Existence of Solutions for Boundary Value Problems of Vibration Equation with Fractional Derivative ()
1. Introduction
Fractional calculus has been applied more and more widely in many fields of science and engineering, many scholars have done a lot of research on it [1] - [8] . When describing fractional Brownian motion in Anomalous diffusion, if time fractional differential operator is introduced, fractional Langevin equation [9]
is obtained. The fractional Langevin equation describes both subdiffusion for
and superdiffusion for
.
Fractional differential equations are also used to describe damped vibrations in viscoelastic media. In [10] , Podlubny studied the initial value problem for the inhomogeneous Bagley-Torvik equation
and the numerical solutions are presented. The numerical solution is in agreement with the analytical solution, obtained with the help of the fractional Green’s function for a three-term fractional differential equation with constant coefficients.
Motivated by the above works, we study the following boundary value problems for a class of vibration differential equation describing the fractional order damped system with signal stimulus
(1.1)
where
is the Caputo fractional derivative of order
,
is continuous. Moreover,
indicates the ratio of inertia force to mass,
is external damping term, namely, dissipative term,
represents the external force.
By using the Laplace transform, the kernel function is obtained. And then, by using the eigenvalue and the improved Leray-Schauder degree, the existence of the solutions to boundary value problem (1.1) is proved, see Theorem 1. So we can investigate the state of the oscillator motion under this system.
2. Preliminaries
In this part, we recall some definitions and lemmas which are critical to the existence result. The definitions of fractional integral and fractional derivative can be found in [11] [12] .
Definition 1. [10] Assume that function
is defined in
, then the Laplace transform of
is defined as
as long as the generalized integral is convergent.
Definition 2. [10] The original
can be restored from the Laplace transform
with the help of the inverse Laplace transform
where
lies in the right half plane of the absolute convergence of the Laplace integral.
Definition 3. [12] Let
. The function
whenever the series converges is called the two-parameter Mittag-Leffler function with parameters
and
.
Lemma 1. [11] Let
. The Laplace transform formula for
is
Let
.
Lemma 2. [12] Let
. The power series
is convergent for all
. In other words,
is an entire function.
Lemma 3. Let
. Then
Proof. By Definition 3 and Lemma 2, we can get
Thus, the lemma can be obtained.
Lemma 4. [11] Let
be a two-parameter Mittag-Leffler function. Then
Denote
Lemma 5. The functions
and
, defined above, have the following properties.
1)
and
are represented by absolutely and uniformly convergent series and
on
;
2)
;
3)
.
Proof. 1) By Lemma 2, we can show
is an entire function. Thus,
is represented by absolutely and uniformly convergent series on
.
Similarly,
is also represented by absolutely and uniformly convergent series on
. And we can easily have
.
2) In view of
is represented by absolutely and uniformly convergent series on
,
.
3) From the definition of
, we have
.
The proof is complete.
Lemma 6. For
is continuous on
and
, the unique solution of
(2.1)
is
(2.2)
where
(2.3)
Proof. By Lemma 1, we have
Apply Laplace transform to both sides of
, we can easily obtain
namely,
(2.4)
If
, we have
By virtue of Lemma 4, we can show
Similarly, if
and
, we have
So (2.4) is equivalent to
(2.5)
Furthermore,
then we can get the inverse Laplace transform for (2.5) is
(2.6)
Because
and
, we can show
(2.7)
Substituting (2.7) into (2.6), we get
where
is defined by (2.3).
On the other hand, by using the above proof, if
satisfies (2.2), we obtain that x satisfies
and
. The proof is complete.
Lemma 7. The eigenfunction of
(2.8)
is
and its corresponding eigenvalue
is the solution of equation
,
where K is a constant and
.
Proof. Let
is the solution of boundary value problem (2.8). Apply Laplace transform to both sides of
, we can easily obtain
(2.9)
If
, we have
By virtue of Lemma 4, we can show, if
,
.
Similarly, if
and
,
So (2.9) is equivalent to
(2.10)
Furthermore, we can get the inverse Laplace transform for (2.10) is
Because
, we can show
The proof is complete.
Lemma 8. The function
defined by (2.3) is continuous on
.
Proof. By the definition of
and Lemma 5, we get
is continuous for
.The proof is complete.
3. The Existence of the Solutions
Let
Throughout this paper, we always suppose that the following conditions are satisfied.
(H1) There exists constant
such that
for any
.
(H2) There exists
such that
, here
satisfies
.
Let
, with the norm
. Obviously,
is a Banach space.
Define the operators
,
By virtue of Lemma 6, the solution of boundary value problem (1.1) is equivalent to the fixed point of the operator A; Boundary value problem (2.8) is equivalent to the following integral equation
(3.1)
Therefore,
, we have
is the eigenvalue of operator T corresponding to the eigenfunction (3.1).
Lemma 9.
is completely continuous.
Proof. Let
. Obviously,
.
Let
such that
as
. So there exists
such that
,
.
Let
. Then
By virtue of Lebesgue’s dominated convergence theorem, we have
so
as
. Hence, the operator A is continuous.
For each x in the bounded area D,
Consequently, the operator A is uniformly bounded.
By the continuity of
on
,
for any
, if
, then we have
.
If
, we obtain
Then, through the Arzela-Ascoli theorem, the operator A is compact on D.
To summarize,
is completely continuous. The proof is complete.
Lemma 10. The operator A is Frechét differentiable at
, and
.
Proof. Since
, then for any
and
, there exists
such that
, for any
. Namely,
. Let
. Then for any
,
.
So we have
.
Thus,
.
For the above
, there exists
such that
, namely,
for
. Thus, we can show
The proof is complete.
Lemma 11. [13] Let
be a bounded open set in infinite dimensional real Banach space E,
and
be completely continuous. Suppose that
,
,
. Then
.
Lemma 12. [14] Let A be a completely continuous operator which is defined on a Banach space E. Assume that 1 is not an eigenvalue of the asymptotic derivative. The completely continuous vector field
is then nonsingular on spheres
of sufficiently large radius
and
where k is the sum of the algebraic multiplicities of the real eigenvalues of
in
.
Generalizing the previous lemmas, we obtain the following result.
Theorem 1. If (H1) and (H2) hold, then boundary value problem (1.1) has at least one nontrivial solution.
Proof. Obviously
is bounded open set, and
. Via Lemma 9, we get
is completely continuous.
Combining (H2) and Lemma 10, we obtain the eigenvalue of
is
. Therefore, through Lemma 12, we get
By Lemma 7, we have
. Therefore,
Through the definition of
and Lemma 5, we get
for
. Considering (H1), for any
,
i.e.
. By Lemma 11, we get
In conclusion,
.
So we get at least one
is a fixed point of the operator A. That is to say, x is one nontrivial solution of nonlinear problem (1.1). The proof is complete.
4. Conclusion
Theorem 1 is the main result of this paper. By Theorem 1, boundary value problem (1.1) has at least one nontrivial solution under the conditions of (H1) and (H2). Because boundary value problem (1.1) has at least one nontrivial solution, we can investigate the state of the oscillator motion under this system in the later research.
Acknowledgements
We thank the Editor and the referee for their comments.