A Survey on Existence Results for Boundary Value Problems of Hilfer Fractional Differential Equations and Inclusions

: This paper is a survey of the recent results of the author for various classes of boundary value problems for Hilfer fractional differential equations and inclusions of fractional order in ( 1,2 ] supplemented with different kinds of nonlocal boundary conditions.


Introduction
Fractional differential equations become another necessary tool in solving real-life problems in different research areas such as mathematical biology, engineering, mechanics, and physics; for example, see the monographs [1][2][3][4][5][6][7][8][9]. Boundary value problems of fractional differential equations and inclusions represent an important class of applied analysis. Most researchers have studied fractional differential equations by taking Caputo or Riemann-Liouville derivative. 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-type fractional order derivative operators were introduced in the literature by Hadamard, Erdelyi-Kober, Katugampola, and others. A generalization of derivatives of both Riemann-Liouville and Caputo was given by R. Hilfer in [10] and is 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 a derivative interpolates between the Riemann-Liouville and Caputo derivative. Fractional differential equations involving Hilfer derivative have many applications; see [11][12][13][14][15][16] and references cited therein.
This survey is devoted to articles published by the author and his collaborators and concern some recent existence and uniqueness results for various classes of boundary value problems for Hilfer fractional differential equations and inclusions of fractional order in (1,2] supplemented with different kinds of nonlocal boundary conditions. The rest of this survey is organized as follows. In Section 2, we introduce some notations and definitions of fractional calculus and multivalued analysis. In the subsequent sections, we present existence and uniqueness results for boundary value problems for Hilfer, ψ-Hilfer fractional, and sequential fractional differential equations and inclusions with a variety of nonlocal boundary conditions, such as multipoint, integral, integral multipoint, integro-multipoint, integro-multistrip-multipoint and Riemann-Stieltjes integral multistrip. We also present existence and uniqueness results for coupled systems of Hilfer and ψ-Hilfer types and Hilfer-Hadamard fractional and sequential fractional differential equations. Note that our goal here is a more complete and comprehensive review, and as such, the choice is made to include as many results as possible to illustrate the progress on the matter. Any proofs (that are rather long) are omitted, for this matter, and the reader is referred to the relative article accordingly.
Let us describe briefly the used methods to obtain our results. In each problem, we first present an auxiliary result concerning a linear variant of the corresponding boundary value problem that is very useful to transform the studied problem into a fixed point problem. A variety of fixed point theorems are then used to establish the existence and uniqueness results. For the single-valued case, the fixed point theorems of Banach, Boyd and Wong, Krasnoselskii, Sadovskii, Isaia and the Leray-Schauder nonlinear alternative were used, while in the multivalued case, the fixed point theorems of Bohnenblust-Karlin, Martelli, Covitz-Nadler and the Leray-Schauder nonlinear alternative for multivalued maps maps were used. For the multivalued case, we present existence results for both cases, convex-valued (upper semicontinuous case), and nonconvex-valued (Lipschitz case) multifunctions. To obtain the existence and uniqueness results for fractional coupled systems, Banach's contraction mapping principle and the Leray-Schauder alternative are used. In each theorem, we indicate the used fixed point theorem.

Preliminaries
In this section, we introduce some notations and definitions of fractional calculus and multivalued analysis. In [10] (see also [11]), another new definition of the fractional derivative, known as the generalized Riemann-Liouville fractional derivative, was suggested; it is defined as Definition 4. The generalized Riemann-Liouville fractional derivative or Hilfer fractional derivative of order α and parameter β of a function u is defined by H D α,β u(t) = I β(n−α) D n I (1−β)(n−α) u(t), where n − 1 < α < n, 0 ≤ β ≤ 1, t > a, D n = d n dt n .

Remark 1.
In the above definition, type β allows D α,β to interpolate continuously between the classical Riemann-Liouville fractional derivative and the Caputo fractional derivative. When β = 0, the Hilfer fractional derivative corresponds to the Riemann-Liouville fractional derivative while when β = 1, the Hilfer fractional derivative corresponds to the Caputo fractional derivative In the following, we recall some notations and results from ψ-Hilfer fractional derivatives.
be a finite or infinite interval of the half-axis (0, ∞) and α > 0. In addition, let ψ(t) be a positive increasing function on (a, b], which has a continuous derivative ψ (t) on (a, b). The ψ-Riemann-Liouville fractional integral of a function f with respect to another function ψ on [a, b] is defined by where Γ(·) represents the Gamma function.

Definition 6 ([2]
). Let ψ (t) = 0 and α > 0, n ∈ N. The Riemann-Liouville derivatives of a function f with respect to another function ψ of order α correspondent to the Riemann-Liouville is defined by represents the integer part of the real number α. This is the greatest integer n such that n ≤ α.
Some preliminaries from the Hilfer-Hadamard fractional derivative are presented next. [2]). The Hadamard fractional integral of order α ∈ R + for a function f : [a, ∞) → R is defined as
Definition 9 (Hadamard fractional derivative [2]). The Hadamard fractional derivative of order α > 0 applied to the function f : [a, ∞) → R is defined as where δ n = (t d dt ) n and [α] denotes the integer part of the real number α.
The Hilfer-Hadamard fractional derivative may be viewed as interpolating the Hadamard fractional derivative. Indeed, for β = 0, this derivative reduces to the Hadamard fractional derivative.
For the basic concepts of multivalued analysis, we refer to [19,20]. A multivalued map G : X → P (X) has a fixed point if there is x ∈ X such that x ∈ G(x).
Furthermore, a Carathéodory function F is called L 1 −Carathéodory if: for all x ∈ R with x ≤ ρ and for a.e. t ∈ [a, b].

Nonlocal Fractional Integral Boundary Conditions
The study of boundary value problems for Hilfer-type fractional differential equations of order α ∈ (1, 2] with nonlocal integral boundary conditions, initiated in [21], is conducted by considering the following problem where H D α,β is the Hilfer fractional derivative of order α, 1 < α < 2, and parameter β, The following lemma deals with a linear variant of the boundary value problem (3).
. Then, the function x is a solution of the boundary value if and only if In view of Lemma 1, we define an operator A : It should be noticed that problem (3) has solutions if and only if the operator A has fixed points.

Existence and Uniqueness Results for the Problem (3)
The existence and uniqueness results for the boundary value problem (3) are given in the following theorems. Theorem 1 is based on Banach's contraction mapping principle, Theorem 2 on Banach contraction mapping principle together with Hölder inequality, and Theorem 3 on Boyd and Wong fixed point theorem for nonlinear contractions. Theorem 1. Assume that: and x, y ∈ R. Then, the boundary value problem (3) has a unique solution on [a, b], provided that Theorem 2. Suppose that f : [a, b] × R → R is a continuous function. In addition, we assume that: ], x, y ∈ R and θ ∈ L 1/σ ([a, b], R + ), σ ∈ (0, 1).
If θ ω < 1, then the boundary value problem (3) has a unique solution on [a, b], where Theorem 3. Let f : [a, b] × R → R be a continuous function satisfying the assumption: continuous and H * the constant defined by Then, the boundary value problem (3) has a unique solution on [a, b].
Two existence results are presented now, based on Krasnoselskii's fixed point theorem and the Leray-Schauder nonlinear alternative, respectively.
Theorem 4. Let f : [a, b] × R → R be a continuous function satisfying (1.1). In addition, we assume that: Then, the boundary value problem (3) has at least one solution on [a, b] provided that Then, the boundary value problem (3) has at least one solution on [a, b].

Existence Results for the Inclusion Problem
The multivalued case of the problem (3), that is, the boundary value problem of Hilfer-type fractional differential inclusions with nonlocal integral boundary conditions were studied in [22], where F : [a, b] × R → P (R) is a multivalued map, P (R) is the family of all nonempty subsets of R, and the other parameters are as in problem (3).
Our existence results for convex-and nonconvex-valued multifunctions, based, respectively, on the Leray-Schauder nonlinear alternative for multivalued maps maps and Covitz and Nadler fixed point theorem for contractive multivalued maps, are as follows.
Theorem 6. Assume that (5.2) holds. In addition, we suppose that: Then, the boundary value problem (4) has at least one solution on [a, b].

Theorem 7.
Assume that the following conditions hold: x)) ≤ m(t)|x −x| for almost all t ∈ [a, b] and x,x ∈ R with m ∈ C([a, b], R + ) and d(0, F(t, 0)) ≤ m(t) for almost all t ∈ [a, b]. Then, the boundary value problem (4) has at least one solution on

Pantograph Fractional Differential Equations and Inclusions with Nonlocal Fractional Integral Boundary Conditions
A new class of boundary value problems of pantograph equations with Hilfer-type fractional differential equations and nonlocal integral boundary conditions of the form were introduced in [23], 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.
The following lemma deals with a linear variant of the boundary value problem (5).
Then, the function x is a solution of the boundary value problem if and only if In view of Lemma 2, we define an operator A : It should be noticed that problem (5) has solutions if and only if the operator A has fixed points.

Existence and Uniqueness Results for the Problem (5)
The existence and uniqueness result for the problem (5), based on Banach's contraction mapping principle, is as follows.
Theorem 8. Assume that: then the boundary value problem (5) has a unique solution on [a, b].
The existence results, based on Krasnoselskii's fixed point theorem and the Leray-Schauder nonlinear alternative, respectively, are given in the following theorems.
a continuous function satisfying (8.1). In addition, we assume that: Then, the boundary value problem (5) has at least one solution on [a, b], provided Theorem 10. Assume that: Then, the boundary value problem (5) has at least one solution on [a, b].

Existence Results for the Inclusion Problem
The multivalued version of the problem (5) were studied also in [23] by considering the following inclusion problem where is a multivalued function and (P (R) is the family of all nonempty subsets of R).
The existence results in the case when F has convex values (the upper semicontinuous case) are given in the next theorems. Theorem 11 is based on the Bohnenblust-Karlin fixed point theorem, Theorem 12 on Martelli's fixed point theorem, and Theorem 13 on the Leray-Schauder nonlinear alternative for multivalued maps.
Theorem 11. Assume that: where ϕ ρ is the function that appears in Definition 11.
Then, the boundary problem (6) has at least one solution on [a, b], provided that: Theorem 12. Assume that the following hypotheses hold: and each x, y ∈ R.
Then, the problem (6) has at least one solution on [a, b].
Theorem 13. Assume that (10.2) and (11.1) hold. In addition, we assume that: (13.1) there exists a continuous nondecreasing function ψ : [0, ∞) → (0, ∞) and a function p ∈ C([a, b], R + ) such that Then, the boundary value problem (6) has at least one solution on [a, b]. Now, we state the existence of solutions for the boundary value problem (6) with a nonconvex-valued right hand side. The proof is based on the Covitz and Nadler fixed point theorem.

Theorem 14.
Assume that the following conditions hold: Then, the boundary value problem (6) has at least one solution on [a, b] if
The following lemma deals with a linear variant of the boundary value problem (7).
if and only if In view of Lemma 3, we define an operator A : It should be noticed that problem (7) has a solution if and only if the operator A has fixed points. (7) Our existence and uniqueness results for the problem (7), based respectively on Banach's contraction mapping principle, Krasnoselskii's fixed point theorem, and the Leray-Schauder nonlinear alternative, are as follows.

Existence and Uniqueness Results for the Problem
Theorem 15. Assume that: , then the boundary value problem (7) has a unique solution on [a, b].
Theorem 16. Let f : [a, b] × R × R → R be a continuous function satisfying (15.1). In addition, we assume that: Then, the boundary value problem (7) has at least one solution on [a, b] provided Then, the boundary value problem (7) has at least one solution on [a, b].

Existence Results for the Inclusion Problem
The corresponding multivalued problem of the form is also studied in [24], where F : [a, b] × R 2 → P (R) is a multivalued map (P (R) is the family of all nonempty subjects of R).
In the case when F has convex values, existence results based on Martelli's fixed point theorem and the Leray-Schauder nonlinear alternative for multivalued maps, respectively, are given below.
Theorem 18. Assume that the following hypotheses hold: Then, the problem (8) and a function p ∈ C([a, b], R + ) such that F(t, x, y) P := sup{|y| : Then, the boundary value problem (8) has at least one solution on [a, b].
The existence result for the boundary value problem (8) with a nonconvex-valued right hand side based on the Covitz and Nadler fixed point theorem, is the following.
Theorem 20. Assume that the following conditions hold:

Nonlocal Integro-Multipoint Boundary Conditions
Existence and uniqueness of solutions were studied in [25] for the following new class of boundary value problems consisting of fractional-order sequential Hilfer-type differential equations supplemented with nonlocal integro-multipoint boundary conditions of the form where H D α,β denote the Hilfer fractional derivative operator of order α, 1 < α < 2 and parameter β, The following lemma concerns a linear variant of the sequential boundary value problem (9).
Then, the function x is a solution of the sequential boundary value problem if and only if In view of Lemma 4, we define an operator A : . It is obvious that the sequential nonlocal boundary value problem (9) has a solution if and only if the operator A has fixed points.
We use the following notations: and 4.1.1. Existence and Uniqueness Results for the Problem (9) The existence and uniqueness results for the problem (9) based on Banach's contraction mapping principle, Krasnoselskii's fixed point theorem, and the Leray-Schauder nonlinear alternative, respectively, are given as follows.
Theorem 21. Assume that: where Ω and Ω 2 are defined by (10) and (11), respectively, and , then the sequential boundary value problem (9) have a unique solution on [a, b].
Theorem 22. Let f : [a, b] × R × R → R be a continuous function satisfying the following assumption: Then, if Ω 1 < 1, the sequential boundary value problem (9) has at least one solution on [a, b].
(23.2) there exists a constant K > 0 such that where Ω and Ω 1 are defined by (10) and (11), respectively, and L 1 is defined in Theorem 21.
Then, the sequential boundary value problem (9) has at least one solution on [a, b].

Existence Results for the Inclusion Problem
The authors of [25] studied the corresponding multivalued problem where Let us discuss first the case when the multivalued F has convex values. The existence results, based on Martelli's fixed point theorem and the Leray-Schauder nonlinear alternative for multivalued maps, respectively, are the following.
We state an auxiliary lemma that plays a key role to transform the problem (13) into a fixed point problem.

Lemma 5.
Let h ∈ C([a, b], R) and Then, x ∈ C([a, b], R) is a solution of the linear boundary value problem if and only if

Having in mind Lemma 5, we introduce an operator
Obviously, the problem (13) is equivalent to the fixed point problem: x = Ax.
We use the following notations: and

Existence and Uniqueness Results for the Problem (13)
For the problem (13), we state the following existence and uniqueness results, based on Krasnoselskii's fixed point theorem and Banach's contraction mapping principle, respectively.
Theorem 27. Let f : [a, b] × R → R be a continuous function satisfying the conditions: Then, the sequential Hilfer fractional boundary value problem (13) has at least one solution on [a, b] provided that LQ 1 < 1, where Q 1 is given by (15).  (14) and (15), respectively.

Existence Results for the Inclusion Problem
The multivalued version of the problem (13) is also studied in [26] by considering the following inclusion problem: where F : [a, b] × R → P (R) is a multivalued map (P (R) is the family of all nonempty subsets of R). Now, we consider the multivalued problem (16).
In the first existence result, based on the Leray-Schauder nonlinear alternative for multivalued maps, F has convex values and is L 1 -Carathéodory.
where Q and Q 1 are given by (14) and (15), respectively.

Then, the sequential Hilfer inclusion fractional boundary value problem (16) has at least one solution on [a, b].
In the second existence result for the problem (16), based on the Covitz and Nadler fixed point theorem, F is a nonconvex-valued multivalued map.
Theorem 30. Assume that the following conditions hold: where Q and Q 1 are given by (14) and (15), respectively.

Riemann-Stieltjes Integral Multistrip Boundary Conditions
In [27], a new class of sequential Hilfer-type boundary value problems for fractional differential equations involving Riemann-Stieltjes integral multistrip boundary conditions of the form were discussed, where H D α,β denotes the Hilfer fractional derivative operator of order α, 1 < α < 2 and parameter β, 0 ≤ β ≤ 1, f : [a, b] × R → R is a continuous function, b a x(s)dH(s) is the Riemann-Stieltjes integral with respect to the function H : [a, b] → R, a ≥ 0, k, µ i ∈ R, a < η i < ξ i ≤ b, i = 1, 2, . . . , n. Lemma 6. Let a ≥ 0, 1 < α < 2, γ = α + 2β − αβ, h ∈ C([a, b], R), and Then, the function x is a solution of the sequential boundary value problem if and only if In view of Lemma 6, we define an operator A : It is obvious that the sequential boundary value problem (17) has solutions if and only if the operator A has fixed points.
We use the notations: and

Existence and Uniqueness Results for the Problem (17)
The existence and uniqueness results for the problem (17), based on Banach's contraction mapping principle, Krasnoselskii's fixed point theorem, and the Leray-Schauder nonlinear alternative, respectively, are given as follows.
Theorem 31. Assume that: where Ω and Ω 1 are defined by (19) and (20), respectively, then the sequential Hilfer boundary value problem (17) has a unique solution on [a, b].
Theorem 32. Let f : [a, b] × R → R be a continuous function such that: where Ω 1 is defined in (19), the sequential Hilfer boundary value problem (17) has at least one solution on [a, b].

Theorem 33.
Let Ω 1 < 1. In addition, we assume that: nondecreasing and continuous function and p ∈ C([a, b], R + ); Then, the sequential Hilfer boundary value problem (17) has at least one solution on [a, b].

Existence Results for the Inclusion Problem
The authors of [27] also covered the multivalued case of the problem (17) by considering the following sequential inclusion boundary value problem where F : [a, b] × R → P (R) is a multivalued function and (P (R) is the family of all nonempty subsets of R).
x(s)ds, and there exists a a.e. on [a, b] and The following existence result is based on the Leray-Schauder nonlinear alternative for multivalued maps.
Then, the sequential Hilfer boundary value problem (21) has at least one solution on [a, b].
The following lemma deals with a linear variant of the problem (23).

Lemma 7.
Let Λ = 0. For z ∈ C([a, b], R), the function x is a solution of the linear problem if and only if By Lemma 7, we introduce an operator A : Notice that a fixed point of the operator A is a solution of the problem (23). We set the following notation: Our existence results are stated below. Theorem 35 is based on Sadovski's fixed point theorem and Theorem 36 on Isaia's fixed point theorem.
Theorem 36. Assume that: If = δH < 1, then the ψ-Hilfer fractional boundary value problem (23) has at least one solution x ∈ C([a, b], R) and the set of solutions is bounded in C([a, b], R).
In order to transform the problem (28) into a fixed point problem, we provide the following auxiliary lemma, which concerns a linear variant of the boundary value problem (28).
if and only if x satisfies the integral equation where For the sake of convenience, we use the following notations: In view of Lemma 8, an operator A : . It should be noticed that the problem (28) has solutions if and only if the operator A has fixed points.
In the first result, based on Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the problem (28).
Next, we present two existence results, based on Krasnoselskii's fixed point theorem and the Leray-Schauder nonlinear alternative, respectively.
Then, the problem (28) has at least one solution on [a, b].

Multipoint Boundary Conditions
The study of existence and uniqueness of solutions for a new class of boundary value problems of sequential ψ-Hilfer-type fractional differential equations with multipoint boundary conditions of the form were initiated in [30], where H D α,β;ψ is the ψ-Hilfer fractional derivative of order α, 1 < α < 2 and parameter β, We first give an auxiliary lemma concerning a linear variant of the boundary value problem (33).
Then, the function x ∈ C([a, b], R) is a solution of the boundary value problem if and only if In view of Lemma 9, we define an operator A : It should be noticed that the sequential boundary value problem (33) has a solution if and only if the operator A has fixed points.
In the following, we use the notations: and Our first result is an existence and uniqueness result, and the other two are existence results, based on Banach's contraction mapping principle, Krasnoselskii's fixed point theorem, and the Leray-Schauder nonlinear alternative, respectively.
Theorem 40. Assume that: where Ω and Ω 1 are defined by (34) and (35), respectively, then the boundary value problem (33) has a unique solution on [a, b].
Theorem 41. Let f : [a, b] × R → R be a continuous function such that: Then, the boundary value problem (33) has at least one solution on [a, b] provided that where Ω 1 is given by (35).
Theorem 42. Let f : [a, b] × R → R be a continuous function. Assume that Ω 1 < 1. In addition, we suppose that: Then, the boundary value problem (33) has at least one solution on [a, b].
The following auxiliary lemma concerning a linear variant of the sequential Hilfer boundary value problem (36) plays a fundamental role in establishing the existence and uniqueness results for the given nonlinear problem.
Lemma 10. Let a ≥ 0, 1 < α < 2, 0 ≤ β ≤ 1, γ = α + 2β − αβ be given constants and For a given h ∈ C([a, b], R), the unique solution of the sequential Hilfer linear fractional boundary value problem is given by Next, in view of Lemma 10, we define an operator A : In the sequel, we use the following abbreviations: and The existence and uniqueness results for the problem (36), based on Banach's contraction mapping principle and Krasnoselskii's fixed point theorem, respectively, are as follows.
Theorem 43. Assume that: (43.1) there exists a finite number L > 0 such that, for all t ∈ [a, b] and for all x, y ∈ R, the following inequality is valid: Then, the sequential ψ-Hilfer fractional boundary value problem (36) has a unique solution on [a, b] provided that LΩ + Ω 1 < 1, where Ω and Ω 1 are defined by (37) and (38), respectively.
Theorem 44. Let f : [a, b] × R → R be a continuous function such that: Then, the sequential ψ-Hilfer fractional boundary value problem (36) has at least one solution on [a, b] provided that Ω 1 < 1, where Ω 1 is defined in (38).

Existence Results for the Inclusion Problem
The multivalued case of the problem (36) is also considered in [31], where F : [a, b] × R → P (R) is a multivalued function, and (P (R) is the family of all nonempty subjects of R).
The following existence result is based on the Leray-Schauder nonlinear alternative for multivalued maps.
Then, the sequential Hilfer inclusion fractional boundary value problem (39) has at least one solution on [a, b].
The following lemma deals with a linear variant of the problem (40).
Below are two existence results for the system (40), based on the Leray-Schauder alternative and Krasnoselskii's fixed point theorem, respectively.
Theorem 47. Assume that there exist real constants u i , v i ≥ 0 for i = 1, 2 and u 0 , v 0 > 0 such that for any x i ∈ R, (i = 1, 2), we have: Theorem 48. Assume that f , g : [a, b] × R × R → R are continuous functions satisfying assumption (46.1) in Theorem 46. In addition, we suppose that two positive constants P, Q exist such that for all t ∈ [a, b] and x i , y i ∈ R, i = 1, 2, | f (t, x 1 , x 2 )| ≤ P and |g(t, x 1 , x 2 )| ≤ Q.
then the problem (40) has at least one solution on [a, b].
The following lemma deals with a linear variant of the system (45).
Then, the system (54) has at least one solution on [1, e].