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Existence of Solutions for Fractional Differential Inclusions with Antiperiodic Boundary Conditions
Boundary Value Problems volume 2009, Article number: 625347 (2009)
Abstract
We study the existence of solutions for a class of fractional differential inclusions with anti-periodic boundary conditions. The main result of the paper is based on Bohnenblust- Karlins fixed point theorem. Some applications of the main result are also discussed.
1. Introduction
In some cases and real world problems, fractional-order models are found to be more adequate than integer-order models as fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. The mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electro dynamics of complex medium, polymer rheology, and so forth, involves derivatives of fractional order. In consequence, the subject of fractional differential equations is gaining much importance and attention. For details and examples, see [1–14] and the references therein.
Antiperiodic boundary value problems have recently received considerable attention as antiperiodic boundary conditions appear in numerous situations, for instance, see [15–22].
Differential inclusions arise in the mathematical modelling of certain problems in economics, optimal control, and so forth. and are widely studied by many authors, see [23–27] and the references therein. For some recent development on differential inclusions, we refer the reader to the references [28–32].
Chang and Nieto [33] discussed the existence of solutions for the fractional boundary value problem:

In this paper, we consider the following fractional differential inclusions with antiperiodic boundary conditions

where denotes the Caputo fractional derivative of order
,
Bohnenblust-Karlin fixed point theorem is applied to prove the existence of solutions of (1.2).
2. Preliminaries
Let denote a Banach space of continuous functions from
into
with the norm
Let
be the Banach space of functions
which are Lebesgue integrable and normed by
Now we recall some basic definitions on multivalued maps [34, 35].
Let be a Banach space. Then a multivalued map
is convex (closed) valued if
is convex (closed) for all
The map
is bounded on bounded sets if
is bounded in
for any bounded set
of
(i.e.,
.
is called upper semicontinuous (u.s.c.) on
if for each
the set
is a nonempty closed subset of
, and if for each open set
of
containing
there exists an open neighborhood
of
such that
.
is said to be completely continuous if
is relatively compact for every bounded subset
of
If the multivalued map
is completely continuous with nonempty compact values, then
is u.s.c. if and only if
has a closed graph, that is,
imply
In the following study,
denotes the set of all nonempty bounded, closed, and convex subset of
.
has a fixed point if there is
such that
Let us record some definitions on fractional calculus [8, 11, 13].
Definition 2.1.
For a function the Caputo derivative of fractional order
is defined as

where denotes the integer part of the real number
and
denotes the gamma function.
Definition 2.2.
The Riemann-Liouville fractional integral of order for a function
is defined as

provided the right-hand side is pointwise defined on
Definition 2.3.
The Riemann-Liouville fractional derivative of order for a function
is defined by

provided the right-hand side is pointwise defined on
In passing, we remark that the Caputo derivative becomes the conventional derivative of the function as
and the initial conditions for fractional differential equations retain the same form as that of ordinary differential equations with integer derivatives. On the other hand, the Riemann-Liouville fractional derivative could hardly produce the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations (the same applies to the boundary value problems of fractional differential equations). Moreover, the Caputo derivative for a constant is zero while the Riemann-Liouville fractional derivative of a constant is nonzero. For more details, see [13].
For the forthcoming analysis, we need the following assumptions:
(A1)let be measurable with respect to
for each
, u.s.c. with respect to
for a.e.
, and for each fixed
the set
is nonempty,
(A2)for each there exists a function
such that
for each
with
, and

where depends on
For example, for
we have
and hence
If
then
is not finite.
A function is a solution of the problem (1.2) if there exists a function
such that
a.e. on
and

which, in terms of Green's function , can be expressed as

where

Here, we remark that the Green's function for
takes the form (see [22])

Now we state the following lemmas which are necessary to establish the main result of the paper.
Lemma 2.5 (Bohnenblust-Karlin [36]).
Let be a nonempty subset of a Banach space
, which is bounded, closed, and convex. Suppose that
is u.s.c. with closed, convex values such that
and
is compact. Then G has a fixed point.
Lemma 2.6 ([37]).
Let be a compact real interval. Let
be a multivalued map satisfying
and let
be linear continuous from
then the operator
is a closed graph operator in
3. Main Result
Theorem 3.1.
Suppose that the assumptions and
are satisfied, and

Then the antiperiodic problem (1.2) has at least one solution on
Proof.
To transform the problem (1.2) into a fixed point problem, we define a multivalued map as

Now we prove that satisfies all the assumptions of Lemma 2.6, and thus
has a fixed point which is a solution of the problem (1.2). As a first step, we show that
is convex for each
For that, let
Then there exist
such that for each
we have

Let Then, for each
we have

Since is convex (
has convex values), therefore it follows that
In order to show that is closed for each
let
be such that
in
Then
and there exists a
such that

As has compact values, we pass onto a subsequence to obtain that
converges to
in
Thus,
and

Hence
Next we show that there exists a positive number such that
where
Clearly
is a bounded closed convex set in
for each positive constant
If it is not true, then for each positive number
, there exists a function
with
and

On the other hand, in view of , we have

where we have used the fact that

Dividing both sides of (3.8) by and taking the lower limit as
, we find that
which contradicts (3.1). Hence there exists a positive number
such that
Now we show that is equicontinuous. Let
with
Let
and
then there exists
such that for each
we have

Using (3.8), we obtain

Obviously the right-hand side of the above inequality tends to zero independently of as
Thus,
is equicontinuous.
As satisfies the above assumptions, therefore it follows by Ascoli-Arzela theorem that
is a compact multivalued map.
Finally, we show that has a closed graph. Let
and
We will show that
By the relation
we mean that there exists
such that for each

Thus we need to show that there exists such that for each

Let us consider the continuous linear operator so that

Observe that

Thus, it follows by Lemma 2.6 that is a closed graph operator. Further, we have
Since
therefore, Lemma 2.6 yields

Hence, we conclude that is a compact multivalued map, u.s.c. with convex closed values. Thus, all the assumptions of Lemma 2.6 are satisfied and so by the conclusion of Lemma 2.6,
has a fixed point
which is a solution of the problem (1.2).
Remark 3.2.
If we take where
is a continuous function, then our results correspond to a single-valued problem (a new result).
Applications
As an application of Theorem 3.1, we discuss two cases in relation to the nonlinearity in (1.2), namely,
has (a) sublinear growth in its second variable (b) linear growth in its second variable (state variable). In case of sublinear growth, there exist functions
such that
for each
In this case,
For the linear growth, the nonlinearity
satisfies the relation
for each
In this case
and the condition (3.1) modifies to
In both the cases, the antiperiodic problem (1.2) has at least one solution on
Examples
-
(a)
We consider
and
in (1.2). Here,
Clearly
satisfies the assumptions of Theorem 3.1 with
(condition (3.1). Thus, by the conclusion of Theorem 3.1, the antiperiodic problem (1.2) has at least one solution on
-
(b)
As a second example, let
be such that
and
in (1.2). In this case, (3.1) takes the form
As all the assumptions of Theorem 3.1 are satisfied, the antiperiodic problem (1.2) has at least one solution on
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Acknowledgments
The authors are grateful to the referees for their valuable suggestions that led to the improvement of the original manuscript. The research of V. Otero-Espinar has been partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, Project PGIDIT06PXIB207023PR.
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Ahmad, B., Otero-Espinar, V. Existence of Solutions for Fractional Differential Inclusions with Antiperiodic Boundary Conditions. Bound Value Probl 2009, 625347 (2009). https://doi.org/10.1155/2009/625347
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DOI: https://doi.org/10.1155/2009/625347
Keywords
- Fractional Order
- Fractional Derivative
- Fractional Calculus
- Fractional Differential Equation
- Differential Inclusion