Nonlocal boundary value problems for Hilfer-type pantograph fractional differential equations and inclusions

In this paper, we study boundary value problems, involving the Hilfer fractional derivative, for pantograph fractional differential equations and inclusions supplemented by nonlocal integral boundary conditions. Existence and uniqueness results are obtained by using well-known fixed point theorems for single and multi-valued functions. Examples illustrating our results are also presented.


Introduction
In recent years, the theory of fractional differential equations has played a very important role in a new branch of applied mathematics, which has been utilized for mathematical models in engineering, physics, chemistry, signal analysis, etc. For details and applications we refer the reader to the classical reference texts such as [1][2][3][4][5][6][7]. Fractional differential equations are considered valuable tools to model many real world problems. Boundary value problems of differential equations represent an important class of applied analysis. Most of the researchers have given attention to study fractional differential equations by taking Caputo or Riemann-Liouville derivatives. Engineers and scientists have developed some new models that involve fractional differential equations for which the Riemann-Liouville derivative is not considered appropriate. Therefore certain modifications were introduced to avoid the difficulties and some new types of fractional order derivative operators were introduced in the literature by authors like Caputo, Hadamard, and Erdely-Kober. A generalization of derivatives of both Riemann-Liouville and Caputo was given by Hilfer in [8], known as the Hilfer fractional derivative of order α and a type β ∈ [0, 1], which can be reduced to the Riemann-Liouville and Caputo fractional derivatives when β = 0 and β = 1, respectively. Such derivative interpolates between the Riemann-Liouville and Caputo derivative. Fractional differential equations involving the Hilfer derivative have many applications; see [9][10][11][12] and the references cited therein.
An important class of differential equations containing proportional delays are called pantograph equations. This important class was named after Ockendon and Tayler [13]. Numbers of applications have been studied by many researchers of these equations in applied sciences including biology, physics, economics, and electrodynamics. For more details as regards the aforesaid equations, we refer to [14,15].
Initial value problems involving Hilfer fractional derivatives were studied by several authors; see for example [16][17][18] and the references therein. Nonlocal boundary value problems for the Hilfer fractional derivative were studied in [19]. Initial value problems for pantograph equations with the Hilfer fractional derivative were studied in [15,20].
To the best of our knowledge, there is no work on boundary value problems for pantograph equations with the Hilfer fractional derivative in the literature. This paper comes to fill this gap, by introducing a new class of boundary value problems of pantograph equations with Hilfer-type fractional differential equations and nonlocal integral boundary conditions, of the form (1.1) x(a) = 0, where H D α,β is the Hilfer fractional derivative of order α, 1 < α < 2 and parameter β, 0 ≤ β ≤ 1, f : [a, b] × R × R → R is a continuous function, I δ is the Riemann-Liouville fractional integral of order δ > 0, a ≥ 0, A, B, c ∈ R and 0 < λ < 1.
Existence and uniqueness results are proved by using well-known fixed point theorems. We make use of Banach's fixed point theorem to obtain the uniqueness result, while the nonlinear alternative of Leray-Schauder type [21] and Krasnoselskii's fixed point theorem [22] are applied to obtain the existence results for the problem (1.1)-(1.2).
After that we study the multi-valued version of the problem (1.1)-(1.2) by considering the inclusion problem 3) x(a) = 0,

Preliminaries
In this section, we introduce some notations and definitions of fractional calculus and multi-valued analysis and present preliminary results needed in our proofs later [2,5].

Definition 2.1
The Riemann-Liouville fractional integral of order α > 0 of a continuous function u : [a, ∞) → R, is defined by provided the right-hand side exists on (a, ∞).

Definition 2.2
The Riemann-Liouville fractional derivative of order α > 0 of a continuous function u, is defined by where n = [α] + 1, [α] denotes the integer part of real number α, provided the right-hand side is point-wise defined on (a, ∞).

Definition 2.3
The Caputo fractional derivative of order α > 0 of a continuous function u, is defined by provided the right-hand side is point-wise defined on (a, ∞).
In [8] (see also [9]) another new definition of the fractional derivative was suggested. The generalized Riemann-Liouville fractional derivative is defined as follows.

Definition 2.4
The generalized Riemann-Liouville fractional derivative or the Hilfer fractional derivative of order α and parameter β of a function u is defined by In the following lemma we present the compositional property of Riemann-Liouville fractional integral operator with the Hilfer fractional derivative operator.
The following lemma deals with a linear variant of the boundary value problem (1.1)-(1.2).

3)
if and only if Assume that x is a solution of the nonlocal (2.2)-(2.3). Operating the fractional integral I α on both sides of equation (2.2) and using Lemma 2.6, we obtain where c 0 and c 1 are some real constants. From the first boundary condition x(a) = 0 we can obtain c 0 = 0, since lim t→a (ta) γ -2 = ∞. Then we get Substituting the values of c 1 in (2.5), we obtain the solution (2.4). The converse follows by direct computation. This completes the proof.

Main results for the single-valued problem (1.1)-(1.2)
In view of Lemma 2.7, we define an operator A : C → C by It should be noticed that problem (1.1)-(1.2) has solutions if and only if the operator A has fixed points. In the following, for the sake of convenience, we set We prove existence as well as existence and uniqueness results, for the boundary value problem (1.1)-(1.2) by using well-known fixed point theorems.
Our existence and uniqueness result is based on Banach's fixed point theorem.
where Ω is defined by 2). Applying the Banach contraction mapping principle, we shall show that A has a unique fixed point.
which implies that Ax -Ay ≤ 2LΩ xy . As 2LΩ < 1, A is a contraction. Therefore, we deduce by the Banach contraction mapping principle that A has a fixed point which is the unique solution of the boundary value problem (1.1)-(1.2). The proof is completed.
Example 3.2 Consider the nonlocal boundary value problem for the Hilfer-type pantograph fractional differential equation of the form (3.5) The setting yields γ = 17/10, Λ = 1.872599119 and Ω = 4.129461300. Now, we put which satisfies (H 1 ) as Setting L = 1/9, we obtain 2LΩ ≈ 0.9176580667 < 1 which shows that inequality (3.3) is true. Then, by the conclusion of Theorem 3.1, we deduce that the boundary value problem (3.5) has a unique solution on [1/2, 5/2]. Next we present two existence results. The first is based on the well-known Krasnoselskii fixed point theorem ( [22]). and For any x, y ∈ B ρ , we have This shows that A 1 x + A 2 y ∈ B ρ . It is easy to see, using (3.6), that A 2 is a contraction mapping.
where Ω is defined by (3.2). Proof Let the operator A be defined by (3.1). Firstly, we shall show that A maps bounded sets (balls) into bounded set in C. For a number r > 0, let B r = {x ∈ C : x ≤ r} be a bounded ball in C.
and, consequently, Next we will show that A maps bounded sets into equicontinuous sets of C. Let τ 1 , τ 2 ∈ [a, b] with τ 1 < τ 2 and x ∈ B r . Then we have As τ 2τ 1 → 0, the right-hand side of the above inequality tends to zero independently of x ∈ B r . Therefore, by the Arzelá-Ascoli theorem, the operator A : C → C is completely continuous. The result will follow from the Leray-Schauder nonlinear alternative ( [21]) once we have proved the boundedness of the set of all solutions to equations x = νAx for ν ∈ (0, 1). Let x be a solution. Then, for t ∈ [a, b], and following computations similar to the first step, we have which leads to In view of (H 4 ), there exists M such that x = M. Let us set We see that the operator A :Ū → C is continuous and completely continuous. From the choice of U, there is no x ∈ ∂U such that x = νAx for some ν ∈ (0, 1). Consequently, by the nonlinear alternative of Leray-Schauder type ( [21]), we deduce that A has a fixed point x ∈Ū which is a solution of the boundary value problem (1.1)-(1.2). This completes the proof. Also by L 1 ([a, b], R) we denote the space of functions For a normed space (X, · ), we define P q (X) = {Y ∈ P(X) : Y has the property q}. Thus, for example, P cl,b (X) = {Y ∈ P(X) : Y is closed and bounded}, P cp,c (X) = {Y ∈ P(X) : Y is compact and convex}.
For each y ∈ C([a, b], R), define the set of selections of F by The following lemma will be used in the sequel.

is a closed graph operator in C([a, b], X) × C([a, b], X).
Before stating and proving our main existence results for problem (1.3)-(1.4), we will give the definition of its solution.  F(t, x, y) a.e. on [a, b] such that

The upper semicontinuous case
for all x, y ∈ R with x , y ≤ ρ and for a.e. t ∈ [a, b]; : Let 0 ≤ θ ≤ 1. Then, for each t ∈ [a, b], we have Since F has convex values, that is, S F,x is convex, we have Step C([a, b], R).

N(x) maps bounded sets (balls) into bounded sets in
for some v ρ ∈ S F,x ρ . However, on the other hand, we have Dividing both sides by ρ and taking the lower limit as ρ → ∞, we get which contradicts (4.2). Hence there exists a positive number ρ such that N(B ρ ) ⊆ B ρ .

Step 3. N(x) maps bounded sets into equicontinuous sets of C([a, b], R).
Let x be any element in B ρ and h ∈ N(x), then there exists a function v ∈ S F,x such that, Thus The right-hand side of the above inequality clearly tends to zero independently of x ∈ B ρ as τ 1 → τ 2 . As a consequence of Steps 1-3 together with the Arzelá-Ascoli theorem, we conclude that N : C([a, b], R) → P (C([a, b], R)) is completely continuous.
Step 4. N(x) is closed for each x ∈ C([a, b], R). ([a, b], R). Then u ∈ C([a, b], R) and there exists v n ∈ S F,x n such that, for each t ∈ [a, b], As F has compact values, we pass onto a subsequence (if necessary) to find that v n converges to v in L 1 ([a, b], R). Thus v ∈ S F,x and for each t ∈ [a, b], we have Hence, u ∈ N(x).
Next we show that the operator N is upper semicontinuous. In order to do so, it is enough to establish that N has a closed graph ([25, Proposition 1.2]).
Step 5. N has a closed graph. Let x n → x * , h n ∈ N(x n ) and h n → h * . We need to show that h * ∈ N(x * ). Now h n ∈ N(x n ) implies that there exists v n ∈ S F,x n such that, for each t ∈ [a, b], We must show that there exists v * ∈ S F,x * such that, for each t ∈ [a, b], Consider the continuous linear operator Θ : Observe that h n (t)h * (t) → 0 as n → ∞ and, thus, it follows from Lemma 4.1 that Θ • S F,x is a closed graph operator. Moreover, we have for some v * ∈ S F,x * Hence, we conclude that N is a compact multi-valued map, u.s.c. with convex closed values. As a consequence of Lemma 4.3, we deduce that N has a fixed point which is a solution of the boundary problem (1.3)- (1.4). This completes the proof.
Next, we give an existence result based upon the following form of fixed point theorem which is applicable to completely continuous operators [26]. Lemma 4.5 Let X a Banach space, and T : X → P b,cl,c (X) be a completely continuous multi-valued map. If the set E = {x ∈ X : νx ∈ T(x), ν > 1} is bounded, then T has a fixed point.

Theorem 4.6
Assume that the following hypotheses hold: for some ν > 1 and there exists a function v ∈ S F,x such that For each t ∈ [a, b], we have Taking the supremum over t ∈ [a, b], we get Hence the set E is bounded. As a consequence of Lemma 4.5 we deduce that N has at least one fixed point which implies that the problem (1. Our final existence result in this subsection is based on the Leray-Schauder nonlinear alternative for Kakutani maps ( [21]). with v ∈ S F,x such that, for t ∈ [a, b], we have In view of (A 5 ), we have, for each t ∈ [a, b], Consequently, we have In view of (H 4 ), there exists M such that x = M. Let us set

The lower semicontinuous case
Here we study the case when F is not necessarily convex valued, by applying the nonlinear alternative of Leray-Schauder type together with the selection theorem of Bressan and Colombo [27] for lower semicontinuous maps with decomposable values. Proof It follows from (A 5 ) and (A 6 ) that F is of l.s.c. type [28]. Then, by the selection theorem of Bressan and Colombo [27], there exists a continuous function v : ([a, b], R)) is the Nemytskii operator associated with F, defined as
Observe that x is a solution to the boundary value problem (1.
It can easily be shown that N is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 4.7. So we omit it. This completes the proof.

The Lipschitz case
In this subsection, we prove the existence of solutions for the boundary value problem (1.3)-(1.4) with a non-convex valued right-hand side by applying a fixed point theorem for multi-valued maps due to Covitz and Nadler [29].

Theorem 4.9
Assume that the following conditions hold:  N : C([a, b], R) → P (C([a, b], R)) defined at the beginning of the proof of Theorem 4.4. We show that the operator N satisfies the assumptions of Covitz and Nadler Theorem [29] in two steps.
Step I. N is nonempty and closed for every v ∈ S F,x . Note that the set-valued map F(·, x(·)) is measurable by the measurable selection theorem (e.g., [30,Theorem III.6]) and it admits a measurable selection v : [a, b] → R. Moreover, by the assumption (B 1 ), we have v(t) ≤ m(t) + m(t) x(t) , Theorem 4.9, we deduce that the boundary value problem (4.5) with (4.7) has at least one solution on [3/5, 9/5].

Conclusion
In this paper we initiated the study of a new class of boundary value problems, involving the Hilfer fractional derivative, for pantograph fractional differential equations and inclusions supplemented by nonlocal integral boundary conditions. Existence and uniqueness results are proved in the single-valued case. Banach's fixed point theorem is used to obtain the uniqueness result, while the nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorem are applied to obtain the existence results. For the multi-valued problem we prove existence results for both convex valued and non-convex valued multifunctions. For the case when the multi-valued F has convex values we use the Bohnenblust-Karlin fixed point theorem, Martelli's fixed point theorem and the nonlinear alternative for Kakutani maps. For the lower semicontinuous case the existence result is based on the nonlinear alternative of Leray-Schauder type together with a selection theorem for lower semicontinuous maps with decomposable values. Finally in the case of a possible non-convex valued multi-valued map we use a fixed point theorem for contractive multi-valued maps due to Covitz and Nadler. Examples illustrating the obtained results are also presented.
The results presented in this paper are more general and correspond to several new results corresponding to specific values of the parameters involved in the problem (1.1)-(1.2. For instance, the nonlocal boundary condition given by (1.2) with A = 0, c = 0, B = 0 can be conceived of as a conserved boundary condition as the sum of the values of the continuous unknown function over the given interval of arbitrary length is zero. In other words, our problem becomes an "average type" boundary value problem for Hilfer-type fractional differential equations or inclusions. With B = 1, A = -1, c = 0 our boundary condition (1.2) becomes x(b) = BI δ x(η), etc. In our future work we plan to investigate the existence of solutions for boundary value problems for other kinds of fractional differential equations and boundary conditions.