Fractional-Order Chaotic Systems: Theory and ApplicationsView this Special Issue
A Countable System of Fractional Inclusions with Periodic, Almost, and Antiperiodic Boundary Conditions
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 by . The mappings are proposed to be Lipschitz multivalued mappings. The results are explored according to boundary condition . This type of condition is the generalization of periodic, almost, and antiperiodic types.
Consider the following infinite system:where denotes the Atangana–Baleanu fractional derivative in the Caputo sense of order and is an infinite countable family of Lipschitz continuous multivalued mappings. This means there is an infinite countable sequence of continuous real-valued functions satisfying problems (1) and (2). In this case, we can define the function by . 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  and the control problems for the models descried by parabolic and hyperbolic equations .
The concept of fractional derivative arose before 300 years when L’Hospital asked in 1695 which is addressed in Leibnitz notation for the th derivative “What would happen if the order ?”. 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 , 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 , 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 , 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 . 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.
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 space to be the real sequence Banach space , endowed with the norm
Definition 1. (uniformly convex space). A normed space is called a uniformly convex space if for any , there exists such that if with then .
Lemma 1. The space is uniformly convex for all .
Theorem 1. Adopt such that . Then, the inequalities thereafter hold:(i)Holder inequality: let . Then, and(ii)Minkowski inequality: let . Then, and(iii)Imbedding theorem: let have a finite positive measure and . Then, and
Theorem 2. (compactness in ). Let be a Banach space and . Then, is compact in if and only if(i) is closed and bounded(ii)(iii), then is compact for all
Corollary 1. The subsetsare compact subsets of spaces.
2.2. Basics in Multivalued Maps
Let and be two Banach spaces. A multivalued map is seen as convex (closed) if for every , then is convex (closed) and selected to be completely continuous if is relatively compact for every .
The map is said to be upper semicontinuous if for each closed subset is a closed subset of . This means the set is open for all open sets . It is lower semicontinuous if for each open subset is an open subset of . In other words, seems to be lower semicontinuous as long as the set is open for all open sets .
A map is presented to be measurable multivalued if for every , the function is measurable function.
Given ; then,absorbs the Pompeiu–Hausdorff distance of .
If we adopt 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 , then implies to ).
Definition 2. A multivalued map is known as Caratheodory if(1) is measurable(2)For a.e. is upper semicontinuousIn addition to assumptions (1) and (2), the map is -Caratheodory if for each satisfying and and nondecreasing map for whichfor all .
Definition 3. (nonexpansive and quasi-nonexpansive multimapping). Let be a subset of a metric space and be a multivalued map with (the set of all fixed points of in ). Then, we have the following:(i) is called a nonexpansive mapping if it is contraction according to the metric. This means that, for all , we have(ii) is said to be quasi-nonexpansive if and for all and all , we have(iii) is a closed subset if is closed and convex of .
Definition 4. (hemicompactness). Let , and be defined as in Definition 3. Then, is called hemicompact if(i)For any sequence such that as , there is a subsequence of with (ii) is compact
Theorem 3. (nonempty infinite intersection). For any space , the following statements are equivalent:(i) is compact.(ii)Every decreasing sequence of nonempty closed subsets of has a nonempty intersection(iii)Every collection of nonempty closed subsets of satisfying the finite intersection property has a nonempty intersection
2.3. Fractional Calculus
Definition 5. (Riemann–Liouville integral). For the order , the Riemann–Liouville fractional integral of a function is defined bysince the R.H.S is pointwise defined on .
Definition 6. (Caputo–Fabrizio derivative). CF derivative for the order and is given bywhere is a normalization function such that .
Definition 7. (Mittag–Leffler function). The general form of Mittag–Leffler function of order is written byThe main derivative used in the present paper is the Atangana–Baleanu fractional derivative in the Caputo sense. It is proposed by interchanging the kernel in the Caputo–Fabrizio derivative by the equivalent form via the Mittag–Leffler formula that is . After that, replace and . So, in the spacewe have the following definitions.
Definition 8. (Atangana–Baleanu fractional derivative in the Caputo sense). ABC derivative for the order and is given byThis 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 , the corresponding integral is given by the following definition.
Definition 9. (Atangana–Baleanu fractional integral). AB integral of the order and is given bywhere is the Riemann–Liouville integral of order .
Lemma 2. For , and , the following statements hold: a1: AB integral together with the ABC derivative satisfies the Newton–Leibniz formula a2: they also satisfy the property a3: AB integral is a commutative operator such that, for any two orders , we have
Lemma 3. (antiperiodic solution). The unique solution of the problemis given by
Lemma 4. (periodic/almost periodic solution). The unique solution of the problemis given byand then .
Proof. Similar to the proof of Lemma 3, we haveThe almost periodic/periodic condition leads toUnder this equation, we have three cases: p1: if , then must hold p2: when , both must hold p3: in case , we get So, we can say, in general, that there is an almost periodic/periodic solution if both (31) and (32) hold for every . Now, substituting (31) and (32) to the value of in Lemma 3 with and using the continuity condition, we get . Hence, the periodic/almost periodic solution is given byIt is clear to see that which completes the proof.
2.4. Fixed-Point Theorems
Theorem 4. Let be a uniformly convex real Banach space and be a closed, bounded, convex subset of . Let be nonexpansive multivalued mappings. Then, has a fixed point with
The next theorem is formulated for the infinite countable family of multioperators under the vision of the one-step iterative scheme defined as follows.
Let be a closed, bounded, convex subset of a uniformly convex real Banach space . Let be an infinite countable family of quasi-nonexpansive multivalued mappings with and . Then, for all , the sequence is defined by
Theorem 5. Let be a closed, bounded, convex subset of a uniformly convex real Banach space . Let be a sequence of quasi-nonexpansive and continuous multivalued mappings with and . Let be a sequence defined by (36) with the condition that exist and lie in for all . Assume that one of is hemicompact. Then, the sequence converges strongly.
3. Inferred Results
3.1. Auxiliary Results
Let be defined by . Then, for every multivalued mapping , we have . Define the set-valued maps such as
For the antiperiodic solutions, we define the multioperators for all as follows:where
For the almost periodic/periodic solutions, the multioperators for all are defined bywhere
Consider that the following conditions hold: is a sequence of Caratheodory multivalued mappings that are with(1)The maps are measurable for all .(2)The maps are measurable for all .(3)For all , there exist such that are Lipschitz mappings with constants , respectively. This means whenever . Then, we have the following propositions.
Proposition 1. The set-valued maps are bounded and contraction for all .
Proof. In view of and , we have Thus, we prove the boundedness. To prove the contraction condition for all , consider that . This implies the existence of some subject to where is defined by (41). By using , we can define the sets According to Theorem III.41 in , , and the measurability of the functions and , the sets are also measurable for all . Therefore, the maps are measurable with nonempty closed values. Hence, the measurability of and Proposition 2.1.43 in  drive to the existence of some for which Define by which leads to where by . Using the Akin relation obtained by interchanging the rules of , we conclude that, for all , the operators are contraction.
Proposition 2. For all , define the operators , where is created as the one in Corollary 1. The following statements are all satisfied:(i) for all (ii) are contraction with constants for all if and only if there exist some such that (iii) is a decreasing nested sequence which means for all
Proof. (i)Using Proposition 1 and the definition of , we get, for all , that . This implies . Consequently, , and thus, all for all .(ii)Define the metric map as Consider that there exist some such that . Using the contraction result in Proposition 1, we find Applying , we get the result of the contraction condition for for all . Now, consider that, for all are contraction and . Define the metric Then, where . This contradicts with the assumption that are contraction operators. Hence, we get .(iii)By (ii), we have for all that
Proposition 3. For all , define such as in Proposition 2. Then, a1: for all are quasi-nonexpansive mappings a2: for all are closed subsets of a3: for all are all hemicompact mappings a4:
Proof. a1: following the theorem saying that the continuous image of the compact set is compact itself with applying Proposition 2 (I, II) implies . By Theorem 6 and Definition 3, we get for all . Thus, are all quasi-nonexpansive operators. a2: from (a1) and since is a closed and convex subset of Banach space, we see that are closed subsets of according to Definition 3. a3: due to Definition 4 and the proof of (a1), are hemicompact for all . a4: let ; then, . Hence, which follows . Using Theorem 3, we get .
3.2. Main Results
Theorem 6. Consider that satisfy . Assume that, for all 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 such that .
Theorem 7. Consider that satisfy , , and . Assume that, for all are defined by the same way as (54) with respect to (42) and (43). Moreover, let and .Then, infinite systems (1) and (2) are able to have at least a common periodic solution if and only if there exist some such that .
Proof. Similar to the proof of Theorem 6 but with respect to .
Example 1. Consider the problemThen, we have andIf we take , we find that holds. Furthermore, we havewhich explains that holds with . It is known that . In this case, we need to make sure that for the sake of obtaining . First, by using the ruleswe find thatTaking implies that . Therefore, which means that holds. Finally, since , holds. Applying Theorem 6, there exists at least one solution of antiperiodic type for problems (62) and (63).
Example 2. Assume the following problem:In this problem, we have It follows that , and thus, holds. It is clear to see that holds if we take and then . Since , , and tends to take in order to obtain . Finally, we have which drives to . According to Theorem 7, problems (42), (43), (54), (62), (63), (68), and (69) have at least one periodic solution.
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.
The data used to support the findings of this study are included within the article, and other data used can be obtained from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Both authors contributed equally to this work and read and approved the final manuscript.
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah (Grant no. D-146-130-1439). The authors, therefore, gratefully acknowledge the DSR for the technical and financial support.
K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.