1. Introduction
Fractional differential equations have gained much importance due to their widespread applications in various disciplines of social and natural sciences, and engineering. In recent years, there has been a remarkable development in fractional calculus and fractional differential equations; for instance, see the monographs by Kilbas et al. [
1], Lakshmikantham et al. [
2], Miller and Ross [
3], Podlubny [
4], Samko et al. [
5], Diethelm [
6], Ahmad et al. [
7] and the papers [
8,
9,
10,
11,
12,
13,
14,
15,
16].
In the literature, one can find many works on boundary value problems containing mixed fractional derivatives of different types. In [
17] the authors studied a new class of nonlinear differential equations with Caputo-type fractional derivatives of different orders, and Caputo-type integro-differential boundary conditions:
where
is Caputo fractional derivative of order
is the Riemann–Liouville fractional integral of order
,
are given functions and
In [
18] the authors considered two Caputo–Hadamard type fractional derivatives in a neutral-type differential equation supplemented with Dirichlet boundary conditions:
where
denotes the Caputo–Hadamard fractional derivatives of order
,
and
are appropriate functions.
More recently, in [
19], both Caputo–Hadamard and Hadamard–Caputo fractional derivatives were considered in the boundary values problems:
and
where
and
are the Caputo and Hadamard fractional derivatives of orders
p and
q, respectively,
,
is a continuous function,
and
,
.
Motivated by the above papers, we introduce and investigate a new boundary value problem involving both Riemann–Liouville and Caputo fractional derivatives, and nonlocal fractional integro-differential boundary conditions given by
where
denote the Riemann–Liouville fractional derivative of order
denote the Caputo fractional derivatives of orders
r and
respectively,
is the Riemann–Liouville fractional integral of order
,
and
The main results for the problem (
5), based on Banach contraction mapping principle, Krasnoselskii fixed-point theorem and nonlinear alternative of Leray–Schauder type, are obtained in
Section 3. In
Section 4 we extend our study to the multi-valued analogue of the problem (
5) given by
where
is a multi-valued map (
is the family of all nonempty subsets of
) and all other constants are the same as defined in problem (
5). We derive the existence results for the inclusion boundary value problem (
6) with the aid of standard fixed-point theorems for multi-valued maps. In case of convex-valued right-hand side of the inclusion, we use Leray–Schauder nonlinear alternative for multi-valued maps. In the case of non-convex-valued right-hand side of the inclusion, we apply a fixed-point theorem for multi-valued contractions due to Covitz and Nadler.
Examples illustrating the obtained results are presented in
Section 5, while we recall some basic concepts of fractional calculus, multi-valued analysis and fixed-point theory in
Section 2. We also establish a preliminary result related to the linear variant of the problem (
5) in this section.
Section 6 contains concluding remarks and some interesting discussion for possible extensions.
2. Preliminaries
In this section, we outline some basic concepts of fractional calculus and multi-valued analysis, and state some fixed-point theorems related to our work.
2.1. Fractional Calculus
In this subsection, we recall some basic ideas of fractional calculus [
1,
4] and present known results needed in our forthcoming analysis.
Definition 1. The Riemann–Liouville fractional derivative of order q for a function is defined bywhere denotes the integer part of the real number q, provided the right-hand side is pointwise defined on . Definition 2. The Riemann–Liouville fractional integral of order q for a function is defined byprovided the right-hand side is pointwise defined on . Definition 3. The Caputo derivative of fractional order q for a n-times differential function is defined as Lemma 1. If then the equation is satisfied for
Lemma 2. Let Then the equation is satisfied for
Lemma 3. Let if and if Then the following relations hold:
- (i)
for
- (ii)
if then
- (iii)
Lemma 4. (see [1]) Let . Then, for , the following formula holds:where , and . Lemma 5. (see [1]) Let . Then for holdswhere , and . Lemma 6. Letand Then the unique solution of the linear problemis given by Proof. Applying the Riemann–Liouville fractional integral of order
q to both sides of equation in (
8), and using Lemma 5, we get
where
Next, applying Riemann–Liouville fractional integral of order
r to both sides (
10), we get
where
. From (
11), we have
Using (
11)–(14) in the boundary conditions of (
8), we obtain
which, on substituting in (
11), yields the solution (
9). The converse follows by direct computation. The proof is completed. □
2.2. Multi-Valued Analysis
Let denote the Banach space of continuous functions x from J into with the norm By we denote the Banach space of Lebesgue integrable functions endowed with the norm by
Let be a Banach space. A multi-valued map
- (i)
is convex (closed) valued if is convex (closed) for all
- (ii)
is bounded on bounded sets if is bounded in X for all bounded set B of X, i.e.,
- (iii)
is called upper semi-continuous (u.s.c. for short) on X if for each the set is nonempty, closed subset of X, and for each open set of X containing , there exists an open neighborhood of such that
- (iv)
is said to be completely continuous if is relatively compact for every bounded subset B of
- (v)
has a fixed point if there exists such that
For each
the set of selections for the multi-valued map
F is defined by
In the following, by we denote the set of all nonempty subsets of X which have the property “p”where “p”will be bounded (b), closed (cl), convex (c), compact (cp) etc. Thus Next, we define the graph of G to be the set and recall two useful results regarding closed graphs and upper-semicontinuity.
Lemma 7. ([20] Proposition 1.2) If is u.s.c., then is a closed subset of , i.e., for every sequence and , if , and when , then . Conversely, if G is completely continuous and has a closed graph, then it is upper semi-continuous. Lemma 8. ([21]) Let X be a separable Banach space. Let be an Carathéodory multi-valued map and let Θ be a linear continuous mapping from to . Then the operatoris a closed graph operator in For more details on multi-valued maps and the proof of the known results cited in this section, we refer the interested reader to the books by Deimling [
20], Gorniewicz [
22] and Hu and Papageorgiou [
23].
2.3. Fixed-Point Theorems
In this subsection we collect the fixed-point theorems which are used in the proofs of our main results.
Lemma 9. (Banach fixed-point theorem)
[24] Let X be a Banach space, be closed and is a strict contraction, i.e., for some and all Then F has a unique fixed point in Lemma 10. (Krasnoselskii fixed-point theorem)
[25]. Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (a) whenever ; (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists such that z = Az + Bz. Lemma 11. (Nonlinear alternative for single-valued maps)
[26]. Let E be a Banach space, C be a closed, convex subset of E, U be an open subset of C and . Suppose that is a continuous, compact (that is, is a relatively compact subset of C) map. Then either- (i)
F has a fixed point in , or
- (ii)
there is a (the boundary of U in C) and with
Lemma 12. (Nonlinear alternative for Kakutani maps)
[26]. Let C be a closed convex subset of a Banach space and U be an open subset of C with Suppose that is an upper semi-continuous compact map. Then either- (i)
F has a fixed point in or
- (ii)
there is a and with
Lemma 13. (Covitz and Nadler fixed-point theorem)
[27] Let be a complete metric space. If is a contraction, then . 3. Main Results for Single-Valued Problem (5)
Let
denote the Banach space of all continuous functions from
to
endowed with the norm defined by
. By Lemma 6, we define an operator
by
with
. It should be noticed that problem (
5) has solutions if and only if the operator
has fixed points.
For the sake of convenience, we put
Our first result, dealing with the existence of a unique solution, is based on the Banach contraction mapping principle.
Theorem 1. Let be a continuous function satisfying the Lipschitz condition:there exists a positive constant L such that Ifthen the boundary value problem (5) has a unique solution on where Φ is given by (17). Proof. We transform the problem (
5) into a fixed-point problem,
, where the operator
is defined by (
15). By using the Banach’s contraction mapping principle, we shall show that
has a fixed point which is the unique solution of problem (
5).
We set
and choose
Now, we show that
where
For any
we obtain by the assumption
that
which implies that
. For
and
we have
which, on taking the norm for
yields
. As
, therefore
is a contraction. Hence, by the conclusion of Banach contraction mapping principle, the operator
has a unique fixed point which corresponds to a unique solution of the problem (
5). The proof is completed. □
Next, we prove an existence result for the given problem by using Krasnoselskii fixed-point theorem.
Theorem 2. Assume that is a continuous function satisfying the assumption In addition we suppose that:
, and
Then the boundary value problem (5) has at least one solution on ifwhere Φ is given by (17). Proof. We define
and choose a suitable constant
such that
Furthermore, we define operators
and
on
by
Observe that
. For
, we have
This shows that
. It follows from the assumption
together with (19) that
is a contraction. Since the function
f is continuous, we have that the operator
is continuous. It is easy to verify that
Therefore,
is uniformly bounded on
. Next, we prove the compactness of the operator
. Let us set
. Let
with
Then we have
which is independent of
x and tends to zero as
. Thus,
is equicontinuous. So
is relatively compact on
. Hence, by the Arzelá-Ascoli theorem,
is compact on
. Thus, all the assumptions of Lemma 10 are satisfied. Therefore, the boundary value problem (
5) has at least one solution on
. The proof is completed. □
Remark 1. In the above theorem, we can interchange the role of the operators and to obtain a second result by replacing (19) with the following condition: Now we prove our next existence result by means of Leray–Schauder nonlinear alternative.
Theorem 3. Assume that is a continuous function. In addition, we suppose that:
there exist a continuous nondecreasing functions and a function such that there exists a constant such thatwhere Φ is given by (17).
Then the boundary value problem (5) has at least one solution on Proof. Firstly, we shall show that the operator
defined by (15),
maps bounded sets (balls) into bounded sets in . For a positive number
R, let
be a bounded ball in
. Then, for
we have
which leads to
Secondly, we show that
maps bounded sets into equicontinuous sets of . Let
with
and
. Then, as argued in the proof of Theorem 2, we have
Obviously the right-hand side of the above inequality tends to zero independently of as . Therefore, it follows by the Arzelá-Ascoli theorem that is completely continuous.
The result will follow from the Leray–Schauder nonlinear alternative once it is shown that the set of all solutions to the equation
is bounded for
For that, let
x be a solution of
for
. Then, for
and following the similar computations as in the first step, we have
Consequently, we have
In view of
, there exists
N such that
. Let us set
Please note that the operator
is continuous and completely continuous. From the choice of
U, there is no
such that
for some
. Consequently, by nonlinear alternative of Leray–Schauder type (Lemma 11), we deduce that
has a fixed point in
which is a solution of the boundary value problem (
5). This completes the proof. □
5. Examples
Consider the following nonlinear Riemann–Liouville and Caputo-type fractional boundary value problem
Here , , , , . With these data we find
5.1. Single-Valued Case
- (i).
Please note that
and thus
is satisfied with
Since
by Theorem 1, the boundary value problem (21), with
f given by (
22), has a unique solution on
.
- (ii).
With the function
f given by (
22), we remark that
and
Hence, by Theorem 2, the boundary value problem (21), with
f given by (
22), has at least one solution on
.
- (iii).
It is easy to find that Then by condition with snf we find that
Hence, by Theorem 3, the boundary value problem (21), with
f given by (
23), has at least one solution on
.
5.2. Multi-Valued Case
- (I).
Consider the multi-valued map
given by
Clearly the multi-valued map
F satisfies condition
and that
which yields
and
. Therefore, the condition
is fulfilled. By direct computation, there exists a constant
satisfying condition
. Hence all assumptions of Theorem 4 hold and hence the problem (
21), with
F given by (
24), has at least one solution on
.
- (II).
Let the multi-valued map
be defined by
Choosing
, we can show that
and
for almost all
. In addition, we get
which leads to
. By the conclusion of Theorem 5, the problem (
21), with
F given by (
25), has at least one solution on
.
6. Discussion
Interchanging the position of Riemann–Liouville and Caputo fractional derivatives in problem (
5), we get the following boundary value problem:
In this case the condition
is necessary for the well-posedness of the problem. The solution for the problem (
26) is given by the integral equation
where
Another more general boundary value problem consisting of Riemann–Liouville and Caputo fractional derivatives of neutral type is
where
a continuous function while all other quantities are the same as defined in (
5). Please note that the problem (
5) is a special case of the problem (
28) when
The solution for the problem (
28) is given by
where
We can obtain the existence results for the problem (
28) by following the procedure used in the previous sections.
7. Conclusions
We have developed the existence theory for nonlinear fractional differential equations and inclusions involving both Riemann–Liouville and Caputo fractional derivatives, equipped with nonlocal fractional integro-differential boundary conditions. We applied the fixed-point theorems for single-valued and multi-valued maps to derive the desired results for the given problems. We also discussed the case obtained by interchanging the position of Riemann–Liouville and Caputo fractional derivatives in the original equation in (
5), supplemented with nonlocal integral boundary conditions. Finally, we introduced a neutral-type fractional differential equation containing both Riemann–Liouville and Caputo fractional derivatives subject to the nonlocal fractional integro-differential boundary conditions and provided the outline for obtaining the existence results for this problem. It is imperative to note that the results obtained in this paper are similar to theoretically well-known propagation properties of fractional Schrödinger equation [
30,
31]. Moreover, our results are comparable to parity-time symmetry in a fractional Schrödinger equation [
32] and propagation dynamics of light beam in a fractional Schrödinger equation [
33]. In fact, the work established in the given configuration is new and contributes significantly to the literature on fractional order boundary value problems.