BVP for Hadamard Sequential Fractional Hybrid Differential Inclusions

The study is concerned with the Hadamard sequential fractional hybrid diﬀerential inclusions with two-point hybrid integral boundary conditions. With the help of the Dhage ﬁxed-point theorem for the product of two operators and the Covitz-Nadler ﬁxed-point theorem in the case of fractional inclusions, we obtain the existence results of solutions for Hadamard sequential fractional hybrid diﬀerential inclusions. Finally, two examples are presented to illustrate the main results.


Introduction
Nowadays, with the increasing demand of researchers for the study of natural phenomena, the use of fractional differential operators and fractional differential equations become an effective means to achieve this goal. Compared with integer order operators, fractional operators, which can simulate natural phenomena better, are a class of operators developed in recent years. is kind of operator has been expanded and widely used in modeling real-world phenomena such as biomathematics, electrical circuits, medicine, disease transmission, and control [1][2][3][4][5][6]. Also, some studies in the biological models with fractional-order derivative have been conducted in recent years [7][8][9]. In the past year, fractional differential operators and fractional differential equations have been used in modeling the spread of some viruses, such as Zika virus and mumps virus [10,11]. All of these have enabled researchers to discover the structure of fractional boundary value problems (BVP) and the hereditary nature of their solutions from various aspects. In this regard, many researchers investigated advanced fractional-order modelings and related theoretical results and qualitative behaviors of such fractional-order boundary value problems, see [12][13][14][15][16][17][18][19][20] and the references therein.
ere have been appeared different versions of fractional operators during these years. Much of the work on fractional differential equations only involves either Riemann-Liouville derivative or Caputo derivative [21][22][23][24][25][26][27][28][29]. Guo et al. ( [30,31]) discussed the existence and Hyers-Ulam stability of solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1 < β < 2 and the existence and Hyers-Ulam stability of the almost periodic solution to the fractional differential equation with impulse and fractional Brownian motion under nonlocal condition. Ma et al. [32] investigated the existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space.
However, there is another concept of fractional derivative in the literature which was introduced by Hadamard in 1892 [33]. is derivative is known as Hadamard fractional derivative and differs from aforementioned derivatives in the sense that the kernel of the integral in its definition contains logarithmic function of arbitrary exponent. Many researchers have studied and obtained some results on the existence of solutions of Hadamard fractional differential equations in recent years. Yang ([34, 35]) studied the extremal solutions for a coupled system of nonlinear Hadamard fractional differential equations with Cauchy initial value conditions and the existence and nonexistence of positive solutions for the eigenvalue problems of nonlinear Hadamard fractional differential equations with p-Laplacian operator. Tomar et al. [36] established certain generalized Hermite-Hadamard inequalities for generalized convex functions via local fractional integral.
In 1993, Miller and Ross also defined another type of fractional derivative called sequential derivative, which is a combination of the existing derivative operators. From then on, the attention of some researchers was attracted to finding a connection between the Hadamard fractional derivative and the sequential fractional derivative [37][38][39][40]. In [41], by using the topological degree theory and Leray-Schauder fixed-point theory, Rezapour and Etemad studied the existence of solutions for the following Caputo-Hadamard fractional boundary value problem via mixed multiorder integroderivative conditions: points out the Caputo-Hadamard fractional derivative of order ξ ∈ α, ς, { c 1 , c 2 }, with the notation H I q 1+ standing for the Hadamard fractional integrals of order q ∈ q 1 , q 2 . e function f formulated by f: [1, M] × R ⟶ R is assumed to be continuous on [1, M] × R with respect to its both components.
As a generalization of fractional boundary value problems, hybrid differential problems with different kinds of boundary conditions have received a lot of attention in recent years [42][43][44]. e research in this field started from Dhage and Lakshmikantham in 2010 [45]. ere is a new concept of differential equation in the literature which was introduced by Dhage and Lakshmikantham. ey described this novel differential equation as a hybrid differential equation and investigated the extremal solutions of this new BVP by using some useful fundamental differential inequalities [45]. So far, there are few studies about the existence and various properties of solutions for hybrid boundary value problems of fractional order. In [46], by using a fixed-point theorem due to Dhage, the authors developed some existence theorem for Hadamard-type fractional hybrid differential inclusions problem: is a multivalued map, and P(R) is the family of all nonempty subsets of R. In [47], by using a hybrid fixed-point theorem of Schaefer type for a sum of three operators due to Dhage, the authors investigated the existence of solutions for the nonlocal fractional BVP of hybrid inclusion problem given by where μ: C(J, R) ⟶ R, α ∈ R, 9 ∈ (1, 2], C D 9 0+ is the Caputo derivative, and R I β j 0 + is the Riemann-Liouville integral of order φ > 0, such that φ ∈ β 1 , β 2 , . . . , β m . In [48], by using the well-known Dhage fixed-point theorems for single-valued and set-valued maps, Baleanu and Etemad studied a new fractional hybrid model of thermostat in 2 Journal of Function Spaces which the thermostat controls an amount of heat based on the temperature detected by sensors. is hybrid differential inclusions of Caputo type are illustrated by is the Caputo derivative of fractional order α ∈ 9, 9 − 1 , the function Φ: In [49], the authors investigated the following fractional three-point hybrid problem: where 9 ∈ (2, 3] 9 * > 0, η ∈ (0, 1). e function G: In [50], by using various novel analytical techniques based on α − ψ−contractive mappings, endpoints, and the fixed points of the product operators, the authors investigated a new category of the sequential hybrid inclusion problem with three-point integroderivative boundary conditions: Journal of Function Spaces 3 where 9 ∈ (2, 3], p ∈ (0, 1), p 1 , p 2 , c, ξ > 0, C D Motivated by these problems, in this study, we will study the following Hadamard sequential fractional hybrid differential inclusion with two-point hybrid Hadamard integroboundary conditions: e Hadamard sequential fractional hybrid differential inclusion BVP (7) is modeled with respect to the generalized operators with kernels, including logarithmic functions. In other words, the presented formulation for the given Hadamard sequential fractional hybrid differential inclusion BVP (7) involves two different derivatives in the format of the Hadamard. e supposed abstract fractional hybrid differential inclusion problem (7) with given hybrid integral boundary conditions can describe some mathematical models of real and physical processes in which some parameters are often adjusted to suitable situations. e value of these parameters can change the effects of fractional derivatives and integrals. Moreover, we express that such a Hadamard sequential fractional hybrid differential inclusion BVP is new and enriches the literature on boundary value problems for nonlinear Hadamard fractional differential inclusions. In this way, with the help of Dhage fixed-point theorem and Covitz-Nadler fixed-point theorem in the case of multivalued mapping, we try to find the existence criteria of solutions for the proposed problem (7). e rest of this study is organized as follows. In Section 2, some preliminary facts that we need in the sequel are given. In Section 3, the existence results of solution for system (7) are discussed. In Section 4, two examples are given to prove validity of the results we obtained.
Definition 2 (see [2]). e Hadamard fractional integral of order β for a function g is defined as provided the integral exists.

Lemma 1. For any h ∈ C([1, e], R). A function x ∈ AC([1, e], R) is a solution of the Hadamard sequential fractional hybrid differential equations:
supplemented with the boundary conditions in (7) if and only if it satisfies the following integral equation: where Proof. As argued in [2], the solution of Hadamard differential equation in (11) can be written as where c i , (i � 0, 1) are the unknown arbitrary constants. Making use of the integral boundary conditions given by (7) in (15), we obtain where A i and B i (i � 1, 2) are, respectively, given by (13) and (14), and Journal of Function Spaces Solving (16) for c 0 and c 1 and using notation (13), we find that Substituting the values of c 0 and c 1 in (15), we get the desired solution (12). is completes the proof.
For a normed space □ Definition 4 (see [51]). A multivalued map G: Definition 5 (see [51]). e multivalued map G is bounded Definition 6 (see [51] Definition 7 (see [51]). A multivalued map G is said to be completely continuous if G(B) is relatively compact for every B ∈ P b (X).
Definition 8 (see [51]). A multivalued map G has a fixed point if there is x ∈ X, such that x ∈ G(x). e fixed point set of the multivalued operator G will be denoted by Fix G.
Lemma 2 (see [51]). If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, that is, x n ⟶ x * , y n ⟶ y * , and y n ∈ G(x n ) imply y * ∈ G(x * ).
Lemma 4 (see [53]). Let X be a Banach algebra and A: X ⟶ X be a single-valued and B: X ⟶ P cp,cv (X) be a multivalued operator satisfying the following: Definition 13 (see [51]). A multivalued map G: X ⟶ P (X) is said to be a contraction mapping if there is a constant 0 < λ < 1, such that for every x, y ∈ X, where H d is the Hausdorff metric.
Lemma 5 (see [54]). Let (X, d) be a complete metric space. If N: X ⟶ P cl (X) is a contraction, then Fix N ≠ ∅.

Main Results
In this section, we will study the existence results of solutions for problem (7). First of all, we fix our terminology.
In assumption (H 8 ), H d is the Hausdorff metric, where d is the Euclidean metric in R defined by d(x, y) � |x − y| for x, y ∈ R.

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Observe that N(x) � AxBx. We will show that the operators A and B satisfy all the conditions of Lemma 4. For the sake of convenience, we split the proof into several steps.
Step 2. e multivalued operator B is compact and upper semicontinuous on X, that is, (ii) of Lemma 4 holds.
First, we show that B has convex values. Let w 1 , w 2 ∈ Bx, and then, there are v 1 , v 2 ∈ S G,x , such that where for all t ∈ [1, e]. Hence, θu 1 (t) + (1 − θ)u 2 (t) ∈ Bx, and consequently, Bx is convex for each x ∈ X. As a result, B defines a multivalued operator B: X ⟶ P cv (X).
Next, we show that B maps bounded sets into bounded sets in X. To see this, let Q be a bounded set in X, and then, there exists a real number r > 0, such that ‖x‖ ≤ r, ∀x ∈ Q. Now, for each h ∈ Bx, there exist v ∈ S G,x , such that Journal of Function Spaces 11 en, for each t ∈ [1, e], using (H 2 ), we have Journal of Function Spaces Journal of Function Spaces 13 hence, erefore, B(Q) is uniformly bounded.
Next, we show that B maps bounded sets into equicontinuous sets. For this purpose, we assume that Q be, as above, a bounded set and h ∈ Bx for some x ∈ Q, and then, there exists a v ∈ S G,x , such that us, for any t 1 , t 2 ∈ [1, e], t 2 > t 1 , we have independent of x ∈ Q as t 1 − t 2 ⟶ 0. erefore, B(Q) is an equicontinuous set in X. Now, an application of the Arzela-Ascoli theorem yields that B(Q) is relatively compact.
In our next step, we show that B is upper semicontinuous. By Lemma 2, B will be upper semicontinuous if we prove that it has a closed graph. Let x n ⟶ x * , h n ∈ Bx n , and h n ⟶ h * . en, we need to show that h * ∈ Bx * . Associated with h n ∈ Bx n , there exists v n ∈ S G,x n , such that Journal of Function Spaces us, it suffices to show that there exists v * ∈ S G,x * , such that for each t ∈ [1, e], Let us consider the linear operator Θ: 16

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Notice that the operator Θ is continuous.
which implies that Θ(v n ) ⟶ Θ(v * ) in C( [1, e], R). us, it follows by Lemma 3 that Θ°S G is a closed graph operator. Furthermore, we have h n (t) ∈ Θ(S G,x n ). Since x n ⟶ x * , therefore, we have for some v * ∈ S G,x * . As a result, we have that the operator B is compact and upper semicontinuous.
is is obvious by (H 4 ) since we have and K � M 0 . us, all the conditions of Lemma 4 are satisfied, and a direct application of it yields that either conclusion (i) or conclusion (ii) holds. We show that conclusion (ii) is not possible.

Proof.
e proof is similar to that of eorem 1 and is omitted.
Proof. Observe that the set S G,x is nonempty for each x ∈ C [1, e] by assumption (H 7 ), and thus, G has a measurable selection. We now show that the operator N: C[1, e] ⟶ P(C [1, e]) satisfies the assumptions of Lemma 5. To establish that Nx ∈ P cl (C[1, e]), for each x ∈ C [1, e], let w n n ≥ 1 ⊂ Nx be such that w n ⟶ w as n ⟶ ∞ in C [1, e]. en, w ∈ C [1, e], and there exists v n ∈ S G,x , such that for each t ∈ [1, e], we have with v n (t) ∈ G(t, x(t), H I q x(t)), t ∈ [1, e].
Since G has compact values, therefore, we can pass onto a subsequence (denoted in a same way) to obtain that v n converges to v in L 1 [1, e].
us, v ∈ S G,x , and for each t ∈ [1, e], we have w n (t) ⟶ w(t), where Hence, w ∈ N(x). Next, we show that N is a contraction, that is, where λ 0 is defined in (56), and d 1 is the metric induced by the norm ‖ · ‖ in C [1, e]. For this, let x, x ∈ [1, e] and w 1 ∈ Nx. en, there exists v 1 ∈ S G,x , such that for all t ∈ [1, e], we obtain