We study the existence and multiplicity of solutions for the following fractional boundary value problem:
Consider the fractional boundary value problem (BVP for short) of the following form:
Differential equations with fractional order are generalization of ordinary differential equations to noninteger order. This generalization is not mere mathematical curiosities but rather has interesting applications in many areas of science and engineering such as in viscoelasticity, electrical circuits, and neuron modeling. The need for fractional order differential equations stems in part from the fact that many phenomena cannot be modeled by differential equations with integer derivatives. Such differential equations got the attention of many researchers and considerable work has been done in this regard, see the monographs of Kilbas et al. [
Recently, there are many papers dealing with the existence of solutions (or positive solutions) of nonlinear initial (or singular and nonsingular boundary) value problems of fractional differential equation by the use of techniques of nonlinear analysis (fixed-point theorems [
Variational methods are very powerful techniques in nonlinear analysis and are extensively used in many disciplines of pure and applied mathematics including ordinary and partial differential equations, mathematical physics, gauge theory, and geometrical analysis. The existence and multiplicity of solutions for Hamilton systems, Schrödinger equations, and Dirac equations have been studied extensively via critical point theory, see [
In [
In this paper, in order to establish the existence and multiplicity of solutions for BVP (
For the superquadratic case, we make the following assumptions.
where
Assume that (A1)–(A3) hold and that
For the asymptotically quadratic case, we assume the following. There exists
Our second and third main results read as follows.
Assume that
Assume that there exists
Then BVP (
For the subquadratic case, we give the following multiplicity result.
Assume that
Then BVP (
In this section, we recall some background materials in fractional differential equation and critical point theory. The properties of space
Let
Let
The left and right Caputo fractional derivatives are defined via the above Riemann-Liouville fractional derivatives. In particular, they are defined for the function belonging to the space of absolutely continuous functions, which we denote by
Let
The left and right Riemann-Liouville fractional integral operators have the property of a semigroup, that is,
Define
Let
Let
According to (
Define
Making use of Property
In the following, we will treat BVP (
Let
If
A function
where
Let
By Lemma
If
Let
Let there exist there exists
Then
Let
For
It follows from Lemma
Recall that a sequence
We will first establish the following lemma and then give the proof of Theorem
Assume (A), (A2), and (A3) hold, then the functional
Let
By (A2), there exist positive constants
It follows from
Combining (
On the other hand, by (A3), there exist
By (A), we have
Therefore, we obtain
It follows from (
If
If
Since
By Proposition
Then we obtain
By (A1), there exist
Let
Therefore, we have
It is obvious from the definition of
By (A1), there exist
It follows from (A) that
Therefore, we obtain
Choosing
For
Finally, noting that
The following lemmata are needed in the proof of Theorem
Assume (A5), then for any
The proof is similar to that of Lemma 2 in [
Assume (A), (A2’), (A4), and (A5), then the functional
Suppose that
We only need to show that
By (A2), it follows that there exist constants
By assumption (A), it follows that
Passing to the limit in the last inequality, we get
By virtue of Lemma
We assert that meas
Since
By (A4), we obtain thye following:
By (
By virtue of Lemmas
The proof of Theorem
The functional
By (
The functional
Let
Suppose that
However, from (
Noting that
For any
For every
For any
Since
It is easy to prove that the odd mapping
Since
Otherwise, for any positive integer
Set
Since
By the equivalence of the norms on the finite dimensional space, we have
By (
Thus, there exist
In fact, if not, we have
It implies that
Now let
By (
For any
Choosing
Now from the assertion of Lemma
In this section, we give some examples to illustrate our results.
In BVP (
These show that all conditions of Theorem
By Theorem
In BVP (
Let
If
In BVP (
By Theorem
This paper is partially supported by the NNSF (no. 11171351) of China.