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  -  . When describing fractional Brownian motion in Anomalous diffusion, if time fractional differential operator is introduced, fractional Langevin equation 
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  , 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
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.
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   .
Definition 1.  Assume that function is defined in , then the Laplace transform of is defined as
as long as the generalized integral is convergent.
Definition 2.  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.  Let . The function
whenever the series converges is called the two-parameter Mittag-Leffler function with parameters and .
Lemma 1.  Let . The Laplace transform formula for is
Lemma 2.  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.  Let be a two-parameter Mittag-Leffler function. Then
Lemma 5. The functions and , defined above, have the following properties.
1) and are represented by absolutely and uniformly convergent series and on ;
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
Proof. By Lemma 1, we have
Apply Laplace transform to both sides of , we can easily obtain
If , we have
By virtue of Lemma 4, we can show
Similarly, if and , we have
So (2.4) is equivalent to
then we can get the inverse Laplace transform for (2.5) is
Because and , we can show
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
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
If , we have
By virtue of Lemma 4, we can show, if ,
Similarly, if and ,
So (2.9) is equivalent to
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
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
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
For the above , there exists such that , namely, for . Thus, we can show
The proof is complete.
Lemma 11.  Let be a bounded open set in infinite dimensional real Banach space E, and be completely continuous. Suppose that , , . Then .
Lemma 12.  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.
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.
We thank the Editor and the referee for their comments.
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