Existence Results for ψ-Hilfer Fractional Integro-Differential Hybrid Boundary Value Problems for Differential Equations and Inclusions

Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia


Introduction
Some real-world problems in physics, mechanics, and other fields can be described better with the help of fractional differential equations. So, differential equations of fractional order has recently received a lot of attention and now constitutes a significant branch of nonlinear analysis. 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] and the references cited therein.
Hybrid fractional differential equations have also been studied by several researchers. This class of equations involves the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. Some recent results on hybrid differential equations can be found in a series of papers [15][16][17][18].
We will give a brief history on the subject of hybrid differential and fractional differential equations. In 2010, Dhage and Lakshmikantham [19] initiated the study of the first-order hybrid differential equation where f ∈ CðJ × ℝ, ℝ \ f0gÞ and g ∈ CðJ × ℝ, ℝÞ: They established the existence, uniqueness results, and some fundamental differential inequalities. In 2011, Zhao et al. [15] discussed the following hybrid fractional initial value problem x 0 ð Þ = x 0 ∈ ℝ, where D q is the Riemann-Liouville fractional derivative of order 0 < q < 1,f ∈ CðJ × ℝ, ℝ \ f0gÞ, and g ∈ CðJ × ℝ, ℝÞ: Sun et al. [16] studied the following hybrid fractional boundary value problem where D q is the Riemann-Liouville fractional derivative of order 1 < q < 2,f ∈ Cð½0, 1 × ℝ, ℝ \ f0gÞ, and g ∈ Cð½0, 1 × ℝ, ℝÞ: In [20], the authors studied 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 The main result was obtained by means of a hybrid fixed-point theorem for three operators in a Banach algebra due to Dhage [21]. The existence of solutions for an initial value problem of hybrid fractional integro-differential equations, given by was studied in [22]. Here, D θ is the Caputo fractional derivative of order θ ∈ fα, ωg with 0 < α, ω ≤ 1; A generalization of Krasnoselskii fixedpoint theorem due to Dhage [21] was used in the proof of the existence result. The problem (5) was extended in [23] to boundary value problems of the form where D θ is the Caputo fractional derivative of order θ ∈ fα, ωg with 0 < α ≤ 1, 1 < ω ≤ 2; I β i is the Riemann-Liouville frac-tional integral of order β i > 0, h i ∈ Cð½0, 1 × ℝ, ℝÞ for i = 1, 2, ⋯, n, g ∈ Cð½0, 1 × ℝ, ℝ \ f0gÞ, f ∈ Cð½0, 1 × ℝ, ℝÞ. An existence result is proved via Dhage's [21] fixed-point theorem.
For recent results on hybrid boundary value problems of fractional differential equations and inclusions, we refer to [24][25][26] and references cited therein. In the literature, there do exist several definitions of fractional integrals and derivatives. One of them is the Hilfer fractional derivative, which composites both Riemann-Liouville and Caputo fractional derivatives [27]. Fractional differential equations involving Hilfer derivative have many applications, and we refer to [28] and the references cited therein. There are actual world occurrences with uncharacteristic dynamics such as atmospheric diffusion of pollution, signal transmissions through strong magnetic fields, the effect of the theory of the profitability of stocks in economic markets, the theoretical simulation of dielectric relaxation in glass forming materials, and network traffic. See [29,30] and references cited therein.
In the present work, we study a ψ-Hilfer hybrid fractional integro-differential nonlocal boundary value problem of the form where H D α,ρ;ψ a + is the ψ-Hilfer fractional derivative operator of order α, with 0 < α ≤ 2, 0 ≤ ρ ≤ 1; I n. An existence result is established via a fixed-point theorem for the product of two operators due to Dhage [21].
As a second problem, we investigate the existence of solutions for the following inclusion ψ-Hilfer fractional hybrid integro-differential equations with nonlocal boundary conditions of the form where F : ½a, b × ℝ ⟶ P ðℝÞ is a multivalued map, P ðℝÞ is the family of all subsets of ℝ, and the other quantities are the same as in boundary value problem (8). Here, the existence result is based on a multivalued fixed-point theorem for the product of two operators due to Dhage [38]. The rest of the paper is arranged as follows: in Section 2, we recall some notations, definitions, and lemmas from fractional calculus needed in our study. Also, we prove an auxiliary lemma helping us to transform the hybrid boundary value problem (8) into an equivalent integral equation. The main existence result for the ψ-Hilfer hybrid boundary value problem (8) is contained in Section 3. The obtained result is illustrated by a numerical example. Section 4 is devoted in the study of the inclusion case of the hybrid boundary value problem (8) by considering the multivalued hybrid boundary value problem (9). Some special cases are discussed in Section 5.

Preliminaries
This section is assigned to recall some notation in relation to fractional calculus. We denote by AC n ð½a, b, ℝÞ the n-times absolutely continuous functions given by Definition 1 (see [2]). Let ða, bÞ, ð−∞≤a < b≤∞Þ, be a finite or infinite interval of the half-axis ð0, ∞Þ and α > 0. Also, let ψ be an increasing and positive monotone function on ða, b, having a continuous derivative ψ ′ ðxÞ on ða, bÞ. The ψ-Riemann-Liouville fractional integral of a function f with respect to another function ψ on ½a, b is defined by where Γð·Þ is the Euler Gamma function.
Definition 2 (see [2]). Let ψ′ðtÞ ≠ 0 and α > 0, n ∈ ℕ. The Riemann-Liouville derivatives of a function f with respect to another function ψ of order α is defined by where n = ½α + 1, ½α is represent the integer part of the real number α.
Lemma 4 (see [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.
Then, x is a solution of the ψ-Hilfer hybrid fractional integro-differential nonlocal boundary value problem 3 Advances in Mathematical Physics if and only if x satisfies the equation Proof. Let x ∈ Cð½a, b, ℝÞ be a solution of the problem (18). Applying the operator I α;ψ a + to both sides of (18) and using Lemma 7, we obtain where c 1 , c 2 ∈ ℝ. By using the first boundary condition, xðaÞ = 0, we get the constant c 2 = 0. From the second boundary condition, xðbÞ = mðxÞ, we find that Substituting the value of c 1 and c 2 in (20), we obtain (19). Conversely, by a direct computation, it is easy to show that the solution x given by (19) satisfies the problem (18). The proof of Lemma 8 is completed.☐ Let X = CðJ, ℝÞ be the Banach space of continuous realvalued functions defined on ½a, b, equipped with the norm kxk = sup t∈½a,b jxðtÞj and a multiplication ðxyÞðtÞ = xðtÞyðtÞ, ∀t ∈ ½a, b: Then, clearly, X is a Banach algebra with abovedefined supremum norm and multiplication in it.
Lemma 9 (see [21]). Let S be a nonempty, closed convex, and bounded subset of the Banach algebra X and A : X ⟶ X, B : S ⟶ X two operators such that (a) A is Lipschitzian with a Lipschitz constant k Then, the operator equation x = AxBx has a solution.

Existence Result for the Problem (8)
In view of Lemma 8, we define an operator Q : X ⟶ X by Notice that the problem (8) has solutions if and only if the operator Q has fixed points.

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S is a closed, convex, and bounded subset of the Banach space X. We set sup t∈½a,b jλ i ðtÞj = kλ i k,i = 1, 2, ⋯, m, sup t∈½a,b jμðtÞj = kμk, and sup t∈½a,b jνðtÞj = kνk: Next, we define two more operators A : X ⟶ X and B : S ⟶ X as follows: Clearly, Qx = AxBx. In the next steps, we show that the operators A and B fulfil all the assumptions of Lemma 9. The proof is divided into three steps.☐ Step 1. We show that the operator A is Lipschitzian with Lipschitz constant k, i.e., condition ðaÞ of Lemma 9 is fulfilled. Let x, y ∈ S: Then we have Consequently, Hence, the operator A is Lipschitzian with Lipschitz constant k = kϕk.
Step 2. We show that the condition ðbÞ of Lemma 9 is satisfied, i.e., the operator B is completely continuous on S: First, we will prove its continuity. Let fx j ðtÞg be a sequence of functions in S converging to a function xðtÞ ∈ S: Then, by the Lebesgue dominant theorem, for each t ∈ ½a, b, we have Therefore, 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, kBxk ≤ M, ∀t ∈ ½a, b, which shows that the operator B is uniformly bounded on S: Now, we show that the operator B is equicontinuous. Let t 1 < t 2 and x ∈ S: Then, we have

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As t 2 − t 1 ⟶ 0, the right-hand side tends to zero, independently of x. Thus, B is equicontinuous. Therefore, it follows by Aezelá-Ascoli theorem that B is a completely continuous operator on S: Step 3. We show that the third condition ðcÞ of Lemma 9 is fulfilled. For any y ∈ S, we have which implies kxk ≤ r, and so, x ∈ S: Moreover, by (26), it holds kM < 1 which is fulfilled condition ðdÞ of Lemma 9. Hence, all the conditions of Lemma 9 are satisfied, and consequently, the operator equation xðtÞ = AxðtÞBxðtÞ has at least one solution in S: Therefore, there exists a solution of the ψ-Hilfer hybrid fractional integro-differential nonlocal boundary value problem (8) in ½a, b: The proof is finished. Now, we present an example of ψ-Hilfer hybrid fractional integro-differential boundary value problem to illustrate our main result.
Definition 15. A function x ∈ ACð½a, b, ℝÞ is a solution of the problem (9) if xðaÞ = 0, xðbÞ = mðxÞ, and there exists function v ∈ L 1 ð½a, b, ℝÞ such that vðtÞ ∈ Fðt, xðtÞÞ a.e. on ½ a, b and Theorem 16. Assume that (A 1 ) and (A 3 ) hold. In addition, we suppose that (B 1 ) F : ½a, b × ℝ ⟶ P cp,cv ðℝ + Þ is L 1 -Carathéodory multivalued map; (B 2 ) The functions g and h i , i = 1, 2, ⋯, n satisfy condition ðA 2 Þ; (B 3 ) There exists a continuous function q ∈ Cð½a, b, ℝ + Þ such that Then, the nonlocal ψ-Hilfer hybrid inclusion boundary value problem (9) has at least one solution on ½a, b, provided that Proof. To transform the boundary value problem (9) into a fixed-point problem, by using Lemma 8, we define a multivalued operator Q 1 : X ⟶ P ðXÞ as 7 Advances in Mathematical Physics Next, we introduce the operator A : X ⟶ X as in (27) and the multivalued operator B 1 : X ⟶ P ðXÞ by We will show that the operators A and B 1 satisfy the hypotheses of Lemma 14. The proof is given in a series of steps.
Step 3. The operator B 1 is completely continuous and upper semicontinuous on X: Let S be a bounded set of X: Then, there exists a constant r such that kxk ≤ r, for all x ∈ S: We prove first that the operator B 1 is completely continuous. Let x ∈ B 1 ðSÞ: Then, there exists v ∈ S F,x such that for any x ∈ S: Then, we have for all t ∈ ½a, b, which implies that kxk ≤ M 1 . Therefore, the operator B 1 is uniformly bounded on X: Next, we show that B 1 ðSÞ is an equicontinuous set in X: Let t 1 , t 2 ∈ ½a, b with t 1 < t 2 and x ∈ B 1 ðSÞ. Then, we have , v ∈ S F,x , t ∈ a, b ½ :