Two sequential fractional hybrid differential inclusions

The main objective of this paper is to concern with a new category of the sequential hybrid inclusion boundary value problem with three-point integro-derivative boundary conditions. In this direction, we employ various novel analytical techniques based on α-ψ-contractive mappings, endpoints, and the fixed points of the product operators to obtain the main results. Finally, we provide two examples to illustrate our main results.


Introduction
Every day that passes, the human needs for studying natural phenomena is increasing. One of the possible methods for achieving this goal is using mathematical operators and computer modeling. The fractional operators were developed over the years, and their importance has become more and more apparent to researchers today. Instances of the application of such fractional operators can be found in various sciences such as biomathematics, electrical circuits, medicine, and so on [1][2][3][4][5][6]. All these items have led researchers to find many aspects of the structure of the fractional boundary value problems and hereditary properties of their solutions. In this regard, many researchers investigated advanced fractional modelings [7,8] and related theoretical results and qualitative behaviors of such fractional boundary value problems [9][10][11][12][13][14][15][16][17][18][19][20][21][22].
There have been appeared different versions of fractional operators during these years. In monograph [23], Miller and Ross defined another type of fractional derivative called sequential derivatives, which ares a combination of the existing derivative operators. Later, the attention of some researchers was attracted to finding a connection between the usual Riemann-Liouville derivative and the sequential fractional derivative [24,25]. These useful results have led to publishing some papers on the sequential fractional boundary value problems (see, e.g., [26][27][28][29][30]). In 2015, Alsaedi et al. studied the sequential problem ⎧ ⎨ ⎩ ( c D α * 0 + + k c D α * -1 0 + )w(s) = g(s, w(s)), s ∈ [0, 1], w(0) = 0, w (0) = 0, w(ξ ) = a R I where α * ∈ (2,3], ξ ∈ (η, 1), with η, β > 0, a, k ∈ R + , and g : [0, 1] × R → R is a continuous function [31]. On the other hand, hybrid differential problems with different kinds of boundary conditions have gained extensive attention of many researchers [32][33][34]. This area begins with a joint work presented by Dhage and Lakshmikantham [35] in 2010. The authors addressed a novel differential equation entitled a hybrid differential equation and investigated the extremal solutions of this new BVP by deriving some useful fundamental differential inequalities [35]. Later, Zhao et al. [36] gave an abstract extension for the problem mentioned in [35] to fractional order and defined a boundary value problem of fractional hybrid differential equations. Until now, the limited research papers have been published on various properties of solutions for hybrid boundary value problems of fractional order. In 2016, Ahmad et al. [37]performed an important existence analysis for the nonlocal fractional BVP of hybrid inclusion problem given by

Preliminaries
Let > 0. The definition of the Riemann-Liouville integral of a function w : [0, +∞) → R is of the form of R I 0 + w(s) = s 0 (s-r) -1 Γ ( ) w(r) dr, provided that the value of the integral is finite [40,41]. Let ∈ (n -1, n) be such that n = [ ] + 1. For a function w ∈ AC (n) R ([0, +∞)), the fractional derivative of Caputo type is given by c D 0 + w(s) = s 0 (s-r) n--1 Γ (n-) w (n) (r) dr, provided that the integral is finite-valued [40,41]. Moreover, for a sufficiently smooth function w : [0, +∞) → R, the sequential fractional derivative is defined by where = ( 1 , 2 , . . . , n ) is a multiindex [23]. Note that, in general, the sequential derivative operator D 0 + can be Riemann-Liouville, Caputo, Hadamard, Caputo-Hadamard, or any other version of derivative operators. In this research, we employ the sequential derivative of Caputo type defined as follows. For n -1 < < n, the Caputo sequential fractional derivative for a sufficiently smooth function w : [0, +∞) → R is given by is the Riemann-Liouville fractional integral of order n - [40]. It has been verified that the general solution for the homogeneous differential equation c D 0 + w(s) = 0 is given by w(s) =m 0 +m 1 s +m 2 s 2 + · · · +m n-1 s n-1 and k=0m k s k = w(s) +m 0 +m 1 s +m 2 s 2 + · · · +m n-1 s n-1 , wherem 0 , . . . ,m n-1 are real numbers with n = [ ] + 1 [23]. Here we recall some important and required properties in the Banach spaces. For this purpose, consider the normed space (W, · W ). For convenience, we denote by P(W), P cls (W), P bnd (W), P cmp (W), and P cvx (W) the collections of all subsets, all closed subsets, all bounded subsets, all compact subsets, and all convex subsets of W, respectively. The element w * ∈ W is a fixed point for a given set-valued map S : W → P(W) whenever w * ∈ S(w * ) [42]. We denote the family of all fixed points of S by FIX (S) [42]. The Pompeiu-Hausdorff metric where d W (A 1 , a 2 ) = inf a 1 ∈A 1 d W (a 1 , a 2 ) and d W (a 1 , A 2 ) = inf a 2 ∈A 2 d W (a 1 , a 2 ) [42]. A setvalued map S : W → P cls (W) is Lipschitzian with positive constantλ if PH d W (S(w), S(w )) ≤λd W (w, w ) for all w, w ∈ W. A Lipschitz map S is a contraction ifλ ∈ (0, 1) [42]. A map S is said to be completely continuous if S(K) is relatively compact for each K ∈ P bnd (W), whereas S : 42,43]. Also, S is an upper semicontinuous if for every w * ∈ W, the set S(w * ) belongs to P cls (W) and for each open set U containing S(w * ), there is a neighborhood O * 0 of w * such that that S(O * 0 ) ⊆ U [42]. We construct the graph of the set-valued map S : W → P cls (Z) by Graph(S) = {(w, z) ∈ W × Z : z ∈ S(w)}. The Graph(S) is closed if for two arbitrary convergent sequences {w n } n≥1 in W and {z n } n≥1 in Z with w n → w 0 , z n → z 0 , and z n ∈ S(w n ), we have the inclusion z 0 ∈ S(w 0 ) [42,43]. In view of [42], it is deduced that if the set-valued map S : W → P cls (Z) is upper semicontinuous, then Graph(S) is a closed subset of W × Z. On the contrary, if S is completely continuous and closed (i.e., has a closed graph), then S is upper semicontinuous [42]. In addition, S has convex values if S(k) ∈ P cvx (W) for each w ∈ W. Furthermore, a collection of selections of S at point [42,43]. Note that if S is an arbitrary set-valued map, then for each w ∈ C W ([0, 1]), we have (SEL) S,w = ∅ whenever dim(W) < ∞ [42]. We say that S : [42,43]). Besides, a Carathéodory set-valued map S : 1] {|q| : q ∈ S(s, w)} ≤ ϕ μ (s) for all |w| ≤ μ and for almost every s ∈ [0, 1] [42,43].
Samet et al. [44] introduced a new collection of nondecreasing nonnegative functions By considering the properties of these functions it is obvious that ψ(s) < s for all s > 0 [44]. Later, Mohammadi et al. [45] constructed a new structure for set-valued maps with the following definition. [45]. In addition, we say that W has property (C α ) if for every convergent [46]. Besides, we say that S has an approximate endpoint property if inf w∈W sup z∈Sw d W (w, z) = 0 [46]. We need the following results.
be an L 1 -Carathéodory set-valued map, and let Ξ : having the closed-graph property.

Theorem 2 ([48])
Let W be a Banach algebra. Assume that the following statements for a single-valued map Φ * 1 : W → W and a set-valued map Φ * 2 : W → P cmp,cvx (W) are valid: .  1] |w(s)|. Note that (W, · W ) with multiplication given by (w · w )(s) = w(s)w (s) is a Banach algebra. Now consider the nonzero constants Lemma 5 Letã ∈ W. Then w 0 is a solution function for the fractional sequential hybrid differential equation with three-point hybrid integro-derivative boundary conditions if and only if w 0 is a solution of the integral equation whereΩ 1 ,Ω 2 , and * are given in (5).
Proof First, suppose that w 0 is a solution for the sequential hybrid equation (6). Then By taking the th-order Riemann-Liouville integral on this equality, we obtain = R I 0 +ã(s) +m 0 +m 1 s +m 2 s 2 . Hence The last equality yields p 1 [ On the other hand, taking the first-order Caputo derivative of Equation (9) with respect to s, we get Now we multiply both sides of the last equality by e p 2 s : Finally, in view of three-point hybrid boundary conditions, we obtainm 0 = 0, If we insert the valuesm 0 ,m 1 , andm 2 into Equation (10), then w 0 (s) = ζ s, w 0 (s), R I γ 0 + w 0 (s) This shows that the function w 0 satisfies the integral equation (8). For the next part, it is easy to check that w 0 is a solution for the sequential BVP (6)-(7) whenever w 0 satisfies the integral equation (8).
Now, in the following theorem, we deal with some useful estimates.
Proof (A1) First of all, an easy computation yields We further have Now combining the obtained results, we get This completes the proof of (A1).
(A2) Similarly to Equation (12), we have The proof of this estimate is similar to that of (A2) and so is omitted.
Proof Let w ∈ W. Consider the following family of selections of the operator S: for someθ ∈ (SEL) S,w . Then we can express the product equation G(w) = Φ * 1 wΦ * 2 w. The main objective herein is to show that Φ * 1 and Φ * 2 satisfy all assumptions of Theorem 2. We first proceed by proving that the operator Φ * 1 is Lipschitzian on W. Let w 1 , w 2 ∈ W be arbitrary elements. Under hypothesis (Hyp1), it follows that ). In the subsequent step, we will check that the set-valued map Φ * 2 is convex-valued. Let w 1 , w 2 ∈ Φ * 2 w. We chooseθ 1 ,θ 2 ∈ (SEL) S,w such that for almost all s ∈ [0, 1]. Since S has convex values, (SEL) S,w is a convex set. By this point we find that λθ 1 (s) + (1λ)θ 2 (s) ∈ (SEL) S,w for any s ∈ [0, 1], and so Φ * 2 w belongs to P cvx (W) for all w ∈ W. Now we prove the complete continuity of Φ * 2 on W. We need to prove the equicontinuity and uniform boundedness of the set Φ * 2 (W). To this aim, we first show that Φ * 2 maps each bounded set to a bounded subset of W. Forε ∈ R + , consider the bounded ball (1e -p 2 ) p 2 Γ ( ) + p +ξ -1 (p 2 p + e -p 2 p -1) where M is given in (14). Consequently, v ≤ M θ L 1 , which means that Φ * 2 (W) is uniformly bounded. Now we claim that the operator Φ * 2 maps each bounded set to an equicontinuous subset. Let w ∈ Vε and v ∈ Φ * 2 w. Selectθ ∈ (SEL) S,w such that Thus we observe that the limit of the right-hand side is zero without considering w ∈ Vε as s 1 → s 2 . Therefore by the Arzelà-Ascoli theorem we conclude that Φ * 2 : C R ([0, 1]) → P(C R ([0, 1]) is completely continuous. We further prove that Φ * 2 has a closed graph, which implies the upper semicontinuity of this operator. Let w n ∈ Vε and v n ∈ Φ * 2 w n be such that w n → w * and v n → v * . We show the inclusion v * ∈ Φ * 2 w * . For each index n ≥ 1 and v n ∈ Φ * 2 w n , we selectθ n ∈ (SEL) S,w n such that Hence by Theorem 2, Ξ • (SEL) S has a closed graph. Since v n ∈ Ξ ((SEL) S,w n ) and Thus v * ∈ Φ * 2 w * , and so Φ * 2 has a closed graph. This implies that Φ * 2 is upper semicontinuous. On the other hand, note that the operator Φ * 2 has compact values. Hence Φ * 2 is a compact and upper semicontinuous. In the next step, in addition to hypothesis (Hyp3), by a similar argument we get . It is clear that l * ˆ < 1 2 . We see that all three assumptions of Theorem 2 are satisfied for the operators Φ * 1 and Φ * 2 . Now we only need to show that one of conditions (a) or (b) is valid. We claim that the invalid condition is (b). To observe this, by Theorem 2 and hypothesis (Hyp4) we may assume that w is an arbitrary element belonging to O * with w = q. Obviously, α 0 w(s) ∈ Φ * 1 w(s)Φ * 2 w(s) for any α 0 > 1. We select the corresponding functionθ ∈ (SEL) S,w . Then for each α 0 > 1, we have . According to (13), the impossibility of condition (b) of Theorem 2 follows. Consequently, we have w ∈ Φ * 1 wΦ * 2 w. Hence the existence of a fixed point for the operator G is proved, and thus the sequential hybrid inclusion BVP (1)-(2) has a solution. This completes the proof.
In this position, we continue our process to reach the existence results for the sequential nonhybrid inclusion BVP (3)-(4) by using two new theoretical theorems.
Proof In a similar manner, each fixed point of the operator K : W → P(W) given by (15) is a solution of the sequential inclusion BVP (3)-(4). Due to assumption (Hyp5), the measurability of the set-valued map s → S(s, w(s)) is clear, and so it is closed-valued for all w ∈ W. Hence S has a measurable selection, and (SEL) S,w = ∅. Now we want to prove that K(w) is a closed subset of W for all w ∈ W. To this end, we consider a sequence {w n } n≥1 of K(w) such that w n → w. For each n, chooseθ n ∈ (SEL) S,w such that for all s ∈ [0, 1]. Hence w ∈ K(w), and so K has closed values. By the assumptions of the theorem we know that S is a compact set-valued map. Thus we can easily check that the set K(w) is bounded for all w ∈ W. In this position, we are going to prove that the operator K is an α-ψ-contraction. To see this, define the nonnegative function α : W × W → [0, ∞) by α(w, w ) = 1 ifζ (w(s), w (s)) ≥ 0 and α(w, w ) = 0 otherwise. Let w, w ∈ W and z 1 ∈ K(w ). Chooseθ 1 ∈ (SEL) S,w such that for all w, w ∈ W such thatζ (w(s), w (s)) ≥ 0 for s ∈ [0, 1]. Therefore there is h ∈ S(s, w(s)) such that |θ 1 (s) -h| ≤ σ (s)ψ(|w(s)w (s)|) 1 M σ . We further introduce the new set-valued

Conclusions
Nowadays we need to study more natural phenomena to obtain more abilities for modeling. The fractional operators were developed over the years, and today their importance has become more and more apparent to researchers. In this way, it is necessary to design different and complicated modelings by utilizing the fractional differential problems. In this work, we review sequential fractional hybrid differential inclusions with three-point integro-derivative boundary value conditions. We employ some analytical tools to study the existence results corresponding to problems (1)- (2) and (3)-(4). We use some notions such as approximate endpoint, (C α ), and the compactness property in this regard. Finally, we provide two examples to illustrate our main results.