςx) <1k.

(vii)   ρ is a function of 1k-contractive factor, that is, for any sequence {ςx}x≥1 in ϝ+, from and after certain term, it is decreasing and 0≤supx∈Nρ(ςx)<1k.

Definition 2.5. [24] A relation ⪯ on a nonempty set L is called a partial order if it is

(i)   reflexive;

(ii)   antisymmetric;

(iii)   transitive.

A set L together with a partial ordering ⪯ is called a partially ordered set (or pset) and is denoted by (L,⪯L).

Definition 2.6. [24] Let L be a nonempty set and (L, ⪯) be a pset. Then, any two elements ς,ω∈L are said to be comparable if either ς⪯ω or ω⪯ς.

Definition 2.7. [24] A pset (L,⪯L) is called:

(i)   a lattice, if ς∨ω∈L, ς∧ω∈L for any ς,ω∈L;

(ii)   a complete lattice, if ⋁∇∈L, ⋀∇∈L for any ∇⊆L;

(iii)   distributive lattice if ς∨(ω∧ξ)=(ς∨ω)∧(ς∨ξ), ς∧(ω∨ξ)=(ς∧ω)∨(ς∧ξ), for any ς,ω,ξ∈L.

A pset L is called a complete lattice if for every doubleton {ς,ω} in L, either sup{ς,ω}=ς⋁ω or inf{ς,ω}=ς⋀ω exists.

Definition 2.8. [10] An L-fuzzy set (L-fset) ∇ on a nonempty set ℧ is a function with domain ℧ and whose range lies in a complete distributive lattice L with top and bottom elements 1L and 0L, respectively.

Denote the class of all L-fuzzy sets on a nonempty set ℧ by L℧ (to depict a mapping :℧⟶L).

Definition 2.9. [10] The τL-level set of an L-fset ∇ is denoted by [∇]τL and is defined as follows:

[∇]τL={{ς∈℧:0L⪯L∇(ς)}¯,if τL=0L{ς∈℧:τL⪯L∇(ς)},ifτL∈L∖{0L}.

Definition 2.10. [11,12] Let ℧ be a nonempty set and Y a MS. Then, Ψ:℧⟶LY is called an Lmap. The function value Ψ(ς)(ω) is called the degree of membership of ω in Ψ(ς). For any two Lmaps Υ,Ψ:℧⟶LY, a point u∈℧ is called an L-fuzzy invariant point of Υ if there exists τL∈L∖{0L} such that u∈[Υu]τL. A point u is known as a common L-fuzzy invariant point of Υ and Ψ if u∈[Υu]τL∩[Ψu]τL.

Consistent with Rashid et al. [11,12], let (℧,μ) be a MS and consider τL∈L∖{0L} such that [∇]τL,[△]τL∈CB(℧). Then,

pτL(∇,△)=infς∈[∇]τL,ω∈[△]τLμ(ς,ω).

DτL(∇,△)=Υ([∇]τL,[△]τL).

p(∇,△)=supτLpτL(∇,△).

μL∞(∇,△)=supτLDτL(∇,△).

Definition 2.11. [19,25] Let Δ,Θ,Λ:℧⟶℧ be self-mappings and Ψ:℧⟶L℧ be an Lmap. An element u is called a coincidence point of Δ,Θ,Λ, and Ψ if Δu=Θu=Λu∈[Ψu]τL. If Δ=Θ=Λ=I℧ (the identity mapping on ℧), then u=Δu=Θu=Λu∈[Ψu]τL gives an L-fuzzy invariant point of Ψ.

We represent the set of all L-fuzzy invariant points of Ψ and the family of coincidence points of Δ,Θ,Λ and Ψ by ℒℱix(Ψ), and 𝒞𝒪𝒫(Δ,Θ,Λ,Ψ), respectively.

3  Main Results

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

Theorem 3.1. Let (℧,μ) be a complete MS, Ψ:℧⟶L℧ an Lmap, Δ,Θ,Λ:℧⟶℧ be continuous self-mappings and ρ:ϝ+⟶[0,1)=I∖{1} be a 𝒟-function. Suppose that:

(ax1) for every ς∈℧, there exists τL∈L∖{0L} such that [Ψς]τL is a nonempty closed and bounded subset of ℧;

(ax2) for every ς∈℧, {Δω=Θω=Λω  for all ω∈[Ψς]τL}⊆[Ψς]τL;

(ax3) there exist mappings δ,ν,h:℧⟶ϝ+ such that

Υ([Ψς]τL,[Ψω]τL)≤ρ(μ(ς,ω))[ζ1μ(ς,ω)+ζ2μ(ς,[Ψς]τL)+ζ3μ(ω,[Ψω]τL)]+δ(Δω)μ(Δω,[Ψς]τL)+ν(Θω)μ(Θω,[Ψς]τL)+h(Λω)μ(Λω,[Ψς]τL),

for all ς,ω∈℧, where ζ1,ζ2,ζ3∈ϝ+ with ζ1+ζ2+ζ3<1.

Then, 𝒞𝒪𝒫(Δ,Θ,Λ,Ψ)∩ℒℱix(Ψ)≠∅.

Proof. By (ax2), we notice that for every ς∈℧, μ(Δω,[Ψς]τL)=μ(Θω,[Ψς]τL)=μ(Λω,[Ψς]τL)=0 for all ω∈[Ψς]τL. So, for every ς∈℧, it if follows from (ax3) that for all ω∈[Ψς]τL,

Υ([Ψς]τL,[Ψω]τL)≤ρ(μ(ς,ω))[ζ1μ(ς,ω)+ζ2μ(ς,[Ψς]τL)+ζ3μ(ω,[Ψω]τL).(1)

Further, for every ω∈[Ψς]τL, μ(ω,[Ψω]τL)≤Υ([Ψς]τL,[Ψω]τL). Hence, for each ς∈℧, (1) gives

μ(ω,[Ψω]τL)≤ρ(μ(ς,ω))[ζ1μ(ς,ω)+ζ2μ(ς,[Ψς]τL)+ζ3μ(ω,[Ψω]τL)≤ρ(μ(ς,ω))[ζ1μ(ς,ω)+ζ2μ(ς,[Ψς]τL)]1−ζ3ρ(μ(ς,ω))≤ρ(μ(ς,ω))[ζ1μ(ς,ω)+ζ2μ(ς,[Ψς]τL)].(2)

Let ς0∈℧ and choose ς1∈[Ψς0]τL. If μ(ς0,ς1)=0, then ς0=ς1∈[Ψς0]τL, that is, ς0∈ℒℱix(Ψ) and the proof is finished. Otherwise, if μ(ς0,ς1)>0, then consider a function ρ:ϝ+⟶I∖{1} defined as ρ(t)=1+ρ(t)2. By Lemma 2.1, ρ is a 𝒟-function and 0≤ρ(t)<ρ(t)<1 for all t∈ϝ+. From (2), it follows that

μ(ς1,[Ψς1]τL)≤ρ(μ(ς0,ς1))[ζ1μ(ς0,ς1)+ζ2μ(ς0,[Ψς0]τL)]<ρ(μ(ς0,ς1))[ζ1μ(ς0,ς1)+ζ2μ(ς0,ς1)]=ρ(μ(ς0,ς1))[(ζ1+ζ2)μ(ς0,ς1)].(3)

Given that ζ1+ζ2+ζ3<1, there exists η∈(0,1) such that ζ1+ζ2<η=1−ζ3<1. Thus, (3) can be represented as:

μ(ς1,[Ψς1]τL)<ηρ(μ(ς0,ς1))μ(ς0,ς1)<ρ(μ(ς0,ς1))μ(ς0,ς1).(4)

From (4), we claim that there exists ς2∈[Ψς1]τL such that

μ(ς1,ς2)<ρ(μ(ς0,ς1))μ(ς0,ς1).(5)

Assume that this claim is not true, that is, μ(ς1,ς2)≥ρ(μ(ς0,ς1))μ(ς0,ς1). Then, we get

μ(ς1,ς2)≥infγ∈[Ψς1]τLμ(ς1,γ)≥ρ(μ(ς0,ς1))μ(ς0,ς1),

that is, μ(ς1,[Ψς1]τL)≥ρ(μ(ς0,ς1))μ(ς0,ς1), contradicting (4). Now, if μ(ς1,ς2)=0, then ς1=ς2∈Ψς1 and so ς1∈ℒℱix(Ψ). Otherwise, there exists ς3∈[Ψς2]τL such that

μ(ς2,ς3)<ρ(μ(ς1,ς2))μ(ς1,ς2).(6)

Let τx=μ(ςx−1,ςx) for every x∈N. On similar steps as above, we can construct a sequence {ςx}x∈N in ℧ with ςx∈[Ψςx−1]τL for every x∈N and

τx+1<ρ(τx)<τx.(7)

Given that ρ is a 𝒟-function, then, utilizing Lemma 2.2, yields 0≤supx∈Nρ(τx)<supx∈N ρ(τx)<1. Hence, 0<supx∈Nρ(τx)=sup{1+ρ(τx)2:x∈N}<1. Take ξ:=supx∈Nρ(τx), then 0<ξ<1. Since ρ(t)<1 for all t∈ϝ+, then, by (7), {τx}x∈N is a nonincreasing sequence. Whence, for every x∈N, we see that

τx+1<ρ(τx)≤ξτx.(8)

Hence, it if follows from (8) that

μ(ςx,ςx+1)=τx+1≤ξτx≤⋯≤ξxτ1=ξxμ(ς0,ς1).(9)

For any m,x,x0∈N with m>x>x0, by (9), we get

μ(ςm,ςx)≤∑j=xm−1μ(ςj,ςj+1)≤∑j=xm−1ξjμ(ς0,ς1)≤∑j=x∞ξjμ(ς0,ς1)≤ξx1−ξμ(ς0,ς1)⟶0  (as  x⟶∞).

Thus, limx⟶∞{μ(ςm,ςx):m>x}=0. This proves that {ςx}x∈N is a Cauchy sequence in ℧. Whence, there exists u∈℧ such that ςx⟶u as x⟶∞. Since ςx∈[Ψςx−1]τL for every x∈N, it if follows from condition (ax2) that for every x∈N,

Δςx=Θςx=Λςx∈[Ψςx−1]τL.(10)

Employing the defining property of Δ,Θ and Λ, leads to

u=limx⟶∞Δςx=limx⟶∞Θςx=limx⟶∞Λςx=limx⟶∞Δu=limx⟶∞Θu=limx⟶∞Λu.

We propose that u∈[Ψu]τL. Suppose the contrary that μ(u,[Ψu]τL)>0. Since the function ς⟼μ(ς,[Ψu]τL) is continuous, then, from condition (ax3), we obtain

μ(u,[Ψu]τL)=limx⟶∞μ(ςx,[Ψu]τL)≤limx⟶∞Υ([Ψςx−1]τL,[Ψu]τL)≤limx⟶∞{ρ(μ(ςx−1,u))[ζ1μ(ςx−1,u)+ζ2μ(ςx−1,Ψςx−1)+ζ3μ(u,Ψu)]+δ(Δu)μ(Δu,[Ψςx−1]τL)+ν(Θu)μ(Θu,[Ψςx−1]τL)+h(Λu)μ(Λu,[Ψςx−1]τL)}<limx⟶∞{ρ(μ(ςx−1,u))[ζ1μ(ςx−1,u)+ζ2μ(ςx−1,ςx)+ζ3μ(u,[Ψu]τL)]+δ(Δu)μ(Δu,ςx)+ν(Θu)μ(Θu,ςx)+h(Λu)μ(Λu,ςx)}<ζ3ρ(μ(u,u))μ(u,[Ψu]τL)<ζ3μ(u,[Ψu]τL),

a contradiction for all ζ3∈(0,1). Hence, μ(u,[Ψu]τL)=0. Since [Ψu]τL is closed, it must be the case that u∈[Ψu]τL. By condition (ax2), Δu=Θu=Λu∈[Ψu]τL. Consequently, u∈𝒞𝒪𝒫(Δ,Θ,Λ,Ψ)∩ℒℱix(Ψ).

Definition 3.1. [26] A nonempty subset ∇ of ℧ is called proximal if, for every ς∈℧, there exists ζ1∈∇ such that μ(ς,ζ1)=μ(ς,∇).

We denote the class of all bounded proximal subsets of ℧ by 𝒫br(℧). Since every proximal set is closed (see [26]), it follows that 𝒫br(℧)⊆CB(℧). Now, take

𝒞𝒫(℧)={∇∈L℧:[∇]τL∈𝒫br(℧),  for each τL∈L∖{0L}}.

In what follows, we study another coincidence point result in connection with μL∞-metric for L-fuzzy sets. It is pertinent to point out that the L-fuzzy invariant point results in the setting of μL∞-metric is of great importance in analyzing Hausdorff dimensions. These dimensions help us to understand the notions of ε∞ -space which is of tremendous importance in higher energy physics (see, e.g., [27,28]).

Theorem 3.2.Let (℧,μ) be a complete MS, Ψ:℧⟶𝒞𝒫(℧) an Lmap, Δ,Θ,Λ:℧⟶℧ be continuous self-mappings and ρ:ϝ+⟶I∖{1} be a 𝒟-function. Assume that:

(ax1) for every ς∈℧, {Δω=Θω=Λω for all  ω∈Ψς}⊆Ψς;

(ax2) there exist three mappings δ,ν,h:℧⟶ϝ+ such that

μL∞(Ψς,Ψω)≤ρ(μ(ς,ω))[ζ1μ(ς,ω)+ζ2p(ς,Ψς)+ζ3p(ω,Ψω)]+δ(Δω)p(Δω,Ψς)+ν(Θω)p(Θω,Ψς)+h(Λω)p(Λω,Ψς),

for all ς,ω∈℧, where ζ1,ζ2,ζ3∈ϝ+ with ζ1+ζ2+ζ3<1.

Then, 𝒞𝒪𝒫(Δ,Θ,Λ,Ψ)∩ℒℱix(Ψ)≠∅.

Proof. Let ς∈℧ be arbitrary, and define a function τL:℧⟶L∖{0L} by τL(ς):=τL=1L. Then, by hypothesis, [Ψς]1L∈CB(℧). Whence, for all ς,ω∈℧, we see that

D1L(Ψς,Ψω)≤μL∞(Ψς,Ψω)≤ρ(μ(ς,ω))[ζ1μ(ς,ω)+ζ2p(ς,Ψς)+ζ3p(ω,Ψω)]+δ(Δω)p(Δω,Ψς)+ν(Θω)p(Θω,Ψς)+h(Λω)p(Λω,Ψς).

Since [Ψς]1L⊆[Ψς]τL∈CB(℧), it follows that μ(ς,[Ψς]τL)≤μ(ς,[Ψς]1L) for every τL∈L∖{0L}. Hence, p(ς,Ψς)≤μ(ς,[Ψς]1L) for all ς∈℧. Consequently,

Υ([Ψς]1L,[Ψω]1L)≤ρ(μ(ς,ω))[ζ1μ(ς,ω)+ζ2μ(ς,[Ψς]1L)+ζ3μ(ω,[Ψω]1L)]+δ(Δω)μ(Δω,[Ψς]1L)+ν(Θω)μ(Θω,[Ψς]1L)+h(Λω)μ(Λω,[Ψς]1L).

Hence, Theorem 3.1 can be applied to find u∈℧ such that u∈𝒞𝒪𝒫(Δ,Θ,Λ,Ψ)∩ℒℱix(Ψ).

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

Example 3.1. Let L={p,q,r,ν,s,m,x,w} be such that p⪯L,s⪯Lr⪯Lw, p⪯Lν⪯Lq⪯Lw, s⪯Lm⪯Lw, ν⪯Lm⪯Lw, x⪯Lq⪯Lw; and each element of the doubletons {r,m},{m,q},{s,x},{x,ν} are not comparable. Then, (L,⪯L) is a complete distributive lattice. Let η∞ be the space of all bounded sequences endowed with the supremum norm ‖.‖∞, and let {ex} be the canonical basis of η∞. Let {ζx}x∈N be a sequence of positive real numbers satisfying ζ1=ζ2 and ζx+1<ζx for all x ≥ 2 (for example, take ζ1=12 and ζx=1x,x≥2). It follows that {ζx}x∈N is convergent. Put vx=ζxex for all x∈N and let ℧={vx}x∈N be a bounded and complete subset of η∞. Then (℧,‖.‖∞) is a complete MS, and ‖vx−vm‖∞=ζx if m > x.

Consider an Lmap Ψ:℧⟶L℧ defined as follows:

Ψ(v1)(t)=Ψ(v2)(t)={w,if t∈{v1,v2}p,ift=vx+1,x>2,

Ψ(v3)(t)=Ψ(v4)(t)=⋯=Ψ(vx)(t)={r,if t∈{v1,v2}w,ift=vx+1.

Then, for every x∈N and ζL(vx)=w, we see that

[Ψvx]ζL={{v1,v2},if  w∈{v1,v2}{vx+1},if  w∉{v1,v2}.

Further, define the mappings Δ,Θ,Λ:℧⟶℧ as

Δvx=Θvx=Λvx={v2,if  x∈{1,2}vx+1,if  x>2.

Then, we observe that the following hold:

(ax1) for every ς∈℧, there exists ζL∈L∖{0L}: [Ψς]ζL∈CB(℧);

(ax2) for every ς∈℧, {Δω=Θω=Λω∈[Ψς]ζL}⊆[Ψς]ζL;

and 𝒞𝒪𝒫(Δ,Θ,Λ,Ψ)∩ℒℱix(Ψ)={v1,v2}.

To prove that Δ,Θ and Λ are continuous, it is enough to show the nonexpansiveness of Δ,Θ and Λ. So, we check the cases

(i)   ‖Δv1−Δv2‖∞=0<ζ1=‖v1−v2‖∞;

(iii)   ‖Δv1−Δvm‖∞=ζ2=ζ1=‖v1−vm‖∞ for any m > 2;

(iv)   ‖Δv2−Δvm‖∞=ζ2=‖v2−vm‖∞ for any m > 2;

(vi)   ‖Δvx−Δvm‖∞=ζx+1<ζx=‖vx−vm‖∞ for any m > 2 and m > x.

This shows that ‖Δς−Δω‖∞≤‖ς−ω‖∞ for all ς,ω∈℧, indicating the nonexpansiveness of Δ. Since Δ=Θ=Λ, then Δ,Θ and Λ are continuous.

Define ρ:ϝ+⟶I∖{1} as

ρ(t)={ζx+2ζx,if  t=ζx for some x∈N15,otherwise.

Since limr⟶t+supρ(r)=15<1 for all t∈ϝ+, then ρ is a 𝒟-function. Also, define the mappings δ,ν,h:℧⟶℧ by

δ(vx)=ν(vx)=h(vx)={0,if x∈{1,2}x,ifx>2.

Now, we claim that

Υ∞([Ψς]ζL,[Ψω]ζL)≤ρ(‖ς−ω‖∞)[ζ1‖ς−ω‖∞+ζ2‖ς−[Ψς]ζL‖∞+ζ3‖ω−[Ψω]ζL‖∞]+δ(Δω)‖Δω−[Ψς]ζL‖∞+ν(Θω)‖Θω−[Ψς]ζL‖∞+h(Λω)‖Λω−[Ψς]ζL‖∞(11)

for all ς,ω∈℧ and ζ1,ζ2,ζ3∈ϝ+ with ζ1+ζ2+ζ3<1, where Υ∞ is the Hausdorff metric induced by the norm ‖.‖∞.

To verify (11), we check the possibilities:

Case 1. If x = 1, m = 2 and ζ1=16,ζ2=ζ3=0, we see that

ρ(‖v1−v2‖∞)(ζ1‖v1−v2‖∞+ζ2‖v1−[Ψv1]ζL‖∞+ζ3‖v2−[Ψv2]ζL‖∞)+δ(Δv2)‖Δv2−[Ψv1]ζL‖∞+ν(Θv2)‖Θv2−[Ψv1]ζL‖∞+h(Λv2)‖Λv2−[Ψv1]ζL‖∞=ζ36>0=Υ∞([Ψv1]ζL,[Ψv2]ζL).

Case 2. For x = 1, m > 2 and ζ1=13,ζ2=ζ3=0, we see that

ρ(‖v1−vm‖∞)(ζ1‖v1−vm‖∞+ζ2‖v1−[Ψv1]ζL‖∞+ζ3‖vm−[Ψvm]ζL‖∞)+δ(Δvm)‖Δvm−[Ψv1]ζL‖∞+ν(Θvm)‖Θvm−[Ψv1]ζL‖∞+h(Λvm)‖Λvm−[Ψv1]ζL‖∞=ζ33+3(m+1)ζ1>ζ1=Υ∞([Ψv1]ζL,[Ψvm]ζL).

Case 3. For x = 2, m > 2 and ζ1=18,ζ2=ζ3=0, we see that

ρ(‖v2−vm‖∞)(ζ1‖v2−vm‖∞+ζ2‖v2−[Ψv2]ζL‖∞+ζ3‖vm−[Ψvm]ζL‖∞)+δ(Δvm)‖Δvm−[Ψv2]ζL‖∞+ν(Θvm)‖Θvm−[Ψv2]ζL‖∞+h(Λvm)‖Λvm−[Ψv2]ζL‖∞=ζ48+3(m+1)ζ2>ζ2=Υ∞([Ψv2]ζL,[Ψvm]ζL).

Case 4. For x > 2, m > x and ζ1=14,ζ2=ζ3=0, we see that

ρ(‖vx−vm‖∞)(ζ1‖vx−vm‖∞+ζ2‖vx−[Ψvx]ζL‖∞+ζ3‖vm−[Ψvm]ζL‖∞)+δ(Δvm)‖Δvm−[Ψvx]ζL‖∞+ν(Θvm)‖Θvm−[Ψvx]ζL‖∞+h(Λvm)‖Λvm−[Ψvx]ζL‖∞=ζx+24+3(m+2)ζx+1>ζx+1=Υ∞([Ψvx]ζL,[Ψvm]ζL).

Thus, following cases (1)−(4), we have demonstrated that (11) is valid. Hence, all the assumptions of Theorem 3.1 are agreed with. Whence, 𝒞𝒪𝒫(Δ,Θ,Λ,Ψ)∩ℒℱix(Ψ)≠∅.

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

μ1∞(Ψ(v1),Ψ(vm))=ζ1>λζ1=λ‖v1−vm‖∞

for all λ∈(0,1) and m > 2. Hence, the main result of Heilpern [2] is not useable in this example to obtain any fuzzy invariant point of Ψ.

Similarly, let ζL=w and consider a multivalued mapping F:℧⟶CB(℧) defined by Fv=[Ψv]w for all v∈℧. Then, we see that

Υ(Ψv1,Ψvm)=Υ([Ψv1]w,[Ψvm]w)=ζ1>λζ1=λ‖v1−vm‖∞

for all λ∈(0,1) and m > 2. Whence, the principal result of Nadler [3] is not useful here to obtain the invariant point of Ψ.

Also, we note that

Υ([Ψv1]ζL,[Ψvm]ζL)=ζ1>ζ3=ρ(‖v1−vm‖∞)‖v1−vm‖∞

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

Fig. 1 represents the Lattice in Example 3.1.

Figure 1: The lattice in example 3.1

Consider a nonempty subset A of ℧ and the mapping Ψ:℧⟶℧. We recall that A is Ψ-invariant if Ψ(A)⊆A. Now, we come up with some observations from Theorems 3.1 and 3.2.

Corollary 3.1. Let (℧,μ) be a complete MS, Ψ:℧⟶L℧ be an Lmap, Δ:℧⟶℧ be a continuous self-mapping and ρ:ϝ+⟶I∖{1} be a 𝒟-function. Suppose that:

(i)   for every ς∈℧, there exists τL∈L∖{0L} such that [Ψς]τL∈CB(℧);

(ii)   [Ψς]τL is Δ-invariant (i.e. Δ([Ψς]τL)⊆[Ψς]τL) for every ς∈℧;

(iii)   there exists δ:℧⟶ϝ+ such that

Υ([Ψς]τL,[Ψω]τL)≤ρ(μ(ς,ω))[ζ1μ(ς,ω)+ζ2μ(ς,[Ψς]τL)+ζ3μ(ω,[Ψω]τL)]+δ(Δω)μ(Δω,[Ψς]τL),

for all ς,ω∈℧ and ζ1,ζ2,ζ3∈ϝ+ with ζ1+ζ2+ζ3<1.

Then, 𝒞𝒪𝒫(Δ,Ψ)∩ℒℱix(Ψ)≠∅.

Proof. Define the mappings ν,h:℧⟶ϝ+ as ν(ς)=h(ς)=0 for all ς∈℧ in Theorem 3.1.

The next observation follows from Corollary 3.1.

Corollary 3.2. Let (℧,μ) be a complete MS, Ψ:℧⟶L℧ be an Lmap, Δ:℧⟶℧ be continuous and ρ:ϝ+⟶I∖{1} be a 𝒟-function. Suppose that

(i)   for every ς∈℧, there exists τL∈L∖{0L} such that [Ψς]τL∈CB(℧);

(ii)   [Ψς]τL is Δ-invariant (i.e. Δ([Ψς]τL)⊆[Ψς]τL) for every ς∈℧;

(iii)   there exist ξ≥0 and δ^:℧⟶[0,ξ] such that

Υ([Ψς]τL,[Ψω]τL)≤ρ(μ(ς,ω))[ζ1μ(ς,ω)+ζ2μ(ς,[Ψς]τL)+ζ3μ(ω,[Ψω]τL)]+δ^(Δω)μ(Δω,[Ψς]τL),

for all ς,ω∈℧ and ζ1,ζ2,ζ3∈ϝ+ with ζ1+ζ2+ζ3<1.

Then, 𝒞𝒪𝒫(Δ,Ψ)∩ℒℱix(Ψ)≠∅.

Corollary 3.3.Let (℧,μ) be a complete MS, Ψ:℧⟶L℧ be an Lmap, Δ:℧⟶℧ be continuous and ρ:ϝ+⟶I∖{1} be a 𝒟-function. Suppose that

(i)   for every ς∈℧, there exists τL∈L∖{0L} such that [Ψς]τL∈CB(℧);

(ii)   [Ψς]τL is Δ-invariant (i.e., Δ([Ψς]τL)⊆[Ψς]τL) for every ς∈℧;

(iii)   there exists ξ≥0 such that

Υ([Ψς]τL,[Ψω]τL)≤ρ(μ(ς,ω))[ζ1μ(ς,ω)+ζ2μ(ς,[Ψς]τL)+ζ3μ(ω,[Ψω]τL)]+ξμ(Δω,[Ψς]τL),

for all ς,ω∈℧ and ζ1,ζ2,ζ3∈ϝ+ with ζ1+ζ2+ζ3<1.

Then, 𝒞𝒪𝒫(Δ,Ψ)∩ℒℱix(Ψ)≠∅.

Proof. Take δ^:℧⟶[0,ξ] as δ^(ς)=ξ for all ς∈℧ in Corollary 3.2.

Corollary 3.4. Let (℧,μ) be a complete MS, Ψ:℧⟶L℧ be an Lmap and ρ:ϝ+⟶I∖{1} be a 𝒟-function. Suppose that there exists δ:℧⟶ϝ+ such that for every ς∈℧, there exists τL∈L∖{0L} with [Ψς]τL∈CB(℧), and

Υ([Ψς]τL,[Ψω]τL)≤ρ(μ(ς,ω))[ζ1μ(ς,ω)+ζ2μ(ς,[Ψς]τL)+ζ3μ(ω,[Ψω]τL)]+δ(ω)μ(ω,[Ψς]τL),(12)

for all ς,ω∈℧ and ζ1,ζ2,ζ3∈ϝ+ with ζ1+ζ2+ζ3<1.

Then, ℒℱix(Ψ)≠∅.

Proof. Take Δ:=I℧, the identity mapping on ℧ in Corollary 3.1.

Corollary 3.5. Let (℧,μ) be a complete MS, Ψ:℧⟶𝒞𝒫(℧) be an Lmap. Suppose that

μL∞(Ψ(ς),Ψ(ω))≤γμ(ς,ω),(13)

for all ς,ω∈℧ and for some γ∈(0,1). Then, ℒℱix(Ψ)≠∅.

Proof. First, note that Υ([Ψς]τL,[Ψω]τL)≤μL∞(Ψ(ς),Ψ(ω)) for all ς,ω∈℧ and τL∈L. Now, taking ζ2=ζ3=0, ρ(t)=tζ1(1+t),ζ1∈(0,1) and δ(t)=0 for all t∈Ω in Corollary 3.4 completes the proof.

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:

{CD0+ρW(t)=Λρ−βvWI−λ1WK−μ¯vW+γ1ρI+γ2ρIK+θ1vWK,CD0+ρWK(t)=λ1vWK−μ¯vWK−θ1vWK,CD0+ρI(t)=βvWI−γ1ρ−τK1ρ−μ¯vI+λ2vIK+θ2vIK,CD0+ρIK(t)=λ2vIK−μ¯vIK−θ2ρ−γ2ρ−τK2vIK,CD0+ρK(t)=μvI−ϕvK,(14)

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 WK(t), an infective population that is free from lock-down I(t), an infective population that is not free from lock-down IK(t), and a cumulative density of the lock-down program K(t). For the rest of the parameters and numerical simulations in addition to a few new existence results of (14), one can refer to [14,23]. The above model is reformulated as:

{CD0+ρW(t)=Θ1(t,W,W,WK,I,IK,K),CD0+ρWK(t)=Θ2(t,W,WK,I,IK,K),CD0+ρI(t)=Θ3(t,W,WK,I,IK,K),CD0+ρIK(t)=Θ4(t,W,WK,I,IK,K),CD0+ρK(t)=Θ5(t,W,WK,I,IK,K),(15)

where

{Θ1(t,W,W,WK,I,IK,K)=Λρ−βvWI−λ1WK−μ¯vW+γ1ρI+γ2ρIK+θ1vWK,Θ2(t,W,WK,I,IK,K)=λ1vWK−μ¯vWK−θ1vWK,Θ3(t,W,WK,I,IK,K)=βvWI−γ1ρ−τK1ρ−μ¯vI+λ2vIK+θ2vIK,Θ4(t,W,WK,I,IK,K)=λ2vIK−μ¯vIK−θ2ρ−γ2ρ−τK2vIK,Θ5(t,W,WK,I,IK,K)=μvI−ϕvK.(16)

Consequently, the model (14) takes the form:

{CD0ρϑ(t)=ν(t,ϑ(t)),t∈Ω=[0,b],  0<ρ<1ϑ(0)=ϑ0≥0,(17)

with the conditions:

{ϑ(t)=(W,WK,I,IK,K)tr,ϑ(0)=(W0,WK0,I0,IK0,K0)tr,ν(t,ϑ(t))=(Θi(t,W,WK,I,IK,K))tr,i=1,…,5,(18)

where (.)tr denotes the transpose operation.

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

{CD0ρϑ(t)∈V(t,ϑ(t)),t∈Ω=(0,δ)ϑ(0)=ϑ0≥0,(19)

where V:Ω×R⟶P(R) is a set-valued map (P(R) is the power set of R). We establish existence conditions for solutions to the inclusion problem (19) for which the right-hand side is non-convex by applying an invariant point theorem for Lmaps. For some related recent applications of crisp mathematical techniques, we refer to [29–32] and some citations in there. We now recall some needed concepts of fractional calculus and set-valued analysis as follows.

Definition 4.1. [33] Let ρ>0 and δ∈L′([0,δ],ℝ). Then, the Riemann-Liouville fractional integral order ρ for a function δ is defined as

I0+ρδ(t)=1Γ(ρ)∫0t(t−τ)ρ−1μτ,t>0,

where Γ(.) is the gamma function given by Γ(ρ)=∫0∞τρ−1e−τμτ.

Definition 4.2. [33] Let x−1<ρ<x,x∈N, and δ∈Cx(0,δ). Then, the Caputo fractional derivative of order ρ for a function δ is defined as

CD0+ρδ(t)=1Γ(x−ρ)∫0t(t−τ)x−ρ−1δx(τ)μτ,t>0.

Lemma 4.1. [33] Let ℜ(ρ)>0,x=[ℜ(ρ)]+1, and δ∈ACx(0,δ). Then

(I0+ρCD0+ρδ)(t)=δ(t)−∑k=1m(D0+kδ)(0+)k!.

In particular, if 0<ρ≤1, then (I0+ρCD0+ρδ)(t)=δ(t)−δ(0).

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

ϑ(t)=ϑ0+I0+ρν(t,ϑ(t))=ϑ0+1Γ(ρ)∫0t(t−τ)ρ−1ν(τ,ϑ(τ))μτ.(20)

Let ℧=C(Ω,R) denotes the Banach space of all continuous functions ϑ from Ω to R equipped with the norm defined as

‖ϑ‖=sup{|ϑ(t)|:t∈Ω=[0,δ]},

where

|ϑ(t)|=|W(t)|+|WK(t)|+|I(t)|+|IK(t)|+|K(t)|

and W,WK,I,IK,K∈℧.

Definition 4.3. [34] Let ℧ be a nonempty set. A single-valued mapping δ:℧⟶℧ is called a selection of a set-valued mapping V:℧⟶P(℧), if δ(ϑ)∈V(ϑ) for every ϑ∈℧.

For each ϑ∈℧, we take the set of all selections of V by

WV,ϑ={δ∈L′(Ω,ℝ):δ(t)∈V(t,ϑ(t))   for almost every (a.e.)     t∈Ω}.

Let (L,⪯L) be a complete distributive lattice and τLV:℧⟶L∖{0L} be an arbitrary mapping, where V:Ω×R⟶P(R) is a crisp set-valued map. For each ϑ∈℧, define an Lmap Ψ(ϑ):℧⟶L as

Ψ(ϑ)(t)={τLV(ϑ),if t∈V(t,ϑ(t))0L,otherwise.

Then, the set of all selections of V can be regarded as the set of all selections of an Lmap Ψ, denoted by WL,ϑ, and is defined as

WL,ϑ={δ∈L′(Ω,ℝ):δ(t)   ∈   [Ψϑ]τL for a.e. t∈Ω}.

Definition 4.4. A function ϑ∈ C ′ (Ω,ℝ) is a solution of problem (19) if there is a function φ∈L′(Ω,ℝ) with φ(t)∈V(t,ϑ(t))a.e. on Ω such that

ϑ(t)=ϑ0+1Γ(ρ)∫0t(t−τ)ρ−1φ(τ)μτ(21)

and ϑ(0)=ϑ0≥0.

Definition 4.5. [34] A set-valued mapping V:Ω⟶P(R) with a nonempty compact convex values is said to be Lebesgue measurable, if for every ϖ∈R, the function t⟼μ(ϖ,V(t))=inf{|ϖ−ζ|:ζ∈V(t)} is Lebesgue measurable.

Definition 4.6. An Lmap Ψ:Ω⟶LΩ with nonempty compact convex cut set is said to be Lebesgue measurable, if for every τL∈L, the function t⟼μ(ϖ,[Ψ(t)]τL)=inf{|ϖ−ζ|:ζ∈[Ψ(t)]τL} is Lebesgue measurable.

Let 𝒦(℧) represents the family of all nonempty compact subsets of ℧. The following is our main contribution in this section.

Theorem 4.1. Assume that the following conditions are satisfied:

(N1) V:Ω×R⟶𝒦(R) is such that V(.,ϑ):Ω⟶𝒦(R) is Lebesgue measurable for every ϑ∈R;

(N2) there exists a continuous function h:Ω⟶ϝ+ such that for all ϑ,ω∈R,

Υ(V(t,ϑ),V(t,ω))≤h(t)|ϑ−ω|,

for almost all t∈Ω and μ(0,V(t,0))≤h(t) for almost all t∈Ω.

Then, the differential inclusion (19) has at least one solution in Ω, provided that Φ‖h‖<1, where Φ=bρΓ(ρ+1) and Γ(.) is the gamma function.

Proof. We start by resolving (19) into an L-fuzzy invariant point problem. Accordingly, let ℧=C(Ω,R) and τL:℧⟶L∖{0L} be a mapping. For each ϑ∈℧, let the mapping ℶϑ:Ω⟶R be given as

ℶϑ(t)=ϑ0+1Γ(ρ)∫0t(t−τ)ρ−1φ(τ)μτ,φ∈WL,ϑ.

Then, define an Lmap ΘV:℧⟶L℧ as follows:

ΘV(ϑ)(l)={τL(ϑ),if l(t)=ℶϑ(t)0L,otherwise.

By setting τL(ϑ):=τL for all ϑ∈℧, there exists τL∈L∖{0L} such that

[ΘV(ϑ)]τL={l∈℧:l(t)=ℶϑ(t)=ϑ0+1Γ(ρ)∫0t(t−τ)ρ−1φ(τ)μτ,φ∈WL,ϑ}.

It is clear that the L-fuzzy invariant points of ΘV are solutions to problem (19). Now, we prove that ΘV satisfies all the conditions of Corollary 3.5. For this, the following cases are examined:

Case I. [ΘV(ϑ)]τL is nonempty and closed for every φ∈WL,ϑ. Given that V(.,ϑ(.)) is Lebesgue measurable, then ΘV(ϑ) is also Lebesgue measurable. Whence, by the Lebesgue measurable selection theorem (see, e.g., [34]), ΘV(ϑ) admits a Lebesgue measurable selection φ:Ω⟶R. Furthermore, by condition (N2), |φ(t)|≤h(t)+h(t)|ϑ(t)|, that is, φ∈L′(Ω,ℝ) and hence V is integrably bounded. Thus, WV,ϑ and WL,ϑ are nonempty. Now, we show that [ΘV(ϑ)]τL is closed for every ϑ∈℧. Let {ϑx}x∈N⊆[ΘV(ϑ)]τL be such that ϑx⟶u(x⟶∞) in ℧. Then u∈℧ and there exists φx∈WL,ϑx such that for every t∈Ω,

ϑx(t)=ϑ0+1Γ(ρ)∫0t(t−τ)ρ−1φx(τ)μτ.

Since V has compact values, it follows that [ΘV(ϑ)]τL∈𝒦(℧) for every τL∈L. Then, we move onto a subsequence to obtain that φx converges to u∈L′(Ω,ℝ). Whence, u∈WL,ϑ and for every t∈Ω, we see that

ϑx(t)⟶u(t)=ϑ0+1Γ(ρ)∫0t(t−τ)ρ−1φ(τ)μτ.

Hence, u∈[ΘV(ϑ)]τL.

Case 2. Next, we prove that (13) is satisfied. Recall that for every t∈Ω and τL∈L∖{0L}, we see that V(t,ϑ(t))=[ΘV(ϑ)]τL. Accordingly, for every ϑ,ω∈℧, let ϑ,ω∈℧ and l1∈[ΘV(ϑ)]τL. Then, there exists φ1(t)∈[ΘV(ϑ)]τL for some τL∈L such that for every t∈Ω,

l1(t)=ϑ0+1Γ(ρ)+∫0t(t−τ)ρ−1φ1(τ)μτ.(22)

By (N2), Υ([ΘV(ϑ)]τL,[ΘV(ω)]τL)≤h(t)‖ϑ−ω‖. Whence, there exists ρ∈[ΘV(ω)]τL for some τL∈L such that

|l1(t)−ρ(t)|≤h(t)|ϑ(t)−ω(t)|,t∈Ω.

Define Ξ:Ω⟶P(R) by Ξ(t)={t∈R:|l1(t)−ρ(t)|≤h(t)|ϑ(t)−ω(t)|}. Since the crisp set-valued map Ξ(t)∩[ΘV(ω)]τL is Lebesgue measurable for every τL∈L∖{0L} (see [34]), there exists a function φ2 which is a Lebesgue measurable selection of Ξ. Thus, φ2(t)∈[ΘV(ω)]τL, and for every t∈Ω, we see that |φ1(t)−φ2(t)|≤h(t)|ϑ(t)−ω(t)|. For each t∈Ω, take

l2(t)=ϑ0+1Γ(ρ)∫0t(t−τ)ρ−1φ2(τ)μτ.(23)

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

|l1(t)−l2(t)|≤1Γ(ρ)∫0t(t−τ)ρ−1[|φ1(τ)−φ2(τ)|]μτ≤1Γ(ρ)∫0t(t−τ)ρ−1[h(t)|ϑ(t)−ω(t)|||]μτ≤bρΓ(ρ+1)‖h‖‖ϑ−ω‖=Φ‖h‖‖ϑ−ω‖.

Whence, ‖l1−l2‖≤Φ‖h‖‖ϑ−ω‖. On similar steps, interchanging the roles of ϑ and ω, we obtain

Υ([ΘV(ϑ)]τL,[ΘV(ω)]τL)≤Φ‖h‖‖ϑ−ω‖=γ‖ϑ−ω‖.(24)

Taking supremum over all of τL∈L∖{0L} in (24), gives

μL∞(Ψ(ϑ),Ψ(ω))≤γ‖ϑ−ω‖=γμ(ϑ,ω),

for all ϑ,ω∈℧. Thus, all the hypotheses of Corollary 3.5 are satisfied. It follows that ΘV has at least one L-fuzzy invariant point in ℧, which corresponds to the solution of problem 19.

5  Conclusions

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 ℳ𝒯-function. Theorem 3.1 is an L-fuzzy extension of the invariant point results studied by Berinde-Berinde [35], Du [22], Mizoguchi-Takahashi [36], Nadler [3], Reich [37], Rus [38], and a handful of others in the related domains. In addition to the aforementioned work, Theorem 3.2 is an improvement of the principal idea of Heilpern [2]. As a consequence of the theorems obtained herein, a few corollaries can be derived from existing results. A comparative example (Example 3.1) is provided to support our abstractions and indicate the preeminence of the proposed ideas. In Theorem 14, an existence result for a nonlinear fractional differential inclusion model for COVID-19 was presented by utilizing the idea of μL∞-metric for L-fuzzy sets.

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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