Definition 2.1 A function x is said to be a solution of BVP (1.1) if and satisfies BVP (1.1). In addition, x is said to be a nontrivial solution if for and x is solution of BVP (1.1).
For the convenience of the reader, we present some definitions, lemmas, and basic results that will be used later. These and other related results and their proofs can be found, for example, in [6–9].
Definition 2.2 (see )
. Suppose that
then the α
th Riemann-Liouville fractional integral is defined by
whenever the right-hand side is defined. Similarly, with
, we define the α
th Riemann-Liouville fractional derivative to be
where is the unique positive integer satisfying and .
Lemma 2.1 (see )
Lemma 2.2 (see )
Assume thatwith a fractional derivative of order, then, where, (). Herestands for the standard Riemann-Liouville fractional integral of orderanddenotes the Riemann-Liouville fractional derivative as Definition 2.1.
Lemma 2.3 If
, then the boundary value problem
has the unique solution
whereis the Green function of BVP
By applying Lemma 2.2, we may reduce (2.1) to an equivalent integral equation
and (2.4), we have
. Consequently, a general solution of (2.3) is
By (2.5) and Lemma 2.1, we have
So, from (2.6), we have
, combining with (2.7), we obtain
into (2.5), the unique solution of the problem (2.1) is
The proof is completed. □
, we have
This completes the proof. □
Now let us consider the following modified problem of BVP (1.1)
Lemma 2.5 Let, . Then (2.9) can be transformed into (1.1). Moreover, ifis a solution of the problem (2.9), then the functionis a solution of the problem (1.1).
into (1.1), by Lemmas 2.1 and 2.2, we can obtain that
and also . It follows from that . Using , , (2.9) is transformed into (1.1).
be a solution for the problem (2.9). Then, from Lemma 2.1, (2.9) and (2.10), one has
which implies that
. Thus from (2.10), for
, we have
Moreover, it follows from the monotonicity and property of
Consequently, is a solution of the problem (1.1). □
Now let us define an operator
Clearly, the fixed point of the operator T is a solution of BVP (2.9); and consequently is also a solution of BVP (1.1) from Lemma 2.5.
Lemma 2.6is a completely continuous operator.
Proof Noticing that is continuous, by using the Ascoli-Arzela theorem and standard arguments, the result can easily be shown. □
Lemma 2.7 (see )
Let X be a real Banach space, Ω be a bounded open subset of X, where, is a completely continuous operator. Then, either there exists, such that, or there exists a fixed point.