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

(is 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 by ABCD []i(t)] ∈Yi(t, ]i(t) 􏼈 􏼉 ∞ i�1). (e mappings yi(t, ]i(t) 􏼈 􏼉 ∞ i�1) are proposed to be Lipschitz multivalued mappings. (e results are explored according to boundary condition σ]i(0) � c]i(ρ), σ, c ∈ R. (is type of condition is the generalization of periodic, almost, and antiperiodic types.


Introduction
Consider the following infinite system: where ABC D α denotes the Atangana-Baleanu fractional derivative in the Caputo sense of order α ∈ (0, 1] and y i i∈N is an infinite countable family of Lipschitz continuous multivalued mappings. is means there is an infinite countable sequence of continuous real-valued functions ] i (t) i∈N satisfying problems (1) and (2). In this case, we can define the function V(t): [0, ρ] ⟶ R N by V(t) � (] i (t)) i∈N .
is 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][2][3][4] and the mentioned references therein). en, many scholars were attracted to develop these problems into the ones associated with fractional derivatives. For instance, see the required results in [5][6][7][8] and references cited therein. e 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]. e concept of fractional derivative arose before 300 years when L'Hospital asked in 1695 which is addressed in Leibnitz notation for the nth 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.
e 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.
is 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][16][17][18] and the references therein.
In [19], we studied how to generate the differential equations and inclusions by one form (we call them as equiinclusion problems). en, 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. is means we hatch a generalization of previous literature studies in the infinite system field. is 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 ABC D α , 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 ℓ p (R N ), 1 < p < ∞. eorems 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. e results are affected by hemicompactness, compactness, contraction properties, and the one-step iterative scheme.

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

Real Sequence Banach Space.
Define the space V to be the real sequence Banach space ℓ p (R N ), 1 < p < ∞, endowed with the norm en, from [29][30][31], we have the following facts.

Basics in Multivalued
Maps. e concept of multioperators is related to the multivalued maps. So, it is required to show some facts about them and their properties. ese facts are confirmed in [32][33][34][35][36][37].
Let (E, ‖.‖) and (H, ‖.‖) be two Banach spaces. A multivalued map A: E ⟶ P cl (E) is seen as convex (closed) if for every e ∈ E, then A(e) is convex (closed) and selected to be completely continuous if A(B) is relatively compact for every B ∈ P b (E).
e map A is said to be upper semicontinuous if for each is an open subset of E. In other words, A seems to be lower semicontinuous as long as the set e ∈ E: Given C, B ∈ P cl (E); then, absorbs the Pompeiu-Hausdorff distance of C, B.
If we adopt A 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 ] n ⟶ ] * and y n ⟶ y * , then y n ∈ A(] n ) implies to y * ∈ A(] * )).

Definition 2. A multivalued map
In addition to assumptions (1) and (2), the map A is Definition 3 (nonexpansive and quasi-nonexpansive multimapping). Let K be a subset of a metric space E and . en, we have the following: (i) T is called a nonexpansive mapping if it is contraction according to the metric. is means that, for all v 1 , v 2 ∈ K, we have (ii) T is said to be quasi-nonexpansive if Fix(T) ≠ ∅ and for all w ∈ Fix(T) and all v ∈ K, we have Definition 4 (hemicompactness). Let K, E, and T be defined as in Definition 3. en, T is called hemicompact if (iii) Every collection W n |n ∈ N of nonempty closed subsets of W satisfying the finite intersection property has a nonempty intersection

Fractional Calculus.
In this part, we give definitions related to the used derivative, its history, the corresponding integral, and some properties [38][39][40].
Definition 6 (Caputo-Fabrizio derivative). CF derivative for the order α ∈ [0, 1] and η(ς) ∈ H 1 (a, b) is given by where M(α) is a normalization function such that Definition 7 (Mittag-Leffler function). e general form of Mittag-Leffler function E α of order α is written by Complexity e main derivative used in the present paper is the Atangana-Baleanu fractional derivative in the Caputo sense. It is proposed by interchanging the kernel exp(− (α(ς − s)/1 − α)) in the Caputo-Fabrizio derivative by the equivalent form via the Mittag-Leffler formula that is ∞ r�0 ((− a(t − s)) r /r!), a � (α/1 − α). After that, replace r! by Γ(αr + 1) and (− a(t − s)) r by (− a(t − s)) αr . So, in the space we have the following definitions.  H 1 (a, b) is given by is 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][42][43] and references therein). Depending on the constant M(α), the corresponding integral is given by the following definition. ∈ H 1 (a, b) is given by

Lemma 3 (antiperiodic solution). e unique solution of the problem
is given by Proof. By applying AB I α to both sides of (23), we get which implies Now, applying (24), we find that

Fixed-Point eorems.
is section is surveyed by some fixed-point theorems investigated in the uniformly convex real Banach space [44,45]. en, T has a fixed point x ∈ K with x ∈ Tx. e next theorem is formulated for the infinite countable family of multioperators under the vision of the one-step iterative scheme defined as follows.
Let K be a closed, bounded, convex subset of a uniformly convex real Banach space V. Let T i : K ⟶ P b (K) be an infinite countable family of quasi-nonexpansive multivalued mappings with ∩ ∞ i�1 Fix(T i ) ≠ ∅ and p ∈ ∩ ∞ i�1 Fix(T i ). en, for all n ∈ N, the sequence V n is defined by Theorem 5. Let K be a closed, bounded, convex subset of a uniformly convex real Banach space V. Let T i : K ⟶ P b (K) be a sequence of quasi-nonexpansive and continuous multivalued mappings with ∩ ∞ i�1 Fix(T i ) ≠ ∅ and p ∈ ∩ ∞ i�1 Fix(T i ). Let V i be a sequence defined by (36) with the condition that lim n⟶∞ σ n,r and lim n⟶∞ σ n,n exist and lie in [0, 1) for all r ∈ 0 { } ∪ N. Assume that one of T i is hemicompact. en, the sequence V i converges strongly. t)) i∈N . en, for every multivalued mapping V(t)). Define the set-valued maps S Y i ,V such as

Auxiliary Results
For the antiperiodic solutions, we define the multioperators Υ i : ℓ p ⟶ R for all i ∈ N as follows: where For the almost periodic/periodic solutions, the multioperators Π i : ℓ p ⟶ R for all i ∈ N are defined by where Consider that the following conditions hold: with (1) e maps t ⟶ y i (t, V(t)) are measurable for all V ∈ V.
. en, we have the following propositions.

Proposition 1.
e set-valued maps Υ i (V) are bounded and contraction for all i ∈ N.
Proof. In view of (H 1 ) and (H 4 ), we have us, we prove the boundedness. To prove the contraction condition for all i ∈ N, con- where Δ is defined by (41). By using (H 2 ), we can define the sets According to eorem III.41 in [46], (H 1 ), and the measurability of the functions ε 1 i (t) and erefore, the maps t ⟶ Υ i (t) ∩ Y i (t, V(t)) are measurable with nonempty closed values. Hence, the measurability of Υ i (t) and Proposition 2.1.43 in [47] drive to the existence of some which leads to where ΞL i < 1 by (H 3 ). Using the Akin relation obtained by interchanging the rules of V 1 and V 2 , we conclude that, for all i ∈ N, the operators Υ i are contraction.
□ Proposition 2. For all n ∈ N, define the operators T n : K x → P(K x ), where K x is created as the one in Corollary 1. e following statements are all satisfied: , and thus, all T n (V) ∈ P b (K x ) for all i ∈ N.
(ii) Define the metric map G d as (55) . . , n. Using the contraction result in Proposition 1, we find Applying (H 3 ), we get the result of the contraction condition for T n for all n ∈ N. Now, consider that, for all n ∈ N, T n are contraction and ] i ∉ Υ i (V). Define the metric en, 6 Complexity where (n + Z) > 1. is contradicts with the assumption that T n are contraction operators. Hence, we get ] i ∈ Υ i (V). (iii) By (ii), we have for all n ∈ N that □ Proposition 3. For all n ∈ N, define T n such as in Proposition 2. en, a1: for all n ∈ N, T n are quasi-nonexpansive mappings a2: for all n ∈ N, Fix(T n ) are closed subsets of K x a3: for all n ∈ N, T n are all hemicompact mappings a4: ∩ ∞ i�1 Fix(T n ) ≠ ∅ Proof a1: following the theorem saying that the continuous image of the compact set is compact itself with applying Proposition 2 (I, II) implies T n (K x ) ⊂ P cp (K x ). By eorem 6 and Definition 3, we get Fix(T n ) ≠ ∅ for all n ∈ N. us, T n are all quasi-nonexpansive operators. a2: from (a1) and since K x is a closed and convex subset of Banach space, we see that Fix(T n ) are closed subsets of K x according to Definition 3. a3: due to Definition 4 and the proof of (a1), T n are hemicompact for all n ∈ N. a4: let p ∈ Fix(T n+1 ); then, p ∈ T n+1 (p)⊆T n (p). Hence, p ∈ T n (p) which follows Fix(T n+1 )⊆Fix(T n ). Using eorem 3, we get ∩ ∞ i�1 Fix(T n ) ≠ ∅. . Assume that, for all n ∈ N, T n are defined by (40), (41), and (54). Also, let Then, infinite systems (1) and (2) are able to have a common antiperiodic solution if and only if there exist some Ψ Y i (t) ∈ Υ i (V) such that ] i � Ψ Y i (t).
Proof. Define the sequence V n n ≥ 1 by (36)- (38). Under the vision of eorem 5, Propositions 1-3 explain the existence of the common solution to infinite systems (1) and (2). □ Theorem 7. Consider that Y i : J × V ⟶ P cp (R) satisfy (H 1 ), (H 2 ), and (H 4 ). Assume that, for all n ∈ N, T n are defined by the same way as (54) with respect to (42) and (43). Moreover, let and (H 5 ) i≥1 L i < ∞ and Z � Ξ ∞ i�1 L i < 1. en, infinite systems (1) and (2) are able to have at least a common periodic solution if and only if there exist some Proof. Similar to the proof of eorem 6 but with respect to Ξ.