1. Introduction
Fractional order models, providing excellent description of memory and hereditary processes, are more adequate than integer order ones. Some recent contributions to fractional differential equations and inclusions have been carried out, see the monographs [
1,
2,
3,
4,
5,
6,
7,
8], and the references cited therein. The study of fractional differential equations or inclusions with anti-periodic boundary problems, which are applied in different fields, such as physics, chemical engineering, economics, populations dynamics and so on, have recently received considerable attention, see the references ([
9,
10]) and papers cited therein.There are several definitions of fractional differential derivatives and integrals, such like Caputo type, Rimann-Liouville type, Hadamard type and Erdelyi-Kober type and so on. In order to develop the fractional calculus, some different and special form of differential operators are chosen, for example, see [
11,
12,
13,
14,
15] and the references therein. The
order
-Caputo fractional derivative was first introduced by Almeida in [
3]. Some properties, like semigroup law, Taylor’s Theorem, Fermat’s Thorem, etc., were presented. This newly defined fractional derivative could model more accurately the process using differential kernels for the fractional operator. In 2018, Samet and Aydi in [
16] considered the following fractional differential equation with anti-periodic boundary conditions:
where
is the
-Caputo fractional derivative of order
and
is a given function. A Lyapunov-type inequality is established for problem (1). The authors also give some examples to illustrate the applications of their main results.
Very recently, Chen et al. in [
10] studied the following anti-periodic boundary problem involving the Riesz-Caputo derivative
where
is a Riesz-Caputo derivative, which can reflect both the past and the future nonlocal memory effects and
is a continuous function with respect to
and
Some existence results of solutions are given based on the Lipschitz condition, the growth condition and the comparison condition. Most of the present work are concerned with fractional differential equations or inclusions involving Riemann-Liouville or Caputo fractional derivative, merely reflecting the past or future memory effect. Riesz derivative is a two-sided fractional operator, whose advantage is that it could reflect both the past and the future memory effects. We take anomalous diffusion problem for example. The fractional differential equation with the Riesz derivative is adopted to describe the anomalous diffusion problem, in which the Riesz derivative stands for the nonlocality and the dependence on path of the diffusion concentration. Some applications of Riesz derivative about anomalous diffusion, we refer the reader to [
17,
18]. Another typical example is stocks. According to the price trend of the past and future time, investors would buy or sell a stock at an agreed-on price within a period of time. This process depends on both past state and its development in the future, which is the characteristic of Riesz derivative. There are some other applications of this derivative, and we refer the reader to [
10,
19,
20]. In 2009, Ahamad and Otero-Espinar [
1] investigated the following fractional inclusions with anti-periodic boundary conditions
where
is the standard Caputo derivative of order
is a multivalued map,
is the family of all subsets of
Some sufficient conditions for the existence of solutions are given by means of Bohnenblust-Karlin fixed point theorem.
Inspired by the above-mentioned works, in this paper, we are concerned with the following anti-periodic fractional inclusions with
-Riesz-Caputo derivative:
where
is the
-Riesz-Caputo fractional derivative of order
, and
is a multivalued map. Sufficient conditions for the existence of solutions are given in view of the fixed point theorems for multi-valued mapping. The aim of this paper is to develop the calculus of fractional derivatives. We shall combine the two definitions of Riesz-Caputo derivative and
-Caputo fractional derivative. Then we investigate the existence of solutions of anti-periodic inclusions (4). The rest of this paper is organized as follows. We first present some basic definitions of fractional calculus,
-Caputo derivative, Riesz-Caputo derivative and multi-valued maps, and then a new definition of
-Riesz-Caputo fractional derivative of order
is given. In
Section 3, the main results on the existence of solutions for anti-periodic boundary value problem (4) are provided. We present two examples in order to illustrate our main results in last section. Our results generalize some published known results. There is no literature to research the fractional differential inclusions with
-Riesz-Caputo fractional derivative. If we take
where
is a given continuous function, then the problem (4) corresponds to the single-valued problem (1). If we take
where
is a given continuous function, then the problem (4) corresponds to the single-valued problem (2).
2. Preliminaries
In this section, we recall some notation, definitions and preliminaries about fractional calculus [
6,
7,
21],
-Caputo fractional calculus [
3,
4,
5,
22,
23], and Riesz or Riesz-Caputo fractional derivative [
17,
18,
19].
Definition 1 ([
6])
. The left Caputo fractional derivative order α () of a function is given by that is, Similarly, the right Caputo fractional integral order α () of a function is given by Definition 2 ([
6])
. The fractional left, right and Riemann-Liouville integrals of order are defined asLet be a given function such that Definition 3 ([
3])
. The fractional left, right integral of order of a function with respect to ψ are defined by Definition 4 ([
3])
. The left, right ψ-Caputo fractional derivative of order α () of a function are defined as Remark 1. Consider the Riemann-Liouville and Hadamard fractional operators are obtained.
Inspired by the above definitions, we shall present a new definition of -Riesz-Caputo fractional derivative of order which is a combination of -Caputo fractional derivative and Riesz-Caputo fractional derivative.
Definition 5. Let . For , the ψ-Riesz-Caputo fractional derivative of order α () could be defined byIf we take it follows from (7)–(9) that the classic Riesz-Caputo derivative fractional order α () of a function is given bywhich is defined as in [19]. For convenience, denote The following are definitions and properties concerning multi-valued maps [
24,
25,
26,
27,
28] which will be used in the remainder of this paper.
Definition 6 ([
28])
. A multivalued map :- (a)
denote the set as the graph of G,is measurable. - (b)
if is called Lipschitz if and only if there exists such that - (c)
if is called contraction if and only if it is Lipschitz with
- (d)
G is said to be measurable if for every the function
Definition 7 ([
26])
. Assume that is a multivalued map with nonempty compact values. Denote a multivalued operator associated with F asfor a.e. is a closed interval from a to Definition 8 ([
26])
. Assume that Y is a separable metric space and is a multivalued operator. If N is lower semi-continuous(l.s.c.) and has nonempty closed and decomposable values, we say N has a property (BC). Definition 9 ([
28])
. For each denote the selection set of F as Definition 10 ([
28])
. Let The Pompeiu-Hausdorff distance of is defined bywhere Property 1 ([
24])
. Let G be a completely continuous multi-valued map with nonempty compact values, then T is u.s.c. ⟺ G has a closed graph. The following lemmas play important roles in the proof of our main results.
Lemma 1 ([
28])
. (Nonlinear alternative for Kakutani maps ). Assume that E is a Banach space, C is a closed convex subset of E, and U is an open subset of C with . Let be a upper semicontinuous compact map. Then either- (i)
F has a fixed point in , or
- (ii)
there exist a and satisfying .
Lemma 2 ([
29])
. Let be a complete metric space. If is a contraction, then Lemma 3 ([
30])
. Let X be a Banach space, and be a Carathédory set-valued map with and let be a linear continuous mapping. Then the set-valued map defined byis a closed graph operator in Lemma 4 ([
20])
. Assume that Y is a separable metric space and is a multivalued operator with the property (BC). Then there exists a continuous single-valued function satisfying for every i.e., N has a continuous selection. Lemma 5. If and , thenFrom (10) and Lemma 2.1 in [15], for , and , we have thatBy (11), similar to the proof of Lemma 2.2 in [10], we have the following lemma. Lemma 6. Assume that . A function given byis a unique solution of the following anti-periodic boundary value problem As the same argument of Lemma 2.1 in [
16], we can easily obtain the following result, which plays a very important role in proving the main results.
Lemma 7. If , and for each , thenandMoreover, we have Lemma 8. If with , and , then the problemcould be transformed into the following problemA nontrivial solution to (18) is given by where and . Proof. We introduce the function
, defined by
In virtue of (16), one has
From boundary condition (17) and condition
, we have that
Therefore, the problem (17) could be transformed into problem (18). By virtue of Lemma 6, we obtain is a nontrivial solution to (18).
From Lemma 8, we can easily know that
is a unique solution of problem (17). ☐
3. Main Results
We pose the following hypotheses:
is Carathéodory and it has nonempty compact and convex values;
there exist a continuous nondecreasing function
and a function
satisfying
is such that, for every is measurable.
There exists
for almost all
such that
with
for almost all
is a nonempty compact-valued multivalued map such that
- (a)
is is measurable.
- (b)
is lower semicontinuous for each
Now we are in the position to state our main results. The first theorem is dealing with the Carathéodory case.
Theorem 1. Assume that – hold. Moreover, if there exists a constant such thatThen (4) has at least one solution on Proof. The operator
is defined as follows:
We divide the proof into 5 parts, which shows that T satisfies all the conditions of Lemma 1.
Part (i).T maps the bounded sets into bounded sets of
. Set
which is a bounded ball in
, then for
there exists
such that
Part (ii).T maps bounded set into equicontinuous sets. Let
where
is a bounded set in
, for
we have
independent of
as
the right side hand of above inequality tends to
According to the Ascoli-Arzelá Theorem,
T is completely continuous.
Part (iii).T has a closed graph. Set
and
Then, we shall show that
For
there exist
such that
Hence, it suffices to show that there exists
such that for each
Define the continuous linear the operator
:
We have
as
Thus, in light of Lemma 3,
is a closed graph operator. Furthermore, we have
By
we obtain
for some
Part (iv).T is convex for each Since is convex, it is obviously true.
Part (v). We show that there exists a open set
, with
for any
and all
Let
Then for
there exists
such that
A similar discussion as in part (i), we have
By (23), there exists
M such that
Let
It is clear that the operator is upper semicontinuous and completely continuous. If we choose U properly, for some there is no such that Thus, by means of Lemma 1, we can get the conclusion that thereexists a fixed point , that is, it is a solution of problem (4). We complete the proof. ☐
We shall give the second theorem which is concerned with the Lipschitz case.
Theorem 2. Suppose that the conditions – are satisfied. Moreover, ifthen problem (4) has at least a solution on . Proof. By (22), we define the operator
as follows:
Obviously, the fixed point of T is the solution of (4). Our aim is to prove that the operator T satisfies all the conditions in Lemma 2. The proof will be given in two claims.
Claim 1. For each
the operator
T is closed. Let
be such that
in
Then
, and there exists
such that for each
For
F has compact values, we get a subsequence
which converges to
Thus,
and for each
one has
Therefore,
Claim 2. We shall show that there exists
such that
Let
and
There exists
such that for each
By
, there exists
such that
is defined as
The multivalued operator
is measurable, so there exits a measurable selection for
. We denote this function as
For each
one has
Then, we define for each
it follows that
Interchanging
u and
yields
Thus, T is a contraction by . Since Lemma 2, we conclude that T admits a fixed point which is a solution to problem (4). ☐
The third theorem is about the lower semicontinuous case.
Theorem 3. Assume that – hold, if is also satisfied, then the anti-periodic boundary problem (4) has at least one solution on
Proof. It is clear that F is of l.s.c. type as condition is satisfied. By means of Lemma 4, there exists a continuous function such that for all
Next, we shall consider the following problem
Note that if
is a solution to (42), then
u is a solution to the problem (4). we define the operator
as
We transform the problem (42) into a fixed point problem. Obviously, the operator is continuous and completely continuous. As the remainder of the proof is similar to that of Theorem 1, we omit it here. ☐
Remark 2. If we take where is a given continuous function, then the problem (4) corresponds to the single-valued problem (1).
Remark 3. If we take where is a given continuous function, then the problem (4) corresponds to the single-valued problem (2).