1. Introduction
Let
. We consider the uniqueness and existence for the following nonlinear integro-differential Equation (NI-D equation) with the boundary condition:
where
is an unknown function,
is a constant,
,
is the Liouville–Caputo fractional derivative of order
, and
is the Riemann–Liouville fractional integral of order
. In particular, for
, Equation (
1) was found to be
Fractional differential and integral equations provide powerful tools in describing and modeling many phenomenons in various fields of science and engineering, such as control theory, porous media, memory and electromagnetics [
1,
2,
3,
4]. There has been a great deal of research published on the existence and uniqueness of fractional differential and integral equations involving Riemann–Liouville or Liouville–Caputo operators with initial conditions or boundary value problems [
5,
6,
7,
8,
9,
10,
11,
12,
13].
In 2022, Rezapour et al. [
14] investigated the existence of solutions for a category of the multi-point boundary value problem involving a
p-Laplacian differential operator with the generalized fractional derivatives depending on another function. The authors in [
15] considered the existence, uniqueness and stability of a positive solution in relation to a fractional version of a variable order thermostat model equipped with nonlocal boundary values in the Caputo sense using Guo–Krasnoselskii’s fixed point theorem on cones.
In 2021, Turab et al. [
16] looked into the existence of solutions for a class of nonlinear boundary value problems on a hexasilinane graph with applications in chemical formulas. The authors in [
17] dealt with the existence and Ulam–Hyers stability (UHs) of Caputo-type fuzzy fractional differential equations (FFDEs) with time-delays by applying Schauder’s fixed point theorem and a hypothetical condition. In 2017, Sun et al. [
18] studied the existence and uniqueness for the following system of FDEs with a boundary value based on Banach’s contractive principle (BCP) and the Laray–Schauder fixed point theorem (L-SFTP):
where
are real constants and
is a continuous function. Their work relies on Lemma 2.4 in the paper, which states the following:
Lemma 1. Let , where denotes the integer part of κ and let be continuous. A function is a solution to the FDE with the boundary valueif and only if However, the authors consider this lemma is to be incorrect, and the term
should not appear in the lemma. Indeed,
plus the boundary condition
only implies that
To move forward, we begin by introducing several differential and integral operators, a Banach space
, the Mittag–Leffler function (the M-L function) as well as Babenko’s approach (BA) in
Section 2. Then, we present sufficient conditions for the existence and uniqueness of the solutions using the BCP and the L-SFPT, with illustrative examples to show the applications of the main theorems in
Section 3. Finally, we summarize the entire paper in
Section 4.
2. Preliminaries
We define the Banach space
for
as
with the norm
Clearly, .
The Riemann–Liouville fractional integral
of order
is defined for function
as (see [
1,
2])
if the integral exists. In particular,
from [
19]. In fact,
where
is the Dirac delta function, which is an identity in terms of convolution.
Let
. The Liouville–Caputo derivative of fractional order
of function
is defined as
if the integral exists.
The two-parameter Mittag–Leffler function [
3] is defined by
BA [
20] is a useful instrument in solving differential and integral equations with initial conditions by treating bounded integral operators as normal variables. The method itself is close to the Laplace transform while dealing with differential equations with constant coefficients, but it can be applied to differential and integral equations with variable coefficients [
21,
22]. Evidently, it is always necessary to prove the convergence of solution series, otherwise the solution is not well-defined. To demonstrate this technique in detail, we present the following example to solve Abel’s integral equation, as well as Lemma 2, which will play an important role in the subsequent section to define the nonlinear mappings.
Consider Abel’s integral equation for
and a constant
c
where
is a continuous function. Clearly,
Treating the factor
as a normal variable, we come to
which is well-defined.
The following lemma is another application of BA.
Lemma 2. Let be continuous. A function is a solution of the FDE with the boundary valueif and only if is a solution of the fractional integral equationwhere . Proof. Applying the operator
to both sides of the equation
and using the condition
for
, we find
Dedifferentiating the above equation
times and setting
, we derive that
Therefore,
by noting that
. In summary, we have
Using BA (treating the factor
as a variable), we come to
utilizing
We note that all the above steps are reversible since BA is.
It remains to be shown that all series on the right-hand side of Equation (
3) are convergent in terms of the norm in
. Clearly,
by the fact that
. Similarly, we have
This completes the proof of Lemma 2. □
Remark 1. (i) In particular, for , the FDE with boundary valuehas the solution (ii) Clearly, the FDE with boundary valuecan be solved along the same lines. This is clearly a generalization of Equation (2). (iii) Lemma 2 still holds for using the same computation.
The following theorems will be used in
Section 3 to study the existence and uniqueness.
Theorem 1. (Banach’s contractive principle). If is a contraction mapping on a complete metric space , then there is exactly one solution of for .
Theorem 2. (Leray–Schauder’s alternative).
Consider the continuous and compact function T of a Banach space S into itself. The boundedness ofimplies that T has a fixed point. 3. Existence and Uniqueness of Solutions
Theorem 3. Assume that is a continuous function satisfying the Lipschitz condition for a constant :where . Furthermore, we suppose Then, the FD system (1) has a unique solution in the space . Proof. From Lemma 2, we define a nonlinear mapping Y over the space
as
Clearly,
is continuous on
since
and
is continuous. Moreover,
from the proof of Lemma 2. Clearly,
which infers that
Thus
is a mapping from
to itself. To prove that
is contractive, we notice that, for
Clearly, for
, we derive that
Regarding
,
by using
from Lemma 2. Finally,
It follows from the above that
Since
,
is contractive. By BCP, the FD system (
1) has a unique solution in the space
. This completes the proof of Theorem 3. □
Example 1. The following FDE with boundary valuehas a unique solution in the space . Proof. Clearly, the function
satisfies
Hence, we compute the
value in Theorem 3 as
Therefore . By Theorem 3, the FD system has a unique solution in . This completes the proof of Example 1. □
We are now ready to present the following theorem regarding existence of solutions to the FDE (
1).
Theorem 4. Assume is a continuous and bounded function and Then, the FDE (1) with a boundary value has at least one solution in the space . Proof. Again, we consider the nonlinear mapping
from
to itself by
We first claim (i) that
is continuous. Indeed, we find from the proof of Theorem 3
and we deduce that
is continuous since
is continuous.
(ii) maps bounded sets to bounded sets in .
Since
is bounded, there exists a constant
such that
for all
and
. Let
be a bounded set in
. Then, there exists
such that
for all
. Clearly, from the above,
which is uniformly bounded.
(iii) is completely continuous from to itself.
By the Arzela–Ascoli theorem, it remains to be shown that
is equicontinuous over the bounded set
. For
and
, suppose that
. Then, we have
Let us estimate
first. Clearly,
It follows that, for all
,
By the mean value theorem,
where
. This derives that
Let us consider the term
. Evidently,
and
Like the term
, we have
and
as well as
In summary,
follows similarly. Therefore,
is completely continuous.
(iv) Finally, we prove that the set
is bounded. For any
,
. This infers that
Hence,
is bounded. By L-SFPT, the FDE (
1) with boundary value has at least one solution in the space
. This completes the proof of Theorem 4. □
Example 2. The following FDE with boundary valuehas at least one solution in the space . Proof. Clearly,
is a bounded function over
. By Theorem 4, we need to evaluate the value of
by the online calculator. Therefore, by Theorem 4, the fractional value problem has a solution in the space
. □
Remark 2. Clearly, fromin Theorem 3, we imply thatin Theorem 4. However, a Lipschitz function over may not be a bounded function. Conversely, a bounded function over may not be a Lipschitz function. Furthermore, it seems difficult to study the uniqueness of Example 2, since we cannot claim thatis a Lipschitz function due to the factor . As a final example, in the following, we discuss both the existence and uniqueness of a solution simultaneously.
Example 3. The following FDE with boundary valuehas a unique solution in the space . Proof. Clearly, the function
is bounded and satisfies the condition
It follows from Example 1 that
. Then,
in Theorem 4 by Remark 2. Hence, the system has a solution in the space
by Theorem 4. In addition, the solution is unique by Theorem 3. □
4. Conclusions
We studied the uniqueness and existence of solutions to the nonlinear two-term fractional integro-differential equation with a boundary condition by using Babenko’s approach, the Mittag–Leffler function, Banach’s contractive principle and the Laray–Schauder fixed point theorem. The current work also indicated key errors in the paper (Applied Mathematics, 2017, 8, 312–323) in handling a one-term differential equation. Furthermore, we provided three examples to demonstrate the application of our main theorems using the online Mittag–Leffler calculator. Clearly, it would be interesting and challenging to study the same system with a variable coefficient .
Author Contributions
C.L., methodology, writing—original draft preparation, supervision and project administration. R.S. (Reza Saadati),writing—original draft preparation and project administration. R.S. (Rekha Srivastava), writing–original draft preparation and project administration. J.B., methodology and writing—original draft preparation. All authors have read and agreed to the published version of the manuscript.
Funding
Chenkuan Li is supported by the Natural Sciences and Engineering Research Council of Canada (Grant No. 2019-03907).
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
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