1. Introduction
Fractional differential equations appear in the mathematical modeling of many real-world phenomena occurring in engineering and scientific disciplines, for instance, see References [
1,
2,
3,
4,
5,
6]. Mathematical models based on fractional-order integral and differential operators yield more insight into the characteristics of the associated phenomena, as such operators are nonlocal in nature, in contrast to classical ones. In particular, coupled systems of fractional-order differential equations have received great attention in view of their great utility in handling and comprehending practical issues, such as the synchronization of chaotic systems [
7,
8], anomalous diffusion [
9], and ecological effects [
10]. For recent theoretical results on the topic, we refer the reader to a series of papers [
11,
12,
13,
14,
15,
16,
17,
18] and the references cited therein.
Recently, in Reference [
19], the authors discussed existence and the uniqueness of solutions for sequential Caputo and Hadamard fractional differential equations subject to separated boundary conditions as
where
and
are the Caputo and Hadamard fractional derivatives of orders
p and
q, respectively,
, starting at a point
,
is a continuous function and given constants
,
.
In this paper, we established the existence criteria for a coupled system of sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions as:
where
and
are notations of the Caputo and Hadamard fractional derivatives of orders
and
, respectively,
,
, the nonlinear continuous functions
,
,
,
,
. Meanwhile, the different definitions of Caputo and Hadamard fractional derivatives that appeared in System (
2) are proposed to study the existence theory of solutions of a fractional differential system using a variety of fixed-point theorems. A special case, when
,
, in differential Equation (
2) can be presented as:
which is mixed type of ordinary differential equations and boundary conditions.
The rest of this paper is organized as follows:
Section 2 aims to recall basic definitions and lemmas used in this paper. Section is devoted to the main results concerning the existence and uniqueness of solutions for for System (
2). The Leray–Schauder alternative and Krasnoselskii’s fixed-point theorem were applied to prove existence, while the uniqueness result was obtained via the Banach contraction mapping principle. Some illustrative examples are presented in
Section 4.
3. Main Results
Let , , be the Banach space of all continuous functions form to . Space endowed with the norm is a Banach space. In addition, let with the norm It is obvious that product space is a Banach space with the norm
Now, for brevity, we use the notations:
and
where
. We also use this one for a single fractional integral operator of the Riemann–Liouville and Hadamard types of orders
and
, respectively.
In view of Lemma 3, we define two operators
by
where
and
For computational convenience, we set
Note that all information of Problem (
2) is contained in constants
,
, which are used to establish the following existence theorems. Banach’s contraction mapping principle is applied in the first result to prove the existence and uniqueness of solutions of System (
2).
Theorem 1. Suppose that are continuous functions. In addition, we assume that:
- (H1)
there exist constants , such that for all and and
Then, System (2) has a unique solution on if Proof. Define
and
, such that
Now, we show that the set
where
For
we have that
By direct computation, we get
and
Consequently, it follows that
which implies
. Next, we show that operator
is contraction mapping. For any
we obtain
Therefore, we get the following inequality:
From Inequalities (24) and (25), it yields
As
therefore
is a contraction operator. By applying Banach’s fixed-point theorem, operator
has a unique fixed point in
. Hence, there exists a unique solution of Problem (
2) on
. The proof is completed. ☐
Now, we prove our second existence result via the Leray–Schauder alternative.
Lemma 4. (Leray-Schauder alternative) [22]. Let be a completely continuous operator. Let Then, either set is unbounded, or F has at least one fixed point.
Theorem 2. Assume that there exist real constants for and , such that for any we haveIf and where are given in Equations (
19)–(22)
, then Problem (
2)
has at least one solution on Proof. By continuity of functions
on
operator
is continuous. Now, we show that the operator
is completely continuous. Let
be bounded. Then, there exist two positive constants,
and
, such that
Then, for any
we have
which yields
In addition, we obtain that
Hence, from the above inequalities, we get that set
is uniformly bounded. Next, we prove that set
is equicontinuous. For any
and
such that
we have
Analogously, we can get the following inequality:
Hence, set
is equicontinuous. By applying the Arzelá–Ascoli theorem, set
is relatively compact, which implies that operator
is completely continuous. Lastly, we show that set
,
is bounded. Now, let
then we obtain
, which yields, for any
Then, we have
which imply that
Thus, we get the inequality
where
which shows that set
is bounded. Therefore, by applying Lemma 4, operator
has at least one fixed point in
. Therefore, we deduce that Problem (
2) has at least one solution on
The proof is complete. ☐
The last-existence theorem is based on Krasnoselskii’s fixed-point theorem.
Lemma 5. (Krasnoselskii’s fixed-point theorem) [23] Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let be operators, such that (i) where , (ii) A is compact and continuous, and (iii) B is a contraction mapping. Then, there exists , such that . Theorem 3. Assume that are continuous functions satisfying assumption in Theorem 1. In addition we suppose and there exist two positive constants such that for all and Ifthen the problem (
2)
has at least one solution on Proof. Let
be a ball, where a constant
To apply Lemma 5, we decompose operator
into four operators
, and
on
as
Note that
and
. In addition, observe that ball
is a closed, bounded, and convex subset of Banach space
Now, we show that
for satisfying condition
of Lemma 5. Setting
and using Condition (27), we then have
Furthermore, we can find that
That yields
To show that operator
is a contraction mapping satisfying condition
of Lemma 5, for
, we have
and
It follows Form (29) and (30) that
which is a contraction by inequality in (28). Therefore, condition
of Lemma 5 is satisfied. Finally we show that operator
satisfied the condition
of Lemma 5. By applying the continuity of functions
on
we can conclude that operator
is continuous. For each
one has
and
Then, we obtain the following fact
which implies that set
is uniformly bounded. Next, we show that set
is equicontinuous. For
, such that
, and for any
we prove that
Thus,
tends to zero as
Therefore, set
is equicontinuous. By applying the Arzelá–Ascoli theorem, operator
is compact on
By application of Lemma 5, there exists
, such that
. Therefore, Problem (
2) has at least one solution on
This completes the proof. ☐
Example 1. Consider the following coupled system of sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions: Here , , , , , , , , , , , , , . From the given information, we find that , and , which lead to , , ,
(i) Let two nonlinear functions
f,
be given by
Since
and
we obtain
By the conclusion of Theorem 1, Problem (31) with Problems (32) and (33) have a unique solution on
.
(ii) Now consider functions
f,
defined by
It is easy to verify that and As and , by applying Theorem 2, we get that System (31) with Systems (34) and (35) have at least one solution on .
(iii) Define functions
f,
by
We have
,
and
and
Then we obtain Using Theorem 3, the problem (31) with (36) and (37) has at least one solution on . Observe that the inequality and, thus, Condition (23) is not satisfied. Therefore, Theorem 1 cannot be applied for this case.