Abstract
This paper investigates the existence of solutions for fractional differential inclusions
of order
MSC: 34A60, 34A08.
Keywords:
fractional differential inclusions; antiperiodic; integral boundary conditions; fixed point theorems1 Introduction
The topic of fractional differential equations and inclusions has recently emerged as a popular field of research due to its extensive development and applications in several disciplines such as physics, mechanics, chemistry, engineering, etc.[15]. An important characteristic of a fractionalorder differential operator, in contrast to its integerorder counterpart, is its nonlocal nature. This feature of fractionalorder operators (equations) is regarded as one of the key factors for the popularity of the subject. As a matter of fact, the use of fractionalorder operators in the mathematical modeling of several real world processes gives rise to more realistic models as these operators are capable of describing memory and hereditary properties. For some recent results on fractional differential equations, see [622] and the references cited therein, whereas some recent work dealing with fractional differential inclusions can be found in [2328].
In this paper, we study a boundary value problem of fractional differential inclusions with antiperiodic type integral boundary conditions given by
where
The present work is motivated by a recent paper [22], where the authors considered (1.1) with F as a singlevalued map. The existence of solutions for problem (1.1) has been discussed for the cases when the righthand side is convex as well as nonconvex valued. The first result is based on the nonlinear alternative of LeraySchauder type, whereas the second result is established by combining the nonlinear alternative of LeraySchauder type for singlevalued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values. In the third result, we use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler. Though the methods used are well known, their exposition in the framework of problem (1.1) is new. We recall some preliminary facts about fractional calculus and multivalued maps in Section 2, while the main results are presented in Section 3.
2 Preliminaries
2.1 Fractional calculus
Let us recall some basic definitions of fractional calculus [13].
Definition 2.1 Let
where
Definition 2.2 The RiemannLiouville fractional integral of order ν is defined as
provided the integral exists.
Definition 2.3 A function
In the sequel, the following lemma plays a pivotal role.
Lemma 2.4 ([22])
For a given
where
2.2 Basic concepts of multivalued analysis
Let us begin this section with some basic concepts of multivalued maps [29,30].
Let
• convex (closed) valued if
• bounded on bounded sets if
• upper semicontinuous (u.s.c.) on
• completely continuous if
Remark 2.5 If the multivalued map
Definition 2.6 The multivalued map
Definition 2.7 A multivalued map
is measurable.
Let
Let
Definition 2.8 A multivalued map
For each
Let
Definition 2.9 Let Y be a separable metric space. A multivalued operator
Let
which is called the Nemytskii operator associated with F.
Definition 2.10 Let
Let
where
Definition 2.11 A multivalued operator
3 Existence results
3.1 The Carathéodory case
We recall the following lemmas to prove the existence of solutions for problem (1.1) when the multivalued map F in (1.1) is of Carathéodory type.
Lemma 3.1 (Nonlinear alternative for Kakutani maps) [32]
LetEbe a Banach space, letCbe a closed convex subset ofE, letUbe an open subset ofC, and
(i) Fhas a fixed point in
(ii) there is an
Lemma 3.2 ([33])
Let
is a closed graph operator in
Theorem 3.3Suppose that
(H_{1})
(H_{2}) there exists a continuous nondecreasing function
(H_{3}) there exist continuous nondecreasing functions
(H_{4}) there exists a constant
where
Then the boundary value problem (1.1) has at least one solution on
Proof Define the operator
for
In the second step, we show that
Then for
Thus,
Now we show that
Obviously, the righthand side of the above inequality tends to zero independently
of
In our next step, we show that
Thus it suffices to show that there exists
Let us consider the continuous linear operator
Observe that
Thus, it follows by Lemma 3.2 that
for some
Finally, we show there exists an open set
and using the computations of the second step above, we have
Consequently, we have
In view of (H_{4}), there exists M such that
Note that the operator
3.2 The lower semicontinuous case
This section deals with the case when F is not necessarily convex valued. Our strategy to deal with this problem is based on the nonlinear alternative of LeraySchauder type together with the selection theorem of Bressan and Colombo for lower semicontinuous maps with decomposable values.
Lemma 3.4 (Bressan and Colombo [34])
LetYbe a separable metric space, and let
Theorem 3.5Assume that (H_{2}), (H_{3}), (H_{4}) and the following condition hold:
(H_{4})
(a)
(b)
then the boundary value problem (1.1) has at least one solution on
Proof It follows from (H_{2}) and (H_{4}) that F is of l.s.c. type. Then from Lemma 3.4, there exists a continuous function
Consider the problem
Observe that if
It can easily be shown that
3.3 The Lipschitz case
Here we show the existence of solutions for problem (1.1) with a nonconvex valued righthand side by applying a fixed point theorem for multivalued maps due to Covitz and Nadler [35].
Lemma 3.6 ([35])
Let
Theorem 3.7Assume that the following conditions hold:
(A_{1})
(A_{2})
(A_{3}) There exist constants
Then the boundary value problem (1.1) has at least one solution on
Proof Observe that the set
As F has compact values, we pass onto a subsequence (if necessary) to obtain that
Hence,
Next we show that there exists
Let
By (H_{3}), we have
So, there exists
Define
Since the multivalued operator
For each
Thus,
Hence,
Analogously, interchanging the roles of x and
Since
Example 3.8
Consider the following boundary value problem of fractional differential inclusions:
where
For
Thus,
with
and
Clearly,
we find that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each of the authors, BA, SKN and AA, contributed to each part of this work equally and read and approved the final version of the manuscript.
Acknowledgements
The authors are grateful to the anonymous referees for their useful comments. The research of B. Ahmad and A. Alsaedi was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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