# A Countable System of Fractional Inclusions with Periodic, Almost, and Antiperiodic Boundary Conditions

Authors: Ahmed Salem; Aeshah Al-Dosari
Journal: Complexity
 (2021)

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## Abstract

This article is dedicated to the existence results of solutions for boundary value problems of inclusion type. We suggest the infinite countable system to fractional differential inclusions written byDαABCνit∈Yit,νiti=1∞. The mappings yit,νiti=1∞ are proposed to be Lipschitz multivalued mappings. The results are explored according to boundary condition σνi0=γνiρ,σ,γ∈ℝ. This type of condition is the generalization of periodic, almost, and antiperiodic types.


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## Body

## 1. Introduction

Consider the following infinite system:(1)DαABCνit∈yit,νjtj=1∞,i∈ℕ,t∈0,ρ,(2)σνi0=γνiρ,σ,γ∈ℝ,where DαABC denotes the Atangana–Baleanu fractional derivative in the Caputo sense of order α∈0,1 and yii∈ℕ is an infinite countable family of Lipschitz continuous multivalued mappings. This means there is an infinite countable sequence of continuous real-valued functions νiti∈ℕ satisfying problems (1) and (2). In this case, we can define the function Vt:0,ρ⟶ℝℕ by Vt=νiti∈ℕ. This function denotes the sequence of solutions for the given system.In the field of infinite systems, the research to fractional differential problems started via ordinary derivatives (see [1–4] and the mentioned references therein). Then, many scholars were attracted to develop these problems into the ones associated with fractional derivatives. For instance, see the required results in [5–8] and references cited therein. The importance of the infinite system was arising naturally in the description of physical problems such as stochastic (stochastic metapopulation) models [9, 10], models descried by the Becker–Döring cluster [11, 12], and optimal pursuit equations [13] and the control problems for the models descried by parabolic and hyperbolic equations [14].The concept of fractional derivative arose before 300 years when L’Hospital asked in 1695 which is addressed in Leibnitz notation for thenth derivative “What would happen if the order n=1/2?”. So, the idea of fractional derivative started by operators with power kernel (Riemman and Caputo derivatives). It has been industrialized due to complexities associated with the heterogeneous phenomenon. The fractional differential operators are capable of capturing the behavior of multifaceted media as they have diffusion processes. In this field, many researchers have paid attention in several ways to develop these derivatives. For instance, they found the ways for the development to new ones without the problem of nonsingularity (Caputo–Fabrizio with the exponential kernel) and then without nonlocality (Atangana–Baleanu with the Mittag–Leffler kernel). Mittag–Leffler kernel in the AB derivative helps to understand the beginning and the end of a considered phenomenon which is due to the memory effect of the Mittag–Leffler function. Additionally, in some works, it was proved that the AB derivative can generate chaotic behaviors in some linear and nonlinear systems for certain values taken by the derivative order. In other situations, some researchers have shown that the fractional derivatives lost some of the basic properties that usual derivatives have such as the product rule and the chain rule. For the sake of solving this problem, they investigated the conformable fractional derivative. This derivative is important to develop the Lie symmetry analysis for differential equations involving different fractional derivatives such as the Caputo–Fabrizio derivative and Atangana–Baleanu derivative. In fact, the development of fractional calculus theory matches with the development of analytical and numerical methods for solving fractional differential equations and systems. For showing this importance, we refer to see [15–18] and the references therein.In [19], we studied how to generate the differential equations and inclusions by one form (we call them as equi-inclusion problems). Then, we studied the solvability of this form. Next, in [20], we generated the fractional differential equation at resonance on the half line into the inclusion one and explored the existence results of positive solutions for this problem. After research and reading about the infinite system topic, we find that all proven treatises are linked only to single-valued operators. So, what we refer here is to survey the infinite fractional differential system proposed with multivalued mappings. This means we hatch a generalization of previous literature studies in the infinite system field. This draws a way to new and important extents for the infinite system theory of nonlinear analysis. Furthermore, it would be useful to express some descriptions of complicated physical phenomena. We have a strong anticipation that the partial differential inclusion which leads to the infinite system of fractional differential inclusions would be very influential to make a fundamental shift in the theory of complicated neural sets, huge and stochastic branching processes, and the theory of dissociation of polymers. Also, by the inclusion system with DαABC, we think that modeling and computations will be performed to explore deep and manifold aspects of mixed convective flow of nanofluids and random flow processes of so many fluids.Our work is concerned with the existence of antiperiodic, periodic, and almost periodic solutions for problem (1) in the Banach space ℓpℝℕ,1<p<∞. Theorems used here give us sufficient conditions for the existence of common solutions to the infinite family of quasi-nonexpansive multivalued operators in the uniformly convex real Banach space. The results are affected by hemicompactness, compactness, contraction properties, and the one-step iterative scheme.It is worth remarkable to mention that the field of studying the existence and uniqueness of solutions to fractional differential equations has drawn attention of many contributors [21–28].


## 2. Preliminaries

We present this section with some needed definitions, facts, lemmas, theorems, and auxiliary results used to start the main theorems.


### 2.1. Real Sequence Banach Space

Define the spaceV to be the real sequence Banach space ℓpℝℕ,1<p<∞, endowed with the norm(3)Vℓp=∑i=1∞νip1/p,V=νii∈ℕ.Then, from [29–31], we have the following facts.Definition 1.
(uniformly convex space). A normed spaceV is called a uniformly convex space if for any ε∈0,2, there exists δ=δε>0 such that if v1,v2∈V with v1=v2=1andv2−v1≥ε, then 1/2v1+v2≤1−δ.Lemma 1.
The spaceℓp is uniformly convex for all 1≤p<∞.Theorem 1.
Adopt1≤p,q<∞ such that 1/p+1/q=1. Then, the inequalities thereafter hold:(i)
Holder inequality: letη∈ℓpandη′∈ℓq. Then, ηη′∈ℓ1 and(4)ηη′ℓ1≤ηℓpη′ℓq.(ii)
Minkowski inequality: letη,η′∈ℓp. Then, η+η′∈ℓp and(5)η+η′ℓp≤ηℓp+η′ℓp.(iii)
Imbedding theorem: letΩ have a finite positive measure and 1≤q≤p≤∞. Then, ℓqΩ⊆ℓpΩ and(6)ηℓp≤ηℓq.Theorem 2.
(compactness inℓp). Let E be a Banach space and K⊂ℓpE. Then, K is compact in ℓpE if and only if(i)
K is closed and bounded(ii)
∑r=n∞vrp1/p⟶0asn⟶∞(iii)
IfDr:ℓpE⟶Edefined for allvr∈ℓpEbyDrvrr≥1=vr, then DrK is compact for all r≥1Corollary 1.
The subsets(7)Kx=vrr≥1∈ℓp|vr≤xr,xr⟶0asr⟶0,1≤p≤∞are compact subsets of ℓp spaces.


### 2.2. Basics in Multivalued Maps

The concept of multioperators is related to the multivalued maps. So, it is required to show some facts about them and their properties. These facts are confirmed in [32–37].LetE,. and H,. be two Banach spaces. A multivalued map A:E⟶PclE is seen as convex (closed) if for every e∈E, then Ae is convex (closed) and selected to be completely continuous if AB is relatively compact for every B∈PbE.The mapA is said to be upper semicontinuous if for each closed subset W⊂E,A−1W is a closed subset of E. This means the set e∈E:Ae⊆O is open for all open sets O⊂E. It is lower semicontinuous if for each open subset Z⊂E,A−1Z is an open subset of E. In other words, A seems to be lower semicontinuous as long as the set e∈E:Ae∩O≠∅ is open for all open sets O⊂E.A mapA:0,ε⟶PclE is presented to be measurable multivalued if for every e∈E, the function τ⟶de,Aτ=infde,a:a∈Aτ is ℒ−measurable function.GivenC,B∈PclE; then,(8)hC,B=HdC,B=dHC,B=maxsupc∈Cdc,B,supb∈BdC,babsorbs the Pompeiu–Hausdorff distance of C,B.If we adoptA as a completely continuous function with nonempty compact values, then it is upper semicontinuous if and only if its graph is closed (i.e., if νn⟶ν∗andyn⟶y∗, then yn∈Aνn implies to y∗∈Aν∗).Definition 2.
A multivalued mapA:a,b×ℝ⟶Pℝ is known as Caratheodory if(1)
∀rn∈ℝ,n∈ℕ,τ⟶Aτ,rn is measurable(2)
For a.e.τ∈a,b,rn⟶Aτ,rn is upper semicontinuous
In addition to assumptions (1) and (2), the map A is L1-Caratheodory if for each k>0,∃ϕk∈L1a,b satisfying supτ≥0ϕkτ<+∞ and ϕk>0 and nondecreasing map Ł for which(9)Aτ,rn=supa:aτ∈Aτ,rn≤ϕkτŁr,r=rn,for all r<k,n∈ℕ,τ∈a,b.Definition 3.
(nonexpansive and quasi-nonexpansive multimapping). LetK be a subset of a metric space E and T:K⟶Pcl,bK be a multivalued map with FixT=v∈K|Tv=v (the set of all fixed points of T in K). Then, we have the following:(i)
T is called a nonexpansive mapping if it is contraction according to the metric. This means that, for all v1,v2∈K, we have(10)HdTv2,Tv1≤v2−v1E.(ii)
T is said to be quasi-nonexpansive if FixT≠∅ and for all w∈FixT and all v∈K, we have(11)HdTv,Tw≤v−wE.(iii)
FixT is a closed subset if K is closed and convex of E.Definition 4.
(hemicompactness). LetK,E, and T be defined as in Definition 3. Then, T is called hemicompact if(i)
For any sequencevrr∈ℕ∈K such that dvr,Tvr⟶0 as r⟶0, there is a subsequence vrs of vr with vrs⟶w∈K(ii)
T is compactTheorem 3.
(nonempty infinite intersection). For any spaceW, the following statements are equivalent:(i)
W is compact.(ii)
Every decreasing sequenceWn|n∈ℕ of nonempty closed subsets of W has a nonempty intersection(12)i.eWn+1⊆Wn,∀n∈ℕ⇒∩n≥1Wn≠∅.(iii)
Every collectionWn|n∈ℕ of nonempty closed subsets of W satisfying the finite intersection property has a nonempty intersection(13)i.e∩s=1mWns≠∅⇒∩n≥1Wn≠∅.


### 2.3. Fractional Calculus

In this part, we give definitions related to the used derivative, its history, the corresponding integral, and some properties [38–40].Definition 5.
(Riemann–Liouville integral). For the orderζ>0, the Riemann–Liouville fractional integral of a function hτ:0,∞⟶ℝ is defined by(14)Iζhτ=1Γς∫0ττ−ςζ−1hςdςsince the R.H.S is pointwise defined on 0,∞.Definition 6.
(Caputo–Fabrizio derivative). CF derivative for the orderα∈0,1 and ης∈H1a,b is given by(15)DαCFης=2−αMα21−α∫aςe−ας−s/1−αη′sds,where Mα is a normalization function such that M0=M1=1.Definition 7.
(Mittag–Leffler function). The general form of Mittag–Leffler functionEα of order α is written by(16)Eαv=∑r=0∞vrΓαr+1.
The main derivative used in the present paper is the Atangana–Baleanu fractional derivative in the Caputo sense. It is proposed by interchanging the kernelexp−ας−s/1−α in the Caputo–Fabrizio derivative by the equivalent form via the Mittag–Leffler formula that is ∑r=0∞−at−sr/r!,a=α/1−α. After that, replace r!byΓαr+1 and −at−srby−at−sαr. So, in the space(17)H1Ω=ης|ης,Dρης∈L2Ω,∀ρ≤1,we have the following definitions.Definition 8.
(Atangana–Baleanu fractional derivative in the Caputo sense). ABC derivative for the orderα∈0,1 and νς∈H1a,b is given by(18)DαABCνς=Mα1−α∫aςEα−ας−sα1−αν′sds.
This derivative is related to the fading memory concept and frequently used to discuss and analyze the real-world phenomena such as fluid and nanofluid models (see [41–43] and references therein). Depending on the constant Mα, the corresponding integral is given by the following definition.Definition 9.
(Atangana–Baleanu fractional integral). AB integral of the orderα∈0,1 and νς∈H1a,b is given by(19)IαABνt=1−αMανt+αMαIaαRνt,where IαR is the Riemann–Liouville integral of order α.Lemma 2.
Forα∈0,1,t∈a,b, and νt∈AC1a,b, the following statements hold:a1: AB integral together with the ABC derivative satisfies the Newton–Leibniz formula(20)Ia+αABDa+αABCνt=νt−νa.a2: they also satisfy the property(21)D0+αABCI0+αABνt=νt−νoEα−αtα1−α.a3: AB integral is a commutative operator such that, for any two ordersα,β∈0,1, we have(22)Ia+αABIa+βABνt=1−αMα+αMαIaαR1−βMβ+βMβIaβRνt=1−βMβ+βMβIaβR1−αMα+αMαIaαRνt=Ia+βABIa+αABνt.Lemma 3.
(antiperiodic solution). The unique solution of the problem(23)DαABCνt=εt,t∈0,ρ,(24)σν0=γνρ,σ,γ∈ℝ,withσ≠γ,α∈0,1,is given by(25)νt=1−αMαεt+αMαΓα∫0tt−sα−1εsds+γσ−γ1−αMαερ−σγε0+αMαΓα∫0ρρ−sα−1εsds.Proof.
By applyingIαAB to both sides of (23), we get(26)νt=IαABεt+c,which implies(27)ν0=1−αMαε0+c,νρ=IαABερ+c.
Now, applying (24), we find that(28)c=γσ−γ1−αMαερ−σγε0+αMαΓα∫0ρρ−sα−1εsds.Lemma 4.
(periodic/almost periodic solution). The unique solution of the problem(29)DαABCνt=εt,t∈0,ρ,ν0≊νρ,α∈0,1,is given by(30)νt=IαABεt+cwhenever(31)ε0≊ερ,(32)∫0ρρ−sα−1Γαεsds≊0,and then c≊−1−α/Mαε0≊−1−α/Mαερ.Proof.
Similar to the proof of Lemma3, we have(33)νt=IαABεt+c.
The almost periodic/periodic condition leads to(34)1−αερ−ε0+αΓα∫0ρρ−sα−1εsds≊0.
Under this equation, we have three cases:p1: ifα=0, then ερ≊ε0 must holdp2: whenα∈0,1, both ερ≊ε0and∫0ρρ−sα−1/Γαεsds≊0 must holdp3: in caseα=1, we get ∫0ρεsds≊0
So, we can say, in general, that there is an almost periodic/periodic solution if both (31) and (32) hold for every α∈0,1. Now, substituting (31) and (32) to the value of c=cσ in Lemma 3 with σ⟶γ and using the continuity condition, we get limσ⟶γc=−1−α/Mαε0=c|σ=γ. Hence, the periodic/almost periodic solution is given by(35)νt=IαABεt−1−αMαε0=1−αMαεt−ε0+αMαΓα∫0tt−sα−1εsds.
It is clear to see thatν0≊0≊νρ which completes the proof.


### 2.4. Fixed-Point Theorems

This section is surveyed by some fixed-point theorems investigated in the uniformly convex real Banach space [44, 45].Theorem 4.
LetV be a uniformly convex real Banach space and K be a closed, bounded, convex subset of V. Let T:K⟶PcpK be nonexpansive multivalued mappings. Then, T has a fixed point x∈K with x∈Tx.
The next theorem is formulated for the infinite countable family of multioperators under the vision of the one-step iterative scheme defined as follows.
LetK be a closed, bounded, convex subset of a uniformly convex real Banach space V. Let Ti:K⟶PbK be an infinite countable family of quasi-nonexpansive multivalued mappings with ∩i=1∞FixTi≠∅ and p∈∩i=1∞FixTi. Then, for all n∈ℕ, the sequence Vn is defined by(36)V1∈K,Vn+1=σn,0Vn+∑r=1nσn,rVn,r,(37)Vn,r∈TrVn,such thatp−Vn,r=dp,TrVn,(38)σn,r⊂0,1,with∑r=0nσn,r=1,r∈ℕ.Theorem 5.
LetK be a closed, bounded, convex subset of a uniformly convex real Banach space V. Let Ti:K⟶PbK be a sequence of quasi-nonexpansive and continuous multivalued mappings with ∩i=1∞FixTi≠∅ and p∈∩i=1∞FixTi. Let Vi be a sequence defined by (36) with the condition that limn⟶∞σn,randlimn⟶∞σn,n exist and lie in 0,1 for all r∈0∪ℕ. Assume that one of Ti is hemicompact. Then, the sequence Vi converges strongly.


## 2.1. Real Sequence Banach Space

Define the spaceV to be the real sequence Banach space ℓpℝℕ,1<p<∞, endowed with the norm(3)Vℓp=∑i=1∞νip1/p,V=νii∈ℕ.Then, from [29–31], we have the following facts.Definition 1.
(uniformly convex space). A normed spaceV is called a uniformly convex space if for any ε∈0,2, there exists δ=δε>0 such that if v1,v2∈V with v1=v2=1andv2−v1≥ε, then 1/2v1+v2≤1−δ.Lemma 1.
The spaceℓp is uniformly convex for all 1≤p<∞.Theorem 1.
Adopt1≤p,q<∞ such that 1/p+1/q=1. Then, the inequalities thereafter hold:(i)
Holder inequality: letη∈ℓpandη′∈ℓq. Then, ηη′∈ℓ1 and(4)ηη′ℓ1≤ηℓpη′ℓq.(ii)
Minkowski inequality: letη,η′∈ℓp. Then, η+η′∈ℓp and(5)η+η′ℓp≤ηℓp+η′ℓp.(iii)
Imbedding theorem: letΩ have a finite positive measure and 1≤q≤p≤∞. Then, ℓqΩ⊆ℓpΩ and(6)ηℓp≤ηℓq.Theorem 2.
(compactness inℓp). Let E be a Banach space and K⊂ℓpE. Then, K is compact in ℓpE if and only if(i)
K is closed and bounded(ii)
∑r=n∞vrp1/p⟶0asn⟶∞(iii)
IfDr:ℓpE⟶Edefined for allvr∈ℓpEbyDrvrr≥1=vr, then DrK is compact for all r≥1Corollary 1.
The subsets(7)Kx=vrr≥1∈ℓp|vr≤xr,xr⟶0asr⟶0,1≤p≤∞are compact subsets of ℓp spaces.


## 2.2. Basics in Multivalued Maps

The concept of multioperators is related to the multivalued maps. So, it is required to show some facts about them and their properties. These facts are confirmed in [32–37].LetE,. and H,. be two Banach spaces. A multivalued map A:E⟶PclE is seen as convex (closed) if for every e∈E, then Ae is convex (closed) and selected to be completely continuous if AB is relatively compact for every B∈PbE.The mapA is said to be upper semicontinuous if for each closed subset W⊂E,A−1W is a closed subset of E. This means the set e∈E:Ae⊆O is open for all open sets O⊂E. It is lower semicontinuous if for each open subset Z⊂E,A−1Z is an open subset of E. In other words, A seems to be lower semicontinuous as long as the set e∈E:Ae∩O≠∅ is open for all open sets O⊂E.A mapA:0,ε⟶PclE is presented to be measurable multivalued if for every e∈E, the function τ⟶de,Aτ=infde,a:a∈Aτ is ℒ−measurable function.GivenC,B∈PclE; then,(8)hC,B=HdC,B=dHC,B=maxsupc∈Cdc,B,supb∈BdC,babsorbs the Pompeiu–Hausdorff distance of C,B.If we adoptA as a completely continuous function with nonempty compact values, then it is upper semicontinuous if and only if its graph is closed (i.e., if νn⟶ν∗andyn⟶y∗, then yn∈Aνn implies to y∗∈Aν∗).Definition 2.
A multivalued mapA:a,b×ℝ⟶Pℝ is known as Caratheodory if(1)
∀rn∈ℝ,n∈ℕ,τ⟶Aτ,rn is measurable(2)
For a.e.τ∈a,b,rn⟶Aτ,rn is upper semicontinuous
In addition to assumptions (1) and (2), the map A is L1-Caratheodory if for each k>0,∃ϕk∈L1a,b satisfying supτ≥0ϕkτ<+∞ and ϕk>0 and nondecreasing map Ł for which(9)Aτ,rn=supa:aτ∈Aτ,rn≤ϕkτŁr,r=rn,for all r<k,n∈ℕ,τ∈a,b.Definition 3.
(nonexpansive and quasi-nonexpansive multimapping). LetK be a subset of a metric space E and T:K⟶Pcl,bK be a multivalued map with FixT=v∈K|Tv=v (the set of all fixed points of T in K). Then, we have the following:(i)
T is called a nonexpansive mapping if it is contraction according to the metric. This means that, for all v1,v2∈K, we have(10)HdTv2,Tv1≤v2−v1E.(ii)
T is said to be quasi-nonexpansive if FixT≠∅ and for all w∈FixT and all v∈K, we have(11)HdTv,Tw≤v−wE.(iii)
FixT is a closed subset if K is closed and convex of E.Definition 4.
(hemicompactness). LetK,E, and T be defined as in Definition 3. Then, T is called hemicompact if(i)
For any sequencevrr∈ℕ∈K such that dvr,Tvr⟶0 as r⟶0, there is a subsequence vrs of vr with vrs⟶w∈K(ii)
T is compactTheorem 3.
(nonempty infinite intersection). For any spaceW, the following statements are equivalent:(i)
W is compact.(ii)
Every decreasing sequenceWn|n∈ℕ of nonempty closed subsets of W has a nonempty intersection(12)i.eWn+1⊆Wn,∀n∈ℕ⇒∩n≥1Wn≠∅.(iii)
Every collectionWn|n∈ℕ of nonempty closed subsets of W satisfying the finite intersection property has a nonempty intersection(13)i.e∩s=1mWns≠∅⇒∩n≥1Wn≠∅.


## 2.3. Fractional Calculus

In this part, we give definitions related to the used derivative, its history, the corresponding integral, and some properties [38–40].Definition 5.
(Riemann–Liouville integral). For the orderζ>0, the Riemann–Liouville fractional integral of a function hτ:0,∞⟶ℝ is defined by(14)Iζhτ=1Γς∫0ττ−ςζ−1hςdςsince the R.H.S is pointwise defined on 0,∞.Definition 6.
(Caputo–Fabrizio derivative). CF derivative for the orderα∈0,1 and ης∈H1a,b is given by(15)DαCFης=2−αMα21−α∫aςe−ας−s/1−αη′sds,where Mα is a normalization function such that M0=M1=1.Definition 7.
(Mittag–Leffler function). The general form of Mittag–Leffler functionEα of order α is written by(16)Eαv=∑r=0∞vrΓαr+1.
The main derivative used in the present paper is the Atangana–Baleanu fractional derivative in the Caputo sense. It is proposed by interchanging the kernelexp−ας−s/1−α in the Caputo–Fabrizio derivative by the equivalent form via the Mittag–Leffler formula that is ∑r=0∞−at−sr/r!,a=α/1−α. After that, replace r!byΓαr+1 and −at−srby−at−sαr. So, in the space(17)H1Ω=ης|ης,Dρης∈L2Ω,∀ρ≤1,we have the following definitions.Definition 8.
(Atangana–Baleanu fractional derivative in the Caputo sense). ABC derivative for the orderα∈0,1 and νς∈H1a,b is given by(18)DαABCνς=Mα1−α∫aςEα−ας−sα1−αν′sds.
This derivative is related to the fading memory concept and frequently used to discuss and analyze the real-world phenomena such as fluid and nanofluid models (see [41–43] and references therein). Depending on the constant Mα, the corresponding integral is given by the following definition.Definition 9.
(Atangana–Baleanu fractional integral). AB integral of the orderα∈0,1 and νς∈H1a,b is given by(19)IαABνt=1−αMανt+αMαIaαRνt,where IαR is the Riemann–Liouville integral of order α.Lemma 2.
Forα∈0,1,t∈a,b, and νt∈AC1a,b, the following statements hold:a1: AB integral together with the ABC derivative satisfies the Newton–Leibniz formula(20)Ia+αABDa+αABCνt=νt−νa.a2: they also satisfy the property(21)D0+αABCI0+αABνt=νt−νoEα−αtα1−α.a3: AB integral is a commutative operator such that, for any two ordersα,β∈0,1, we have(22)Ia+αABIa+βABνt=1−αMα+αMαIaαR1−βMβ+βMβIaβRνt=1−βMβ+βMβIaβR1−αMα+αMαIaαRνt=Ia+βABIa+αABνt.Lemma 3.
(antiperiodic solution). The unique solution of the problem(23)DαABCνt=εt,t∈0,ρ,(24)σν0=γνρ,σ,γ∈ℝ,withσ≠γ,α∈0,1,is given by(25)νt=1−αMαεt+αMαΓα∫0tt−sα−1εsds+γσ−γ1−αMαερ−σγε0+αMαΓα∫0ρρ−sα−1εsds.Proof.
By applyingIαAB to both sides of (23), we get(26)νt=IαABεt+c,which implies(27)ν0=1−αMαε0+c,νρ=IαABερ+c.
Now, applying (24), we find that(28)c=γσ−γ1−αMαερ−σγε0+αMαΓα∫0ρρ−sα−1εsds.Lemma 4.
(periodic/almost periodic solution). The unique solution of the problem(29)DαABCνt=εt,t∈0,ρ,ν0≊νρ,α∈0,1,is given by(30)νt=IαABεt+cwhenever(31)ε0≊ερ,(32)∫0ρρ−sα−1Γαεsds≊0,and then c≊−1−α/Mαε0≊−1−α/Mαερ.Proof.
Similar to the proof of Lemma3, we have(33)νt=IαABεt+c.
The almost periodic/periodic condition leads to(34)1−αερ−ε0+αΓα∫0ρρ−sα−1εsds≊0.
Under this equation, we have three cases:p1: ifα=0, then ερ≊ε0 must holdp2: whenα∈0,1, both ερ≊ε0and∫0ρρ−sα−1/Γαεsds≊0 must holdp3: in caseα=1, we get ∫0ρεsds≊0
So, we can say, in general, that there is an almost periodic/periodic solution if both (31) and (32) hold for every α∈0,1. Now, substituting (31) and (32) to the value of c=cσ in Lemma 3 with σ⟶γ and using the continuity condition, we get limσ⟶γc=−1−α/Mαε0=c|σ=γ. Hence, the periodic/almost periodic solution is given by(35)νt=IαABεt−1−αMαε0=1−αMαεt−ε0+αMαΓα∫0tt−sα−1εsds.
It is clear to see thatν0≊0≊νρ which completes the proof.


## 2.4. Fixed-Point Theorems

This section is surveyed by some fixed-point theorems investigated in the uniformly convex real Banach space [44, 45].Theorem 4.
LetV be a uniformly convex real Banach space and K be a closed, bounded, convex subset of V. Let T:K⟶PcpK be nonexpansive multivalued mappings. Then, T has a fixed point x∈K with x∈Tx.
The next theorem is formulated for the infinite countable family of multioperators under the vision of the one-step iterative scheme defined as follows.
LetK be a closed, bounded, convex subset of a uniformly convex real Banach space V. Let Ti:K⟶PbK be an infinite countable family of quasi-nonexpansive multivalued mappings with ∩i=1∞FixTi≠∅ and p∈∩i=1∞FixTi. Then, for all n∈ℕ, the sequence Vn is defined by(36)V1∈K,Vn+1=σn,0Vn+∑r=1nσn,rVn,r,(37)Vn,r∈TrVn,such thatp−Vn,r=dp,TrVn,(38)σn,r⊂0,1,with∑r=0nσn,r=1,r∈ℕ.Theorem 5.
LetK be a closed, bounded, convex subset of a uniformly convex real Banach space V. Let Ti:K⟶PbK be a sequence of quasi-nonexpansive and continuous multivalued mappings with ∩i=1∞FixTi≠∅ and p∈∩i=1∞FixTi. Let Vi be a sequence defined by (36) with the condition that limn⟶∞σn,randlimn⟶∞σn,n exist and lie in 0,1 for all r∈0∪ℕ. Assume that one of Ti is hemicompact. Then, the sequence Vi converges strongly.


## 3. Inferred Results

### 3.1. Auxiliary Results

LetVt:C0,ρ⟶ℝℕ be defined by Vt=νiti∈ℕ. Then, for every multivalued mapping yi, we have yit,ν1t,…,νit,…=yit,Vt. Define the set-valued maps SYi,V such as(39)SYi,V=εit|εit∈yit,Vt∩L1J,ℝ,J=0,ρ.For the antiperiodic solutions, we define the multioperatorsϒi:ℓp⟶ℝ for all i∈ℕ as follows:(40)ϒiVt=ΨYit|ΨYit=Δεit,εit∈SYi,V¯,where(41)Δεit=1−αMαεit+αMαΓα∫0tt−sα−1εisds+γσ−γ1−αMαεiρ−σγεi0+αMαΓα∫0ρρ−sα−1εisds.For the almost periodic/periodic solutions, the multioperatorsΠi:ℓp⟶ℝ for all i∈ℕ are defined by(42)ΠiVt=ΘYit|ΘYit=Δ¯εit,εit∈SYi,V¯,where(43)Δ¯εit=1−αMαεit−εi0+αMαΓα∫0tt−sα−1εisds.Consider that the following conditions hold:H1yi is a sequence of L1−Caratheodory multivalued mappings that are(44)yi:J×V⟶Pcpℝ,with(1)
The mapst⟶yit,Vt are measurable for all V∈V.(2)
The mapsV⟶yit,Vt are measurable for all t∈J.(3)
For alli∈ℕ, there exist ψi∈CJ,ℝ+ such thatH2yi are Lipschitz mappings with constants Li∈ℝ+, respectively. This means(46)HdYit,V1t,Yit,V2t≤LiV1−V2,∀i∈ℕ.ℋ3∑i≥1Li<∞andZ=Ξ∑i=1∞Li<1 whenever(47)Ξ=21−αMα+1+γσ−γραMαΓα.ℋ4νi/Ξψi∗νi>1,ψi∗=supt∈Jψit.Then, we have the following propositions.Proposition 1.
The set-valued mapsϒiV are bounded and contraction for all i∈ℕ.Proof.
In view ofℋ1 and ℋ4, we have(48)ϒiV≤1−αMα+tαMαΓα+γσ−γ1−σγ1−αMα+ραMαΓαYit,Vt≤Ξψi∗νi≤νi.Thus, we prove the boundedness.To prove the contraction condition for alli∈ℕ, consider that ΨYi1t∈ϒiV2. This implies the existence of some εi1t∈SYi,V2¯ subject to(49)ΨYi1t=Δεi1t,whereΔ is defined by (41). By using ℋ2, we can define the sets(50)ϒ¯it=ε∈ℝ|εi1t−εt≤LiV1−V2ℓp.According to Theorem III.41 in [46], ℋ1, and the measurability of the functions εi1t and Wt=LiV1−V2ℓp, the sets ϒ¯it are also measurable for all i∈ℕ. Therefore, the maps t⟶ϒ¯it∩Yit,Vt are measurable with nonempty closed values. Hence, the measurability of ϒit and Proposition 2.1.43 in [47] drive to the existence of some εi2∈SYi,V1¯ for which(51)εi1t−εi2t≤LiV1−V2ℓp,i∈ℕ.DefineΨYi2 by(52)ΨYi2t=Δεi2t,which leads to(53)ΨYi1t−ΨYi2t=Δεi1t−Δεi2t≤ΞLiV1−V2ℓp,i∈ℕ,whereΞLi<1 by ℋ3. Using the Akin relation obtained by interchanging the rules of V1andV2, we conclude that, for all i∈ℕ, the operators ϒi are contraction.Proposition 2.
For alln∈ℕ, define the operators Tn:Kx→PKx,(54)TnV=ν1,…,νn−1,νn,ϒn+1V,…=νi,i≤nϒiV,i>n,whereKx is created as the one in Corollary 1. The following statements are all satisfied:(i)
TnV∈PbKx for all n∈ℕ(ii)
Tn are contraction with constants Z=Ξ∑i=1∞Li for all n∈ℕ if and only if there exist some ΨYit∈ϒiV such that νi=ΨYit(iii)
Tnn∈ℕ is a decreasing nested sequence which means Tn+1⊆Tn for all n∈ℕProof.
(i)
Using Proposition1 and the definition of Kx, we get, for all i∈ℕ, that ϒiV≤νi≤xi. This implies ϒiVi≥1∈Kx. Consequently, T1V∈PbKx, and thus, all TnV∈PbKx for all i∈ℕ.(ii)
Define the metric mapGd as(55)GdTnV1,TnV2=∑i=1∞HdϒiV1,ϒiV2.Consider that there exist someΨYit∈ϒiV such that νi=ΨYit,i=1,…,n. Using the contraction result in Proposition 1, we find(56)GdTnV1,TnV2≤Ξ∑i=1∞LiV1−V2ℓp=ZV1−V2ℓp.Applyingℋ3, we get the result of the contraction condition for Tn for all n∈ℕ.Now, consider that, for alln∈ℕ,Tn are contraction and νi∉ϒiV. Define the metric(57)G¯dTnV1,TnV2=∑r=1nν1r−ν2r+∑r=n+1∞HdϒrV1,ϒrV2.Then,(58)G¯dTnV1,TnV2≤nV1−V2ℓp+Ξ∑r=n+1∞LiV1−V2ℓp=n+ZV1−V2ℓp,wheren+Z>1. This contradicts with the assumption that Tn are contraction operators. Hence, we get νi∈ϒiV.(iii)
By (ii), we have for alln∈ℕ that(59)TnV=νii≤nϒiVi>n=ΨYiti≤nϒiVi>n⊇ΨYiti≤n+1ϒiVi>n+1=νii≤n+1ϒiVi>n+1=Tn+1V.Proposition 3.
For alln∈ℕ, define Tn such as in Proposition 2. Then,a1: for alln∈ℕ,Tn are quasi-nonexpansive mappingsa2: for alln∈ℕ,FixTn are closed subsets of Kxa3: for alln∈ℕ,Tn are all hemicompact mappingsa4:∩i=1∞FixTn≠∅Proof.
a1: following the theorem saying that the continuous image of the compact set is compact itself with applying Proposition2 (I, II) implies TnKx⊂PcpKx. By Theorem 6 and Definition 3, we get FixTn≠∅ for all n∈ℕ. Thus, Tn are all quasi-nonexpansive operators.a2: from (a1) and sinceKx is a closed and convex subset of Banach space, we see that FixTn are closed subsets of Kx according to Definition 3.a3: due to Definition4 and the proof of (a1), Tn are hemicompact for all n∈ℕ.a4: letp∈FixTn+1; then, p∈Tn+1p⊆Tnp. Hence, p∈Tnp which follows FixTn+1⊆FixTn. Using Theorem 3, we get ∩i=1∞FixTn≠∅.


### 3.2. Main Results

Theorem 6.
Consider thatYi:J×V⟶Pcpℝ satisfy ℋ1,ℋ2,ℋ3, andℋ4. Assume that, for all n∈ℕ,Tn are defined by (40), (41), and (54). Also, let(60)Ξ=21−αMα+1+γσ−γραMαΓα.Then, infinite systems (1) and (2) are able to have a common antiperiodic solution if and only if there exist some ΨYit∈ϒiV such that νi=ΨYit.Proof.
Define the sequenceVnn≥1 by (36)–(38). Under the vision of Theorem 5, Propositions 1–3 explain the existence of the common solution to infinite systems (1) and (2).Theorem 7.
Consider thatYi:J×V⟶Pcpℝ satisfy ℋ1, ℋ2, and ℋ4. Assume that, for all n∈ℕ,T¯n are defined by the same way as (54) with respect to (42) and (43). Moreover, let(61)Ξ¯=21−αMα+ραMαΓαandℋ5∑i≥1Li<∞andZ¯=Ξ¯∑i=1∞Li<1.
Then, infinite systems (1) and (2) are able to have at least a common periodic solution if and only if there exist some ΘYit∈ΠiV such that νi=ΘYit.Proof.
Similar to the proof of Theorem6 but with respect to Ξ¯.Corollary 2.
Infinite systems (1) and (2) are able to have one or more common almost periodic solutions.


## 3.1. Auxiliary Results

LetVt:C0,ρ⟶ℝℕ be defined by Vt=νiti∈ℕ. Then, for every multivalued mapping yi, we have yit,ν1t,…,νit,…=yit,Vt. Define the set-valued maps SYi,V such as(39)SYi,V=εit|εit∈yit,Vt∩L1J,ℝ,J=0,ρ.For the antiperiodic solutions, we define the multioperatorsϒi:ℓp⟶ℝ for all i∈ℕ as follows:(40)ϒiVt=ΨYit|ΨYit=Δεit,εit∈SYi,V¯,where(41)Δεit=1−αMαεit+αMαΓα∫0tt−sα−1εisds+γσ−γ1−αMαεiρ−σγεi0+αMαΓα∫0ρρ−sα−1εisds.For the almost periodic/periodic solutions, the multioperatorsΠi:ℓp⟶ℝ for all i∈ℕ are defined by(42)ΠiVt=ΘYit|ΘYit=Δ¯εit,εit∈SYi,V¯,where(43)Δ¯εit=1−αMαεit−εi0+αMαΓα∫0tt−sα−1εisds.Consider that the following conditions hold:H1yi is a sequence of L1−Caratheodory multivalued mappings that are(44)yi:J×V⟶Pcpℝ,with(1)
The mapst⟶yit,Vt are measurable for all V∈V.(2)
The mapsV⟶yit,Vt are measurable for all t∈J.(3)
For alli∈ℕ, there exist ψi∈CJ,ℝ+ such thatH2yi are Lipschitz mappings with constants Li∈ℝ+, respectively. This means(46)HdYit,V1t,Yit,V2t≤LiV1−V2,∀i∈ℕ.ℋ3∑i≥1Li<∞andZ=Ξ∑i=1∞Li<1 whenever(47)Ξ=21−αMα+1+γσ−γραMαΓα.ℋ4νi/Ξψi∗νi>1,ψi∗=supt∈Jψit.Then, we have the following propositions.Proposition 1.
The set-valued mapsϒiV are bounded and contraction for all i∈ℕ.Proof.
In view ofℋ1 and ℋ4, we have(48)ϒiV≤1−αMα+tαMαΓα+γσ−γ1−σγ1−αMα+ραMαΓαYit,Vt≤Ξψi∗νi≤νi.Thus, we prove the boundedness.To prove the contraction condition for alli∈ℕ, consider that ΨYi1t∈ϒiV2. This implies the existence of some εi1t∈SYi,V2¯ subject to(49)ΨYi1t=Δεi1t,whereΔ is defined by (41). By using ℋ2, we can define the sets(50)ϒ¯it=ε∈ℝ|εi1t−εt≤LiV1−V2ℓp.According to Theorem III.41 in [46], ℋ1, and the measurability of the functions εi1t and Wt=LiV1−V2ℓp, the sets ϒ¯it are also measurable for all i∈ℕ. Therefore, the maps t⟶ϒ¯it∩Yit,Vt are measurable with nonempty closed values. Hence, the measurability of ϒit and Proposition 2.1.43 in [47] drive to the existence of some εi2∈SYi,V1¯ for which(51)εi1t−εi2t≤LiV1−V2ℓp,i∈ℕ.DefineΨYi2 by(52)ΨYi2t=Δεi2t,which leads to(53)ΨYi1t−ΨYi2t=Δεi1t−Δεi2t≤ΞLiV1−V2ℓp,i∈ℕ,whereΞLi<1 by ℋ3. Using the Akin relation obtained by interchanging the rules of V1andV2, we conclude that, for all i∈ℕ, the operators ϒi are contraction.Proposition 2.
For alln∈ℕ, define the operators Tn:Kx→PKx,(54)TnV=ν1,…,νn−1,νn,ϒn+1V,…=νi,i≤nϒiV,i>n,whereKx is created as the one in Corollary 1. The following statements are all satisfied:(i)
TnV∈PbKx for all n∈ℕ(ii)
Tn are contraction with constants Z=Ξ∑i=1∞Li for all n∈ℕ if and only if there exist some ΨYit∈ϒiV such that νi=ΨYit(iii)
Tnn∈ℕ is a decreasing nested sequence which means Tn+1⊆Tn for all n∈ℕProof.
(i)
Using Proposition1 and the definition of Kx, we get, for all i∈ℕ, that ϒiV≤νi≤xi. This implies ϒiVi≥1∈Kx. Consequently, T1V∈PbKx, and thus, all TnV∈PbKx for all i∈ℕ.(ii)
Define the metric mapGd as(55)GdTnV1,TnV2=∑i=1∞HdϒiV1,ϒiV2.Consider that there exist someΨYit∈ϒiV such that νi=ΨYit,i=1,…,n. Using the contraction result in Proposition 1, we find(56)GdTnV1,TnV2≤Ξ∑i=1∞LiV1−V2ℓp=ZV1−V2ℓp.Applyingℋ3, we get the result of the contraction condition for Tn for all n∈ℕ.Now, consider that, for alln∈ℕ,Tn are contraction and νi∉ϒiV. Define the metric(57)G¯dTnV1,TnV2=∑r=1nν1r−ν2r+∑r=n+1∞HdϒrV1,ϒrV2.Then,(58)G¯dTnV1,TnV2≤nV1−V2ℓp+Ξ∑r=n+1∞LiV1−V2ℓp=n+ZV1−V2ℓp,wheren+Z>1. This contradicts with the assumption that Tn are contraction operators. Hence, we get νi∈ϒiV.(iii)
By (ii), we have for alln∈ℕ that(59)TnV=νii≤nϒiVi>n=ΨYiti≤nϒiVi>n⊇ΨYiti≤n+1ϒiVi>n+1=νii≤n+1ϒiVi>n+1=Tn+1V.Proposition 3.
For alln∈ℕ, define Tn such as in Proposition 2. Then,a1: for alln∈ℕ,Tn are quasi-nonexpansive mappingsa2: for alln∈ℕ,FixTn are closed subsets of Kxa3: for alln∈ℕ,Tn are all hemicompact mappingsa4:∩i=1∞FixTn≠∅Proof.
a1: following the theorem saying that the continuous image of the compact set is compact itself with applying Proposition2 (I, II) implies TnKx⊂PcpKx. By Theorem 6 and Definition 3, we get FixTn≠∅ for all n∈ℕ. Thus, Tn are all quasi-nonexpansive operators.a2: from (a1) and sinceKx is a closed and convex subset of Banach space, we see that FixTn are closed subsets of Kx according to Definition 3.a3: due to Definition4 and the proof of (a1), Tn are hemicompact for all n∈ℕ.a4: letp∈FixTn+1; then, p∈Tn+1p⊆Tnp. Hence, p∈Tnp which follows FixTn+1⊆FixTn. Using Theorem 3, we get ∩i=1∞FixTn≠∅.


## 3.2. Main Results

Theorem 6.
Consider thatYi:J×V⟶Pcpℝ satisfy ℋ1,ℋ2,ℋ3, andℋ4. Assume that, for all n∈ℕ,Tn are defined by (40), (41), and (54). Also, let(60)Ξ=21−αMα+1+γσ−γραMαΓα.Then, infinite systems (1) and (2) are able to have a common antiperiodic solution if and only if there exist some ΨYit∈ϒiV such that νi=ΨYit.Proof.
Define the sequenceVnn≥1 by (36)–(38). Under the vision of Theorem 5, Propositions 1–3 explain the existence of the common solution to infinite systems (1) and (2).Theorem 7.
Consider thatYi:J×V⟶Pcpℝ satisfy ℋ1, ℋ2, and ℋ4. Assume that, for all n∈ℕ,T¯n are defined by the same way as (54) with respect to (42) and (43). Moreover, let(61)Ξ¯=21−αMα+ραMαΓαandℋ5∑i≥1Li<∞andZ¯=Ξ¯∑i=1∞Li<1.
Then, infinite systems (1) and (2) are able to have at least a common periodic solution if and only if there exist some ΘYit∈ΠiV such that νi=ΘYit.Proof.
Similar to the proof of Theorem6 but with respect to Ξ¯.Corollary 2.
Infinite systems (1) and (2) are able to have one or more common almost periodic solutions.


## 4. Examples

Example 1.
Consider the problem(62)D2/3ABCνn∈12nrνn1+∑s=1rνsr=1n,(63)3V0=Vρ.
Then, we haveYnt,Vt=1/2nrνn/1+∑s=1rνsr=1n and(64)Yn≤12nνn.
If we takeψnt=1/2n=ψn∗, we find that ℋ1 holds. Furthermore, we have(65)HdYnt,V1t,Ynt,V2t≤12nν1n−ν2n=12nν1n−ν2nP1/p≤12nV1−V2ℓp,which explains that ℋ2 holds with Ln=1/2n. It is known that ∑n=1∞Ln=∑n=1∞1/2n=1<∞. In this case, we need to make sure that Ξ<1 for the sake of obtaining ℋ3. First, by using the rules(66)Mα=22−α,ΓαΓ1−α=πsinαπ,we find that(67)M23=32,Γ13≈2.6789385347077476337,1M2/3Γ2/3≈0.0086,1+γσ−γραMαΓα≈0.0129ρ2/3,21−αMα=49.
Takingρ2/3≤2/90.0129 implies that ρ∈0,10.5881117. Therefore, Ξ≤8/9<1 which means that ℋ3 holds. Finally, since ψn∗Ξ≤8/92n<1,n≥1, ℋ4 holds. Applying Theorem 6, there exists at least one solution of antiperiodic type for problems (62) and (63).Example 2.
Assume the following problem:(68)D1ABCνn∈12+rnsinνn1+∑s=1rνsr=1n,(69)V0=Vρ.
In this problem, we haveα=Mα=1,(70)Ynt,Vt=12+rnsinνn1+∑s=1rνsr=1n,Yn≤13nνn.
It follows thatψnt=1/3n=ψn∗, and thus, ℋ1 holds. It is clear to see that ℋ2 holds if we take Ln=1/3n and then ∑n=1∞Ln=1/2<1. Since α=1, Ξ¯=ρ, and Z¯=ρ/2 tends to take ρ∈0,2 in order to obtain ℋ5. Finally, we have 1/Ξψn∗=3n/ρ>3/2>1 which drives to ℋ6. According to Theorem 7, problems (42), (43), (54), (62), (63), (68), and (69) have at least one periodic solution.Remark 1.
If we take problems (42), (43), (54), (62), (63), (68), and (69) with the condition V0≈Vρ, then by the same results in Example 2, we get the existence result of the almost periodic solution.


## 5. Conclusion

By this work, we connected between three sides of generalization: first, using the generalization of fractional differential operators with the Mittag–Leffler kernel; second, generating the infinite countable system of equations by multivalued mappings; third, using the general form of periodic, almost periodic, and antiperiodic boundary conditions. We explain how to obtain the exact solutions in the three cases according to the boundary conditions. Consequently, we show the sufficient conditions for the existence results to different solutions and give some related examples to the main theorems. These results have strong impacts to give an even better description of the dynamics of real-world problems (in particular, the dynamics complex systems). These results also have practical extensions to understand the complex phenomena related to multifaceted media, chaotic behaviors, fluctuations, nanofluids, and heterogeneous phenomenon. Next time, we will work to study the generalization of the conjugate value problem with one of the most important derivatives.



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