A class of impulsive vibration equation with fractional derivatives

Xue Wang; Xiping Liu; Mei Jia; Xue Wang; Xiping Liu; Mei Jia

doi:10.3934/math.2021120

AIMS Mathematics

2021, Volume 6, Issue 2: 1965-1990. doi: 10.3934/math.2021120

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Research article

A class of impulsive vibration equation with fractional derivatives

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received: 25 October 2020 Accepted: 30 November 2020 Published: 07 December 2020
MSC : 26A33, 34A08, 34A37, 34B15

In this paper, we study a class of second order impulsive vibration equation containing fractional derivatives. By using monotone iterative technique, some new results on the multiplicity for solutions of the equations under nonlinear boundary conditions are obtained, and the properties of the solutions are discussed. Finally, the practicability of our results is discussed through a concrete example.

Keywords:

Citation: Xue Wang, Xiping Liu, Mei Jia. A class of impulsive vibration equation with fractional derivatives[J]. AIMS Mathematics, 2021, 6(2): 1965-1990. doi: 10.3934/math.2021120

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Abstract

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

$\begin{equation} \begin{cases} x''(t) - \lambda {}^cD_{0^ + }^\alpha x(t) = f\big(t, x(t), x'(t)\big), \;t \in J', \; \\ \Delta x(t)|_{t = t_k } = I_k \big(x(t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ \Delta x'(t)|_{t = t_k } = Q_k \big(x(t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ g_0 \big(x(0), x(1)\big) = 0, \;g_1 \big(x'(0), x'(1)\big) = 0, \end{cases} \end{equation}$

(1.1)

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

$E_{\alpha, \beta}(z) = \sum\limits_{j = 0}^{\infty}\frac{z^{j}} {\Gamma(j\alpha+\beta)},$

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

$E_{\alpha , \beta }^{(k)} (z) = \sum\limits_{j = 0}^\infty {\frac{{z^j \Gamma (k + j + 1)}}{{\Gamma (j + 1)\Gamma \big(\alpha (k + j) + \beta \big)}}}.$

Proof. By Definition 2.1 and Lemma 2.1, we get

$\begin{align*} E_{\alpha , \beta }^{(k)} (z) & = \frac{{d^k }}{{dz^k }}E_{\alpha , \beta } (z) = \sum\limits_{j = 0}^\infty {\frac{{d^k }}{{dz^k }}} (\frac{{z^j }}{{\Gamma (j\alpha + \beta )}}) = \sum\limits_{j = k}^\infty {\frac{{\Gamma (j + 1)z^{j - k} }}{{\Gamma (j - k + 1)\Gamma (\alpha j + \beta )}}}\\ & = \sum\limits_{j = 0}^\infty {\frac{{z^j \Gamma (k+j+1)}}{{\Gamma (j+1)\Gamma \big(\alpha (k + j) + \beta \big)}}}. \end{align*}$

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

$\begin{align*} {}^cD_{0^ + }^\beta x(t) - \lambda {}^cD_{0^ + }^\alpha x(t) = 0\;(t > 0) \end{align*}$

has its linearly independent solutions given by

$\begin{equation} x_j (t) = t^j E_{\beta - \alpha , j + 1} (\lambda t^{\beta - \alpha } ) - \lambda t^{\beta - \alpha + j} E_{\beta - \alpha , \beta - \alpha + j + 1} (\lambda t^{\beta - \alpha } )\;\;(j = 0, 1, \cdots , n - 1), \end{equation}$

(2.1)

$\begin{equation} x_j (t) = t^j E_{\beta - \alpha , j + 1} (\lambda t^{\beta - \alpha } )\;(j = n, \cdots , l - 1). \end{equation}$

(2.2)

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

${}^cD_{0^ + }^\beta x(t) - \lambda {}^cD_{0^ + }^\alpha x(t) = h(t)\;(t > 0)$

is given by

$x(t) = \int_0^t {(t - s)^{\beta - 1} E_{\beta - \alpha , \beta } \big(\lambda (t - s)^{\beta - \alpha }\big)} h(s){\rm{d}}s + \sum\limits_{j = 0}^{l - 1} {c_j x_j (t)},$

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

$\left\{ \begin{array}{l} x''(t) - \lambda {}^cD_{0^ + }^\alpha x(t) = h(t), \;t \in J, \\ x(\xi ) = x_0 , \;x'(\xi ) = x_1 , \;\xi \in J \\ \end{array} \right.$

has a unique solution, which is given by

$\begin{align} x(t) = & \int_0^t {(t - s)E_{2 - \alpha , 2} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} - \int_0^\xi {(\xi - s)E_{2 - \alpha , 2} \big(\lambda (\xi - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + x_0\\ &+ \frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) - \xi E_{2 - \alpha , 2} (\lambda \xi ^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda \xi ^{2 - \alpha } )}}\Big(x_1 - \int_0^\xi {E_{2 - \alpha , 1}\big(\lambda(\xi - s)^{2 - \alpha }\big)h(s)} {\rm{d}}s\Big), \end{align}$

(2.3)

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

$\begin{align} x'(t) = \int_0^t {E_{2 - \alpha , 1} \big(\lambda (t - s)^{2 - \alpha } \big)h(s)} {\rm{d}}s + \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{E_{2 - \alpha , 1}(\lambda \xi ^{2 - \alpha } )}}\Big(x_1 - \int_0^\xi {E_{2 - \alpha , 1} \big(\lambda (\xi - s)^{2 - \alpha } \big)h(s)} {\rm{d}}s\Big). \end{align}$

(2.4)

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

$x''(t) - \lambda {}^cD_{0^ + }^\alpha x(t) = h(t)$

is given by

$\begin{equation*} x(t) = \int_0^t {(t - s)E_{2 - \alpha , 2} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + c_1 tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) + c_0, \end{equation*}$

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

$\begin{equation*} x'(t) = \int_0^t {E_{2 - \alpha , 1} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + c_1 E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } ). \end{equation*}$

Therefore,

$\begin{align*} &x(\xi ) = \int_0^\xi {(\xi - s)E_{2 - \alpha , 2}\big(\lambda (\xi - s)^{2 - \alpha }\big)h(s){\rm{d}}s} + c_1 \xi E_{2 - \alpha , 2} (\lambda \xi ^{2 - \alpha } ) + c_0 = x_0, \\ &x'(\xi ) = \int_0^\xi {E_{2 - \alpha , 1} \big(\lambda (\xi - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + c_1 E_{2 - \alpha , 1} (\lambda \xi ^{2 - \alpha } ) = x_1. \end{align*}$

We get

$\begin{align*} &c_1 = \frac{1}{{E_{2 - \alpha , 1} (\lambda \xi ^{2 - \alpha } )}}\Big(x_1 - \int_0^\xi {E_{2 - \alpha , 1} \big(\lambda (\xi - s)^{2 - \alpha } \big)h(s){\rm{d}}s} \Big), \\ &c_0 = x_0 - \int_0^\xi {(\xi - s)E_{2 - \alpha , 2} \big(\lambda (\xi - s)^{2 - \alpha } \big)h(s){\rm{d}}s} - c_1 \xi E_{2 - \alpha , 2} (\lambda \xi ^{2 - \alpha } ). \end{align*}$

$\begin{align*} x(t) = & \int_0^t {(t - s)E_{2 - \alpha , 2} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} - \int_0^\xi {(\xi - s)E_{2 - \alpha , 2} (\lambda (\xi - s)^{2 - \alpha } )h(s){\rm{d}}s} + x_0\\ &+ c_1\big(tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) - \xi E_{2 - \alpha , 2} (\lambda \xi ^{2 - \alpha } )\big)\\ = &\int_0^t {(t - s)E_{2 - \alpha , 2} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} - \int_0^\xi {(\xi - s)E_{2 - \alpha , 2} \big(\lambda (\xi - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + x_0\\ &+ \frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) - \xi E_{2 - \alpha , 2} (\lambda \xi ^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda \xi ^{2 - \alpha } )}}\Big(x_1 - \int_0^\xi {E_{2 - \alpha , 1} \big(\lambda (\xi - s)^{2 - \alpha } \big)h(s)} {\rm{d}}s\Big), \end{align*}$

and

$x'(t) = \int_0^t {E_{2 - \alpha , 1} \big(\lambda (t - s)^{2 - \alpha } \big)h(s)} {\rm{d}}s + \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda \xi ^{2 - \alpha } )}}\Big(x_1 - \int_0^\xi {E_{2 - \alpha , 1} \big(\lambda (\xi - s)^{2 - \alpha } \big)h(s)} {\rm{d}}s\Big).$

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

$x(t_k)-x(t_k-\varepsilon) = x'(\xi_k)\varepsilon,$

and

$x'_-(t_k) = \mathop {\lim }\limits_{\varepsilon\to 0^+} \frac{{x(t_k)-x(t_k-\varepsilon)}}{\varepsilon} = \mathop {\lim }\limits_{\varepsilon\to 0^+} \frac{x'(\xi_k)\varepsilon}{\varepsilon} = x'(t_k^-), \;k = 1, 2, \cdots, m.$

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

$\begin{align} x'(t_k) = x'_-(t_k ) = x'(t_k^-), \;k = 1, 2, \cdots , m. \end{align}$

(2.5)

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

$\alpha _1 \underline \prec A\alpha _1 , \;A\beta _1 \prec \beta _1 , \;\alpha _2 \prec A\alpha _2 , \;A\beta _2 \underline \prec \beta _2.$

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

$\alpha _1 \underline \prec x_1 \prec \prec \beta _1 , \;\;\alpha _2 \prec \prec x_2 \underline \prec \beta _2 , \;\alpha _2 \underline {\not \prec } x_3 \underline {\not \prec } \beta _1.$

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

$\begin{equation} \left\{ \begin{array}{l} x''(t) - \lambda {}^cD_{0^ + }^\alpha x(t) = h(t), \;t \in J', \; \\ \Delta x(t)|_{t = t_k } = p_k , \;k = 1, \;2, \cdots , \;m, \\ \Delta x'(t)|_{t = t_k } = q_k , \;k = 1, \;2, \cdots , \;m, \\ m_1 x(0) + m_2 x(1) = \gamma _0 , \;n_1 x'(0) + n_2 x'(1) = \gamma _1 \; \\ \end{array} \right. \end{equation}$

(3.1)

has a unique solution, which is given by

$\begin{equation} x(t) = \int_0^1 {G(t, s)h(s){\rm{d}}s} + \varphi (t) + \Big(\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} - \frac{{m_2 E_{2 - \alpha , 2} (\lambda )}}{{(m_1 + m_2 )(n_1 + n_2 E_{2 - \alpha , 1} (\lambda ))}}\Big)\gamma _1 + \frac{{\gamma _0 }}{{m_1 + m_2 }}, \;t\in J, \end{equation}$

(3.2)

where

$\begin{align} G(t, s) = \left\{ \begin{array}{l} (t - s)E_{2 - \alpha , 2} \big(\lambda (t - s)^{2 - \alpha } \big) - \frac{{n_2 tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} \\ \;\;\;\; + \frac{{m_2 n_2 E_{2 - \alpha , 2} (\lambda )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{(m_1 + m_2 )(n_1 + n_2 E_{2 - \alpha , 1} (\lambda ))}} - \frac{{m_2 (1 - s)E_{2 - \alpha , 2} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{m_1 + m_2 }}, \;0 \le s \le t \le 1, \\ \frac{{m_2 n_2 E_{2 - \alpha , 2} (\lambda )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{(m_1 + m_2 )(n_1 + n_2 E_{2 - \alpha , 1} (\lambda ))}} - \frac{{n_2 tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} \\ \;\;\;\; - \frac{{m_2 (1 - s)E_{2 - \alpha , 2} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{m_1 + m_2 }}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0 \le t < s \le 1, \end{array} \right. \end{align}$

(3.3)

$\begin{align} \varphi (t) = & \sum\limits_{0 < t_i < t}^{} {p_i } - \frac{{m_2 }}{{m_1 + m_2 }}\sum\limits_{i = 1}^m {p_i } + \sum\limits_{0 < t_i < t} {\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) - t_i E_{2 - \alpha , 2} (\lambda t_i ^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}} q_i\\ &+ \sum\limits_{i = 1}^m {\Big(\frac{{m_2 n_2 E_{2 - \alpha , 2} (\lambda )E_{2 - \alpha , 1} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}} - \frac{{m_2 \big(E_{2 - \alpha , 2} (\lambda ) - t_i E_{2 - \alpha , 2} (\lambda t_i^{2 - \alpha } )\big)}}{{(m_1 + m_2 )E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}}\\ &- \frac{{n_2 E_{2 - \alpha , 1} (\lambda )tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )}}{{\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}\Big)q_i, \;t\in J. \end{align}$

(3.4)

Furthermore,

$\begin{align} x'(t) = \int_0^1 {G'_t (t, s)h(s)} {\rm{d}}s + \varphi '(t) + \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}\gamma _1, \;t\in J, \end{align}$

(3.5)

$\begin{align} &G'_t (t, s) = \left\{ \begin{array}{l} E_{2 - \alpha , 1} \big(\lambda (t - s)^{2 - \alpha } \big) - \frac{{n_2 E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}, \;0 \le s \le t \le 1, \\ - \frac{{n_2 E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}, \;0 \le t < s \le 1, \\ \end{array} \right. \end{align}$

(3.6)

and

$\begin{align} \varphi '(t) = \sum\limits_{0 < t_i < t} {\frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}} q_i - \sum\limits_{i = 1}^m {\frac{{n_2 E_{2 - \alpha , 1} (\lambda )E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}q_i }, \;t\in J. \end{align}$

(3.7)

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

$\left\{ \begin{array}{l} x''(t) - \lambda {}^cD_{0^ + }^\alpha x(t) = h(t), \\ x(0) = c_0 , \;x'(0) = c_1 \\ \end{array} \right.$

has a unique solution

$x(t) = \int_0^t {(t - s)E_{2 - \alpha , 2} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + c_1 tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) + c_0,$

and

$x'(t) = \int_0^t {E_{2 - \alpha , 1} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + c_1 E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } ), \;c_0 , \;c_1 \in \mathbb{R}.$

$\begin{align*} &x(t_1^ - ) = \int_0^{t_1 } {(t_1 - s)E_{2 - \alpha , 2} \big(\lambda (t_1 - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + c_1 t_1 E_{2 - \alpha , 2} (\lambda t_1 ^{2 - \alpha } ) + c_0, \\ &x'(t_1^ - ) = \int_0^{t_1 } {E_{2 - \alpha , 1} \big(\lambda (t_1 - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + c_1 E_{2 - \alpha , 1} (\lambda t_1 ^{2 - \alpha } ). \end{align*}$

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

$\begin{align*} x(t) = & \int_0^t {(t - s)E_{2 - \alpha , 2} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} - \int_0^{t_1 } {(t_1 - s)E_{2 - \alpha , 2} \big(\lambda (t_1 - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + x(t_1^ + )\\ &+ \frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) - t_1 E_{2 - \alpha , 2} (\lambda t_1 ^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_1 ^{2 - \alpha } )}}\Big(x'(t_1^ + ) - \int_0^{t_1 } {E_{2 - \alpha , 1} \big(\lambda (t_1 - s)^{2 - \alpha } \big)h(s)} {\rm{d}}s\Big)\\ = &\int_0^t {(t - s)E_{2 - \alpha , 2} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} - \int_0^{t_1 } {(t_1 - s)E_{2 - \alpha , 2} (\lambda (t_1 - s)^{2 - \alpha } )h(s){\rm{d}}s}\\ &+ x(t_1 ^ - ) + p_1 + \frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) - t_1 E_{2 - \alpha , 2} (\lambda t_1 ^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_1 ^{2 - \alpha } )}}\Big(x'(t_1^ - ) + q_1 - \int_0^{t_1 } {E_{2 - \alpha , 1} \big(\lambda (t_1 - s)^{2 - \alpha } \big)h(s)} {\rm{d}}s\Big)\\ = & \int_0^t {(t - s)E_{2 - \alpha , 2} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} - \int_0^{t_1 } {(t_1 - s)E_{2 - \alpha , 2} (\lambda (t_1 - s)^{2 - \alpha } )h(s){\rm{d}}s} + p_1\\ &+ \int_0^{t_1 } {(t_1 - s)E_{2 - \alpha , 2} \big(\lambda (t_1 - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + c_1 t_1 E_{2 - \alpha , 2} (\lambda t_1 ^{2 - \alpha } ) + c_0\\ &+ \frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) - t_1 E_{2 - \alpha , 2} (\lambda t_1 ^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_1 ^{2 - \alpha } )}}\Big(\int_0^{t_1 } {E_{2 - \alpha , 1} \big(\lambda (t_1 - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + c_1 E_{2 - \alpha , 1} (\lambda t_1 ^{2 - \alpha } )\\ &+ q_1 - \int_0^{t_1 } {E_{2 - \alpha , 1} \big(\lambda (t_1 - s)^{2 - \alpha } \big)h(s)} {\rm{d}}s\Big)\\ = &\int_0^t {(t - s)E_{2 - \alpha , 2} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )c_1 + c_0 + p_1\\ &+ \frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) - t_1 E_{2 - \alpha , 2} (\lambda t_1 ^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_1 ^{2 - \alpha } )}}q_1, \end{align*}$

and

$\begin{align*} x'(t) = &\int_0^t {E_{2 - \alpha , 1} \big(\lambda (t - s)^{2 - \alpha } \big)h(s)} {\rm{d}}s + \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_1 ^{2 - \alpha } )}}\Big(x'(t_1^ - ) + q_1\\ &- \int_0^{t_1 } {E_{2 - \alpha , 1} \big(\lambda (t_1 - s)^{2 - \alpha } \big)h(s)} {\rm{d}}s\Big)\\ = &\int_0^t {E_{2 - \alpha , 1} \big(\lambda (t - s)^{2 - \alpha } \big)h(s)} {\rm{d}}s + E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )c_1 + \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_1 ^{2 - \alpha } )}}q_1. \end{align*}$

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

$\begin{align*} x(t) = & \int_0^t {(t - s)E_{2 - \alpha , 2} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )c_1 + c_0 + \sum\limits_{0 < t_i < t}^{} {p_i }\\ &+ \sum\limits_{0 < t_i < t}^{} {\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) - t_i E_{2 - \alpha , 2} (\lambda t_i ^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}} q_i, \end{align*}$

and

$x'(t) = \int_0^t {E_{2 - \alpha , 1} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )c_1 + \sum\limits_{0 < t_i < t}^{} {\frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}q_i }.$

Hence,

$\begin{align*} x(1) = &\int_0^1 {(1 - s)E_{2 - \alpha , 2} \big(\lambda (1 - s)^{2 - \alpha } \big)h(s){\rm{d}}s + } E_{2 - \alpha , 2} (\lambda )c_1 + c_0 + \sum\limits_{i = 1}^m {p_i }\\ &+ \sum\limits_{i = 1}^m {\frac{{E_{2 - \alpha , 2} (\lambda ) - t_i E_{2 - \alpha , 2} (\lambda t_i^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}} q_i, \\ x'(1) = &\int_0^1 {E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)h(s){\rm{d}}s + } E_{2 - \alpha , 1} (\lambda )c_1 + \sum\limits_{i = 1}^m {\frac{{E_{2 - \alpha , 1} (\lambda )}}{{E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}q_i }. \end{align*}$

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

$\begin{align*} \left\{ \begin{array}{l} - (m_1 + m_2 )c_0 - m_2 E_{2 - \alpha , 2} (\lambda )c_1 = m_2 \int_0^1 {(1 - s)E_{2 - \alpha , 2} \big(\lambda (1 - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + m_2 \sum\limits_{i = 1}^m {p_i } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+ m_2 \sum\limits_{i = 1}^m {\frac{{E_{2 - \alpha , 2} (\lambda ) - t_i E_{2 - \alpha , 2} (\lambda t_i^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}q_i } - \gamma _0, \\ - (n_1 + n_2 E_{2 - \alpha , 1} (\lambda ))c_1 = n_2 \int_0^1 {E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + \sum\limits_{i = 1}^m {\frac{{n_2 E_{2 - \alpha , 1} (\lambda )}}{{E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}q_i } - \gamma _1. \\ \end{array} \right. \end{align*}$

$\begin{align*} c_0 = &\int_0^1 {\Big( - \frac{{m_2 (1 - s)E_{2 - \alpha , 2} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{m_1 + m_2 }} + \frac{{m_2 n_2 E_{2 - \alpha , 2} (\lambda )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}}\Big)h(s){\rm{d}}s} \\ & - \frac{{m_2 }}{{m_1 + m_2 }}\sum\limits_{i = 1}^m {p_i } + \sum\limits_{i = 1}^m {\Big(\frac{{m_2 n_2 E_{2 - \alpha , 2} (\lambda )E_{2 - \alpha , 1} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}} \\ & - \frac{{m_2 \big(E_{2 - \alpha , 2} (\lambda ) - t_i E_{2 - \alpha , 2} (\lambda t_i^{2 - \alpha } )\big)}}{{(m_1 + m_2 )E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}\Big)q_i - \frac{{m_2 E_{2 - \alpha , 2} (\lambda )\gamma _1 }}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}} + \frac{{\gamma _0 }}{{m_1 + m_2 }}, \end{align*}$

$c_1 = - \frac{{n_2 }}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}\Big(\int_0^1 {E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)h(s)} {\rm{d}}s + \sum\limits_{i = 1}^m {\frac{{E_{2 - \alpha , 1} (\lambda )}}{{E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}} q_i \Big) + \frac{{\gamma _1 }}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}.$

Therefore,

$\begin{align*} x(t) = & \int_0^t {(t - s)E_{2 - \alpha , 2} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s} + \int_0^1 {\Big(\frac{{m_2 n_2 E_{2 - \alpha , 2} (\lambda )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}}}\\ &- \frac{{n_2 tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} - \frac{{m_2 (1 - s)E_{2 - \alpha , 2} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{m_1 + m_2 }}\Big)h(s){\rm{d}}s\\ &+ \sum\limits_{0 < t_i < t}^{} {p_i } - \frac{{m_2 }}{{m_1 + m_2 }}\sum\limits_{i = 1}^m {p_i } + \sum\limits_{0 < t_i < t} {\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) - t_i E_{2 - \alpha , 2} (\lambda t_i ^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}} q_i\\ & + \sum\limits_{i = 1}^m {\Big(\frac{{m_2 n_2 E_{2 - \alpha , 2} (\lambda )E_{2 - \alpha , 1} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}} - \frac{{m_2 \big(E_{2 - \alpha , 2} (\lambda ) - t_i E_{2 - \alpha , 2} (\lambda t_i ^{2 - \alpha } )\big)}}{{(m_1 + m_2 )E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}}\\ & - \frac{{n_2 E_{2 - \alpha , 1} (\lambda )tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )}}{{\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}\Big)q_i + \frac{{\gamma _0 }}{{m_1 + m_2 }}\\ & + \Big(\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} - \frac{{m_2 E_{2 - \alpha , 2} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}}\Big)\gamma _1\\ = &\int_0^1 {G(t, s)h(s){\rm{d}}s + \varphi (t)} + \Big(\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} - \frac{{m_2 E_{2 - \alpha , 2} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}}\Big)\gamma _1 + \frac{{\gamma _0 }}{{m_1 + m_2 }}, \;t \in [0, 1]. \end{align*}$

And

$\begin{align*} x'(t) = & \int_0^t {E_{2{\rm{ - }}\alpha , 1} \big(\lambda (t - s)^{2 - \alpha } \big)h(s){\rm{d}}s - \int_0^1 {\frac{{n_2 E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )E_{2 - \alpha , 1} (\lambda (1 - s)^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}h(s)} {\rm{d}}s}\\ &+ \sum\limits_{0 < t_i < t} {\frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}q_i } - \sum\limits_{i = 1}^m {\frac{{n_2 E_{2 - \alpha , 1} (\lambda )E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}} q_i + \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}\gamma _1\\ = & \int_0^1 G'_t (t, s)h(s){\rm{d}} s + \varphi '(t) + \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}\gamma _1, \;t \in [0, 1]. \end{align*}$

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

$G'_t (t, s) = \frac{{\partial G(t, s)}}{{\partial t}} = E_{2 - \alpha , 1} \big(\lambda (t - s)^{2 - \alpha } \big) - \frac{{n_2 E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} > 0.$

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

$\begin{align*} G(s, s) = &\frac{{m_2 n_2 E_{2 - \alpha , 2} (\lambda )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}} - \frac{{n_2 sE_{2 - \alpha , 2} (\lambda s^{2 - \alpha } )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} \\ &- \frac{{m_2 (1 - s)E_{2 - \alpha , 2} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{m_1 + m_2 }} > 0. \end{align*}$

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

$\frac{{\partial G(t, s)}}{{\partial t}} = - \frac{{n_2 E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{n_1 - n_2 E_{2 - \alpha , 1} (\lambda )}} > 0,$

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

$G(0, s) = \frac{{m_2 n_2 E_{2 - \alpha , 2} (\lambda )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}} - \frac{{m_2 (1 - s)E_{2 - \alpha , 2}\big(\lambda (1 - s)^{2 - \alpha }\big)}}{{m_1 + m_2 }} > 0.$

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

$\big(E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )\big)' = \Big(\sum\limits_{k = 0}^\infty {\frac{{(\lambda t^{2 - \alpha } )^k }}{{\Gamma \big((2 - \alpha )k + 1\big)}}}\Big)' = \sum\limits_{k = 1}^\infty {\frac{{(2 - \alpha )k\lambda ^k t^{(2 - \alpha )k - 1} }}{{\Gamma \big((2 - \alpha )k + 1\big)}}} \ge 0, \;t \in [0, 1].$

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

$\frac{{\partial ^2 G(t, s)}}{{\partial t^2 }} = \Big(E_{2 - \alpha , 1} \big(\lambda (t - s)^{2 - \alpha } \big)\Big)' - \frac{{n_2 E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)\big(E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )\big)'}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} \ge 0.$

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

$G'_t (s, s) = 1 - \frac{{n_2 E_{2 - \alpha , 1} (\lambda s^{2 - \alpha } )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} > 0,$

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

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

$\frac{{\partial G^2 (t, s)}}{{\partial t^2 }} = - \frac{{n_2 E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)\big(E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )\big)'}}{{n_1 - n_2 E_{2 - \alpha , 1} (\lambda )}} \ge 0,$

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

$G'_t (0, s) = - \frac{{n_2 E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{n_1 - n_2 E_{2 - \alpha , 1} (\lambda )}} > 0,$

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

$\;\varphi (t) \ge 0, \;\;\varphi '(t) \ge 0, \;t\in J_k.$

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

$\begin{equation} \left\{ \begin{array}{l} x''(t) - \lambda {}^cD_{0^ + }^\alpha x(t) = f\big(t, u(t), u'(t)\big), \;t \in J', \; \\ \Delta x(t)|_{t = t_k } = I_k \big(u(t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ \Delta x'(t)|_{t = t_k } = Q_k \big(u(t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ m_1 x(0) + m_2 x(1) = g_0 \big(u(0), u(1)\big) + m_1 u(0) + m_2 u(1): = \gamma _{u, 0} , \; \\ n_1 x'(0) + n_2 x'(1) = g_1 \big(u'(0), u'(1)\big) + n_1 u'(0) + n_2 u'(1): = \gamma _{u, 1} . \\ \end{array} \right. \end{equation}$

(4.1)

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

$\begin{align*} x(t) = &\int_0^1 {G(t, s)f\big(s, u(s), u'(s)\big){\rm{d}}s} + \varphi _u (t)\\ &+ \Big(\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} - \frac{{m_2 E_{2 - \alpha , 2} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}}\Big)\gamma _{u, 1} + \frac{{\gamma _{u, 0} }}{{m_1 + m_2 }}, \;t\in J, \end{align*}$

and

$x'(t) = \int_0^1 {G'_t (t, s)f\big(s, u(s), u'(s)\big)} {\rm{d}}s + \varphi '_u (t) + \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}\gamma _{u, 1}, \;t\in J,$

where

$\begin{align} \varphi _u (t) = & \sum\limits_{0 < t_i < t}^{} {I_i \big(u(t_i )\big)} - \frac{{m_2 }}{{m_1 + m_2 }}\sum\limits_{i = 1}^m {I_i \big(u(t_i )\big)} + \sum\limits_{0 < t_i < t} {\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) - t_i E_{2 - \alpha , 2} (\lambda t_i ^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}} Q_i \big(u(t_i )\big)\\ &+ \sum\limits_{i = 1}^m {\Big(\frac{{m_2 n_2 E_{2 - \alpha , 2} (\lambda )E_{2 - \alpha , 1} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}} - \frac{{m_2 \big(E_{2 - \alpha , 2} (\lambda ) - t_i E_{2 - \alpha , 2} (\lambda t_i^{2 - \alpha } )\big)}}{{(m_1 + m_2 )E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}}\\ &- \frac{{n_2 E_{2 - \alpha , 1} (\lambda )tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )}}{{\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}\Big)Q_i \big(u(t_i )\big), \;t\in J \end{align}$

(4.2)

and

$\begin{align} \varphi '_u (t) = \sum\limits_{0 < t_i < t} {\frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}} Q_i \big(u(t_i )\big) - \sum\limits_{i = 1}^m {\frac{{n_2 E_{2 - \alpha , 1} (\lambda )E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}Q_i } \big(u(t_i )\big), \;t\in J. \end{align}$

(4.3)

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

$\begin{align*} Tu(t) = &\int_0^1 {G(t, s)f\big(s, u(s), u'(s)\big){\rm{d}}s} + \varphi _u (t)\nonumber\\ &+ \Big(\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} - \frac{{m_2 E_{2 - \alpha , 2} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}}\Big)\gamma _{u, 1} + \frac{{\gamma _{u, 0} }}{{m_1 + m_2 }}, \;t\in J. \end{align*}$

By Lemma 3.1 and (3.5),

$\begin{align*} (Tu)'(t) = \int_0^1 {G'_t (t, s)f\big(s, u(s), u'(s)\big)} {\rm{d}}s + \varphi '_u (t) + \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}\gamma _{u, 1}. \end{align*}$

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

$\mathop {\lim }\limits_{n \to \infty } {\rm{|}}f\big(t, u_n (t), u'_n (t)\big) - f\big(t, u(t), u'(t)\big){\rm{|}} = 0, \;t\in J,$

$\mathop {\lim }\limits_{n \to \infty } |I_k \big(u_n (t_k )\big) - I_k \big(u(t_k )\big)| = 0, \;\mathop {\lim }\limits_{n \to \infty } {\rm{|}}Q_k \big(u_n (t_k )\big) - Q_k \big(u(t_k )\big){\rm{|}} = 0, \;k = 1, 2, \cdots, m,$

$\mathop {\lim }\limits_{n \to \infty } {\rm{|}}\gamma _{u_n , 0} - \gamma _{u, 0} {\rm{|}} = 0, \;\mathop {\lim }\limits_{n \to \infty } {\rm{|}}\gamma _{u_n , 1} - \gamma _{u, 1} {\rm{|}} = 0.$

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

$\begin{align*} &|Tu_n (t) - Tu(t)|\\ = &{\Big{|}}\int_0^1 {G(t, s)\Big(f\big(s, u_n (s), u'_n (s)\big) - f\big(s, u(s), u'(s)\big)\Big)} \;{\rm{d}}s + \big(\varphi _{u_n } (t) - \varphi _u (t)\big)\\ &+ \Big(\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} - \frac{{m_2 E_{2 - \alpha , 2} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}}\Big)(\gamma _{u_n , 1} - \gamma _{u, 1} ) + \frac{{\gamma _{u_n , 0} - \gamma _{u, 0} }}{{m_1 + m_2 }}{\Big{|}}\\ \le &\int_0^1 {G(1, s){\rm{\big|}}f\big(s, u_n (s), u'_n (s)\big) - f\big(s, u(s), u'(s)\big){\rm{\big|}}} \;{\rm{d}}s\\ &+ \frac{{|m_1 + m_2 | + |m_2 |}}{{|m_1 + m_2 |}}\sum\limits_{i = 1}^m {\big|I_i \big(u_n (t_i )\big) - I_i \big(u(t_i )\big)\big|}\\ &+ \frac{{(|m_1 | + 2|m_2 |)\big(|n_1 | + 2|n_2 |E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 2} (\lambda )}}{{|m_1 + m_2 ||n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}\sum\limits_{i = 1}^m {\big|Q_i \big(u_n (t_i )\big) - Q_i \big(u(t_i )\big)\big|}\\ &+ \frac{{(|m_1 | + 2|m_2 |)E_{2 - \alpha , 2} (\lambda )}}{{|m_1 + m_2 ||n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}{\rm{|}}\gamma _{u_n , 1} - \gamma _{u, 1} {\rm{|}} + \frac{{{\rm{|}}\gamma _{u_n , 0} - \gamma _{u, 0} {\rm{|}}}}{{|m_1 + m_2 |}}\to 0 \;(n \to \infty), \end{align*}$

and

$\begin{align*} &|(Tu_n )'(t) - (Tu)'(t)|\\ = &\Big|\int_0^1 {G'_t (t, s)\Big(f\big(s, u_n (s), u'_n (s)\big) - f\big(s, u(s), u'(s)\big)\Big)} {\rm{d}}s + \big(\varphi '_{u_n } (t) - \varphi '_u (t)\big)\\ &+ \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}(\gamma _{u_n , 1} - \gamma _{u, 1} )\Big|\\ \le& \int_0^1 {G'_t (1, s)} \big|f\big(s, u_n (s), u'_n (s)\big) - f\big(s, u(s), u'(s)\big)\big|{\rm{d}}s\\ &+ \sum\limits_{i = 1}^m {\frac{{(|n_1 | + 2|n_2 |E_{2 - \alpha , 1} (\lambda ))E_{2 - \alpha , 1} (\lambda )}}{{|n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}\big|Q_i \big(u_n (t_i )\big) - Q_i \big(u(t_i )\big)\big|}\\ &+ \frac{{E_{2 - \alpha , 1} (\lambda )}}{{|n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}|\gamma _{u_n , 1} - \gamma _{u, 1} | \to 0 \;\;(n \to \infty). \end{align*}$

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 $t \in J$ ,

${\rm{|}}f\big(t, u(t), u'(t)\big){\rm{|}} \le L, \;\;|\gamma _{u, 0} | \le L, \;|\gamma _{u, 1} | \le L,$

$\;{\rm{|}}I_k \big(u(t)\big){\rm{|}} \le L, \;{\rm{|}}Q_k \big(u(t)\big){\rm{|}} \le L, \;k = 1, \;2, \; \cdots , \;m.$

Then

$\begin{align*} |\varphi _u (t)| \le \frac{{m(|m_1 | + 2|m_2 |)\Big(|n_1 |\big(1 + E_{2 - \alpha , 2} (\lambda )\big) + |n_2 |E_{2 - \alpha , 1} (\lambda )\big(1 + 2E_{2 - \alpha , 2} (\lambda )\big)\Big)}}{{|m_1 + m_2 ||n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}L, \end{align*}$

$\begin{align*} |\varphi '_u (t)| \le \frac{{m\big(|n_1 | + 2|n_2 |E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda )}}{{|n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}L. \end{align*}$

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

$\begin{align*} {\rm{|}}Tu(t){\rm{|}} = &{\rm{\Big|}}\int_0^1 {G(t, s)f\big(s, u(s), u'(s)\big)} {\rm{d}}s + \varphi _u (t) + \frac{{\gamma _{u, 0} }}{{m_1 + m_2 }}\\ &+ \Big(\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} - \frac{{m_2 E_{2 - \alpha , 2} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}}\Big)\gamma _{u, 1} \Big|\\ \le& \Big(\int_0^1 {G(1, s)} {\rm{d}}s + \frac{{(|m_1 | + 2|m_2 |)E_{2 - \alpha , 2} (\lambda ) + |n_1 | + |n_2 |E_{2 - \alpha , 1} (\lambda )}}{{|m_1 + m_2 ||n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}\\ &+ \frac{{m(|m_1 | + 2|m_2 |)\Big(|n_1 |\big(1 + E_{2 - \alpha , 2} (\lambda )\big) + |n_2 |E_{2 - \alpha , 1} (\lambda )\big(1 + 2E_{2 - \alpha , 2} (\lambda )\big)\Big)}}{{|m_1 + m_2 ||n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}\Big)L, \end{align*}$

and

$\begin{align*} {\rm{|(}}Tu)'(t){\rm{|}} = &{\rm{\Big|}}\int_0^1 {G'_t (t, s)f(s, u(s), u'(s))} {\rm{d}}s + \varphi '_u (t) + \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}\gamma _{u, 1} \Big|\\ \le &\Big(\int_0^1 {G'_t (1, s){\rm{d}}s} + \frac{{m\big(|n_1 | + 2|n_2 |E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda ) + E_{2 - \alpha , 1} (\lambda )}}{{|n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}\Big)L. \end{align*}$

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

$|G(t'_1 , s) - G(t'_2 , s)| < \frac{\varepsilon }{{2L}}.$

Denote

$G_0 (t, s) = - \frac{{n_2 tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )E_{2 - \alpha , 1} \big(\lambda (1 - s)^{2 - \alpha } \big)}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}.$

By (3.6),

$\begin{equation*} G'_t (t, s) = \left\{ \begin{array}{l} E_{2 - \alpha , 1} \big(\lambda (t - s)^{2 - \alpha } \big) + G_0 (t, s), \;0 \le s \le t \le 1, \\ G_0 (t, s), \;0 \le t < s \le 1. \\ \end{array} \right. \end{equation*}$

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

$\begin{align*} |G_0 (t'_1 , s) - G_0 (t'_2 , s)| < \frac{\varepsilon }{{3L}} \end{align*}$

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

$|t'_1 E_{2 - \alpha , 2} (\lambda {t'_1 }^{2 - \alpha } ) - t'_2 E_{2 - \alpha , 2} (\lambda {t'_2}^{2 - \alpha } )| < \frac{{|n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|\varepsilon }}{{2\Big(m\big(|n_1 | + 2|n_2 |E_{2 - \alpha , 1} (\lambda)\big) + 1\Big)L}},$

and

$|E_{2 - \alpha , 1} (\lambda {t'}_1 ^{2 - \alpha } ) - E_{2 - \alpha , 1} (\lambda {t'_2} ^{2 - \alpha } )| < \frac{{|n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|\varepsilon }}{{3\Big(m\big(|n_1 | + 2|n_2 |E_{2 - \alpha , 1} (\lambda )\big)+1\Big)L}},$

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

$\begin{align*} |\varphi _u (t'_1 ) - \varphi _u (t'_2 )| \le& \sum\limits_{i = 1}^m {\big|t'_1 E_{2 - \alpha , 2} (\lambda {t'_1 }^{2 - \alpha } ) - t'_2 E_{2 - \alpha , 2} (\lambda {t'_2} ^{2 - \alpha } )\big||Q_i (u(t_i ))|}\\ & + \sum\limits_{i = 1}^m {\frac{{|n_2 |E_{2 - \alpha , 1} (\lambda )|t'_1 E_{2 - \alpha , 2} (\lambda {t'_1 }^{2 - \alpha } ) - t'_2 E_{2 - \alpha , 2} (\lambda {t'_2} ^{2 - \alpha } )|}}{{|n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}|Q_i (u(t_i ))|}\\ \le& \frac{{mL\big(|n_1 | + 2|n_2 |E_{2 - \alpha , 1} (\lambda )\big)}}{{|n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}\big|t'_1 E_{2 - \alpha , 2} (\lambda {t'_1 }^{2 - \alpha } ) - t'_2 E_{2 - \alpha , 2} (\lambda {t'_2} ^{2 - \alpha } )\big|, \end{align*}$

and

$\begin{align*} |\varphi '_u (t'_1 ) - \varphi '_u (t'_2 )| \le& \sum\limits_{i = 1}^m {\big|E_{2 - \alpha , 1} (\lambda {t'_1} ^{2 - \alpha } ) - E_{2 - \alpha , 1} (\lambda{ t'_2 }^{2 - \alpha } )\big||Q_i (u(t_i ))|}\\ & + \sum\limits_{i = 1}^m {\frac{{|n_2 |E_{2 - \alpha , 1} (\lambda )|E_{2 - \alpha , 1} (\lambda {t'_1} ^{2 - \alpha } ) - E_{2 - \alpha , 1} (\lambda{ t'_2 }^{2 - \alpha } )|}}{{|n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}|Q_i (u(t_i ))|}\\ \le& \frac{{mL\big(|n_1 | + 2|n_2 |E_{2 - \alpha , 1} (\lambda )\big)}}{{|n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}\big|E_{2 - \alpha , 1} (\lambda {t'_1} ^{2 - \alpha } ) - E_{2 - \alpha , 1} (\lambda{ t'_2 }^{2 - \alpha } )\big|. \end{align*}$

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

$|E_{2-\alpha, 1}\big(\lambda(t'_1-s)^{2-\alpha}\big)-E_{2-\alpha, 1}\big(\lambda(t'_2-s)^{2-\alpha}\big)| < \frac{\varepsilon}{6L}.$

Then

$\begin{align*} &\Big|\int_0^{t'_1 } {E_{2 - \alpha , 1} \big(\lambda (t'_1 - s)^{2 - \alpha } \big)f\big(s, u(s), u'(s)\big)} {\rm{d}}s - \int_0^{t'_2 } {E_{2 - \alpha , 1} \big(\lambda (t'_2 - s)^{2 - \alpha } \big)f\big(s, u(s), u'(s)\big)} {\rm{d}}s\Big|\\ \le& \Big|\int_0^{t'_1 } {E_{2 - \alpha , 1} \big(\lambda (t'_1 - s)^{2 - \alpha } \big)f\big(s, u(s), u'(s)\big)} {\rm{d}}s - \int_0^{t'_1 } {E_{2 - \alpha , 1} \big(\lambda (t'_2 - s)^{2 - \alpha } \big)f\big(s, u(s), u'(s)\big)} {\rm{d}}s\\ & - \int_{t'_1 }^{t'_2 } {E_{2 - \alpha , 1} \big(\lambda (t'_2 - s)^{2 - \alpha } \big)f\big(s, u(s), u'(s)\big)} {\rm{d}}s\Big|\\ \le& L\int_0^{t'_1 } {\big|(E_{2 - \alpha , 1} \big(\lambda (t'_1 - s)^{2 - \alpha } \big)} - E_{2 - \alpha , 1} \big(\lambda (t'_2 - s)^{2 - \alpha } )\big)\big|{\rm{d}}s + L\Big|\int_{t'_1 }^{t'_2 } {E_{2 - \alpha , 1} \big(\lambda (t'_2 - s)^{2 - \alpha } \big)} {\rm{d}}s\Big|\\ \le& L\big|(E_{2 - \alpha , 1} \big(\lambda (t'_1 - s)^{2 - \alpha } \big) - E_{2 - \alpha , 1} \big(\lambda (t'_2 - s)^{2 - \alpha } )\big)\big| + L|E_{2 - \alpha , 1} (\lambda )| |t'_2 - t'_1 |\\ < & \frac{\varepsilon }{3}. \end{align*}$

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

$\begin{align*} &{\rm{|}}Tu(t'_1 ) - Tu(t'_2 ){\rm{|}}\\ \leq & \int_0^1 {\big|G(t'_1 , s) - G(t'_2 , s)\big|\big|f(s, u(s), u'(s))}\big| {\rm{d}}s + \big|\varphi _u (t'_1 ) - \varphi _u (t'_2 )\big|\\ &+ \frac{{\big|t'_1 E_{2 - \alpha , 2} (\lambda {t'}_1 ^{2 - \alpha } ) - t'_2 E_{2 - \alpha , 2} (\lambda {t'}_2 ^{2 - \alpha } )\big|}}{{|n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}\gamma _{u, 1}\\ \le& L\int_0^1 {\big|G(t'_1 , s) - G(t'_2 , s)\big|} {\rm{d}}s + \big|\varphi _u (t'_1 ) - \varphi _u (t'_2 )\big|\\ &+ \frac{L}{{|n_1 + n_2 E_{2 - \alpha , 1} (\lambda )|}}\big|t'_1 E_{2 - \alpha , 2} (\lambda {t'_1} ^{2 - \alpha } ) - t'_2 E_{2 - \alpha , 2} (\lambda {t'_2 }^{2 - \alpha } )\big|\\ < & \frac{\varepsilon }{2}+\frac{{\Big(m\big(|n_1 | + 2|n_2 |E_{2 - \alpha , 1} (\lambda )\big) + 1\Big)L}}{{|n_1 + n_2 E_{2 - \alpha } (\lambda )|}}\big|t'_1 E_{2 - \alpha , 2} (\lambda {t'_1} ^{2 - \alpha } ) - t'_2 E_{2 - \alpha , 2} (\lambda {t'_2 }^{2 - \alpha } )\big|\\ < &\frac{\varepsilon }{2} + \frac{\varepsilon }{2} = \varepsilon, \end{align*}$

and

$\begin{align*} &{\rm{|(}}Tu)'(t'_1 ) - (Tu)'(t'_2 )|\\ = & \Big|\int_0^{t'_1 } {E_{2 - \alpha , 1} \big(\lambda (t'_1 - s)^{2 - \alpha } \big)f\big(s, u(s), u'(s)\big)} {\rm{d}}s - \int_0^{t'_2 } {E_{2 - \alpha , 1} \big(\lambda (t'_2 - s)^{2 - \alpha } \big)f\big(s, u(s), u'(s)\big)} {\rm{d}}s\\ &+ \int_0^1 {\big(G_0 (t'_1 , s) - G_0 (t'_1 , s)\big)f\big(s, u(s), u'(s)\big)} {\rm{d}}s + \varphi '_u (t'_1 ) - \varphi '_u (t'_2 )\\ &+ \frac{{\gamma _{u, 1}}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}\big(E_{2 - \alpha , 1} (\lambda {t'}_1 ^{2 - \alpha } ) - E_{2 - \alpha , 1} (\lambda{ t'_2 }^{2 - \alpha } )\big)\Big|\\ \le& \Big|\int_0^{t'_1 } {E_{2 - \alpha , 1} \big(\lambda (t'_1 - s)^{2 - \alpha } \big)f\big(s, u(s), u'(s)\big)} {\rm{d}}s - \int_0^{t'_2 } {E_{2 - \alpha , 1} \big(\lambda (t'_2 - s)^{2 - \alpha } \big)f\big(s, u(s), u'(s)\big)} {\rm{d}}s\Big|\\ & + L\big|G_0 (t'_1 , s) - G_0 (t'_2 , s)\big| + \frac{{\Big(m\big(|n_1 | + 2|n_2 |E_{2 - \alpha , 1} (\lambda )\big) + 1\Big)L}}{{|n_1 + n_2 E_{2 - \alpha } (\lambda )|}}\big|E_{2 - \alpha , 1} (\lambda {t'_1} ^{2 - \alpha } ) - E_{2 - \alpha , 1} (\lambda {t'_2 }^{2 - \alpha } )\big|\\ < & \frac{\varepsilon }{3} + \frac{\varepsilon }{3} + \frac{\varepsilon }{3} = \varepsilon. \end{align*}$

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 $y_1\le y_2$ , $z_1\le z_2\in\mathbb{R}$ ,

$g_0 (z_2 , y_2 ) - g_0 (z_1 , y_1 ) \ge - m_1 (z_2 - z_1 ) - m_2 (y_2 - y_1 ),$

and

$g_1 (z_2 , y_2 ) - g_1 (z_1 , y_1 ) \ge - n_1 (z_2 - z_1 ) - n_2 (y_2 - y_1 ).$

(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 $y_1\le y_2$ , $z_1\le z_2\in\mathbb{R}$ ,

$g_0 (z_2 , y_2 ) - g_0 (z_1 , y_1 ) \le - m_1 (z_2 - z_1 ) - m_2 (y_2 - y_1 ),$

and

$g_1 (z_2 , y_2 ) - g_1 (z_1 , y_1 ) \ge - n_1 (z_2 - z_1 ) - n_2 (y_2 - y_1 ).$

(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 $y_1\le y_2$ , $z_1\le z_2\in\mathbb{R}$ ,

$g_0 (z_2 , y_2 ) - g_0 (z_1 , y_1 ) \ge - m_1 (z_2 - z_1 ) - m_2 (y_2 - y_1 ),$

and

$g_1 (z_2 , y_2 ) - g_1 (z_1 , y_1 ) \le - n_1 (z_2 - z_1 ) - n_2 (y_2 - y_1 ).$

(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 $y_1\le y_2$ , $z_1\le z_2\in\mathbb{R}$ ,

$g_0 (z_2 , y_2 ) - g_0 (z_1 , y_1 ) \le - m_1 (z_2 - z_1 ) - m_2 (y_2 - y_1 ),$

and

$g_1 (z_2 , y_2 ) - g_1 (z_1 , y_1 ) \le - n_1 (z_2 - z_1 ) - n_2 (y_2 - y_1 ).$

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

$\left\{ \begin{array}{l} x''(t) - \lambda {}^cD_{0^ + }^\alpha x(t) \ge 0, \;t \in J', \; \\ \Delta x(t)|_{t = t_k } \ge 0, \;k = 1, \;2, \cdots , \;m, \\ \Delta x'(t)|_{t = t_k } \ge 0, \;k = 1, \;2, \cdots , \;m, \\ m_1 x(0) + m_2 x(1) \ge 0, \;n_1 x'(0) + n_2 x'(1) \ge 0. \end{array} \right.$

(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

$\left\{ \begin{array}{l} x''(t) - \lambda {}^cD_{0^ + }^\alpha x(t) \ge 0, \;t \in J', \; \\ \Delta x(t)|_{t = t_k } \ge 0, \;k = 1, \;2, \cdots , \;m, \\ \Delta x'(t)|_{t = t_k } \ge 0, \;k = 1, \;2, \cdots , \;m, \\ m_1 x(0) + m_2 x(1) \le 0, \;n_1 x'(0) + n_2 x'(1) \ge 0. \end{array} \right.$

(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

$\left\{ \begin{array}{l} x''(t) - \lambda {}^cD_{0^ + }^\alpha x(t) \ge 0, \;t \in J', \; \\ \Delta x(t)|_{t = t_k } \ge 0, \;k = 1, \;2, \cdots , \;m, \\ \Delta x'(t)|_{t = t_k } \ge 0, \;k = 1, \;2, \cdots , \;m, \\ m_1 x(0) + m_2 x(1) \le 0, \;n_1 x'(0) + n_2 x'(1) \le 0. \end{array} \right.$

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

$\left\{ \begin{array}{l} x''(t) - \lambda {}^cD_{0^ + }^\alpha x(t) = h(t), \;t \in J', \\ \Delta x(t)|_{t = t_k } = p_k , \;k = 1, \;2, \cdots , \;m, \\ \Delta x'(t)|_{t = t_k } = q_k , \;k = 1, \;2, \cdots , \;m, \\ m_1 x(0) + m_2 x(1) = \gamma _0 , \;n_1 x'(0) + n_2 x'(1) = \gamma _1, \end{array} \right.$

has a unique solution

$x(t) = \int_0^t {G(t, s)h(s){\rm{d}}s} + \varphi (t) + \Big(\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} - \frac{{m_2 E_{2 - \alpha , 2} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}}\Big)\gamma _1 + \frac{{\gamma _0 }}{{m_1 + m_2 }}, \;t\in J,$

and

$x'(t) = \int_0^1 {G'_t (t, s)h(s)} {\rm{d}}s + \varphi '(t) + \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}\gamma _1, \;t\in J.$

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 $t \in J$ ,

$\begin{align*} f\big(t, u_2 (t), u'_2 (t)\big) -& f\big(t, u_1 (t), u'_1 (t)\big) \ge 0, \nonumber\\ I_k \big(u_2 (t_k )\big) - I_k \big(u_1 (t_2 )\big) \ge 0, \;&Q_k \big(u_2 (t_k )\big) - Q_k \big(u_2 (t_k )\big) \ge 0. \end{align*}$

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)

$\begin{equation} f\big(t, u_2 (t), u'_2 (t)\big)-f\big(t, u_1 (t), u'_1 (t)\big) > 0, t \in [a, b]. \end{equation}$

(4.4)

By (H3.1), we can get that

$\begin{align*} \gamma _{u_2 , 0} - \gamma _{u_1 , 0} = &g_0 \big(u_2 (0), u_2 (1)\big) - g_0 \big(u_1 (0), u_1 (1)\big)\nonumber \\ &+(m_1 u_2 (0) + m_2 u_2 (1)) - (m_1 u_1 (0) + m_2 u_1 (1))\nonumber \\ \ge& 0, \\ \gamma _{u_2 , 1} - \gamma _{u_1 , 1} = &g_1 \big(u_2 (0), u_2 (1)\big) - g_1 \big(u_1 (0), u_1 (1)\big)\nonumber \\ &+\big(n_1 u_2 (0) + n_2 u_2 (1)\big) - \big(n_1 u_1 (0)n_2 u_1 (1)\big)\nonumber \\ \ge& 0. \end{align*}$

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

$\begin{align*} &\varphi _{u_2 } (t) - \varphi _{u_1 } (t)\\ = &\sum\limits_{0 < t_i < t}^{} {\Big(I_i \big(u_2 (t_i )\big) - I_i \big(u_1 (t_i )\big)\Big)} - \frac{{m_2 }}{{m_1 + m_2 }}\sum\limits_{i = 1}^m {\Big(I_i \big(u_2 (t_i )\big) - I_i \big(u_1 (t_i )\big)\Big)}\\ &+ \sum\limits_{0 < t_i < t} {\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } ) - t_i E_{2 - \alpha , 2} (\lambda t_i ^{2 - \alpha } )}}{{E_{2 - \alpha , 1} \big(\lambda t_i ^{2 - \alpha } \big)}}\Big(Q_i \big(u_2 (t_i )\big) - Q_i \big(u_1 (t_i )\big)\Big)}\\ &+ \sum\limits_{i = 1}^m {\Big( - \frac{{m_2 \big(E_{2 - \alpha , 2} (\lambda ) - t_i E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )\big)}}{{(m_1 + m_2 )E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}} - \frac{{n_2 E_{2 - \alpha , 1} (\lambda )\big(tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )\big)}}{{\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}}\\ & + \frac{{m_2 n_2 E_{2 - \alpha , 2} (\lambda )E_{2 - \alpha , 1} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}\Big)\Big(Q_i \big(u_2 (t_i )\big) - Q_i \big(u_1 (t_i )\big)\Big)\\ \ge &\sum\limits_{0 < t_i < t}^{} {\Big(I_i \big(u_2 (t_i )\big) - I_i \big(u_1 (t_i )\big)\Big)} - \frac{{m_2 }}{{m_1 + m_2 }}\sum\limits_{i = 1}^m {\Big(I_i \big(u_2 (t_i )\big) - I_i \big(u_1 (t_i )\big)\Big)}\\ &+ \sum\limits_{i = 1}^m {\frac{{m_2 n_2 E_{2 - \alpha , 2} (\lambda )E_{2 - \alpha , 1} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda )}}\Big(Q_i \big(u_2 (t_i )\big) - Q_i \big(u_1 (t_i )\big)\Big)}\\ \ge& 0, \end{align*}$

and

$\begin{align*} &\varphi '_{u_2 } (t) - \varphi '_{u_1 } (t)\nonumber\\ = &\sum\limits_{0 < t_i < t} {\frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{E_{2 - \alpha , 1} (\lambda t_i ^{2 - \alpha } )}}}\Big(Q_i \big(u_2 (t_i )\big) - Q_i \big(u_1 (t_i )\big)\Big)\nonumber\\ &- \sum\limits_{i = 1}^m {\frac{{n_2 E_{2 - \alpha , 1} (\lambda )E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{(n_1 + n_2 E_{2 - \alpha , 1} (\lambda ))E_{2 - \alpha , 1} (\lambda t_i^{2 - \alpha } )}}\Big(Q_i \big(u_2 (t_i )\big) - Q_i \big(u_1 (t_i )\big)\Big)}\nonumber\\ \ge& \sum\limits_{0 < t_i < t} {\frac{1}{{E_{2 - \alpha , 1} (\lambda )}}}\Big(Q_i \big(u_2 (t_i )\big) - Q_i \big(u_1 (t_i )\big)\Big)\nonumber\\ &- \sum\limits_{i = 1}^m {\frac{{n_2 E_{2 - \alpha , 1} (\lambda )}}{{\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)E_{2 - \alpha , 1} (\lambda )}}\Big(Q_i \big(u_2 (t_i )\big) - Q_i \big(u_1 (t_i )\big)\Big)}\nonumber\\ \ge& 0. \end{align*}$

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

$\begin{align*} Tu_2 (t) - Tu_1 (t) = &\int_0^1 {G(t, s)\Big(f\big(s, u_2 (s), u'_2 (s)\big) - f\big(s, u_1 (s), u'_1 (s)\big)\Big)} {\rm{d}}s + \big(\varphi _{u_2 } (t) - \varphi _{u_1 } (t)\big)\\ &+ \Big(\frac{{tE_{2 - \alpha , 2} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}} - \frac{{m_2 E_{2 - \alpha , 2} (\lambda )}}{{(m_1 + m_2 )\big(n_1 + n_2 E_{2 - \alpha , 1} (\lambda )\big)}}\Big)(\gamma _{u_2 , 1} - \gamma _{u_1 , 1} )\\ &+\frac{{\gamma _{u_2 , 0} - \gamma _{u_1 , 0} }}{{m_1 + m_2 }}\\ \ge& \int_a^b {G(0, s)\Big(f\big(s, u_2 (s), u'_2 (s)\big) - f\big(s, u_1 (s), u'_1 (s)\big)\Big)} {\rm{d}}s\\ > & 0, \end{align*}$

and

$\begin{align*} (Tu_2 )'(t) - (Tu_1 )'(t) = &\int_0^1 {G'_t (t, s)\Big(f\big(s, u_2 (s), u'_2 (s)\big) - f\big(s, u_1 (s), u'_1 (s)\big)\Big)} {\rm{d}}s + \big(\varphi '_{u_2 } (t) - \varphi '_{u_1 } (t)\big)\\ &+ \frac{{E_{2 - \alpha , 1} (\lambda t^{2 - \alpha } )}}{{n_1 + n_2 E_{2 - \alpha , 1} (\lambda )}}(\gamma _{u_2 , 1} - \gamma _{u_1 , 1} )\\ \ge &\int_a^b {G'_t (0, s)\Big(f\big(s, u_2 (s), u'_2 (s)\big) - f\big(s, u_1 (s), u'_1 (s)\big)\Big)} {\rm{d}}s\\ > &0. \end{align*}$

Thus,

$Tu_2 - Tu_1 \in\mathring {P}$

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

$\begin{align} \left\{ \begin{array}{l} z''_i (t) - \lambda {}^cD_{0^ + }^\alpha z_i (t) \le f\big(t, z_i (t), z'_i (t)\big), \;t \in J', \; \\ \Delta z_i (t)|_{t = t_k } \le I_k \big(z_i (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ \Delta z'_i (t)|_{t = t_k } \le Q_k \big(z_i (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ g_0 \big(z_i (0), z_i (1)\big) \ge 0, \;g_1 \big(z'_i (0), z'_i (1)\big) \ge 0, \end{array} \right. \end{align}$

(4.5)

and

$\begin{align} \left\{ \begin{array}{l} y''_i (t) - \lambda {}^cD_{0^ + }^\alpha y_i (t) \ge f\big(t, y_i (t), y'_i (t)\big), \;t \in J', \; \\ \Delta y_i (t)|_{t = t_k } \ge I_k \big(y_i (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ \Delta y'_i (t)|_{t = t_k } \ge Q_k \big(y_i (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ g_0 \big(y_i (0), y_i (1)\big) \le 0, \;g_1 \big(y'_i (0), y'_i (1)\big) \le 0, \end{array} \right. \end{align}$

(4.6)

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

$z_1 (t) \le x_1 (t) < y_1 (t), \;z_2 (t) < x_2 (t) \le y_2 (t), \;z_2 (t)\not \le x_2 (t)\not \le y_1 (t), \;t \in J.$

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

$\begin{align*} \left\{ \begin{array}{l} (Tz_1 )''(t) - \lambda {}^cD_{0^ + }^\alpha (Tz_1 )(t) = f\big(t, z_1 (t), z'_1 (t)\big), t \in J', \; \\ \Delta (Tz_1 )(t)|_{t = t_k } = I_k \big(z_1 (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ \Delta (Tz_1 )'(t)|_{t = t_k } = Q_k \big(z_1 (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ m_1 (Tz_1 )(0) + m_2 (Tz_1 )(1) = g_0 \big(z_1 (0), z_1 (1)\big) + m_1 z_1 (0) + m_2 z_1 (1), \; \\ n_1 (Tz_1 )'(0) + n_2 (Tz_1 )'(1) = g_1 \big(z'_1 (0), z'_1 (1)\big) + n_1 z'_1 (0) + n_2 z'_1 (1). \end{array} \right. \end{align*}$

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

$\begin{align*} x''(t) - \lambda {}^cD_{0^ + }^\alpha x(t) & = \big((Tz_1 )''(t) - z''_1 (t)\big) - \lambda \big({}^cD_{0^ + }^\alpha (Tz_1 )(t) - {}^cD_{0^ + }^\alpha z_1 (t)\big)\\ & = \;\big((Tz_1 )''(t) - \lambda {}^cD_{0^ + }^\alpha (Tz_1 )(t)\big) - \big(z''_1 (t) - \lambda {}^cD_{0^ + }^\alpha z_1 (t)\big) \\ & = f\big(t, z_1 (t), z'_1 (t)\big) - \big(z''_1 (t) - \lambda {}^cD_{0^ + }^\alpha z_1 (t)\big) \\ &\ge 0, \;t \in J'. \end{align*}$

$\begin{align*} \Delta x(t)|_{t = t_k } = \Delta Tz_1 (t)|_{t = t_k } - \Delta z_1 (t)|_{t = t_k } = I_k \big(z_1 (t_k )\big) - \Delta z_1 (t)|_{t = t_k } \ge 0.\\ \Delta x'(t)|_{t = t_k } = \Delta (Tz_1 )'(t)|_{t = t_k } - \Delta z'_1 (t)|_{t = t_k } = Q_k \big(z_1 (t_k )\big) - \Delta z'_1 (t)|_{t = t_k } \ge 0. \end{align*}$

$\begin{align*} m_1 x(0) + m_2 x(1) & = m_1 \big((Tz_1 )(0) - z_1 (0)\big) + m_2 \big((Tz_1 )(1) - z_1 (1)\big)\\ & = m_1 \big((Tz_1 )(0) + m_2 (Tz_1 )(1)\big) - \big(m_1 z_1 (0) + m_2 z_1 (1)\big)\\ & = g_0 \big(z_1 (0), z_1 (1)\big) + \big(m_1 z_1 (0) + m_2 z_1 (1)\big) - \big(m_1 z_1 (0) + m_2 z_1 (1)\big)\\ & = g_0 \big(z_1 (0), z_1 (1)\big)\\ &\ge 0. \end{align*}$

Similarly, we can get that

$n_1 x'(0) + n_2 x'(1)\ge 0.$

By (1) in Lemma 4.3, we have

$x(t) = (Tz_1 )(t) - z_1 (t) \ge 0, \;x'(t) = (Tz_1 )'(t) - z'_1 (t) \ge 0, \;t\in J.$

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

$z_1 \underline \prec x_1 \prec \prec y_1 , \;z_2 \prec \prec x_2 \underline \prec y_2 , \;z_2 \underline {\not \prec } x_3 \underline {\not \prec } y_1.$

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

$z_1 (t) \le x_1 (t) < y_1 (t), \;z_2 (t) < x_2 (t) \le y_2 (t), \;z_2 (t)\not \le x_2 (t)\not \le y_1 (t), \;t \in J.$

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

$\begin{align*} \left\{ \begin{array}{l} z''_i (t) - \lambda {}^cD_{0^ + }^\alpha z_i (t) \le f\big(t, z_i (t), z'_i (t)\big), \;t \in J', \; \\ \Delta z_i (t)|_{t = t_k } \le I_k \big(z_i (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ \Delta z'_i (t)|_{t = t_k } \le Q_k \big(z_i (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ g_0 \big(z_i (0), z_i (1)\big) \le 0, \;g_1 \big(z'_i (0), z'_i (1)\big) \ge 0, \end{array} \right. \end{align*}$

and

$\begin{align*} \left\{ \begin{array}{l} y''_i (t) - \lambda {}^cD_{0^ + }^\alpha y_i (t) \ge f\big(t, y_i (t), y'_i (t)\big), \;t \in J', \; \\ \Delta y_i (t)|_{t = t_k } \ge I_k \big(y_i (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ \Delta y'_i (t)|_{t = t_k } \ge Q_k \big(y_i (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ g_0 \big(y_i (0), y_i (1)\big) \ge 0, \;g_1 \big(y'_i (0), y'_i (1)\big) \le 0, \end{array} \right. \end{align*}$

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

$z_1 (t) \le x_1 (t) < y_1 (t), \;z_2 (t) < x_2 (t) \le y_2 (t), \;z_2 (t)\not \le x_2 (t)\not \le y_1 (t), \;t \in J.$

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

$\begin{align*} \left\{ \begin{array}{l} z''_i (t) - \lambda {}^cD_{0^ + }^\alpha z_i (t) \le f\big(t, z_i (t), z'_i (t)\big), \;t \in J', \; \\ \Delta z_i (t)|_{t = t_k } \le I_k \big(z_i (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ \Delta z'_i (t)|_{t = t_k } \le Q_k \big(z_i (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ g_0 \big(z_i (0), z_i (1)\big) \ge 0, \;g_1 \big(z'_i (0), z'_i (1)\big) \le 0, \end{array} \right. \end{align*}$

and

$\begin{align*} \left\{ \begin{array}{l} y''_i (t) - \lambda {}^cD_{0^ + }^\alpha y_i (t) \ge f\big(t, y_i (t), y'_i (t)\big), \;t \in J', \; \\ \Delta y_i (t)|_{t = t_k } \ge I_k \big(y_i (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ \Delta y'_i (t)|_{t = t_k } \ge Q_k \big(y_i (t_k )\big), \;k = 1, \;2, \cdots , \;m, \\ g_0 \big(y_i (0), y_i (1)\big) \le 0, \;g_1 \big(y'_i (0), y'_i (1)\big) \ge 0, \end{array} \right. \end{align*}$

$z_1 (t) \le x_1 (t) < y_1 (t), \;z_2 (t) < x_2 (t) \le y_2 (t), \;z_2 (t)\not \le x_2 (t)\not \le y_1 (t), \;t \in J.$

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

$z_1 (t) \le x_1 (t) < y_1 (t), \;z_2 (t) < x_2 (t) \le y_2 (t), \;z_2 (t)\not \le x_2 (t)\not \le y_1 (t), \;t \in J.$

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

$\begin{align} \left\{ \begin{array}{l} x''(t) - \frac{1}{{100}}{}^cD_{0^ + }^{\frac{1}{2}} x(t) = \frac{{2t^3 }}{\pi }\arctan (x(t)+x'(t)), \;t\in (0, \frac{1}{2})\cup(\frac{1}{2}, 1), \\ \Delta x(t)|_{t = \frac{1}{2}} = \frac{1}{2}x(\frac{1}{2}), \;\Delta x'(t)|_{t = \frac{1}{2}} = \frac{1}{209}x(\frac{1}{2}), \\ - \cos (2\pi x(0)) + 0.02x(1) - 0.52 = 0, \\ - 6\pi \cos (\pi x'(0)) +\pi x'(1) - 6\pi = 0, \end{array} \right. \end{align}$

(5.1)

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

$\begin{align*} g_0(x, y) = -\cos (2\pi x)+0.02y-0.52, \;g_1(x, y) = -6\pi\cos(\pi x)+\pi y-6\pi. \end{align*}$

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

$\begin{align*} g_0 (x_2 , y_2 ) - g_0 (x_1 , y_1 ) & = - \cos (2\pi x_2 ) + \cos (2\pi x_1 ) + 0.02(y_2 - y_1 ) \\ &\ge - 2\pi (x_2 - x_1 ) + 0.02(y_2 - y_1 ), \\ g_1 (x_2 , y_2 ) - g_1 (x_1 , y_1 )& = - 6\pi \cos (\pi x_2 ) + 6\pi \cos (\pi x_1 ) + \pi (y_2 - y_1 ) \\ &\ge - 6\pi (x_2 - x_1 ) + \pi (y_2 - y_1 ), \end{align*}$

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

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

$\begin{align*} z_1 (t) = \left\{ \begin{array}{l} \frac{3}{2} + t, \;t \in [0, \;\frac{1}{2}], \\ 2 + t, \;t \in (\frac{1}{2}, \;1], \end{array} \right. y_1 (t) = \left\{ \begin{array}{l} 4 +\frac{3}{2}t + \frac{1}{8}t^2 + \frac{1}{4}t^4 , \;t \in [0, \;\frac{1}{2}], \\ 9 +\frac{3}{2}t+ \frac{1}{6}t^2 + \frac{1}{4}t^4 , \;t \in (\frac{1}{2}, \;1], \end{array} \right. \end{align*}$

$\begin{align*} z_2 (t) = \left\{ \begin{array}{l} \frac{{23}}{2} + 3t, \;t \in [0, \;\frac{1}{2}], \\ 12 + 3t, \;t \in (\frac{1}{2}, \;1], \end{array} \right. y_2 (t) = \left\{ \begin{array}{l} 16+\frac{7}{2}t+ \frac{2}{5}t^2 + \frac{1}{6}t^4 , \;t \in [0, \;\frac{1}{2}], \\ 33 +\frac{7}{2}t+ \frac{3}{4}t^2 + \frac{1}{6}t^4 , \;t \in (\frac{1}{2}, \;1]. \end{array} \right. \end{align*}$

We can easily get that

$\begin{align*} z''_1 (t) - \frac{1}{{100}}{}^cD_{0^ + }^{\frac{1}{2}} z_1 (t) - \frac{{2t^3 }}{\pi }\arctan (z_1 (t) + z'_1 (t)) = \left\{ \begin{array}{l} - \frac{{\sqrt t }}{{50\sqrt \pi }} - \frac{{2t^3 \arctan (\frac{5}{2} + t)}}{\pi } < 0, \;t \in (0, \frac{1}{2}), \\ - \frac{{\sqrt t }}{{50\sqrt \pi }} - \frac{{2t^3 \arctan (3 + t)}}{\pi } < 0, \;t \in (\frac{1}{2}, 1), \end{array} \right. \end{align*}$

$\begin{align*} &y''_1 (t) - \frac{1}{{100}}{}^cD_{0^ + }^{\frac{1}{2}} y_1 (t) - \frac{{2t^3 }}{\pi }\arctan (y_1 (t) + y'_1 (t))\\ {\rm{ = }}&\left\{ \begin{array}{l} \frac{1}{4} + 3t^2 - \frac{{\sqrt t (315 + 35t + 96t^3 )}}{{10500\sqrt \pi }} - \frac{{2t^3 \arctan (\frac{{11}}{2} + \frac{{7t}}{4} + \frac{{t^2 }}{8} + t^3 + \frac{{t^4 }}{4})}}{\pi } > 0, \;\;t \in [0, \frac{1}{2}], \\ \frac{{84000\sqrt \pi \sqrt t (1 + 9t^2 ) - 7560t - 840t^2 - 2304t^4 - 35\sqrt 2 \big(\sqrt {t(2t - 1)} + 4\sqrt {t^3 (2t - 1)} \big)}}{{252000\sqrt \pi \sqrt t }}\\ \;\;\;- \frac{{2t^3 \arctan (\frac{{21}}{2} +\frac{11t}{6} + \frac{{t^2 }}{6} + t^3 + \frac{{t^4 }}{4})}}{\pi } > 0, \;\;t \in (\frac{1}{2}, 1], \end{array} \right. \end{align*}$

$\begin{align*} z''_2 (t) - \frac{1}{{100}}{}^cD_{0^ + }^{\frac{1}{2}} z_2 (t) - \frac{{2t^3 }}{\pi }\arctan (z_2 (t) + z'_2 (t)) = \left\{ \begin{array}{l} - \frac{{3\sqrt t }}{{50\sqrt \pi }} - \frac{{2t^3 \arctan (\frac{{29}}{2} + 3t)}}{\pi } < 0, \;t \in (0, \frac{1}{2}), \\ - \frac{{3\sqrt t }}{{50\sqrt \pi }} - \frac{{2t^3 \arctan (15 + 3t)}}{\pi } < 0, \;t \in (\frac{1}{2}, 1), \end{array} \right. \end{align*}$

and

$\begin{align*} &y''_2 (t) - \frac{1}{{100}}{}^cD_{0^ + }^{\frac{1}{2}} y_2 (t) - \frac{{2t^3 }}{\pi }\arctan (y_2 (t) + y'_2 (t))\\ {\rm{ = }}&\left\{ \begin{array}{l} \frac{4}{5} + 2t^2 - \frac{{\sqrt t (735 + 112t + 64t^3 )}}{{10500\sqrt \pi }} - \frac{{2t^3 \arctan (\frac{{39}}{2} + \frac{{43t}}{{10}} + \frac{{2t^2 }}{5} + \frac{{2t^3 }}{3} + \frac{{t^4 }}{6})}}{\pi } > 0, \;\;t \in [0, \frac{1}{2}], \\ \frac{{21000\sqrt \pi \sqrt t (3 + 4t^2 )-2940t - 448t^2 - 256t^4 - 49\sqrt 2 \big(\sqrt {t(2t - 1)} + 4\sqrt {t^3 (2t - 1)} \big)}}{{42000\sqrt \pi \sqrt t }} \\ \;\;\;- \frac{{2t^3 \arctan (\frac{{73}}{2} + 5t + \frac{{3t^2 }}{4} + \frac{{2t^3 }}{3} + \frac{{t^4 }}{6})}}{\pi } > 0, \;\;t \in (\frac{1}{2}, 1]. \end{array} \right. \end{align*}$

$\begin{align*} \left\{ \begin{array}{l} z''_1 (t) - \frac{1}{{100}}{}^cD_{0^ + }^{\frac{1}{2}} z_1 (t) < \frac{{2t^3 }}{\pi }\arctan (z_1 (t) + z'_1 (t)), \; t\in(0, \frac{1}{2})\cup(\frac{1}{2}, 1), \\ \Delta z_1 (t)|_{t = \frac{1}{2}} = z_1 (\frac{1}{2}^ + ) - z_1 (\frac{1}{2}^ - ) = \frac{1}{2} < \frac{1}{2} z_1 (\frac{1}{2}) = 1, \\ \Delta z'_1 (t)|_{t = \frac{1}{2}} = z'_1 (\frac{1}{2}^ + ) - z'_1 (\frac{1}{2}^ - ) = 0 < \frac{1}{209}z_1 (\frac{1}{2}) = \frac{2}{209}, \\ - \cos (2\pi z_1 (0)) + 0.02z_1 (1) - 0.52 > 0, \\ - 6\pi \cos (\pi z'_1 (0)) + \pi z'_1 (1) - 6\pi > 0, \end{array} \right. \end{align*}$

$\begin{align*} \left\{ \begin{array}{l} y''_1 (t) - \frac{1}{{100}}{}^cD_{0^ + }^{\frac{1}{2}} y_1 (t) > \frac{{2t^3 }}{\pi }\arctan (y_1 (t) + y'_1 (t)), \; t\in(0, \frac{1}{2})\cup(\frac{1}{2}, 1), \\ \Delta y_1 (t)|_{t = \frac{1}{2}} = y_1 (\frac{1}{2}^ + ) - y_1 (\frac{1}{2}^ - ) = \frac{481}{96} > \frac{1}{2}y_1 (\frac{1}{2}) = \frac{307}{128}, \\ \Delta y'_1 (t)|_{t = \frac{1}{2}} = y'_1 (\frac{1}{2}^ + ) - y'_1 (\frac{1}{2}^ - ) = \frac{1}{24} > \frac{1}{209}y_1 (\frac{1}{2}) = \frac{307}{13376}, \\ - \cos (2\pi y_1 (0)) + 0.02y_1 (1) - 0.52 < 0, \\ - 6\pi \cos (\pi y'_1 (0)) + \pi y'_1 (1) - 6\pi < 0, \end{array} \right. \end{align*}$

$\begin{align*} \left\{ \begin{array}{l} z''_2 (t) - \frac{1}{{100}}{}^cD_{0^ + }^{\frac{1}{2}} z_2 (t) < \frac{{2t^3 }}{\pi }\arctan (z_2 (t) + z'_2 (t)), \; t\in(0, \frac{1}{2})\cup(\frac{1}{2}, 1), \\ \Delta z_2 (t)|_{t = \frac{1}{2}} = z_2 (\frac{1}{2}^ + ) - z_2 (\frac{1}{2}^ - ) = \frac{1}{2} < \frac{1}{2} z_2 (\frac{1}{2}) = \frac{13}{2}, \\ \Delta z'_2 (t)|_{t = \frac{1}{2}} = z'_2 (\frac{1}{2}^ + ) - z'_2 (\frac{1}{2}^ - ) = 0 < \frac{1}{209}z_2 (\frac{1}{2}) = \frac{13}{209}, \\ - \cos (2\pi z_2 (0)) + 0.02z_2 (1) - 0.52 > 0, \\ - 6\pi \cos (\pi z'_2 (0)) + \pi z'_2 (1) - 6\pi > 0 \end{array} \right. \end{align*}$

and

$\begin{align*} \left\{ \begin{array}{l} y''_2 (t) - \frac{1}{{100}}{}^cD_{0^ + }^{\frac{1}{2}} y_2 (t) > \frac{{2t^3 }}{\pi }\arctan (y_2 (t) + y'_2 (t)), \; t\in(0, \frac{1}{2})\cup(\frac{1}{2}, 1), \\ \Delta y_2 (t)|_{t = \frac{1}{2}} = y_2 (\frac{1}{2}^ + ) - y_2 (\frac{1}{2}^ - ) = \frac{1367}{80} > \frac{1}{2}y_2 (\frac{1}{2}) = \frac{8573}{960}, \\ \Delta y'_2 (t)|_{t = \frac{1}{2}} = y'_2 (\frac{1}{2}^ + ) - y'_2 (\frac{1}{2}^ - ) = \frac{7}{20} > \frac{1}{209}y_2 (\frac{1}{2}) = \frac{8573}{100320}, \\ - \cos (2\pi y_2 (0)) + 0.02y_2 (1) - 0.52 < 0, \\ - 6\pi \cos (\pi y'_2 (0)) + \pi y'_2 (1) - 6\pi < 0. \end{array} \right. \end{align*}$

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,

$z_1 (t) \le x_1 (t) < y_1 (t), \;z_2 (t) < x_2 (t) \le y_2 (t), \;z_2 (t)\not \le x_3 (t)\not \le y_1 (t), \;t \in [0, 1].$

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.

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AIMS Mathematics

A class of impulsive vibration equation with fractional derivatives

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. The solution of linear impulsive vibration equation

4. Existence of three solutions

5. Discussion on applicability of the main results

6. Conclusion

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

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Catalog

Abstract

1. Introduction

2. Preliminaries

3. The solution of linear impulsive vibration equation

4. Existence of three solutions

5. Discussion on applicability of the main results

6. Conclusion

Acknowledgments

Conflict of interest

References

AIMS Mathematics

A class of impulsive vibration equation with fractional derivatives

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. The solution of linear impulsive vibration equation

4. Existence of three solutions

5. Discussion on applicability of the main results

6. Conclusion

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

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Catalog

Abstract

1. Introduction

2. Preliminaries

3. The solution of linear impulsive vibration equation

4. Existence of three solutions

5. Discussion on applicability of the main results

6. Conclusion

Acknowledgments

Conflict of interest

References