An Existence Result for ψ -Hilfer Fractional Integro-Differential Hybrid Three-Point Boundary Value Problems

: In this research work, we study a new class of ψ -Hilfer hybrid fractional integro-differential boundary value problems with three-point boundary conditions. An existence result is established by using a generalization of Krasnosel’ski ˘ i’s ﬁxed point theorem. An example illustrating the main result is also constructed.


Introduction
Differential equations of fractional order have recently received a lot of attention and now constitute a significant branch of nonlinear analysis, because some real world problems in physics, mechanics and other fields can be described better with the help of fractional differential equations. Numerous monographs have appeared devoted to fractional differential equations, for example, see [1][2][3][4][5][6][7][8]. Recently, differential equations and inclusions equipped with various boundary conditions have been widely investigated by many researchers (see [9][10][11][12][13][14][15][16][17][18] and the references cited therein).
Hybrid fractional differential equations have also been studied by several researchers. Hybrid fractional differential equations involve the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. Hybrid systems play a key role in embedded control systems that interact with the physical situation. Time-and event-based behaviors are more accurately described by hybrid models as such models help to deal with challenging design requirements in the design of control systems. Examples include automotive control [19], mobile robotics [20], the process industry [21], real-time software verification [22], transportation systems [23], and manufacturing [24].
In 2011, Zhao et al. [26] investigated the hybrid fractional initial value problem and Sun et al. [27] discussed fractional boundary value problems containing hybrid equations.
In [31], the authors proved the existence of solutions for a nonlocal boundary value problem of hybrid fractional integro-differential equations given by where D α is the Caputo fractional derivative of order α with 1 < α ≤ 2, I β i is the Riemann-Liouville fractional integral of order β i > 0 and functions The main result was proved by using of a hybrid fixed point theorem for three operators in a Banach algebra from Dhage [32]. The existence of solutions of hybrid fractional integro-differential equations with initial conditions, given by was studied in [33]. Here, D χ is the Caputo fractional derivative of order χ ∈ {α, ω} with 0 < α, ω ≤ 1, I β i is the Riemann-Liouville fractional integral of order β i > 0, . A generalization of Krasnosel'skiȋ's fixed point theorem ( [32,34]) was applied to prove the existence result. The problem (3) was extended to higher order fractional derivatives in [35] as a boundary value problem where D χ is the Caputo fractional derivative of order χ ∈ {α, ω} with 0 < α ≤ 1, 1 < ω ≤ 2, Dhage's fixed point theorem [32] was used to obtain an existence result. For recent papers on hybrid boundary value problems of fractional differential equations and inclusions, we refer to [36][37][38] and references cited therein.
In the present work, we study a three-point ψ-Hilfer hybrid fractional integro-differential nonlocal boundary value problem of the form where H D ω,ρ;ψ a is the ψ-Hilfer fractional derivative operator of order ω ∈ {α, p}, with and θ ∈ R. An existence result is established via a generalization of the Krasnosel'skiȋ fixed point theorem ( [32,34]).
The rest of the paper is organized as follows: In Section 2, we recall some notations, definitions, and lemmas from fractional calculus needed in our study. We also prove an auxiliary lemma helping us to transform the hybrid boundary value problem (6) into an equivalent integral equation. The main existence result for the ψ-Hilfer hybrid boundary value problem (6) is contained in Section 3. The obtained result is illustrated by a numerical example.

Preliminaries
This section defines some notation in relation to fractional calculus.
where Γ(·) is the Euler gamma function.
The Riemann-Liouville derivative of a function f with respect to another function ψ of order α is defined by where n = [α] + 1, [α] represents the integer part of the real number α.

Lemma 1 ([2]
). Let α, β > 0. Then, we have the following semigroup property given by Next, we present the ψ-fractional integral and derivatives of a power function. 40]). Let α > 0, υ > 0 and t > a. Then, we have is continuous for each x ∈ R and x satisfies the fractional differential equation and the boundary conditions in (6).
. . , n satisfy boundary value problem (6) and z ∈ C([a, b], R). Then, x is a solution of the ψ-Hilfer hybrid fractional integro-differential boundary value problem of the form: if and only if x satisfies the equation (14). Applying the ψ-Riemann-Liouville fractional integral operator I α;ψ a to both sides of (14) and using Lemma 3, we obtain where c 0 ∈ R. By using the boundary condition, H D p,ρ;ψ a x(a) = 0, we obtain the constant c 0 = 0. Thus, we have Inserting the ψ-Riemann-Liouville fractional integral operator I p;ψ a into both sides of (17) and using Lemma 3, we obtain where c 1 , c 2 ∈ R. From the boundary condition x(a) = 0, we obtain c 2 = 0, while from the boundary condition x(b) = θx(ξ), we find that Substituting the constants c 1 and c 2 into (18), we obtain the integral equation in (15) as desired.
Conversely, by a direct computation, it is easy to show that the solution x given by (15) satisfies the problem (14). The proof of Lemma 4 is completed.
Let X = C([a, b], R) be the Banach space of continuous real-valued functions defined on [a, b], equipped with the norm x = sup t∈[a,b] |x(t)| and a multiplication (xy)(t) = x(t)y(t), ∀t ∈ [a, b]. Then, clearly, X is a Banach algebra with the above-defined supremum norm and multiplication in it.
Lemma 5 ([32,34]). Let S be a nonempty, convex, closed, and bounded set such that S ⊂ X, and let A : X → X and B : S → X be two operators which satisfy the following: (i) A is contraction, (ii) B is completely continuous, and (iii) x = Ax + By, ∀y ∈ S ⇒ x ∈ S.
Then, there exists a solution of the operator equation x = Ax + Bx.

Existence Result
In view of Lemma 4, we define an operator Q : X → X by Notice that the problem (6) has solutions if and only if the operator Q has fixed points.
Proof. Firstly, we consider a subset S of X defined by S = {x ∈ X : x ≤ r}, where r is given by Observe that S is a closed, convex, and bounded subset of the Banach space X. Now, we set sup t∈[a,b] |λ i (t)| = λ i , i = 1, 2, . . . , n, sup t∈[a,b] |µ(t)| = µ , sup t∈[a,b] |ν(t)| = ν . Let us define three operators H, F , G : X → X such that Then, we have In addition, we obtain Moreover, we obtain |Gx(t) − Gy(t)| = |g(t, x(t)) − g(t, y(t))| ≤ φ x − y , and |Gx(t)| ≤ ν . Now we define two operators A : X → X and B : S → X as follows: and Clearly, Qx = Ax + Bx. In the next steps, we show that the operators A and B fulfill all the assumptions of Lemma 5. The proof is divided into three steps: Step 1. The operator A is a contraction mapping. For any x, y ∈ S, we have
Step 2. The operator B is completely continuous on S. First, we will prove that B is continuous. Let {x n } be a sequence of functions in S converging to a function x ∈ S. By Lebesgue domination theorem, for each t ∈ [a, b],, we have Therefore, the operator B is a continuous operator on S. Next, we show that the operator B is uniformly bounded on S. For any x ∈ S, we have Hence, Bx ≤ M, x ∈ S, which shows that the operator B is uniformly bounded on S. Finally, we show that the operator B is equicontinuous. Let t 1 < t 2 and x ∈ S. Then, we have As t 2 − t 1 → 0, the right-hand side of the above inequality tends to zero, independently of x. Thus, B is equicontinuous. Therefore, it follows by Aezelà-Ascoli theorem that B is a completely continuous operator on S.
Step 3. We show that the third condition (iii) of Lemma 5 is fulfilled. For any y ∈ S, we have which implies x ≤ r, and so x ∈ S. Hence, all the conditions of Lemma 5 are satisfied, and consequently the operator equation x(t) = Ax(t) + Bx(t) has at least one solution in S. Therefore, there exists a solution of the ψ-Hilfer hybrid fractional integro-differential boundary value problem (6) in [a, b]. The proof is finished.

Special Cases
The problem (6) considered in the present work is general in the sense that it includes the following classes of new boundary value problems of ψ-Hilfer fractional differential equations.
(I) Let g(t, x) = 1 and h i (t, x) = 0, i = 1, 2, . . . , n for all t ∈ [a, b] and x ∈ R. Then, the problem (6)  Therefore, the main result of this paper also includes the existence results for the solutions of the abovementioned ψ-Hilfer boundary value problems of fractional differential equations as special cases.

Conclusions
In this paper, we studied a new class of ψ-Hilfer hybrid fractional integro-differential boundary value problems with three-point boundary conditions. By using a generalization of Krasnosel'skiȋ's fixed point theorem, we proved an existence result. An example is presented to illustrate our main result. Some special cases are also discussed.