1. Introduction
Fractional-order differentiations and integrations are more accurate tools in expressing real-world problems as compared to integer-order differentiations and integrations. Thus, the theory of fractional differential equations has attracted a lot of attention from many researchers for their wide applications to various fields, such as in physics, bioengineering, electrochemistry, and so on; see [
1,
2,
3] and related references therein. The interested reader is referred to the monographs [
4,
5,
6,
7,
8,
9,
10,
11] for the basic theory of fractional calculus and fractional differential equations.
In the literature, there are several definitions of derivatives and integrals of arbitrary orders. For instance, Kilbas et al. in [
5] introduced fractional integrals and fractional derivatives concerning another function. In a recent paper, Almeida [
12] introduced the so-called
-Caputo fractional operator. Numerous interesting results concerning the existence, uniqueness, and stability of initial value problems and boundary value problems for fractional differential equations with
-Caputo fractional derivatives by applying different types of fixed-point techniques were obtained in [
13,
14,
15].
Hilfer in [
16] generalized both Riemann–Liouville and Caputo fractional derivatives, known as
the Hilfer fractional derivative. We refer to [
17,
18], and references cited therein, for some properties and applications of the Hilfer fractional derivative and to [
19,
20,
21] for initial value problems involving Hilfer fractional derivatives.
Fractional differential equations involving Hilfer derivative have many applications, and we refer to [
22] 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, network traffic, and so on. See [
23,
24] and references cited therein.
In [
25], the authors initiated the study of nonlocal boundary value problems for the Hilfer fractional derivative, by studying the boundary value problem of Hilfer-type fractional differential equations with nonlocal integral boundary conditions
where
is the Hilfer fractional derivative of order
,
and parameter
,
,
is the Riemann–Liouville fractional integral of order
,
,
and
Several existence and uniqueness results were proved by using a variety of fixed point theorems.
In a series of papers [
26,
27,
28,
29,
30], nonlocal boundary value problems involving Hilfer fractional derivatives were studied, with a variety of boundary conditions. Thus, the authors in [
26] studied Hilfer Langevin three-point fractional boundary value problems, the authors in [
27] studied pantograph Hilfer fractional boundary value problems with nonlocal integral boundary conditions, the authors in [
28] studied Hilfer fractional boundary value problems with nonlocal integral integro-multipoint boundary conditions, the authors in [
29] studied Hilfer fractional boundary value problems with nonlocal multipoint, fractional derivative multi-order, and fractional integral multi-order boundary conditions, and the authors in [
30] studied sequential Hilfer fractional boundary value problems with nonlocal integro-multipoint boundary conditions.
Systems of Hilfer–Hadamard sequential fractional differential equations were studied in [
31].
In the present paper, motivated by the research going on in this direction, we study a new class of boundary value problems of sequential Hilfer-type fractional differential equations involving integral multi-point boundary conditions of the form
Here,
is the
-Hilfer fractional derivative operator of order
,
and parameter
,
,
,
is a continuous function,
,
,
,
,
and
is a positive increasing function on
, which has a continuous derivative
on
.
We also cover the multi-valued case of the problem (
3) by considering the following inclusion problem:
where
is a multi valued function, and (
is the family of all nonempty subjects of
).
At the end of this section, we mention that the remaining part of the paper will be organized as follows. In
Section 2, we recall some basic concepts of fractional calculus. In
Section 3, we prove first a lemma relating a linear variant of the problem in (
3) with an integral equation. Moreover, the existence and uniqueness results are established, in the single valued case, by using fixed point theorems. We obtain the existence of a unique solution via Banach’s contraction mapping principle, while Krasnosel’skiĭ’s fixed point theorem is applied to obtain the existence result for the sequential Hilfer fractional boundary value problem (
3). In
Section 4, an existence result is proved for the sequential Hilfer inclusion boundary value problem (
4), via Leray–Schauder nonlinear alternative for multi-valued maps. Illustrative examples for the main results are provided.
2. Preliminaries
This section is assigned to recall some notation in relation to fractional calculus.
Throughout the paper,
denotes the Banach space of all continuous functions from
into
with the norm defined by
We denote by
the
n-times absolutely continuous functions given by
We recall here that a function is called absolutely continuous if, for every , there exists such that implies for all mutually disjoint intervals in A function if and only if f is Lebesgue almost everywhere differentiable with derivative which belongs to such that for all
Definition 1 ([
5])
. Let , be a finite or infinite interval of the half-axis and . In addition, let be a positive increasing function on , which has a continuous derivative on . The ψ-Riemann–Liouville fractional integral of a function f with respect to another function ψ on is defined bywhere represents the Gamma function. Definition 2 ([
5])
. Let and , . The Riemann–Liouville derivatives of a function f with respect to another function ψ of order α correspondent to the Riemann–Liouville is defined bywhere , represents the integer part of the real number α. This is the greatest integer n such that Definition 3 ([
32])
. Let with , is the interval such that and two functions such that ψ is increasing and , for all . The ψ-Hilfer fractional derivative of a function f of order α and type is defined bywhere , represents the integer part of the real number α with . Lemma 1 ([
5])
. Let . Then, we have the following semigroup property given by Next, we present the -fractional integral and derivatives of a power function.
Proposition 1 ([
5,
32])
. Let , and . Then, ψ-fractional integral and derivative of a power function are given by- (i)
- (ii)
Lemma 2 ([
32])
. If , , and , thenfor all , where . 3. Existence and Uniqueness Results for Problem (3)
The following auxiliary lemma concerning a linear variant of the sequential Hilfer boundary value problem (
3) plays a fundamental role in establishing the existence and uniqueness results for the given nonlinear problem.
Lemma 3. Let , , be given constants andFor a given , the unique solution of the sequential Hilfer linear fractional boundary value problemis given by Proof. Applying the operator
on both sides of Equation (
12) and using Lemma 2, there exist real numbers
and
such that
since
From the boundary condition
, we see
Then, we get
From
, we obtain
Substituting the values of
in (
15), we obtain the solution (
14). That the function
as defined in formula (
14), solves the boundary value problem in (
12), (13) can be proved by direct computation. This finishes the proof of Lemma 3. □
Remark 1. If and , then (12) reduces towhich is the Riemann–Liouville fractional differential equation, whereIf and , then (12) is transformed to the Hadamard fractional differential equation of the form:where Next, in view of Lemma 3, we define an operator
by
The continuity of
f shows that
is well defined and fixed points of the operator equation
are solutions of the integral Equation (
14) in Lemma 3.
In the sequel, we use the following abbreviations:
and
By using classical fixed point theorems, we establish in the following subsections existence, as well as existence and uniqueness results, for the sequential -Hilfer fractional boundary value problem (3).
In our first result, we prove the existence of a unique solution of the sequential
-Hilfer fractional boundary value problem (3) based on Banach’s fixed point theorem [
33].
Theorem 1. Assume that:
- (H1)
There exists a finite number such that, for all and for all the following inequality is valid:
Then, the sequential ψ-Hilfer fractional boundary value problem (3) has a unique solution on provided thatwhere Ω and are defined by (17) and (18), respectively. Proof. With the help of the operator defined in (16), we transform the sequential -Hilfer fractional boundary value problem (3) into a fixed point problem, By applying the Banach contraction mapping principle, we shall show that has a unique fixed point.
We put
, and choose
such that
Let
. We show that
For any
, we have
and consequently
which implies that
.
Next, we show that
is a contraction. Let
. Then, for
, we have
which implies that
. As
,
is a contraction. Therefore, by the Banach’s contraction mapping principle, we deduce that
has a fixed point. Obviously, this is the unique solution of the sequential
-Hilfer fractional boundary value problem (3). The proof is complete now. □
The next existence result is based on the a classical fixed point theorem due to Krasnosel’skiĭ’s [
34].
Theorem 2. Let be a continuous function such that:
- (H2)
, and .
Then, the sequential ψ-Hilfer fractional boundary value problem (3) has at least one solution on provided that where is defined in (18). Proof. We consider
where
such that
and
We define the operators
,
on
by
and
For any
, we have
Therefore, which shows that It is easy to see, using the condition that is a contraction mapping.
The operator
is continuous because
f is continuous. In addition,
is uniformly bounded on
because we have
The compactness of the operator
is proved now. Let
with
Then, we have
as
and is independent of
Thus,
is an equicontinuous set. Thus,
is relatively compact on
. By the Arzelá–Ascoli theorem,
is compact on
. Applying the Krasnosel’skiĭ’s fixed point theorem, the sequential
-Hilfer fractional boundary value problem (
3) has at least one solution on
. The proof is completed. □
Example 1. Considering the boundary value problems for ψ-Hilfer type sequential fractional differential equation with integral multi-point boundary conditions,where . In problem (
21), specific constants can be chosen:
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
. Such choices lead to constants as
,
,
and
.
(i) Let the function
be given by
Then, we can check the Lipchitz condition of
as
for all
,
. By setting a constant
, we obtain
which claims that inequality (
19) is fulfilled. Therefore, by an application of Theorem 1, the boundary value problem for
-Hilfer type sequential fractional differential equation with integral multi-point boundary conditions (
21) with (
22) has a unique solution on
. Note that the Theorem 2 can not be applied to this problem because the given function in (
22) is unbounded because
.
(ii) Let the function
be defined, for
, by
It is obvious that the function
is bounded by
which satisfy condition
in Theorem 2. Since
, the conclusion of Theorem 2 yields that the problem (
21) with (
22) has at least one solution on
. If
, then (
23) is reduced to
which satisfies the Lipchitz condition
,
, for all
,
. Theorem 1 cannot be used in this case because
.
4. Existence Results for Problem (4)
For details in multi-valued theory, we refer to [
35,
36,
37].
Definition 4. A function is a solution of the problem (4) if and there exists a function such that a.e. on and In the next theorem, we prove the existence of solutions of the sequential Hilfer inclusion fractional boundary value problem (
4) when the multi-valued map
F has convex values assuming that it is
-Carathéodory, that is, (
i)
is measurable for each
; (
ii)
is upper semicontinuous for almost all
; (
iii) for each
, there exists
such that
for all
with
and for a.e.
For each
denote the set of selections of
F by
For a normed space , let The following lemma is used in the sequel.
Lemma 4 ([
38])
. Let be an -Carathéodory multivalued map and let Θ be a linear continuous mapping from to . Then, the operatoris a closed graph operator in Theorem 3. Assume that In addition, we suppose that:
- (A1)
is -Carathéodory multi-valued map;
- (A2)
there exists a nondecreasing and continuous function and a function such that - (A3)
there exists a number such thatwhere Ω and are given in (17) and (18), respectively.
Then, the sequential Hilfer inclusion fractional boundary value problem (4) has at least one solution on Proof. We define an operator
by
for
in order to transform the problem (
4) into a fixed point problem. Clearly, the solutions of the boundary value problem (
4) are fixed points of
Our proof strategy is to show that all conditions of Leray–Schauder nonlinear alternative for multi-valued maps [
39] are satisfied and, consequently, we conclude that sequential Hilfer inclusion fractional boundary value problem (
4) has at least one solution on
We will give the proof in several steps.
Step 1: is convex for all.
For
, there exist
such that
for almost all
. Let
. Then, we have
F has convex values and thus is convex and . Consequently, which proves that is convex-valued.
Step 2: Bounded sets are mapped by into bounded sets in .
Let
For each
, there exists
such that
Then, for
we have
Thus,
Step 3: Bounded sets are mapped by into equicontinuous sets.
Let
with
and
For each
we obtain
as
and is independent of
By the Arzelá–Ascoli theorem, it follows that
is completely continuous.
In the next step, we will prove that
is upper semicontinuous. In order to reach the desired conclusion, we have to recall from [
35], Proposition 1.2 that
a completely continuous operator is upper semicontinuous if it has a closed graph. Therefore, we will show the following result.
Step 4: has a closed graph.
Consider
and
Then, we will show that
From
there exists
such that, for each
We must show that there exists
such that, for each
Consider the linear operator
given by
Observe that
as
By Lemma 4 that
is a closed graph operator. Moreover, we have
Since
we have that
for some
.
Step 5: We show that there exists an open set with for any and all
Let
for some
Then, there exists
with
such that, for
, we have
Following the computation as in Step 2, we have for each
Thus,
or
In view of
, there exists
M such that
. Consider
Note that
is a compact, upper semicontinuous multi-valued map with convex closed values, and there is no
such that
for some
from the choice of
U. By the Leray–Schauder nonlinear alternative for multivalued maps [
39], we deduce that
has a fixed point
, which is a solution of the sequential Hilfer inclusion fractional boundary value problem (
4). This completes the proof. □
Example 2. Consider the boundary value problem for Hilfer type sequential fractional differential inclusion involving integral multi-point boundary conditionswhere the set is defined byand the function . Choosing constants
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
. With the given data, it can be computed that
,
,
,
. It is obvious that the set
satisfies condition
in Theorem 3. In addition, from
we choose functions
and
by
. Then,
and there exists a constant
satisfying inequality (
25) in
of Theorem 3. Thus, we can conclude that the boundary value problem for Hilfer type sequential fractional differential inclusion involving integral multi-point boundary conditions (
27) has at least one solution on
.