1. Introduction
Fractional differential equations frequently appear in different research areas, such as engineering, physics, economics, chemistry, viscoelasticity, robotics, control theory, etc. We refer the reader to the monographs [
1,
2,
3,
4,
5,
6,
7,
8,
9] for a systematic development of fractional calculus and fractional differential equations. In particular, boundary value problems for fractional differential equations constitute an important and interesting area of research in applied analysis. Recent development on the topic contains different types of fractional-order differential operators such as Caputo, Riemann–Liouville, Hadamard, Erdeyl–Kober, Katugampola, etc. In [
10], Hilfer introduced a new derivative, which interpolates between the Riemann–Liouville and Caputo fractional derivatives. For some applications of the Hilfer fractional derivative, and some recent results on initial and boundary value problems, for instance, see [
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24] and the references cited therein. One can find some recent works on Hilfer–Hadamard fractional differential equations in [
25,
26,
27,
28].
In [
29,
30], Katugampola introduced a generalized fractional operator combining Riemann–Liouville and Hadamard fractional operators. These operators were modified by Jarad et al. [
31] to include the Caputo and Caputo–Hadamard fractional derivatives. The authors in [
32] introduced a new type of fractional derivative, the generalized proportional fractional derivative. The work of [
32] was generalized in [
33,
34] by using the concept of the proportional derivative of a function with respect to another function. In [
35], the Hilfer generalized proportional fractional derivative was proposed. For some recent results on Hilfer generalized proportional fractional differential equations, see [
36,
37].
Recently, in [
38], the authors introduced the
-Hilfer generalized proportional fractional derivative of a function with respect to another function and discussed its properties. As an application, the existence and uniqueness of solutions for the following nonlocal problem of order in
were established:
where
is the
-Hilfer generalized proportional fractional derivative,
is the
-Hilfer generalized proportional fractional integral,
and
is a continuous function.
Motivated by the work presented in [
38], in this paper, we introduce and study a nonlocal mixed boundary value problem for a
-Hilfer generalized proportional fractional differential equation of order in
given by
where
denotes the
-Hilfer generalized proportional fractional derivative operator of order
and type
, respectively,
are given constants,
is a given continuous function,
is the generalized proportional fractional integral operator of order
and
,
,
, are given points. We also study the multivalued analogue of Problem (
1).
Notice that Problem (
1) covers a variety of nonlocal mixed boundary value problems, for example,
- •
Problem (
1) reduces to a nonlocal mixed boundary value problem of Hilfer generalized proportional fractional differential equations of order in
by setting
in it;
- •
Problem (
1) corresponds to a nonlocal mixed boundary value problem of
-Hilfer fractional differential equations of order in
by fixing
in it, which is studied in [
39].
Nonlocal conditions are considered to be more plausible than the classical conditions, as they can correctly describe certain features of physical problems. We emphasize that the mixed nonlocal boundary condition considered in Problem (
1) is of more general form, as it includes multipoint, fractional integral multiorder and fractional derivative multiorder contributions.
The rest of the manuscript is arranged as follows.
Section 2 contains some basic notions of fractional derivatives and integrals and some known results useful in the forthcoming analysis. In
Section 3, we first prove an auxiliary result which plays a key role in transforming the given problem into a fixed point problem. Then, by applying Banach’s contraction mapping principle, the existence of a unique solution for Problem (
1) is shown. Three existence results for Problem (
1) are proved by using the fixed point theorems due to Krasnosel’skiĭ and Schaefer and the nonlinear alternative of the Leray–Schauder type. The methods employed to establish the desired results are based on the well-known tools of fixed point theory. Such methods are effectively applied for solving a variety of problems appearing in applied and mathematical sciences such as equilibrium, optimization, economic and variational inequality problems. Moreover, these methods facilitate developing existence theory for initial and boundary value problems. We investigate the existence of solutions for the multivalued analogue of Problem (
1) in
Section 4. Precisely, the existence results for convex- and nonconvex-valued multifunctions involved in the inclusion problem given by (
19) formulated in
Section 4 are respectively established by applying the Leray–Schauder nonlinear alternative and Covitz–Nadler’s fixed point theorem for multivalued maps. It is imperative to note that the multivalued (inclusion) problems are helpful in the investigation of dynamical systems, stochastic processes, queuing networks, climate control, etc. Examples are provided to illustrate the applicability of the results obtained in
Section 3 and
Section 4.
3. Main Results
In this section, we establish existence and uniqueness results for the
-Hilfer generalized proportional fractional boundary value problem given by (
1). Let us first prove an auxiliary lemma concerning a linear variant of the boundary value problem in (
1), which facilitates transforming the given nonlinear problem into an equivalent integral equation. In our case,
and
.
Lemma 6. Let andThen w is a solution of the following linear ψ-Hilfer generalized proportional fractional boundary value problemif and only if Proof. From Lemma 5 with
we have
which implies that
where
and
Using the first boundary condition (
) in (
8) yields
, since
. In consequence, (
8) takes the form:
From (
9), we have
Inserting the above values into the second boundary condition,
, and using the notation in (
5), we obtain
Substituting the value of
in (
9) yields Equation (
7), as desired. The converse of the lemma can be obtained easily, by direct computation. The proof is finished. □
Let be the Banach space of all continuous functions from to endowed with the norm .
In view of Lemma 6, we introduce an operator
associated with Problem (
1) as
In the sequel, we use the notation:
3.1. Uniqueness Result
Here, the existence of a unique solution for the nonlinear
-Hilfer generalized proportional fractional boundary value problem in (
1) is proved via Banach’s contraction mapping principle [
40].
Theorem 1. Suppose that:
There exists a constant such that, for all and
If where Ω
is defined by (11), then the nonlinear ψ-Hilfer generalized proportional fractional boundary value problem in (1) has a unique solution on Proof. Let
By assumption
, we obtain
We verify the hypothesis of Banach’s contraction mapping principle in two steps.
Step I. Let
with
We show that
For
by using the fact that
we have
Consequently,
which means that
Step II. Now we will show that the operator
is a contraction. Let
Then, for any
, we have
Hence,
which shows that the operator
is a contraction, in view of the assumption
Therefore, the operator
, by Banach’s contraction mapping principle, has a unique fixed point, and thus, the
-Hilfer generalized proportional fractional boundary value problem in (
1) has a unique solution on
The proof is completed. □
3.2. Existence Results
In this subsection, we establish three existence results for the
-Hilfer generalized proportional fractional boundary value problem in (
1). The first existence result is based on the fixed point theorem of Krasnosel’skiĭ [
41].
Theorem 2. Let be a continuous function satisfying In addition, we assume that:
There exists a continuous function such that
Then the nonlinear ψ-Hilfer generalized proportional fractional boundary value problem in (1) has at least one solution on provided that Proof. Consider a closed ball
where
and
is given by (
11). We split the operator
defined by (
10) on
to
X as
where
and
For any
we have
Hence
which shows that
We can prove easily with the help of (
13) that the operator
is a contraction mapping. Note that the operator
is continuous since
f is continuous. In addition, since
is uniformly bounded on
Finally, we have to prove that the operator
is completely continuous. For this, let
Then we have
which tends to zero, independently of
as
Thus,
is equicontinuous, and by the Arzelá–Ascoli Theorem, it is compact on
Thus, the hypotheses of Krasnosel’skiĭ’s fixed point theorem [
41] are satisfied, and hence, its conclusion implies that there exists at least one solution for the nonlinear
-Hilfer generalized proportional fractional boundary value problem in (
1) on
which completes the proof. □
Our next existence result relies on Schaefer’s fixed point theorem [
42].
Theorem 3. Let be a continuous function satisfying the assumption:
There exists a real constant such that for all
Then there exists at least one solution for the nonlinear ψ-Hilfer generalized proportional fractional boundary value problem in (1) on Proof. We will give the proof in two steps.
In the next step, we show that the operator
maps bounded sets into bounded sets in
For
, let
Then, for
, we have
which, after taking the norm for
leads to
Finally, we show that bounded sets are mapped into equicontinuous sets by
For
and
we obtain
which tends to zero, independently of
as
Thus, by the Arzelá–Ascoli theorem, the operator
is completely continuous.
Step II. We show that the set
is bounded. Let
. Then
. For any
, we have
. As in Step I, it follows with the aid of the hypothesis
that
Hence,
which means that the set
is bounded. Thus, the operator
has at least one fixed point by Schaefer’s fixed point theorem [
42], which is a solution for the nonlinear ψ-Hilfer generalized proportional fractional boundary value problem in (
1) on
This completes the proof. □
Our last existence result is based on the Leray–Schauder nonlinear alternative [
43].
Theorem 4. Let In addition, the following assumptions hold:
There exist and a continuous nondecreasing function such that
There exists a constant such thatwhere Ω
is defined by (11).
Then the nonlinear ψ-Hilfer generalized proportional fractional boundary value problem in (1) has at least one solution on Proof. Here, we prove only that there exists an open set with for and since the operator is already shown to be completely continuous in Theorem 3.
Let
be such that
for some
Then, for each
we have
Hence, we obtain
which implies that
In view of
, there is no solution
w such that
. Let us set
Observe that the operator
is continuous and completely continuous. Notice that we cannot find any
satisfying
for some
in view of the definition of
In consequence, we deduce that there exists a fixed point
for the operator
by the application of the Leray–Schauder nonlinear alternative [
43], which is indeed a solution of Problem (
1). This finishes the proof. □
3.3. Illustrative Examples for the Single-Valued Case
Consider the following
-Hilfer generalized proportional fractional boundary value problem:
Here
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
and a function
. Next, we can find that
,
,
and
.
(i) Let the nonlinear Lipschitzian function
be presented in the form
It is easy to see that
for all
and
which implies that
Moreover,
. From Theorem 1, we conclude that Problem (
14) with
f defined in (
15) has a unique solution on
.
(ii) Let the function
be given by
Clearly,
holds true, as
for all
,
with
. Observe that the unique solution to Problem (
14) is not possible since
. However, we have
and
Therefore, applying the result in Theorem 2, we deduce that Problem (
14) with
f defined in (
16) has at least one solution on
.
(iii) Consider a nonlinear function
defined by
Since
for all
,
Theorem 3 implies that Problem (
14) with
f defined in (
17) has at least one solution on
.
(iv) Let us consider
as
which is bounded since
Choosing
and
, we have that
and there exists a constant
K such that
satisfying the inequality in
. Hence, by Theorem 4, Problem (
14) with
f defined in (
18) has at least one solution on
.
4. Multivalued Case
In this section, we study the multivalued case of the nonlinear
-Hilfer generalized proportional fractional boundary value problem in (
1), given by
where
is a multivalued map (
denotes the family of all nonempty subsets of
) and the other symbols are the same as defined in Problem (
1).
Let be a normed space. In the following, we denote the classes of all closed, bounded, compact and compact and convex sets in by and respectively.
The set of selections of
F for each
is defined as
For details on multivalued analysis, see [
44,
45,
46].
4.1. Existence Results for Problem (19)
Definition 4. A function is called a solution of the ψ-Hilfer generalized proportional inclusion fractional boundary value problem in (19) if there exists a function with almost everywhere (a.e.) on such that 4.1.1. Case 1: Convex-Valued Multifunctions
Here we consider the case when the multifunction
F has convex values and we prove an existence result for the
-Hilfer generalized proportional inclusion fractional boundary value problem in (
19) by using the nonlinear alternative for Kakutani maps [
43] and a closed graph operator theorem [
47], under the assumption that
F is
-Carathéodory.
Theorem 5. Assume that:
The multifunction is -Carathéodory;
There exist a nondecreasing function and a continuous function such that
There exists a positive number M such thatwhere Ω
is given by (11).
Then the ψ-Hilfer generalized proportional inclusion fractional boundary value problem (19) has at least one solution on Proof. We introduce a multivalued operator
as
We will prove that the operator
N satisfies the hypotheses of the Leray–Schauder nonlinear alternative for Kakutani maps [
43] in several steps.
Let
, be a bounded set in
. For each
and
there exists
such that
For
using the assumption
, we obtain
and consequently
Let
and
Then there exists
such that
Let
. Then
as
independently of
Hence,
by the Arzelá-Ascoli theorem, is completely continuous.
It is obvious, since is convex by the assumption that F has convex values.
Let
,
and
. Then we show that
. Observe that
implies that there exists
such that, for each
, we have
For each
, we must have
such that
We introduce a continuous linear operator
as
Clearly,
as
and consequently, by the closed graph operator theorem [
47],
is a closed graph operator. In addition, we have
and
for some
Thus,
N has a closed graph, which implies that the operator
N is upper semicontinuous, because, by [
44], Proposition 1.2, a completely continuous operator is upper semicontinuous if it has a closed graph.
Step 5.
There exists an open set such that, for any and all
Let
Then there exists
with
such that, for
, we have
Following the computation as in Step 1, for each
we have
In consequence, we obtain
By the assumption
, there exists a positive number
M satisfying
. Let us define a set
Obviously,
is a compact, conve- valued and upper semicontinuous multivalued map. By the definition of
there does not exist any
for some
satisfying
Therefore, the conclusion of the Leray–Schauder nonlinear alternative for Kakutani maps [
43] applies, and hence, the operator
N has a fixed point
Thus, Problem (
19) has at least one solution on
which concludes the proof. □
4.1.2. Case 2: Nonconvex-Valued Multifunctions
Here, we prove the existence of a solution for the
-Hilfer generalized proportional fractional inclusion boundary value problem in (
19) with a nonconvex-valued multivalued map via the fixed point theorem for contractive multivalued maps from Covitz and Nadler [
48].
Definition 5 ([
49])
. Let be a metric space induced from the normed space and be defined bywhere and . Theorem 6. Assume that:
is such that is measurable for each ;
for almost all and with and for almost all .
Then the ψ-Hilfer generalized proportional inclusion fractional boundary value problem in (19) has at least one solution on ifwhere Ω
is given by (11). Proof. We will prove that the operator
defined by (
20), satisfies the hypotheses of Covitz–Nadler’s fixed point theorem for multivalued maps [
48].
The set-valued map
is measurable by the measurable selection theorem ([
50], Theorem III.6), and hence, it admits a measurable selection
. By the assumption
we obtain
that is,
, and hence,
F is integrably bounded. Therefore, we deduce that
.
Now we show that
for each
. For that, let
with
in
Then
and we can find
such that, for each
Then we can obtain a subsequence (if necessary)
converging to
v in
as
F has compact values. Thus,
and, for each
, we have
Thus,
Step II. Here, we establish that there exists
such that
Let
and
. Then there exists
such that, for each
,
By
, we have
Thus, there exists
such that
Let us define
by
Then there exists a function
which is a measurable selection of
since the multivalued operator
is measurable by Proposition III.4 in [
50]. Hence,
, and for each
, we have
. Thus, for each
, we have
In consequence, we obtain
which leads to
On switching the roles of
w and
, we obtain
which shows that
N is a contraction. Consequently, by Covitz–Nadler’s fixed point theorem [
48], the operator
N has a fixed point
w which corresponds to a solution of the
-Hilfer generalized proportional inclusion fractional boundary value problem (
19). The proof is complete. □
4.2. Illustrative Examples for the Multivalued Case
Let us consider the
-Hilfer generalized proportional inclusion:
with the boundary conditions given by (
14). Here, all the parameters are the same, as considered in Problem (
14).
(a) Consider
defined by
and note that
F is an
-Carathéodory, as
Letting
and
, we find that
and there exists a constant
satisfying the inequality in
. Hence, by Theorem 5, the
-Hilfer generalized proportional inclusion in (
21) with the boundary conditions in (
14), and
F defined in (
22), has at least one solution on
.
(b) Let
be defined by
It is easy to check that
F is measurable and satisfies
. Setting
, we obtain
,
and
. Thus, all the assumptions of Theorem 6 are satisfied, and hence, there exists at least one solution for the inclusion in (
21) with the boundary conditions in (
14) and
F defined in (
23) on
.