On fractional hybrid and non-hybrid multi-term integro-differential inclusions with three-point integral hybrid boundary conditions

In this paper, we investigate the existence of solutions for two nonlinear fractional multi-term integro-differential inclusions in two hybrid and non-hybrid versions. The boundary value conditions are in the form of three-point integral hybrid conditions. In this way, we define a new operator based on the integral solution of the given boundary value inclusion problem and then we use assumptions of a Dhage’s fixed point result for this fractional operator in the hybrid case. Also, the approximate endpoint property is applied for the corresponding set-valued maps in the non-hybrid case. Finally, we provide two examples to illustrate our main results.


Introduction
To design powerful computing software, we need strong mathematical tools. In other words, if one can make exact patterns for natural phenomena and processes by using new mathematical formulas and operators, then more flexible algorithms can be written in the software programming based on such relations and formulas. This results in accurate computer calculations with the least error in the shortest time. In this way, many researchers are currently studying various types of advanced mathematical models using fractional differential equations and related inclusion versions with more general boundary value conditions [1][2][3][4][5][6][7][8]. Indeed, they try to model the processes such that they cover many general cases and in this situation; mathematicians would like to solve a wide range of these boundary value problems with advanced and complicated boundary conditions.
Recently, many papers have been published on the existence of solutions for different fractional boundary value problems (see, for example, ). In the last few decades, fractional hybrid differential equations and inclusions with hybrid or non-hybrid boundary value conditions have received a great deal of interest and attention of many researchers (see, for example, [35][36][37][38][39][40][41][42]).
The starting point for this field is related to a joint work of Dhage and Lakshmikantham in 2010. They introduced a new category of nonlinear differential equations, called ordinary hybrid differential equations, and studied the existence of extremal solutions for this boundary value problem by establishing some fundamental differential inequalities [43]. In 2012, Zhao et al. provided an extension for the Dhage's work to fractional order and considered a boundary value problem of fractional hybrid differential equations [44].

Preliminaries
In this section, we recall some definitions and theorems needed in the sequel. Let ω > 0. The fractional Riemann-Liouville integral of a function k : [a, b] → R is defined by provided that the right-hand side integral exists [47,48]. Now, let n -1 < ω < n and n = [ω] + 1. The fractional Caputo derivative of a function k ∈ C (n) ([a, b], R) is defined by provided that the right-hand side integral exists [47,48]. It has been proved that the general solution for the homogeneous fractional differential equation c D ω 0 k(t) = 0 is in the form k(t) = m * 0 + m * 1 t + m * 2 t 2 + · · · + m * n-1 t n-1 and we have where m * 0 , . . . , m * n-1 are some real constants and n = [ω] + 1 [49]. 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 K : X → P(X ) if k * ∈ K(k * ) [50]. The set of all fixed points of the set-valued map K is denoted by FIX (K) [50]. 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 ) [50]. A setvalued map K : X → P cl (X ) is said to be Lipschitz with a Lipschitz constant λ * > 0 whenever we have PH d X (K(k 1 ), K(k 2 )) ≤ λ * d X (k 1 , k 2 ) for all k 1 , k 2 ∈ X . A Lipschitz map K is called a contraction whenever λ * ∈ (0, 1) [50]. We say that the set-valued map K is completely continuous whenever the set K(W ) is relatively compact for every W ∈ P b (X ). A set-valued map K : [0, 1] → P cl (R) is said to be measurable if the function t → d X (υ, K(t)) is measurable for all υ ∈ R [50,51]. We say that the set-valued map K is an upper semi-continuous (u.s.c.) whenever for each k * ∈ X , the set K(k * ) belongs to P cl (X ) and for every open set V containing K(k * ), there exists an open neighborhood U * 0 of k * such that K(U * 0 ) ⊆ V [50]. A real-valued function k : R → R is called upper semicontinuous whenever lim sup n→∞ k(a n ) ≤ k(a) for all sequences {a n } n≥1 with a n → a [50].
The graph of the set-valued map K : X → P cl (Y) is defined by We say that the graph of K 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 ∈ K(k n ), we have s 0 ∈ K(k 0 ) [50,51]. Suppose that the set-valued map K : X → P cl (Y) is upper semi-continuous. Then Graph(K) is a subset of the product space X × Y which is a closed set. Conversely, if the set-valued map K is completely continuous and has a closed graph, then K is upper semi-continuous [ for almost all t ∈ [0, 1] [50,51]. If K is an arbitrary set-valued map, then for each function k ∈ C([0, 1], X ), we have (SEL) K,k = ∅ whenever dim X < ∞ [50]. A set-valued map K : for all |k| ≤ μ and for almost all t ∈ [0, 1] [50,51]. We say that u ∈ X is an endpoint of the set-valued map K : X → P(X ) whenever we have K(k) = {k} [52]. Also, the set-valued map K has an approximate endpoint property whenever inf k∈X sup k * ∈K(k) d X (k, k * ) = 0 [52]. We need the following results.
Then K has a unique endpoint if and only if K has the approximate endpoint property.

Main results
Now, we investigate the existence of solutions for the fractional hybrid and non-hybrid multi-term integro-differential inclusion problems (1)-(2) and (3). Consider the Banach space with four-point integral hybrid boundary value conditions if and only if k * 0 is a solution for the fractional integral equation where 1 and 2 are positive real constants defined as follows: Proof Assume that k * 0 is a solution for the hybrid differential equation (4). Then, there exist constants m * 0 , m * 1 ∈ R such that Then, and so, for each β ∈ (0, 1], we get .
By using three-point integral hybrid boundary value conditions, we obtain By substituting the values of m * 0 and m * 1 into (8), we get This shows that the function k * 0 is a solution for the fractional integral equation (6). Conversely, one can easily prove that k * 0 is a solution for the boundary value problem (4)-(5) whenever k * 0 is a solution function for the fractional integral equation (6).
Proof For every k ∈ X , define the set of selections of the operator K by for almost all t ∈ [0, 1]. Consider the set-valued map H : X → P(X ) defined by for all t ∈ [0, 1]. It is obvious that the function h 0 is a solution for the hybrid multi-term inclusion problem (1)- (2) if and only if h 0 is a fixed point of the operator H. Now, define the single-valued mapping Φ : X → X by and the set-valued map Υ : X → P(X ) by for all t ∈ [0, 1]. Then, we obtain H(k) = (Φk)(Υ k). We show that both operators Φ and Υ satisfy assumptions of Theorem 2. We first prove that the operator Φ is Lipschitz. Let k 1 , k 2 ∈ X . Since for i = 1, 2, . . . , n, we have for all t ∈ [0, 1]. Hence, we get for all k 1 , k 2 ∈ X . This shows that operator Φ is Lipschitz with a Lipschitz constant In this step, we prove that the set-valued map Υ has convex values. Let k 1 , k 2 ∈ Υ k. Choose ϑ 1 , ϑ 2 ∈ (SEL) K,k such that for j = 1, 2 and for almost all t ∈ [0, 1]. Let μ ∈ (0, 1). Then, we have for almost all t ∈ [0, 1]. Since K has convex values, (SEL) K,k is convex-valued. This follows that μϑ 1 (t) + (1μ)ϑ 2 (t) ∈ (SEL) K,k for all t ∈ [0, 1] and so Υ k is a convex set for all k ∈ X . Now, we prove that the operator Υ is completely continuous. In order to do this, we have to prove two equicontinuity and uniform boundedness properties for the set Υ (X ). First, we show that Υ 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 ∈ Υ k, there exists a function ϑ ∈ (SEL) K,k such that for all t ∈ [0, 1]. Then, we have where Λ * is given in (10). Thus, ζ ≤ Λ * Θ L 1 and this means that the set Υ (X ) is uniformly bounded. Now, we prove that the operator Υ maps bounded sets into equicontinuous sets. Let k ∈ V ε * and ζ ∈ Υ k. Choose ϑ ∈ (SEL) K,k such that for all t ∈ [0, 1]. Assume that t 1 , t 2 ∈ [0, 1] with t 1 < t 2 . Then, we have Notice that the right-hand side tends to zero independently of k ∈ V ε * as t 2 → t 1 . By using the Arzelà-Ascoli theorem, the complete continuity of Υ : is deduced. In the following, we prove that Υ has a closed graph and this implies the upper semi-continuity of the operator Υ . Assume that k n ∈ V ε * and ζ n ∈ (Υ k n ) with k n → k * and ζ n → ζ * . We claim that ζ * ∈ (Υ k * ). For every n ≥ 1 and ζ n ∈ (Υ k n ), choose ϑ n ∈ (SEL) K,k n such that for all t ∈ [0, 1]. It is sufficient to show that there exists a function ϑ * ∈ (SEL) K,k * such that for all t ∈ [0, 1]. Define the continuous linear operator Ξ : for all t ∈ [0, 1]. Hence, Thus, by using Theorem 1, it is deduced that the operator Ξ • (SEL) S has a closed graph. Also, since ζ n ∈ Ξ ((SEL) K,k n ) and k n → k * , so there exists ϑ * ∈ (SEL) K,k * such that for all t ∈ [0, 1]. Hence, ζ * ∈ (Υ k * ) and so Υ has a closed graph. This means that the operator Υ is upper semi-continuous. On the other hand, since the operator Υ has compact values, so Υ is a compact and upper semi-continuous operator. By using assumption (C3), we havê Then,ˆ δ * < 1 2 . Now by using Theorem 2 for Υ , we get that one of the conditions, (a) or (b), holds. We first investigate condition (b). By considering Theorem 2 and assumption (C4), assume that k is an arbitrary member of O * with k =r. Then, λ * k(t) ∈ (Φk)(t)(Υ k)(t) for all λ * > 1. Choose the related function ϑ ∈ (SEL) K,k . Then for each λ * > 1, we have for all t ∈ [0, 1]. Thus, one can write ϕ 1 k(t) , . . . , ϕ n k(t)ξ (t, 0, 0, . . . , 0) + ξ (t, 0, 0, . . . , 0) for all t ∈ [0, 1]. Hence, we get According to condition (9), we see that condition (b) of Theorem 2 is impossible. Thus, k ∈ (Φk)(Υ k). Hence, the operator H has a fixed point and so the hybrid multi-term inclusion problem (1)-(2) has a solution.
Here, we investigate the existence of solutions for the non-hybrid inclusion problem (3).

Definition 7
An absolutely continuous function k : [0, 1] → R is a solution for the fractional inclusion problem (3) whenever there exists an integrable function ϑ ∈ L 1 ([0, 1], R) with ϑ(t) ∈ S(t, k(t)) for almost all t ∈ [0, 1] satisfying three-point integral boundary value conditions and For k ∈ X , the set of selections of S is defined by for almost all t ∈ [0, 1]. Define the operator N : X → P(X ) by Now, we prove next result by using the approximate endpoint property for the set-valued map N which is defined in (11).
for all t ∈ [0, 1] and k 1 , (C8) The operator N has the approximate endpoint property, where N is given in (11). Then the fractional non-hybrid inclusion problem (3) has a solution.
Proof We show that the set-valued map N : X → P(X ) has an endpoint. In this way, we first prove that N (k) is a closed set for all k ∈ X . First of all, by using assumption (C6), the set-valued map t → S(t, k(t)) is measurable and it has closed values for all k ∈ X . Thus, S has measurable selection and the set (SEL) S,k is nonempty. Now, we show that N (k) is a closed subset of X for all k ∈ X . Let {k n } n≥1 be a sequence in N (k) with k n → u. For each n, there exists ϑ n ∈ (SEL) S,k such that for almost all t ∈ [0, 1]. Since S is compact set-valued map, we pass into a subsequence (if necessary) to obtain that a subsequence {ϑ n } n≥1 converges to some ϑ ∈ L 1 ([0, 1]). Hence, we have ϑ ∈ (SEL) S,k and for all t ∈ [0, 1]. This implies that k ∈ N (k), and so operator N is closed-valued. Also, N (k) is a bounded set for all k ∈ X because S is a compact set-valued map. Finally, we are going to prove that PH d X (N (k), N (k )) ≤ ψ( kk ) holds. Let k, k ∈ X and z 1 ∈ N (k ). Choose ϑ 1 ∈ (SEL) S,k such that for all t ∈ [0, 1], there exists h * ∈ S(t, k(t)) such that for all t ∈ [0, 1]. Consider a set-valued map B : [0, 1] → P(X ) which is given by Since ϑ 1 and = σ ψ(|kk |) 1 Λ * * are measurable, one can easily check that the set-valued map B(·) ∩ S(·, k(·)) is measurable. Now, choose ϑ 2 (t) ∈ K(t, k(t)) such that for all t ∈ [0, 1]. Hence, one can get Thus, we obtain z 1 - holds for all k, k ∈ X . Since the set-valued map N has approximate endpoint property (C8), Theorem 3 implies that operator N has a unique endpoint, that is, there exists k * ∈ X such that N (k * ) = {k * }. Thus, k * is a solution for the fractional inclusion problem (3).

Examples
Here, we provide two examples to illustrate our main results.

Conclusion
It is known that a lot of natural phenomena and processes in the world are modeled by different types of fractional differential equations and inclusions. This diversity factor in studying complicate fractional differential equations and inclusions increases our ability for exact modeling of more phenomena. This is useful in making modern software which allow for more cost-free testing and less material consumption. In this work, we investigate the existence of solutions for two fractional hybrid and non-hybrid multi-term integrodifferential inclusions with integral hybrid boundary value conditions. In this work, we investigate the existence of solutions for two fractional hybrid and non-hybrid multi-term integro-differential inclusions with integral hybrid boundary value conditions. It is noted that the fractional hybrid multi-term integro-differential inclusion presented in this paper is new in the sense that the boundary value conditions are stated as three-point mixed Caputo integro-derivative hybrid conditions. Also, this hybrid boundary value problem is general and it involves many fractional dynamical systems as special cases. In this way, we use the Dhage's fixed point result and approximate endpoint property for a set-valued map in our proofs. Finally, we give two examples to illustrate our main results. This topic can be used in mathematical modeling of applied problems in science, engineering, and the real-world phenomena [4,28].