1. Introduction
Fractional differential equations (FDEs) have a profound physical background and rich theoretical connotations and have been particularly eye-catching in recent years. Several-order differential equations refer to equations that contain fractional derivatives or fractional integrals. Currently, fractional derivatives and fractions order integrals have a wide range of applications in many disciplines such as physics, biology, chemistry, etc. For more information see [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10].
Several concepts about fractional derivatives such as Liouville, Caputo, Hadamard, Caputo–Fabrizio and Hilfer derivatives have been proposed in recent years. As a result, a wide range of arbitrary order differential equation designs based on these fractional operators have emerged. The qualitative properties of solutions (existence, uniqueness, and stability) have been studied by several researchers. For the applications and latest work regarding to these operators, we suggest readers to [
11,
12,
13,
14,
15].
On the other hand, fractional hybrid differential equations emerge from a wide range of spaces of applied and physical sciences, this class of equations can be used to model and describe non-homogeneous physical events, e.g., in the deflections of a curved beam with a constant or varying cross-section, electromagnétic waves, or gravity-driven streams, etc. Many researchers have recently become interested in a novel class of mathematical modelings based on hybrid fractional differential equations with hybrid or non-hybrid boundary value conditions [
16,
17,
18,
19].
Coupled systems, including FDEs, are important to study because they appear in a wide range of practical applications. We refer to a collection of papers for some theoretical approaches on coupled systems [
20,
21,
22,
23]. Recent works related to our work were done by [
24,
25]. Sitho et. al. [
24] studied the existence of hybrid fractional integrodifferential equations described as follows
and
where
denotes the Caputo fractional derivatives of order
and
denotes the Riemann-Liouville fractional integral of order
The functions
and
are continuous functions. Boutiara et. al. in [
25] studied the existence of solutions for the following coupled system in the sense of
-Caputo fractional operators
Motivated by the novel advancements of hybrid fractional integrodifferential equations and their applications, also by the above argumentations, by means of Dhage’s hybrid fixed point theorem for three operators in a Banach algebra [
26] and Dhage’s helpful generalization of Krasnoselskii’s fixed point theorem [
27], we investigate the existence of a solution for two classes of coupled hybrid fractional integrodifferential equations. The first class described by
where
is the
-Hilfer fractional derivative of order
and type
with respect to an increasing function
with
, for all
are the
-Riemann-Liouville fractional integral of order
. The functions
with
are continuous functions. We prove an existence result for the
-Hilfer hybrid system (
1) using Dhage’s hybrid fixed point theorem for three operators in a Banach algebra [
26].
We will also look at the necessary conditions for the existence of a solution for
-Hilfer hybrid system which described as follows
where
is the
-Hilfer fractional derivative of order
such that
and types
with respect to an increasing function
with
, for all
are the
-Riemann-Liouville fractional integral of order
. The functions
with
are continuous functions that meet certain standards that will be mentioned later. We prove an existence result for the
-Hilfer hybrid system (
2) using a useful generalization of Krasnoselskii’s fixed point theory due to Dhage [
27].
The remainder of the paper is laid out as follows: We will go through some helpful preliminaries in
Section 2. The existence of the solutions for
-Hilfer hybrid system (
1) has been investigated in
Section 3, whereas the existence of the solutions for
-Hilfer hybrid system (
2) has been addressed in
Section 4. We provide a relevant examples in
Section 5 to demonstrate our findings. In the last section, we will provide some last observations about our findings.
3. Existence of Solution for -Hilfer Hybrid Systems (1)
In this section, we will study the existence of solutions for
-Hilfer hybrid system (
1) by means of Dhage’s hybrid fixed point theorem for three operators in a Banach algebra [
26]. To achieve our main results, the following hypotheses must be satisfied
(H
) The functions
,
are continuous functions and there exist two positive functions
with bound
and
respectively, such that for each
we have
and
(H
) The functions
are continuous functions. For all
there exists a functions
with bound
and a continuous nondecreasing function
such that
Lemma 4 ([
26])
. Let be a closed convex and bounded subset of the Banach algebra and let and be three operators such that- (a)
and are Lipschitzian with Lipschitz constant respectively,
- (b)
is compact and continuous,
- (c)
- (d)
where
Then the operator equation has a solution in .
Lemma 5. Let , are continuous functions with If satisfies the ϕ-Hilfer hybrid system (1), then, satisfies the following integral equations Proof. In the beginning, we assume that
is a solution of
-Hilfer hybrid system (
1). We will prove that
satisfies the integral Equation (
3). First, let
Taking the operator
on both sides of the first Equation (
4) and using Lemma 3, we have
By the condition
we obtain
By the same way, we obtain
From (
6) and (
7), we conclude that
satisfies the integral Equation (
3).
Conversely, assume that
satisfies the integral Equation (
3). Applying the operator
of the integral Equation (
6), with replace
ı by 0, we obtain
Next, applying
on both sides of Equation (
6), we have
The proof is completed. □
Theorem 2. Assume that (H), (H) hold. Then, ϕ-Hilfer hybrid system (1) has at least one solution in , provided thatwhere R is the radius of the closed ball set defined in the following proof. Proof. Let us consider a closed ball set
with
where
By Lemma 5, the
-Hilfer hybrid system (
1) is equivalent to Equation (
3). Define the operator
as
where
To use Lemma 4, we define the following operators
by
and
by
Thus, the coupled system of the above hybrid integral Equation (
9) can be written as a system of operator equations as
we will demonstrate that the operators
,
, and
meet all conditions in Lemma 4. This will be completed in the steps that follow.
Step (1): We begin by demonstrating that
and
are Lipschitz on
. For any
. Then via (H1), we have
By the same way, one can obtain
Hence,
is a Lipschitzian on
with
. Next, with the operator
for any
and via (H1), we have
Hence, is a Lipschitzian on with
Step (2):
is compact and continuous on
. For this purpose let
be a sequence in
such that
. Then by the Lebesgue dominated convergence theorem, for all
we have
This means that
is continuous on
Now, we prove that the set
is uniformly bounded in
. For any
, we have
By the same way, we obtain
This proves that
is uniformly bounded on
. Now, we shall show that
is an equicontinuous set in
. Let
and
. Then, we have
On the other hand, by the same way, we obtain
Thus, is an equicontinuous set in
Step (3): Let
and
such that
Therefore, .
Step (4): Finally, we will show that
Since
then, by (
8), one can obtain
with
and
As a result of Lemma 4, we conclude that the operator equation
has at least one solution in
. □
4. Existence of Solution for -Hilfer Hybrid System (2)
In this section, we will study the existence solution of
-Hilfer hybrid system (
2), according to Dhage’s helpful generalization of Krasnoselskii’s fixed point theorem [
27]. We consider the following assumptions to obtain our main results:
(H
) The functions
are continuous and there exist two positive functions
with bound
respectively, such that for each
we have
(H
) The functions
are continuous. For all
there exists a functions
such that
and
Lemma 6 [
27])
. Let be a Banach space and be a closed convex, bounded and nonempty subset of a Banach space . Let and be operators such that- (i)
is completely continuous,
- (ii)
is a contraction,
- (iii)
for all .
Then the operator equation has a solution.
Definition 3. A function is said to be a solution of ϕ-Hilfer hybrid system (2) if and are continuous for each and satisfies ϕ-Hilfer hybrid system and the conditions in (2). Lemma 7. Let such that are continuous functions. If satisfies the ϕ-Hilfer hybrid system (2), then, satisfies the following integral equations Proof. In the beginning, we assume that
is a solution of
-Hilfer hybrid system (
2). We will prove that
satisfies the integral Equation (
10). First, let
Taking the operator
on both sides of the (
11) and using Lemma 3, we have
By the condition
we obtain
Inserting the operator
into both sides of Equation (
12) and using Lemma 3, with semigroup property
we have
By the condition
we obtain
By the same way, we obtain
From (
13) and (
14), we conclude that
satisfies the integral Equation (
10).
Conversely, assume that
satisfies the integral Equation (
10). Applying the operators
and
of the integral Equation (
13), with replace
ı by 0, we obtain
Applying again the operators
and
of the integral Equation (
14), with replace
ı by 0, we obtain
The proof is completed. □
In the following analyses, we use the following notations to keep things simple,
and
where
and
are bound of the functions
and
respectively.
Theorem 3. Assume that (H), (H) hold. Ifthen, the ϕ-Hilfer hybrid system (2) has at least one solution on Proof. Define a closed ball set
with
Define the operator
as
where
To use Lemma 6, we define operators
by
by
and
by
and
Thus, the coupled system of the above hybrid integral Equation (
15) can be written as a system of operator equations as
In the steps that follow, we will show that the operators and obey the claims of Lemma 6.
Step (1):
is completely continuous. The operator
is obviously continuous. For
,
we have
Let
and
Then, we have
By the same way, we obtain
Thus, is equicontinuous. Consequently, is relatively compact on . Hence, by the Arzelá-Ascoli theorem, we conclude that is compact on .
Step (2):
is a contraction mapping. Let
. Then for
we have
By same technique, one can obtain
Step (3): For any
we have
Similarly, one can obtain
Thus,
. Hence, the last condition in Lemma 6 holds. According to above steps together with Lemma 6, we conclude that
-Hilfer hybrid system (
2) has at least one solution on
□
6. Conclusions
Recently, the theory of fractional differential equations has attracted the interest of several researchers in different filed due to its various applications. In particular, those involving generalized fractional operators. It is important that we investigate the fractional systems with generalized Hilfer derivatives since these derivatives cover many systems in the literature and they contain a kernel with different values that generate many special cases.
The existence of solutions for two class
-Hilfer hybrid fractional integrodifferential equations was investigated in this study. The first result was obtained by applying Dhage’s hybrid fixed point theorem for three operators in a Banach algebra [
26], while the second result was reached by applying Dhage’s helpful generalization of Krasnoselskii’s fixed point theorem [
27]. The main conclusions are well-illustrated with examples. The results obtained in this work includes the results of Sitho et al. [
24], Boutiara et al. [
25] and cover many problems which do not study yet.