On a fractional hybrid multi-term integro-differential inclusion with four-point sum and integral boundary conditions

We investigate the existence of solutions for a fractional hybrid multi-term integro-differential inclusion with four-point sum and integral boundary value conditions. By using Dhage’s fixed point results, we prove our main existence result. Finally, we give an example to illustrate our main result.

The starting point for this field was a work of Dhage and Lakshmikantham in 2010 [35]. They introduced a new category of nonlinear differential equation called ordinary hybrid differential equation and studied the existence of extremal solutions for this boundary value problem by establishing some fundamental differential inequalities [35]. In 2012, Zhao et al. provided an extension for Dhage's work to fractional order and considered a boundary value problem of fractional hybrid differential equations [36]. Later, some papers on different properties of solutions for fractional hybrid boundary value problems were published. In 2015, Hilal and Kajouni discussed the existence of extremal solutions for the Caputo hybrid boundary value problem h(t,k(t)) ) = g(t, k(t)) = 0, where t ∈ J = [0, T], p ∈ (0, 1), the functions h : J × R → R \ {0} and g : J × R → R are continuous, and a, b, c ∈ R with a+b = 0 [37]. In 2016, Ahmad et al. studied the existence of solutions for the nonlocal boundary value problem of fractional hybrid inclusion problem ) ∈ G(t, k(t)) = 0, where t ∈ J = [0, 1], c D α 0 denotes the Caputo fractional derivative of order α ∈ (1,2], and I φ 0 is the Riemann-Liouville fractional integral of order φ > 0 with φ ∈ {β 1 , β 2 , . . . , β m } [38].

Preliminaries
Let ω > 0. The Riemann-Liouville fractional integral of a function k : ds provided that the right-hand side integral exists ( [40,41]). Now, let n -1 < ω < n and n = [ω] + 1. The Caputo fractional derivative of a function k ∈ Γ (n-ω) k (n) (s) ds provided that the right-hand side integral exists ( [40,41]). It has been proved that the general solution for the homogeneous fractional differential equation n-1 t n-1 , and we have where m * 0 , . . . , m * n-1 are some real constants and n = [ω] + 1 [42]. Assume that (X , · X ) is a normed space. The set of all subsets of X , the set of all closed subsets of X , the set of all bounded subsets of X , the set of all compact subsets of X , and the set of all convex subsets of X are represented by P(X ), P cl (X ), P b (X ), P cp (X ), and P cv (X ), respectively. We say that k * ∈ X is a fixed point for the set-valued map S : X → P(X ) if k * ∈ S(k * ) [43]. The set of all fixed points of the set-valued map S is denoted by FIX (S) [43]. The Pompeiu-Hausdorff metric PH d : where d X (A 1 , a 2 ) = inf a 1 ∈A 1 d X (a 1 , a 2 ) and d X (a 1 , A 2 ) = inf a 2 ∈A 2 d X (a 1 , a 2 ) [43]. A setvalued map S : X → P cl (X ) is said to be Lipschitz with constant λ * > 0 whenever we have PH d X (S(k 1 ), S(k 2 )) ≤ λ * d X (k 1 , k 2 ) for all k 1 , k 2 ∈ X . A Lipschitz map S is called contraction whenever λ * ∈ (0, 1) [43]. We say that the set-valued map S is completely continuous whenever the set S(W ) is relatively compact for every W ∈ P b (X ). A setvalued map S : is measurable for all υ ∈ R [43,44]. We say that the set-valued map S is upper semicontinuous (u.s.c.) whenever, for each k * ∈ X , the set S(k * ) belongs to P cl (X ), and for every open set V containing S(k * ), there exists an open neighborhood U * 0 of k * such that S(U * 0 ) ⊆ V [43]. The graph of the set-valued map S : X → P cl (Y) is defined by Graph(S) = {(k, s) ∈ X × Y : s ∈ S(k)}. We say that graph of S is a closed set if, for each sequence {k n } n≥1 in X and {s n } n≥1 in Y, k n → k 0 , s n → s 0 and s n ∈ S(k n ), we have s 0 ∈ S(k 0 ) [43,44]. Suppose that the set-valued map S : X → P cl (Y) is upper semi-continuous. Then Graph(S) is a subset of the product space X × Y which is a closed set. Conversely, if the set-valued map S is completely continuous and has a closed graph, then S is upper semicontinuous ( [43], Proposition 2.1). A set-valued map S is convex-valued if S(k) is a convex set for each element k ∈ X . A set of selections of set-valued map S at point k ∈ C([0, 1], R) is defined by for almost all t ∈ [0, 1] [43,44]. If S is an arbitrary set-valued map, then for each function k ∈ C([0, 1], X ), we have (SEL) S,k = ∅ whenever dim X < ∞ [43]. A set-valued map S : [0, 1] × R → P(R) is called Caratheodory whenever t → S(t, k) is a measurable mapping for each function k ∈ R and k → S(t, k) is an upper semi-continuous mapping for almost all t ∈ [0, 1] [43,44]. Moreover, a Caratheodory set-valued map S : [0, 1] × R → P(R) is said to be L 1 -Caratheodory whenever, for each constant μ > 0, there exists a function φ μ ∈ L 1 ([0, 1], R + ) such that S(t, k) = sup t∈ [0,1] {|q| : q ∈ S(t, k)} ≤ φ μ (t) for all |k| ≤ μ and for almost all t ∈ [0, 1] [43,44]. We need the next results.
is an L 1 -Carathéodory set-valued map, and Ξ : Theorem 2 ([46]) Let X be a Banach algebra. Assume that there exist a single-valued map Φ 1 : X → X and a set-valued map Φ 2 : X → P cp,cv (X ) such that (i) Φ 1 is an operator including the Lipschitzian property with a Lipschitz constant l * ; (ii) Φ 2 is an operator including upper semi-continuity and the compactness property;

Main results
Now, we are ready to study the fractional hybrid multi-term inclusion problem (1)- (2). Consider the Banach space 1] |k(t)|. For convenience, consider the constants Here, we prove our first key result.

Lemma 3
Let z ∈ X . Then k 0 is a solution for the fractional hybrid differential equation with four-point hybrid integral boundary value conditions if and only if k 0 is a solution for the integral equation where Λ 0 , . . . , Λ 8 are given in (3).
Proof Assume that k 0 is a solution for hybrid equation (4). Then there exist constants and so By using the four-point hybrid boundary value conditions, we obtain and m * 2 = - By substituting the values m * 0 , m * 1 , and m * 2 in (7), we get This shows that the function k 0 is a solution for integral equation (6). Conversely, one can easily check that k 0 is a solution for problem (4)-(5) whenever k 0 is a solution function for integral equation (6).
Proof For each k ∈ X , define the set of selections of the operator S by for almost all t ∈ [0, 1]. Define G : X → P(X ) by for some ϑ ∈ (SEL) S,k . One can easily check that g 0 is a solution for the hybrid inclusion problem (1)-(2) if and only if g 0 is a fixed point of the operator G. Define the maps Φ 1 : 1 0 k(s) ds) and Φ 2 : X → P(X ) by for some ϑ ∈ (SEL) S,k . Then we obtain G(k) = Φ 1 kΦ 2 k. We prove that Φ 1 and Φ 2 satisfy the assumptions of Theorem 2. We first show that the operator Φ 1 is Lipschitz. Let k 1 , k 2 ∈ X . Assumption (C1) implies that for all t ∈ [0, 1]. Hence, we get Φ 1 k 1 -Φ 1 k 2 X ≤ 2θ * k 1k 2 X for all k 1 , k 2 ∈ X . This means that the operator Φ 1 is Lipschitz with constant 2θ * . Now, we claim that the setvalued map Φ 2 has convex values. Let k 1 , k 2 ∈ Φ 2 k. Choose ϑ 1 , ϑ 2 in (SEL) S,k such that for almost all t ∈ [0, 1]. Let λ ∈ (0, 1). Then we have for almost all t ∈ [0, 1]. Since S has convex values, (SEL) S,k is convex-valued. This gives that λϑ 1 (t) + (1λ)ϑ 2 (t) ∈ (SEL) S,k for all t ∈ [0, 1], and so Φ 2 k is a convex set for all k ∈ X . Now, we prove that the operator Φ 2 is completely continuous. We have to prove the equi-continuity and uniform boundedness of the set Φ 2 (X ). First, we show that Φ 2 maps all bounded sets into bounded subsets of X . For a positive number ε * ∈ R, consider the bounded ball V ε * = {k ∈ X : k X ≤ ε * }. For every k ∈ V ε * and ζ ∈ Φ 2 k, there exists a function ϑ ∈ (SEL) S,k so that for all t ∈ [0, 1]. Then we have where M is given in (9). Thus, ζ ≤ M q L 1 and this shows that the set Φ 2 (X ) is uniformly bounded. Next, we prove that the operator Φ 2 maps bounded sets into equicontinuous sets. Let k ∈ V ε * and ζ ∈ Φ 2 k. Choose ϑ ∈ (SEL) S,k such that Then we have Note that the right-hand side tends to zero independently of k ∈ V ε * as t 2 → t 1 . By using the Arzela-Ascoli theorem, the complete continuity of Φ 2 : C([0, 1], R) → P (C([0, 1], R)) is deduced. Now, we show that Φ 2 has a closed graph, and this follows the upper semicontinuity of the operator Φ 2 . Assume that k n ∈ V ε * and ζ n ∈ Φ 2 k n with k n → k * and ζ n → ζ * . We claim that ζ * ∈ Φ 2 k * . For every n ≥ 1 and ζ n ∈ Φ 2 k n , choose ϑ n ∈ (SEL) S,k n such that for all t ∈ [0, 1]. It is sufficient to show that there exists a function ϑ * ∈ (SEL) S,k * such that Hence, Theorem 1 implies that the operator Ξ • (SEL) S has a closed graph. Since ζ n ∈ Ξ ((SEL) S,k n ) and k n → k * , there exists ϑ * ∈ (SEL) S,k * such that for all t ∈ [0, 1]. Hence, ζ * ∈ Φ 2 k * and so Φ 2 has a closed graph. From this it follows that the operator Φ 2 is upper semi-continuous. Since the operator Φ 2 has compact values, Φ 2 is a compact and upper semi-continuous operator. By using assumption (C3), we havê Put l * = 2θ * . Thenˆ l * < 1 2 . Now, by using Theorem 2 for Φ 2 , we get that one of the conditions, (a) or (b), holds. We claim that condition (b) is impossible. By considering Theorem 2 and assumption (C4), assume that k is an arbitrary element of O * with k =ρ. Then α 0 k(t) ∈ Φ 1 k(t)Φ 2 k(t) for all α 0 > 1. Choose the related function ϑ ∈ (SEL) S,k . Then, for each α 0 > 1, we have for all t ∈ [0, 1]. Thus, one can write for all t ∈ [0, 1]. Hence, we getρ ≤ ξ * M q L 1 1-2θ * M q L 1 . Now, by using (8), we conclude that condition (b) of Theorem 2 is impossible. Thus, k ∈ Φ 1 kΦ 2 k. Hence, the operator G has a fixed point, and so the hybrid inclusion problem (1)-(2) has a solution.
Here, we provide an example to illustrate our main results.

Conclusion
It is known that most natural phenomena are modeled by different types of fractional differential equations and inclusions. This diversity in investigating complicate fractional differential equations and inclusions increases our ability for exact modelings of more phenomena. This is useful in designing modern software which helps us to allow for more cost-free testing and less material consumption. In this work, we study the existence of solutions for a fractional hybrid multi-term integro-differential inclusion problem with four-point sum and integral boundary value conditions. By using Dhage's fixed point results, we prove our main existence result. Finally, we give an example to illustrate our main result.