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:

(A_{1})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,

(A_{2})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.

Definition 2.4 ([16, 33]).

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

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