A class of impulsive vibration equation with fractional derivatives

1.   Introduction

As we all know, the vibration equation is one of the important research topics in mechanics, physics and other disciplines. In the recent decades, researchers have been paying more and more attentions to the fractional differential equations due to its wide applications on mechanics, physical science, biological sciences and engineering disciplines, etc., see ^{[3,6,7,12,13,17,18,19,20,25]} and the references therein. The development of fractional differential equations provides some new theoretical bases for the study of vibration problems. In ^[22], the vibration equations with fractional derivatives are used to describe the vibration behavior of viscoelastic polymers and good results are obtained. The theoretical study of vibration equations with fractional derivative has also been widely concerned. These studies include the mechanical properties, dynamic characteristics of the system and the correlation functions with various influences for the vibration equations with fractional derivatives, and so on see^{[15,16,21,22]}. In addition, mutation often occurs in vibration system. These abrupt changes can be simulated by the impulsive vibration equation. And this simulation are effective in describing the behavior of real system. There have been a large number of references for the study of fractional impulsive differential equations, see ^{[1,4,5,9,11,23,24,26]}.

In this paper, we study a class of second order impulsive vibration equation containing fractional derivatives under the nonlinear boundary conditions

where $0 < \alpha < 1$, $\lambda > 0$ and ${}^cD_{0^ + }^\alpha$ is the Caputo derivative, $0 = t_0 < t_1 < \cdots < t_m < t_{m + 1} = 1$. Set $J = [0, 1], \; J' = J\backslash \left\{ {t_1, \; t_2, \; \cdots, \; t_m } \right\}, \; J_0 = [0, t_1], \; J_k = (t_k, t_{k + 1}], \; k = 1, \; 2, \; \cdots, \; m$. $\Delta x(t_k) = x(t_k^ +) - x(t_k^ -), \; \Delta x'(t_k) = x'(t_k^ +) - x'(t_k^ -)$. $x(t_k^ -), \; x(t_k^ +)$ denote the left limit and the right limit of $x(t)$ at $t = t_k$. $x'(t_k^ -), \; x'(t_k^ +)$ denote the left limit and the right limit of $x'(t)$ at $t = t_k$. Let $x(t_k) = x(t_k^-)$. $f \in C(J \times\mathbb{R}^2, \mathbb{R}), \; I_k, \; Q_k \in C(\mathbb{R}, \mathbb{R}), \; k = 1, 2, \cdots, \ m$. $g_0, \; g_1 \in C(\mathbb{R}^2, \mathbb{R})$ are given nonlinear functions. By using monotone iterative technique, some new results on multiplicity of boundary value problems are obtained, and the properties of the solutions are discussed. Finally, an example is given out to illustrate the applicability of our main results.

2.   Preliminaries

In this section, we present some basic definitions and lemmas, which will be used to prove our main results.

Definition 2.1. (See^[8], P67) Let $\alpha, \; \beta > 0$. The function $E_{\alpha, \beta}$ is defined by

whenever the series converge is called the two parameters Mittag-Leffler function with parameters $\alpha$ and $\beta$.

Lemma 2.1. (See^[8], P68) Consider the two parameters Mittag-Leffler function $E_{\alpha, \beta }$ for some $\alpha, \; \beta > 0$. The power series defining $E_{\alpha, \beta }$ is convergent for all $z \in\mathbb{C}$. In other words, $E_{\alpha, \beta}$ is an entire function.

Lemma 2.2. Let $\alpha, \; \beta > 0, \; k = 0, \; 1, \; 2, \; \cdots, \; z \in\mathbb{R}$. Then

Proof. By Definition 2.1 and Lemma 2.1, we get

Lemma 2.3. (See^[14], P314) Let $0 < \alpha < \beta, \; n-1 < \alpha\le n, \; l - 1 < \beta\le l\; (n, l \in \mathbb{N}, \; n\leq l, \; \lambda\in\mathbb{R})$. Then

has its linearly independent solutions given by

Lemma 2.4. (See^[14], P324) Let $l - 1 < \beta \le l\; (l \in \mathbb{N}), \; 0 < \alpha < \beta$ be such that $\beta - l + 1 \ge \alpha$, $\lambda \in \mathbb{R}$, and $h(t)$ be a given real function defined on $\mathbb{R_+}$. The general solution to the nonhomogeneous linear differential equation

is given by

where $x_j (t)$ are given by (2.1) and (2.2), $c_j$ are arbitrary real constants $(j = 0, \; 1, \; \cdots, \; l - 1)$.

Lemma 2.5. Let $L[0, 1]$ denote the space of Lebesgue integrable functions on [0, 1], $h\in L[0, 1]$ and $0 < \alpha < 1$, then the Cauchy problem of the second order vibration equation with fractional derivative

has a unique solution, which is given by

and $x(t)$ is derivable while its derivative is given by

Proof. In view of Lemma 2.4, for $\beta = 2, \; 0 < \alpha < 1$, the general solution of the equation

is given by

where $c_0, \; c_1 \in \mathbb{R}$, and

Therefore,

We get

So

and

The proof is completed.

Let $PC^1(J) = \{x:J \to\mathbb{R}\; |\; x, \; x'\in C(J', \mathbb{R}), \; x(t_k^ +), \; x(t_k^ -), \; x'(t_k^ +), \; x'(t_k^ -)\; {\rm{exist}}, \; {\rm{and}}\; x(t_k) = x(t_k^ -), \; k = 1, 2, \cdots, m\}$ and endowed with the normal $\|x\| = \max \{ \mathop {\sup }\limits_{t \in [0, 1]} |x(t)|, \mathop {\sup }\limits_{t \in [0, 1]} |x'(t)|\}$. Then $PC^1 (J)$ is a Banach space.

For $x \in PC^1 (J)$, by the Lagrange mean value theorem, there exists $\xi_k\in [t_k-\varepsilon, t_k]$ such that

and

Thus, for $x \in PC^1 (J)$, we denote

Let $P = {\rm{\{ }}x \in PC^1 (J)|\; x(t) \ge 0, \; x'(t) \ge 0, \; t \in J{\rm{\} }}$. It is obvious that $P \subset PC^1 (J)$ is a normal solid cone. We denote $x\underline \prec y \in PC^1 (J)$ if and only if $x(t)\le y(t)$ and $x'(t)\le y'(t)$ on $t\in [0, 1]$, i.e. $y-x \in P$. We denote $x \prec y$ if $x\underline \prec y \in PC^1 (J)$ and $x \ne y$, and $x \prec \prec y$ if $y-x\in \mathring {P}$.

Lemma 2.6. (See^[10], P220, ^[2], P666) Let $E$ be a Banach space, and $P \subset E$ be a normal solid cone. Suppose that there exist $\alpha _1, \; \beta _1, \; \alpha _2, \; \beta _2 \in E$ with $\alpha _1 \prec \beta _1 \prec \alpha _2 \prec \beta _2$ and $A:[\alpha _1, \beta _2] \to E$ is a completely continuous strongly increasing operator such that

Then the operator $A$ has at least three distinct fixed points $x_1, \; x_2, \; x_3$ on $[\alpha _1, \beta _2]$ such that

3.   The solution of linear impulsive vibration equation

In this section, we obtain the solution of the linear impulsive vibration equation and discuss the properties of its kernel function.

Lemma 3.1. For any $p_k, \; q_k \in\mathbb{R}$ $(k = 1, \; 2, \; \cdots, m)$, $m_i, \; n_i \in\mathbb{R}$ $(i = 1, \; 2)$ and $h\in L[0, 1]$, the following boundary value problem of the second order impulsive vibration equation with fractional derivative

has a unique solution, which is given by

where

Furthermore,

and

Proof. For $t \in [0, t_1]$, let $\xi {\rm{ = }}0, \; x(0) = c_0, \; x'(0) = c_1$, by Lemma 2.5, Cauchy problem

has a unique solution

and

So

For $t \in (t_1, t_2]$, let $\xi = t_1, \; x(t_1^ +) = x(t_1^ -) + p_1, \; x'(t_1^ +) = x'(t_1^ -) + q_1$. By Lemma 2.5, we can obtain that

and

For $t \in (t_k, t_{k + 1}]$, let $\xi = t_k, \; x(t_k^ +) = x(t_k^ -) + p_k, \; x'(t_k^ +) = x'(t_k^ -) + q_k, \; k = 2, 3, \cdots, m$. In the same way, we have

and

Hence,

By the boundary conditions $m_1 x(0) + m_2 x(1) = \gamma _0, \; n_1 x'(0) + n_2 x'(1) = \gamma _1$, we can get that

So

Therefore,

And

Therefore, boundary value problem (3.1) has a unique solution $x = x(t)$ which is given by (3.2), and $G(t, s), \; \varphi(t)$ are given by (3.3) and (3.4), respectively. Furthermore, $x'(t)$ is also established.

For convenience, we give out the following hypothesis:

(H1) The constants $m_i, \; n_i \in \mathbb{R}(i = 1, 2)$ satisfy $m_2 (m_1 + m_2) < 0$ and $n_2 \big(n_1 + n_2 E_{2 - \alpha, 1} (\lambda)\big) < 0$.

Lemma 3.2. Suppose that (H1) holds. Then functions $G$ and $\varphi$ defined by (3.3) and (3.4) satisfy the following properties:

(1) $G(t, s)$ is continuous for $t, \; s \in [0, 1].$

(2) $G(t, s) > 0$ for $t, s \in [0, 1]$ and $\mathop {\max }\limits_{t \in [0, 1]} G(t, s) = G(1, s), \; \mathop {\min }\limits_{t \in [0, 1]} G(t, s) = G(0, s)$.

(3) $G'_t(t, s) > 0$ for $t, s \in [0, 1]$ and $\mathop {\max }\limits_{t \in [0, 1]} G'_t (t, s) = G'_t (1, s), \; \mathop {\min }\limits_{t \in [0, 1]} G'_t (t, s) = G'_t (0, s)$.

(4) If $p_k \ge 0, \; q_k \ge 0, k = 1, \; 2, \; \cdots, \; m,$ then $\varphi (t) \ge 0, \; \varphi '(t) \ge 0,$ for $t \in J_k$.

Proof. (1) By the definition of $G(t, s), \; G\in C([0, 1] \times [0, 1])$ is obvious.

(2) By (H1), for $0 \le s \le t \le 1$,

Then $G(s, s) \le G(t, s) \le G(1, s)$, for $s \in [0, 1]$ and $t \in [s, 1]$. And

Hence, $G(t, s) > 0$ for $0 \le s \le t \le 1$ and $G(t, s) \le G(1, s)$ for $t \in [s, 1]$.

For $0 \le t < s \le 1$,

we can get that $G(0, s) \le G(t, s) < G(s, s)$, for $s \in [0, 1]$ and $t \in [0, s)$. And

Hence, $G(t, s) > 0$ for $0 \le t < s \le 1$ and $G(t, s) < G(s, s)$ for $t \in [0, s)$.

Therefore, $G(t, s) > 0$ for any $t, s \in [0, 1]$. And $G(t, s)$ is monotone increasing with respect to $t \in [0, 1]$, so $\mathop {\max }\limits_{t \in [0, 1]} G(t, s) = G(1, s), \; \mathop {\min }\limits_{t \in [0, 1]} G(t, s) = G(0, s)$.

(3) Since

By (H1), for $0 \le s \le t \le 1$,

Then $G'_t (s, s) \le G'_t (t, s) \le G'_t (1, s)$, for any $s \in [0, 1], \; t \in [s, 1]$. Because

we can get that $G'_t (t, s) > 0$ for $0 \le s \le t \le 1$.

For $0 \le t < s \le 1$,

we can get that $G'_t(0, s)\le G'_t(t, s)\le G'_t(s, s)$ for any $s \in [0, 1]$ and $t\in [0, s)$. Since

we have $G'_t (t, s) > 0$ for $0 \le t < s \le 1$.

Therefore, $G'_t (t, s) > 0$ for any $t, s \in [0, 1]$ and $\mathop {\max }\limits_{t \in [0, 1]} G'_t (t, s) = G'_t (1, s), \; \mathop {\min }\limits_{t \in [0, 1]} G'_t (t, s) = G'_t (0, s)$.

(4) If $p_k, q_k \ge 0$, $k = 1, 2, \cdots, m$, by (H1), (3.4) and (3.7), we can easily get that

The proof is completed.

4.   Existence of three solutions

In this section, we will establish the existence results of the solutions for the boundary value problem (1.1).

For any $u \in PC^1 (J),$ we consider the following boundary value problem

By Lemma 3.1, we can get that boundary value (4.1) is equivalent to the following integral equation

and

where

and

We define an operator $T:PC^1 (J) \to PC^1 (J)$ by

By Lemma 3.1 and (3.5),

We can easily get that the following Lemma 4.1 holds.

Lemma 4.1. The function $x = x(t)$ is the solution of boundary value problem (1.1) if and only if $x$ is a fixed point of the operator $T$ in $PC^1 (J)$.

Lemma 4.2. If (H1) holds, then $T:PC^1 (J) \to PC^1 (J)$ is completely continuous.

Proof. Step 1: $T$ is a continuous operator.

Suppose that $\left\{ {u_n } \right\}\subset PC^1 (J)$ and there exists $u \in PC^1 (J)$ such that $\|u_n - u\| \to 0(n \to \infty)$. Then there exists a constant $M > 0$ such that $\|u_n \| \le M, \; \|u\| \le M.$

Since $f \in C(J \times\mathbb{R}^2, \mathbb{R}), \; I_k, \; Q_k \in C(\mathbb{R}, \mathbb{R}), \; k = 1, 2, \cdots, m$, $g_0, \; g_1 \in C(\mathbb{R}^2, \mathbb{R})$ and $\gamma _{u_n, 0}, \; \gamma _{u_n, 1}, \; \gamma _{u, 0}, \; \gamma _{u, 1}, \in \mathbb{R}$, then

By (H1), Lemma 3.2 and Lebesgue dominated convergence theorem, for any $t \in J$, we have

and

So $||Tu_n - Tu|| \to 0$ as $n\to \infty$, which means $T$ is continuous.

Step 2: $T$ is relatively compact.

Let $\Omega \subset PC^1 (J)$ be a bounded set. By the continuity of the functions $f, \; I_k, \; Q_k\; (k = 1, \; 2, \; \cdots, m)$, $g_0$ and $g_1$, there exists a constant $L > 0$, for any $u \in \Omega$ and $t \in J$,

Then

By Lemma 3.2, for any $t \in J$,

and

Therefore, the operator $T(\Omega)$ is uniformly bounded.

Because $G(t, s)$ is continuous on $[0, 1] \times [0, 1]$, then it is uniformly continuous on $[0, 1] \times [0, 1]$. Thus, for any $\varepsilon > 0$, there exists a constant $\delta _1 > 0$ such that for any $s \in [0, 1]$, $t'_1, \; t'_2 \in J_k, \; k = 0, \; 1, \; \cdots, \; m$, whenever $|t'_1 - t'_2 | < \delta _1$, we can get that

Denote

By (3.6),

Similarly, for the $\varepsilon > 0$, there exists a constant $\delta _2 > 0$ such that

whenever $|t'_1 - t'_2 | < \delta _2$ and $t'_1, \; t'_2 \in J_k, k = 0, \; 1, \; \cdots, \; m$.

By the uniformly continuity of functions $tE_{2 - \alpha, 2} (\lambda t^{2 - \alpha })$ and $E_{2 - \alpha, 1} (\lambda t^{2 - \alpha })$ on $t\in[0, 1]$, we can show that for the $\varepsilon > 0$, there exists a constant $\delta _3 > 0$ such that

and

whenever $|t'_1-t'_2| < \delta _3$ and $t'_1, \; t'_2 \in J_k, k = 0, \; 1, \; \cdots, \; m$. Hence, by (4.2) and (4.3), we have

and

By the uniformly continuity of $E_{2 - \alpha, 1} \big(\lambda (t - s)^{2 - \alpha } \big)$ on $D = \left\{ {(t, s)|\; 0 \le s \le t \le 1} \right\}$, for the $\varepsilon > 0$, there exists a constant $0 < \delta_4 < \frac{\varepsilon }{{6LE_{2-\alpha, 1} (\lambda)}}$, for $(t'_1, s), \; (t'_2, s) \in D$, $|t'_1- t'_2| < \delta_4$ and $t'_1, \; t'_2 \in J_k, k = 0, \; 1, \; \cdots, \; m$, we have

Then

We take $\delta = \min\left\{{\delta_1, \delta_2, \delta_3, \delta_4}\right\}$. Therefore, for any $\varepsilon > 0$, there exists a constant $\delta > 0$ such that for $t'_1, \; t'_2 \in J_k, k = 0, \; 1, \; \cdots, \; m$ whenever $|t'_1-t'_2| < \delta$ and any $u\in \Omega$, we can get that

and

Thus, the operator $T(\Omega)$ is equicontinuous on every interval $J_k$.

According to the Arzela-Ascoli theorem, $T(\Omega)$ is relatively compact.

Therefore, $T:PC^1 (J) \to PC^1 (J)$ is completely continuous.

In the following, we give out some hypotheses.

(H2) $f(t, x_1, y_1) \le f(t, x_2, y_2)$ if $t \in J$, $x_1\le x_2$, $y_1\le y_2\in\mathbb{R}$. Furthermore, $f(t, x_1, y_1) < f(t, x_2, y_2)$ if $x_1 < x_2$ and $y_1 \le y_2$.

And for $x_1 \le x_2\in\mathbb{R}$, $I_k (x_1) \le I_k (x_2), \; Q_k (x_1) \le Q_k (x_2), \; k = 1, 2, \cdots, m.$

(H3.1) If $m_1 + m_2 > 0, \; m_2 < 0, \; n_1 + n_2 E_{2 - \alpha, 1} (\lambda) > 0, \; n_2 < 0$, then for $y_1\le y_2$, $z_1\le z_2\in\mathbb{R}$,

and

(H3.2) If $m_1+ m_2 < 0, \; m_2 > 0, \; n_1+n_2E_{2-\alpha, 1}(\lambda) > 0, \; n_2 < 0$, then for $y_1\le y_2$, $z_1\le z_2\in\mathbb{R}$,

and

(H3.3) If $m_1 + m_2 > 0, \; m_2 < 0, \; n_1 + n_2 E_{2 - \alpha, 1} (\lambda) < 0, \; n_2 > 0$, then for $y_1\le y_2$, $z_1\le z_2\in\mathbb{R}$,

and

(H3.4) If $m_1 + m_2 < 0, \; m_2 > 0, \; n_1 + n_2 E_{2 - \alpha, 1} (\lambda) < 0, \; n_2 > 0$, then for $y_1\le y_2$, $z_1\le z_2\in\mathbb{R}$,

and

Remark. It is easy to see that if one of (H3.1)–(H3.4) holds, then (H1) holds.

Lemma 4.3. For $m_i, \; n_i \in\mathbb{R}(i = 1, \; 2)$, if one of the following conditions holds, then $x(t) \ge 0$ and $x'(t) \ge 0$ for $t \in J$.

(1) $m_1 + m_2 > 0, \; m_2 < 0, \; n_1 + n_2 E_{2 - \alpha, 1} (\lambda) > 0, \; n_2 < 0$ and $x \in PC^1 (J)$ satisfies

(2) $m_1+m_2 < 0, \; m_2 > 0, \; n_1+n_2E_{2-\alpha, 1}(\lambda) > 0, \; n_2 < 0$ and $x \in PC^1 (J)$ satisfies

(3) $m_1 + m_2 > 0, \; m_2 < 0, \; n_1 + n_2 E_{2 - \alpha, 1} (\lambda) < 0, \; n_2 > 0$ and $x \in PC^1 (J)$ satisfies

(4) $m_1 + m_2 < 0, \; m_2 > 0, \; n_1 + n_2 E_{2 - \alpha, 1} (\lambda) < 0, \; n_2 > 0$ and $x \in PC^1 (J)$ satisfies

Proof. (1) Denote $x''(t) - \lambda {}^cD_{0^ + }^\alpha x(t) = h(t), \; \Delta x(t)|_{t = t_k } = p_k, \; \Delta x'(t)|_{t = t_k } = q_k, \; k = 1, \; 2, \cdots, \; m, \; m_1 x(0) + m_2 x(1) = \gamma _0, \; n_1 x'(0) + n_2 x'(1) = \gamma _1.$ Then $h(t) \ge 0, \; p_k \ge 0, \; \; q_k \ge 0, \; k = 1, \; 2, \; \cdots, \; m$.

By Lemma 3.1, the following boundary value problem

has a unique solution

and

It's easy to see that $x(t) \ge 0$ and $x'(t) \ge 0$ for $t \in J$.

Similarly, (2)–(4) are easy to be proved.

Lemma 4.4. Suppose (H2) and one of (H3.1)-(H3.4) holds, then $T$ is a strongly increasing operator.

Proof. Here, we will only prove the conclusion when (H3.1) holds, and other situations are similar.

For any $u_1, \; u_2 \in PC^1 (J)$, and $u_1 \prec u_2$ which implies that $u_1 (t) \le u_2 (t), \; u'_1 (t) \le u'_2 (t)$ and $u_1 (t)\not \equiv u_2 (t)$ for $t\in J$. By (H2), for any $t \in J$,

Since $u_1 (t)\not \equiv u_2 (t)$, there exists an interval $[a, b] \subset J$ such that $u_1(t) < u_2(t)$ for $t\in [a, b]$. It follows from (H2)

By (H3.1), we can get that

By (4.2), (4.3), (H3.1) and (H2), we can get that for $t\in J$,

and

By Lemma 3.2, (H2) and (4.4), for any $u_1, \; u_2 \in PC^1 (J)$ and $u_1 \prec u_2$, as $t \in J$, we have

and

Thus,

which implies that $T$ is a strongly increasing operator.

The proof is completed.

Theorem 4.5. If (H2) and (H3.1) hold, there exist $z_i, \; y_i\in PC^1 (J), \; i = 1, \; 2$, with $z_1 \prec y_1 \prec z_2 \prec y_2$ such that

and

where $i = 1, 2$, $z_2$ and $y_1$ are not the solutions of boundary value problem (1.1). Then boundary value problem (1.1) has at least three distinct solutions $x_1, \; x_2, \; x_3 \in [z_1, \; y_2]$ and satisfies

Proof. By Lemma 4.2 and Lemma 4.4, we can get that $T:[z_1, y_2] \to PC^1(J)$ is a completely continuous strongly increasing operator.

By the definition of operator $T$, we can show that

Let $x = Tz_1-z_1$. By (4.5),

Similarly, we can get that

By (1) in Lemma 4.3, we have

Therefore, $z_1 \underline \prec Tz_1$.

It is similar that we can obtain $z_2 \underline \prec Tz_2$. Because $z_2$ is not the solution of boundary value problem (1.1), then $Tz_2 \ne z_2$. Thus, $z_2 \prec Tz_2$.

By using the same method, we can easily get $Ty_1 \prec y_1, \; Ty_2 \underline \prec y_2$.

By using Lemma 2.6, we can get that the operator $T$ has at least three distinct fixed points $x_1, \; x_2, \; x_3 \in [z_1, \; y_2]$, and satisfies

Therefore, by Lemma 4.1, we can obtain that boundary value problem (1.1) has at least three distinct solutions $x_1, \; x_2, \; x_3 \in [z_1, \; y_2]$, and

The proof is completed.

In the similar way, the following three theorems can be established.

Theorem 4.6. If (H2) and (H3.2) hold, there exist $z_i, \; y_i\in PC^1 (J), \; i = 1, \; 2$, with $z_1 \prec y_1 \prec z_2 \prec y_2$ such that

and

where $i = 1, 2$, $z_2$ and $y_1$ are not the solutions of boundary value problem (1.1). Then the boundary value problem (1.1) has at least three distinct solutions $x_1, \; x_2, \; x_3 \in [z_1, \; y_2]$ and satisfies

Theorem 4.7. If (H2) and (H3.3) hold, there exist $z_i, \; y_i\in PC^1 (J), \; i = 1, \; 2$, with $z_1 \prec y_1 \prec z_2 \prec y_2$ such that

and

where $i = 1, 2$, $z_2$ and $y_1$ are not the solutions of boundary value problem (1.1). Then the boundary value problem (1.1) has at least three distinct solutions $x_1, \; x_2, \; x_3 \in [z_1, \; y_2]$ and satisfies

Theorem 4.8. If (H2) and (H3.4) hold, there exist $z_i, \; y_i\in PC^1 (J), \; i = 1, \; 2$, with $z_1 \prec y_1 \prec z_2 \prec y_2$ such that

and

where $i = 1, 2$, $z_2$ and $y_1$ are not the solutions of boundary value problem (1.1). Then the boundary value problem (1.1) has at least three distinct solutions $x_1, \; x_2, \; x_3 \in [z_1, \; y_2]$ and satisfies

5.   Discussion on applicability of the main results

In this section, we discuss the applicability of our main results.

Consider the following the boundary value problem

where $\alpha = \frac{1}{2}, \; \lambda = \frac{1}{{100}}, \; f(t, x, y) = \frac{{2t^3 }}{\pi }\arctan (x+y), \; I_1 (x) = \frac{1}{2}x, \; Q_1 (x) = \frac{1}{209}x$.

Obviously, $f \in C([0, 1] \times \mathbb{R}^2, \mathbb{R}), \; I_1, \; Q_1 \in C(\mathbb{R}, \mathbb{R}), \; g_0, \; g_1 \in C(\mathbb{R}^2, \mathbb{R})$ are nonlinear functions. And $f, \; I_1, \; Q_1$ satisfy (H2).

Let $m_1 = 2\pi, \; m_2 = -0.02, \; n_1 = 6\pi, \; n_2 = -\pi$. Since

then $g_0, \; g_1$ satisfy (H3.1).

For $t \in [0, 1]$, we take

We can easily get that

and

So

and

We can easily get that $z_1 (t), \; z_2 (t)$ satisfy (4.5) and $y_1 (t), \; y_2 (t)$ satisfy (4.6) with $z_1 \prec y_1 \prec z_2 \prec y_2$.

The conditions of Theorem 4.5 are all satisfied. So by Theorem 4.5, the boundary value problem (5.1) has at least three distinct solutions $x_1, \; x_2, \; x_3 \in [z_1, y_2],$ and moreover,

6.   Conclusion

In this work, we investigate the existence of solutions for a class of second order impulsive vibration equation with fractional derivatives. Some sufficient conditions for existence of the multiplicity solutions are established by applying monotone iterative technique. Finally, a concrete example is given to illustrate the wide range of potential applications of our main results.

Further extensions of this paper are to study the motion state of the vibrator in the system described by boundary value problem (1.1) and the existence of solutions to the boundary value problems with other boundary conditions. Moreover, fractional differential equation models such as the rheological model of the fractional derivative and singular systems model of fractional differential equations have real world applications. So, we also can consider using the boundary value problem of impulsive differential equations to simulate the abrupt changes in the systems described by these models.

Acknowledgments

The authors would like to thank college mathematics characteristic pilot team of University of Shanghai for science and technology for its support to this project.

Conflict of interest

The authors declare that they have no conflicts of interest.