1. Introduction
In the last few decades, fractional-order single-valued and multivalued boundary value problems containing different fractional derivatives such as Caputo, Riemann–Liouville, Hadamard, etc., and classical, nonlocal, integral boundary conditions have been extensively studied, for example, see the articles [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12] and the references cited therein.
In the study of variational principles, fractional differential equations involving both left and right fractional derivatives give rise to a special class of Euler–Lagrange equations, for details, see [
13] and the references cited therein. Let us consider some works on mixed fractional-order boundary value problems. In [
14], the authors discussed the existence of an extremal solution to a nonlinear system involving the right-handed Riemann–Liouville fractional derivative. In [
15], a two-point nonlinear higher order fractional boundary value problem involving left Riemann–Liouville and right Caputo fractional derivatives was investigated, while a problem in terms of left Caputo and right Riemann–Liouville fractional derivatives was studied in [
16]. A nonlinear fractional oscillator equation containing left Riemann–Liouville and right Caputo fractional derivatives was investigated in [
17]. In a recent paper [
18], the authors proved some existence results for nonlocal boundary value problems of differential equations and inclusions containing both left Caputo and right Riemann–Liouville fractional derivatives.
Integro-differential equations appear in the mathematical modeling of several real world problems such as, heat transfer phenomena [
19,
20], forced-convective flow over a heat-conducting plate [
21], etc. In [
22], the authors studied the steady heat-transfer in fractal media via the local fractional nonlinear Volterra integro-differential equations. Electromagnetic waves in a variety of dielectric media with susceptibility following a fractional power-law are described by the fractional integro-differential equations [
23].
Motivated by aforementioned applications of integro-differential equations and [
18], we introduce a new kind of integro-differential equation involving right-Caputo and left-Riemann–Liouville fractional derivatives of different orders and right-left Riemann–Liouville fractional integrals and solve it subject to nonlocal boundary conditions. In precise terms, we prove existence and uniqueness of solutions for the problem given by
where
and
denote the right Caputo fractional derivative of order
and the left Riemann–Liouville fractional derivative of order
,
and
denote the right and left Riemann–Liouville fractional integrals of orders
respectively,
are given continuous functions and
. It is imperative to notice that the integro-differential equation in (
1) and (
2) contains mixed type (integral and nonintegral) nonlinearities.
We organize the rest of the paper as follows.
Section 2 contains some preliminary concepts related to our work. In
Section 3, we prove an auxiliary lemma for the linear variant of the problem (
1) and (
2). Then we derive the existence results for the problem (
1) and (
2) by applying a fixed point theorem due to Krasnoselski and Leray–Schauder nonlinear alternative, while the uniqueness result is established via Banach contraction mapping principle. Examples illustrating the main results are also presented.
3. Main Results
In the following lemma, we solve a linear variant of the problem (
1) and (
2).
Lemma 2. Let . Then the linear problemis equivalent to the fractional integral equation:whereand it is assumed that Proof. Applying the left and right fractional integrals
and
successively to the integro-differential equation in (
3), and then using Lemma 1, we get
where
and
are unknown arbitrary constants.
In view of the condition
, it follows from (
7) that
. Inserting
in (
7) and then using the nonlocal boundary conditions
in the resulting equation, we obtain a system of equations in
and
given by
where
Solving the system (
8), we find that
where
is defined by (6). Substituting the values of
and
together with the notations (
5) in (
7), we obtain the solution (
4). The converse follows by direct computation. This completes the proof. □
Let
denote the Banach space of all continuous functions from
equipped with the norm
By Lemma 2, we define an operator
associated with the problem (
1) and (
2) as
Notice that the fixed points of the operator
are solutions of the problem (
1) and (
2).
In the forthcoming analysis, we use the following estimates:
where we have used
In the sequel, we set
where
3.1. Existence Results
In the following, we prove our first existence result for the problem (
1) and (
2), which relies on Krasnoselskii’s fixed point theorem [
24].
Theorem 1. Assumed that:
There exist such that
There exist such that
and , where
Then the problem (1) and (2) has at least one solution on if where Proof. Introduce the ball
where
,
and
Let us split the operator
on
as
, where
Now, we verify that the operators
and
satisfy the hypothesis of Krasnoselskii’s theorem [
24] in three steps.
(i) For
, we have
where we used (
11). Thus
.
(ii) Using (
) and (
), it is easy to show that
which, in view of the condition:
, implies that the operator
is a contraction.
(iii) Continuity of the functions
implies that the operator
is continuous. In addition,
is uniformly bounded on
as
where
and
(
) are defined by (
9) and (
10) respectively.
To show the compactness of
, we fix
,
. Then, for
, we have
which tends to zero independent of
y as
. This shows that
is equicontinuous. It is clear from the foregoing arguments that the operator
is relatively compact on
. Hence, by the Arzelá-Ascoli theorem,
is compact on
.
In view of the foregoing arguments (
i)-(
iii), the hypothesis of the Krasnoselskii’s fixed point theorem [
24] holds true. Thus, the operator
has a fixed point, which implies that the problem (
1) and (
2) has at least one solution on
. The proof is finished. □
Remark 1. If we interchange the roles of the operators and in the previous result, the condition is replaced with the following one:where and are defined by (9), (10) respectively. The following existence result relies on Leray–Schauder nonlinear alternative [
25].
Theorem 2. Suppose that the following conditions hold:
There exist continuous nondecreasing functions such that , and where .
There exist a constant such that
Then, the problem (1) and (2) has at least one solution on Proof. First we show that the operator is completely continuous. This will be established in several steps.
(i) maps bounded sets into bounded sets in
Let
, where
r is a fixed number. Then, using the strategy employed in the proof of Theorem 1, we obtain
(ii) maps bounded sets into equicontinuous sets.
Let
and
, where
is bounded set in
. Then we obtain
Notice that the right-hand side of the above inequality tends to 0 as , independent of . In view of the foregoing arguments, it follows by the Arzelá–Ascoli theorem that is completely continuous.
The conclusion of the Leray–Schauder nonlinear alternative [
25] will be applicable once it is shown that there exists an open set
with
for
and
. Let
such that
for
. As argued in proving that the operator
is bounded, one can obtain that
which can be written as
On the other hand, we can find a positive number
M such that
by assumption (
). Let us set
Clearly,
contains a solution only when
. In other words, there is no solution
such that
for some
. Therefore,
has a fixed point
which is a solution of the problem (
1) and (
2). The proof is finished. □
3.2. Uniqueness Result
Here we prove a uniqueness result for the problem (
1) and (
2) with the aid of Banach contraction mapping principle.
Theorem 3. If the conditions () and () hold, then the problem (1) and (2) has a unique solution on ifwhere and are defined by (9). Proof. In the first step, we show that
where
with
For
and using the condition (
), we have
Similarly, using (
), we get
In view of (
13) and (
14), we obtain
which implies that
, for any
. Therefore,
. Next, we prove that
is a contraction. For that, let
and
. Then, by the conditions
and
, we obtain
From the above inequality, it follows by the assumption
that
is a contraction. Therefore, we deduce by Banach contraction mapping principle that there exists a unique fixed point for the operator
, which corresponds to a unique solution for the problem (
1) and (
2) on
. The proof is completed. □
3.3. Examples
In this subsection, we construct examples to illustrate the existence and uniqueness results obtained in the last two subsections. Let us consider the following problem:
Here
and
Using the given data, it is found that
where
In consequence, we get
where
,
are defined by (
9) and
is given by (
6).
(i) For illustrating Theorem 1, we have
and that
where
and
. Clearly, the hypothesis of Theorem 1 is satisfied and consequently its conclusion applies to the problem (
15).
(ii) In order to explain Theorem 2, we take the following values (instead of (
16)) in the problem (
15):
and note that
,
,
,
,
and
. By the condition (
), we find that
. Thus, all the conditions of Theorem 2 are satisfied and, hence the problem (
15) with
and
given by (
17) has at least one solution on
(iii) It is easy to show that
and
satisfy the conditions (
) and (
) respectively with
and
and that
Thus, all the assumptions of Theorem 3 hold true and hence the problem (
15) has a unique solution on
4. Conclusions
We considered a fractional differential equation involving left Caputo and right Riemann–Liouville fractional derivatives of different orders and a pair of nonlinearities:
(integral type) and
, equipped with four-point nonlocal boundary conditions. Different criteria ensuring the existence of solutions for the given problem are presented in Theorems 1 and 2, while the uniqueness of solutions is shown in Theorem 3. An interesting and scientific feature of the fractional differential Equation (
1) is that the integral type of nonlinearity can describe composition of a physical quantity (like density) over two different arbitrary subsegments of the given domain. In the case of
this composition takes the form
As pointed out in the introduction, fractional differential equations containing mixed (left Caputo and right Riemann–Liouville) fractional derivatives appear as Euler–Lagrange equations in the study of variational principles. So, such equations in the presence of the integral type of nonlinearity of the form introduced in (
1) enhances the scope of Euler–Lagrange equations studied in [
26]. Moreover, the fractional integro-differential Equation (
1) can improve the description of the electromagnetic waves in dielectric media considered in [
23]. As a special case, our results correspond to a three-point nonlocal mixed fractional order boundary value problem by letting
which is indeed new in the given configuration.