1 Introduction

Towards the end of the 19th century Liouville and Riemann mentioned the definition of the fractional derivative which is the generalization of the traditional integer order differential and integral calculus. The fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. The subject of fractional differential equations is gaining much importance and attention because of its extensive applications in many engineering and scientific disciplines such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electro-analytical chemistry, biology, control theory, fitting of experimental data, and so forth. For more details of the basic theory of fractional differential equations, refer to [1–6] and the references therein. In recent decades, the boundary value problems of fractional differential equations have received a great deal of attention. There are a large number of papers dealing with the existence, nonexistence, multiplicity of solutions of boundary value problem for some nonlinear fractional differential equations (see [7–27]).

As we know, many evolutionary processes experience short-time rapid change after undergoing relatively long smooth variation. In order to describe the dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, diseases, and so on, some authors have used an impulsive differential system to describe these kinds of phenomena since the last century. For the theory of impulsive differential equations, the reader can refer to [28–30]. Recently, the boundary value problems of impulsive fractional differential equations have been studied extensively in the literature (see [31–45]). To the best of our knowledge, there are few articles involving the impulsive fractional order differential equations. Therefore, we will study the existence and uniqueness of solutions for the following impulsive integral boundary value problems (BVPs for short) of fractional order differential equations:

$$ \textstyle\begin{cases} {}_{t_{k}}D_{t}^{\alpha}u(t)= f(t,u,u',D^{\alpha-1}u), \quad t\neq t_{k},\\ \Delta D^{\alpha-1}u(t_{k})= I_{k}(u(t_{k})), \quad k=1,\dots,m,\\ u(0)=u'(0)=0, \quad\quad u'(1)=\int_{0}^{\eta}g(s,u(s))\,ds, \end{cases} $$
(1.1)

where \(2<\alpha\leq3\), \(J=[0,1]\), \(J_{0}=[t_{0},t_{1}]\), \(J_{k}=(t_{k},t_{k+1}]\subset J\) (\(k=1,2,\ldots,m\)). \({}_{t_{k}}D_{t}^{\alpha}\) is the Riemann-Liouville fractional derivative of order \(2<\alpha\leq3\). \(f\in C( J\times\mathbb{R}^{3}, \mathbb{R})\), \(I_{k}\in C(\mathbb{R}, \mathbb{R})\), \(0<\eta<1\), \(g\in C(J\times\mathbb{R},\mathbb{R})\), \(0=t_{0}< t_{1}<\cdots<t_{m}<t_{m+1}=1\). \(u(t_{k}^{+})=\lim_{h\rightarrow 0^{+}}u(t_{k}+h)\) and \(u(t_{k}^{-})=\lim_{h\rightarrow 0^{-}}u(t_{k}+h)\) represent the right and left limits of \(u(t)\) at \(t=t_{k}\), respectively. \(u(t_{k}^{-})=u(t_{k})\), \({}_{t_{k}}D_{t}^{\alpha-1}u(t_{k}^{-})= {}_{t_{k}}D_{t}^{\alpha-1}u(t_{k})\). The right-hand limits \(u(t_{k}^{+})\) and \({}_{t_{k}}D_{t}^{\alpha-1}u(t_{k}^{+})\) all exist. \(\Delta D^{\alpha-1}u(t_{k})= {}_{t_{k}}D_{t}^{\alpha-1}u(t_{k}^{+})- {}_{t_{k-1}}D_{t}^{\alpha-1}u(t_{k}^{-})\).

The rest of this paper is organized as follows. In Section 2, we shall introduce some definitions and lemmas to prove our main results. In Section 3, we give some sufficient conditions for the existence of single positive solutions for boundary value problem (1.1). As an application, one interesting example is presented to illustrate the main results in Section 4. Finally, the conclusion is given to simply recall our studied contents and obtained results in Section 5.

2 Preliminaries

Let \(C(J,\mathbb{R})\) be the Banach space of continuous functions from J to \(\mathbb{R}\) with the norm \(\Vert u\Vert _{C}=\sup_{0\leq t\leq1}\vert u(t)\vert \). Now let us to introduce the useful Banach space \(\mathit {PC}^{1}(J,\mathbb{R})\) defined by

$$\begin{aligned} \mathit {PC}^{1}(J,\mathbb{R})&= \bigl\{ u\in C(J, \mathbb{R}): {}_{t_{k}}D_{t}^{\alpha-1}u\bigl(t_{k}^{+} \bigr) \mbox{ and } {}_{t_{k}}D_{t}^{\alpha-1}u \bigl(t_{k}^{-}\bigr) \mbox{ exist with} \\ &\quad {}_{t_{k}}D_{t}^{\alpha-1}u(t_{k}) = {}_{t_{k}}D_{t}^{\alpha-1}u\bigl(t_{k}^{-}\bigr), k=0,1,\dots, m \bigr\} \end{aligned}$$
(2.1)

equipped with the norm \(\Vert u\Vert _{\mathit {PC}^{1}}=\max\{\Vert u\Vert _{C}, \Vert u'\Vert _{C}, \Vert {}_{t_{k}}D_{t}^{\alpha -1}u\Vert _{C}\}\).

Definition 2.1

A function \(u\in \mathit {PC}^{1}(J,\mathbb{R})\) with its Riemann-Liouville derivative of order α existing on J is a solution of (1.1) if it satisfies (1.1).

For convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions and properties can be found in the literature [2, 4, 6].

Definition 2.2

The Riemann-Liouville fractional integral of order \(\alpha>0\) of a function \(u:(a,+\infty)\rightarrow \mathbb{R}\) is given by

$${}_{a}I_{t}^{\alpha}u(t)=\frac{1}{\Gamma(\alpha)} \int _{a}^{t}(t-s)^{\alpha-1}u(s)\,ds, \quad a>0, $$

provided that the right-hand side is point-wise defined on \((a,+\infty)\).

Definition 2.3

The Riemann-Liouville fractional derivative of order \(\alpha>0\) of a continuous function \(u:(a,+\infty)\rightarrow \mathbb{R}\) is given by

$${}_{a}D_{t}^{\alpha}u(t)=\frac{1}{\Gamma(n-\alpha)} \frac {d^{n}}{dt^{n}} \int_{a}^{t}(t-s)^{n-\alpha-1}u(s)\,ds, $$

where \(a>0\), \(n-1<{\alpha}\leq n\), provided that the right-hand side is point-wise defined on \((a,+\infty)\).

Lemma 2.1

Assume that \(u\in C[a,b]\), \(q\geq p\geq0\), then

$$\begin{aligned} {}_{a}D_{t}^{p} {}_{a}I_{t}^{q}u(t)={}_{a}I_{t}^{q-p}u(t), \quad t\in [a,b]. \end{aligned}$$
(2.2)

Lemma 2.2

see [6], pp. 36-39

Let \(\alpha>0\), n denotes the smallest integer greater than or equal to α. Then the following assertions hold.

  1. (i)

    if \(\lambda>-1\), \(\lambda\neq\alpha-i\), \(i=1, 2, \ldots, n+1\), then for \(t\in[a,b]\)

    $$\begin{aligned} {}_{a}D_{t}^{\alpha}(t-a)^{\lambda}= \frac{\Gamma(\lambda+1)}{\Gamma (\lambda-\alpha+1)}(t-a)^{\lambda-\alpha}. \end{aligned}$$
    (2.3)
  2. (ii)

    \({}_{a}D_{t}^{\alpha}(t-a)^{\alpha-i}=0\), \(i=1, 2, \ldots, n\).

  3. (iii)

    \({}_{a}D_{t}^{\alpha}{} _{a}I_{t}^{\alpha }u(t)=u(t)\), for all \(t\in[a,b]\).

  4. (iv)

    \({}_{a}D_{t}^{\alpha}u(t)=0\) if and only if there exists \(c_{i}\in\mathbb{R}\) (\(i=1,2,\ldots,n\)) such that

    $$\begin{aligned} u(t)=c_{1}(t-a)^{\alpha-1}+c_{2}(t-a)^{\alpha-2} +\cdots +c_{n}(t-a)^{\alpha-n}, \quad t\in[a,b]. \end{aligned}$$
    (2.4)
  5. (v)

    For all \(t\in[a,b]\), then

    $$\begin{aligned} {}_{a}I_{t}^{\alpha}{} _{a}D_{t}^{\alpha}u(t) =c_{1}(t-a)^{\alpha-1}+c_{2}(t-a)^{\alpha-2} +\cdots+c_{n}(t-a)^{\alpha-n}+u(t). \end{aligned}$$
    (2.5)

Lemma 2.3

Schauder fixed point theorem; see [46]

If U is a closed bounded convex subset of a Banach space X and \(T :U \rightarrow U\) is completely continuous, then T has at least one fixed point in U.

Lemma 2.4

For a given \(y\in C(J,\mathbb{R})\), a function \(u\in \mathit {PC}^{1}(J,\mathbb{R})\) is a solution of BVP (2.6)

$$ \textstyle\begin{cases} {}_{t_{k}}D_{t}^{\alpha}u(t)= y(t),\quad t\neq t_{k}, 2< \alpha\leq3, \\ \Delta D^{\alpha-1}u(t_{k})= I_{k}(u(t_{k})), \quad k=1,\dots,m,\\ u(0)=u'(0)=0, \quad\quad u'(1)=\int_{0}^{\eta}g(s,u(s))\,ds, \end{cases} $$
(2.6)

if and only if \(u\in \mathit {PC}^{1}(J,\mathbb{R})\) is a solution of the impulsive fractional integral equation

$$\begin{aligned} u(t)&=\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}y(s)\,ds - \frac{t^{\alpha-1}}{\Gamma(\alpha)} \int_{0}^{1}(1-s)^{\alpha-2}y(s)\,ds \\ &\quad{} -\frac{t^{\alpha-1}}{\Gamma(\alpha)}\sum_{t\leq t_{i}}I_{i} \bigl(u(t_{i})\bigr) +\frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta}g\bigl(s,u(s)\bigr)\,ds, \quad 0\leq t\leq1. \end{aligned}$$
(2.7)

Proof

We denote the solution of (2.6) by \(u(t)=u_{k}(t)\) in \([t_{k}, t_{k+1}]\) (\(k=1,2,\ldots,m\)). For \(t\in J_{0}=[0,t_{1}]\), by (2.5), we have

$$\begin{aligned} u_{0}(t)= {}_{0}I_{t}^{\alpha}y(t)+c_{01}t^{\alpha-1}+c_{02}t^{\alpha-2} +c_{03}t^{\alpha-3}. \end{aligned}$$

\(u(0)=u'(0)=0\) implies that \(c_{02}=c_{03}=0\). Applying Lemma 2.2, we get

$$\begin{aligned}& u_{0}(t)= {}_{0}I_{t}^{\alpha}y(t)+c_{01}t^{\alpha-1}, \\& D^{\alpha-1}_{t}u_{0}(t)=D^{\alpha-1}_{t} \bigl[{}_{0}I_{t}^{\alpha }y(t)+c_{01}t^{\alpha-1} \bigr] = \int_{0}^{t}y(s)\,ds+\Gamma(\alpha)c_{01}, \end{aligned}$$

and

$$\begin{aligned} D^{\alpha-1}u\bigl(t_{1}^{+}\bigr)&=D^{\alpha-1}u_{0} \bigl(t_{1}^{+}\bigr) =D^{\alpha-1}u_{0}(t_{1})+I_{1} \bigl(u(t_{1})\bigr) \\ &= \int_{0}^{t_{1}}y(s)\,ds+\Gamma(\alpha)c_{01}+I_{1} \bigl(u(t_{1})\bigr). \end{aligned}$$

For \(t\in J_{1}=(t_{1},t_{2}]\), by (2.5), we get

$$\begin{aligned} u_{1}(t)= {}_{0}I_{t}^{\alpha}y(t)+c_{11}t^{\alpha-1}+c_{12}t^{\alpha-2} +c_{13}t^{\alpha-3} \end{aligned}$$

and

$$\begin{aligned} D^{\alpha-1}_{t}u_{1}(t)= \int_{0}^{t}y(s)\,ds+\Gamma(\alpha)c_{11}. \end{aligned}$$

Noting that \(u(0)=u'(0)=0\) and \(D^{\alpha-1}u_{1}(t_{1})= D^{\alpha-1}u_{0}(t_{1}^{+})\), we derive \(c_{12}=c_{13}=0\) and \(c_{11}=c_{01}+\frac{I_{1}(u(t_{1}))}{\Gamma(\alpha)}\). So we can obtain

$$\begin{aligned} u_{1}(t)= {}_{0}I_{t}^{\alpha}y(t)+ \biggl[c_{01}+\frac {I_{1}(u(t_{1}))}{\Gamma(\alpha)} \biggr]t^{\alpha-1} \end{aligned}$$

and

$$\begin{aligned} D^{\alpha-1}u\bigl(t_{2}^{+}\bigr)=D^{\alpha-1}u_{1} \bigl(t_{2}^{+}\bigr) =D^{\alpha-1}u_{1}(t_{2})+I_{2} \bigl(u(t_{2})\bigr)= \int_{0}^{t_{2}}y(s)\,ds+\Gamma(\alpha )c_{01}+ \sum_{i=1}^{2}I_{i} \bigl(u(t_{i})\bigr). \end{aligned}$$

By the recurrent method, for \(t\in J_{k}=(t_{k},t_{k+1}]\), \(k=2,3,\ldots,m\), we get

$$\begin{aligned} u_{k}(t)&= {}_{0}I_{t}^{\alpha}y(t)+ \Biggl[c_{01}+\frac{1}{\Gamma (\alpha)}\sum_{i=1}^{k}I_{i} \bigl(u(t_{i})\bigr) \Biggr]t^{\alpha-1} \\ &=\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}y(s)\,ds + \Biggl[c_{01}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{k}I_{i} \bigl(u(t_{i})\bigr) \Biggr]t^{\alpha-1} \end{aligned}$$

and

$$\begin{aligned} D^{\alpha-1}u\bigl(t_{k+1}^{+}\bigr)&=D^{\alpha-1}u_{k} \bigl(t_{k+1}^{+}\bigr) =D^{\alpha-1}u_{k}(t_{k+1})+I_{k+1} \bigl(u(t_{k+1})\bigr) \\ &= \int_{0}^{t_{k+1}}y(s)\,ds+\Gamma(\alpha)c_{01}+ \sum_{i=1}^{k+1}I_{i} \bigl(u(t_{i})\bigr). \end{aligned}$$

So, for \(t\in J_{m}=(t_{m},t_{m+1}]\), we have

$$\begin{aligned} u'(t)=u'_{m}(t)=\frac{1}{\Gamma(\alpha-1)} \int_{0}^{t}(t-s)^{\alpha-2}y(s)\,ds +(\alpha-1) \Biggl[c_{01}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{m}I_{i} \bigl(u(t_{i})\bigr) \Biggr]t^{\alpha-2}. \end{aligned}$$

By \(u'(1)=\int_{0}^{\eta}u(s)\psi(s)\,ds\), we have

$$\begin{aligned} \int_{0}^{\eta}g\bigl(s,u(s)\bigr)\,ds= \frac{1}{\Gamma(\alpha-1)} \int _{0}^{1}(1-s)^{\alpha-2}y(s)\,ds +(\alpha-1) \Biggl[c_{01}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{m}I_{i} \bigl(u(t_{i})\bigr) \Biggr], \end{aligned}$$

which implies that

$$\begin{aligned} c_{01}=\frac{1}{\alpha-1} \int_{0}^{\eta}g\bigl(s,u(s)\bigr)\,ds - \frac{1}{\Gamma(\alpha)} \int_{0}^{1}(1-s)^{\alpha-2}y(s)\,ds - \frac{1}{\Gamma(\alpha)}\sum_{i=1}^{m}I_{i} \bigl(u(t_{i})\bigr). \end{aligned}$$

Therefore, for \(t\in J=[0,1]\), we have

$$\begin{aligned} u(t)&=\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}y(s)\,ds + \frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta}g\bigl(s,u(s)\bigr)\,ds \\ &\quad{} -\frac{t^{\alpha-1}}{\Gamma(\alpha)} \Biggl[ \int _{0}^{1}(1-s)^{\alpha-2}y(s)\,ds +\sum _{i=1}^{m}I_{i} \bigl(u(t_{i})\bigr)-\sum_{t_{i}< t}I_{i} \bigl(u(t_{i})\bigr) \Biggr] \\ &=\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}y(s)\,ds - \frac{t^{\alpha-1}}{\Gamma(\alpha)} \biggl[ \int _{0}^{1}(1-s)^{\alpha-2}y(s)\,ds +\sum _{t\leq t_{i}}I_{i}\bigl(u(t_{i})\bigr) \biggr] \\ &\quad{} +\frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta}g\bigl(s,u(s)\bigr)\,ds, \end{aligned}$$

which indicates that u is a solution of (2.7). Conversely, noting that the above derivations are reversible, we assert that if u is a solution of the impulsive fractional integral equation (2.7), then u is also the solution of BVP (2.6). The proof is complete. □

3 Main results

According to Lemma 2.4, we obtain the following lemma.

Lemma 3.1

A function \(u\in \mathit {PC}^{1}(J,\mathbb{R})\) is a solution of BVP (1.1) if and only if \(u\in \mathit {PC}^{1}(J,\mathbb{R})\) is a solution of the impulsive fractional integral equation

$$\begin{aligned} u(t)&=\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha -1}f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\,ds \\ &\quad{} - \biggl[ \int_{0}^{1}(1-s)^{\alpha-2}f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\,ds +\sum _{t\leq t_{i}}I_{i}\bigl(u(t_{i})\bigr) \biggr] \frac{t^{\alpha-1}}{\Gamma(\alpha)} \\ &\quad{} +\frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta}g\bigl(s,u(s)\bigr)\,ds, \quad 0\leq t\leq1. \end{aligned}$$
(3.1)

Define an operator \(T : \mathit {PC}^{1}(J,\mathbb{R})\rightarrow \mathit {PC}^{1}(J,\mathbb {R})\) as follows:

$$\begin{aligned} (Tu) (t)&=\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha -1}f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\,ds \\ &\quad{} - \biggl[ \int_{0}^{1}(1-s)^{\alpha-2}f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\,ds +\sum _{t\leq t_{i}}I_{i}\bigl(u(t_{i})\bigr) \biggr] \frac{t^{\alpha-1}}{\Gamma(\alpha)} \\ &\quad{} +\frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta}g\bigl(s,u(s)\bigr)\,ds, \quad 0\leq t\leq1. \end{aligned}$$
(3.2)

Then BVP (1.1) has a solution if and only if the operator T exists one fixed point.

Lemma 3.2

Assume that \(f\in C( J\times\mathbb{R}^{3}, \mathbb{R})\), and \(g\in C(J\times\mathbb{R}, \mathbb{R})\). Then \(T: \mathit {PC}^{1}(J,\mathbb{R}) \rightarrow \mathit {PC}^{1}(J,\mathbb{R})\) defined by (3.2) is completely continuous.

Proof

Note that T is continuous in view of continuity of f, \(I_{k}\), and g. Now we show that T is uniformly bounded. In fact, let \(\Omega\subset \mathit {PC}^{1}(J,\mathbb{R})\) be bounded, then there exist some positive constants \(l_{i}\) (\(i=1,2,3\)) such that \(\vert f(t,u,u',D^{\alpha-1}u)\vert \leq l_{1}\), \(\vert g(t,u)\vert \leq l_{2}\), \(\vert I_{k}(u)\vert \leq l_{3}\), for all \(u\in\Omega\). Thus for \(u\in\Omega\), we have

$$\begin{aligned}& \begin{aligned} \bigl\vert (Tu) (t)\bigr\vert &\leq\frac{1}{\Gamma(\alpha)} \int _{0}^{t}(t-s)^{\alpha-1}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha -1}u(s)\bigr)\bigr\vert \,ds \\ &\quad{} + \biggl[ \int_{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds +\sum _{t\leq t_{i}}\bigl\vert I_{i} \bigl(u(t_{i})\bigr)\bigr\vert \biggr]\frac{t^{\alpha -1}}{\Gamma(\alpha)} \\ &\quad{} +\frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\ &\leq\frac{2l_{1}+ml_{3}}{\Gamma(\alpha)}+\frac{l_{2}\eta}{\alpha -1}\triangleq M_{1}, \end{aligned} \\& \begin{aligned} \bigl\vert (Tu)'(t)\bigr\vert & = \biggl\vert \frac{1}{\Gamma(\alpha-1)} \int_{0}^{t}(t-s)^{\alpha -2}f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\,ds \\ & \quad {} - \frac{t^{\alpha-2}}{\Gamma(\alpha-1)} \int_{0}^{1}(1-s)^{\alpha -2}f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\,ds \\ & \quad {} - \frac{t^{\alpha-2}}{\Gamma(\alpha-1)}\sum_{t\leq t_{i}}I_{i} \bigl(u(t_{i})\bigr) +t^{\alpha-2} \int_{0}^{\eta}g\bigl(s,u(s)\bigr)\,ds\biggr\vert \\ & \leq \int_{0}^{1}\frac{(1-s)^{\alpha-2}}{\Gamma(\alpha -1)}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds \\ &\quad {} + \int_{0}^{1}\frac{(1-s)^{\alpha-2}}{\Gamma(\alpha-1)}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds \\ & \quad {} +\frac{1}{\Gamma(\alpha-1)}\sum_{t\leq t_{i}} \bigl\vert I_{i}\bigl(u(t_{i})\bigr)\bigr\vert + \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\ & \leq\frac{2l_{1}+ml_{3}}{\Gamma(\alpha-1)}+l_{2}\eta\triangleq M_{2} \end{aligned} \end{aligned}$$

and

$$\begin{aligned}& \bigl\vert D^{\alpha-1}(Tu) (t)\bigr\vert \\ & \quad = \biggl\vert \int_{0}^{t}f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s) \bigr)\,ds - \int _{0}^{1}(1-s)^{\alpha-2}f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\,ds \\ & \quad\quad {} - \sum_{t\leq t_{i}}I_{i}\bigl(u(t_{i}) \bigr)+\Gamma(\alpha-1) \int_{0}^{\eta }g\bigl(s,u(s)\bigr)\,ds\biggr\vert \\ & \quad \leq \int_{0}^{t}\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha -1}u(s) \bigr)\bigr\vert \,ds + \int_{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds \\ & \quad\quad {} +\sum_{t\leq t_{i}}\bigl\vert I_{i}\bigl(u(t_{i})\bigr)\bigr\vert +\Gamma(\alpha -1) \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\ & \quad \leq 2l_{1}+ml_{3}+l_{2}\eta\Gamma(\alpha-1) \triangleq M_{3}, \end{aligned}$$

which means that \(\Vert u\Vert _{\mathit {PC}^{1}}\leq\max\{ M_{1},M_{2},M_{3}\}\), that is, T is uniformly bounded.

Next, we should prove that T is equicontinuous on \(J=[0,1]\). Indeed, for all \(\bar{t}_{1}, \bar{t}_{2}\in[0,1]\) with \(\bar{t}_{1}\leq \bar{t}_{2}\), we have

$$\begin{aligned}& \begin{aligned} \bigl\vert (Tu) (\bar{t}_{2})-(Tu) (\bar{t}_{1})\bigr\vert &=\biggl\vert \int_{\bar{t}_{1}}^{\bar{t}_{2}}(Tu)'(s)\,ds\biggr\vert \leq \int_{\bar {t}_{1}}^{\bar{t}_{2}}\bigl\vert (Tu)'(s)\bigr\vert \,ds \\ &\leq M_{2}(\bar{t}_{2}-\bar{t}_{1}) \rightarrow 0, \quad \mbox{as } \bar {t}_{1}\rightarrow \bar{t}_{2}, \end{aligned} \\ & \begin{gathered} \bigl\vert (Tu)'(\bar{t}_{2})-(Tu)'( \bar{t}_{1})\bigr\vert \\ \quad \leq \frac{1}{\Gamma(\alpha-1)} \int_{0}^{\bar{t}_{1}} \bigl[(\bar {t}_{2}-s)^{\alpha-2}-( \bar{t}_{1}-s)^{\alpha-2} \bigr] \bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s) \bigr)\bigr\vert \,ds \\ \quad\quad{} +\frac{1}{\Gamma(\alpha-1)} \int_{\bar{t}_{1}}^{\bar{t}_{2}}(\bar {t}_{2}-s)^{\alpha-2} \bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds \\ \quad\quad{} +\frac{\bar{t}_{2}^{\alpha-2}-\bar{t}_{1}^{\alpha-2}}{\Gamma (\alpha-1)} \int_{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha -1}u(s)\bigr)\bigr\vert \,ds \\ \quad\quad{} +\frac{\bar{t}_{2}^{\alpha-2}-\bar{t}_{1}^{\alpha-2}}{\Gamma (\alpha-1)}\sum_{\bar{t}_{2}\leq t_{i}}\bigl\vert I_{i}\bigl(u(t_{i})\bigr)\bigr\vert +\frac{\bar{t}_{1}^{\alpha-2}}{\Gamma(\alpha-1)} \sum_{\bar {t}_{1}\leq t_{i}< \bar{t}_{2}}\bigl\vert I_{i} \bigl(u(t_{i})\bigr)\bigr\vert \\ \quad\quad{} +\bigl(\bar{t}_{2}^{\alpha-2}-\bar{t}_{1}^{\alpha-2} \bigr) \int_{0}^{\eta }\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\ \quad \leq \frac{l_{1}}{\Gamma(\alpha)}\bigl(\bar{t}_{2}^{\alpha-1}-\bar {t}_{1}^{\alpha-1}\bigr) +\frac{l_{1}}{\Gamma(\alpha)}(\bar{t}_{2}- \bar{t}_{1})^{\alpha-1} +\frac{l_{1}}{\Gamma(\alpha)}\bigl( \bar{t}_{2}^{\alpha-2}-\bar {t}_{1}^{\alpha-2}\bigr) \\ \quad\quad{} +\frac{ml_{3}}{\Gamma(\alpha-1)}\bigl(\bar{t}_{2}^{\alpha-2}-\bar {t}_{1}^{\alpha-2}\bigr) +\frac{l_{3}}{\Gamma(\alpha-1)}(\bar{t}_{2}- \bar{t}_{1}) +l_{2}\eta\bigl(\bar{t}_{2}^{\alpha-2}- \bar{t}_{1}^{\alpha-2}\bigr)\rightarrow 0, \\ \quad\quad \mbox{as } \bar{t}_{1}\rightarrow \bar{t}_{2}, \end{gathered} \end{aligned}$$

and

$$\begin{aligned} \bigl\vert D^{\alpha-1}(Tu) (\bar{t}_{2})-D^{\alpha-1}(Tu) ( \bar {t}_{1})\bigr\vert &=\biggl\vert \int_{\bar{t}_{1}}^{\bar{t}_{2}}f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s) \bigr)\,ds +\sum_{\bar{t}_{1}\leq t_{i}< \bar{t}_{2}}I_{i} \bigl(u(t_{i})\bigr)\biggr\vert \\ &\leq \int_{\bar{t}_{1}}^{\bar{t}_{2}}\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s) \bigr)\bigr\vert \,ds +\sum_{\bar{t}_{1}\leq t_{i}< \bar{t}_{2}}\bigl\vert I_{i}\bigl(u(t_{i})\bigr)\bigr\vert \\ &\leq l_{1}(\bar{t}_{2}-\bar{t}_{1})+l_{3}( \bar{t}_{2}-\bar {t}_{1})\rightarrow 0, \quad \mbox{as } \bar{t}_{1}\rightarrow \bar{t}_{2}. \end{aligned}$$

Thus, for any \(\varepsilon>0\) (small enough), there exists \(\delta=\delta(\varepsilon)>0\) with independence of \(\bar{t}_{1}\), \(\bar{t}_{2}\) and u such that \(\Vert (Tu)(\bar{t}_{2})-(Tu)(\bar{t}_{1})\Vert _{\mathit {PC}^{1}}<\varepsilon\), whenever \(\vert \bar{t}_{2}-\bar{t}_{1}\vert <\delta\). Therefore, T is equicontinuous on \(J=[0,1]\). According to the Arzela-Ascoli theorem, it follows that \(T : \mathit {PC}^{1}(J,\mathbb{R})\rightarrow \mathit {PC}^{1}(J,\mathbb{R})\) is completely continuous. □

Theorem 3.1

Assume that the conditions \((B_{1})\)-\((B_{3})\) hold.

\((B_{1})\) :

\(f\in C(J\times\mathbb{R}^{3}, \mathbb{R})\), for all \((t,u,v,w), (t, \bar{u}, \bar{v}, \bar{w})\in J\times\mathbb{R}^{3}\), there exist some functions \(\psi_{i}\in L([0,1])\) (\(i=1,2,3\)) such that

$$\begin{aligned} \bigl\vert f(t,u,v,w)-f(t,\bar{u},\bar{v},\bar{w})\bigr\vert &\leq\bigl\vert \psi_{1}(t)\bigr\vert \vert u-\bar{u}\vert \\ &\quad{} +\bigl\vert \psi_{2}(t)\bigr\vert \vert v-\bar {v}\vert + \bigl\vert \psi_{3}(t)\bigr\vert \vert w-\bar {w}\vert . \end{aligned}$$
\((B_{2})\) :

\(I_{k}\in C(\mathbb{R},\mathbb{R})\), for all \(u, v\in \mathbb{R}\), there exist some constants \(L_{k}>0\) such that

$$\bigl\vert I_{k}(u)-I_{k}(v)\bigr\vert \leq L_{k}\vert u-v\vert , \quad k=1,2,\ldots,m. $$
\((B_{3})\) :

\(g\in C(J,\mathbb{R})\), for all \((t,u), (t,v)\in J\times\mathbb{R}\), there exists a function \(\psi\in L([0,1])\) such that

$$\bigl\vert g(t,u)-g(t,v)\bigr\vert \leq\bigl\vert \psi(t)\bigr\vert \vert u-v\vert . $$

If \(\rho=2\int_{0}^{1}[\vert \psi_{1}(s)\vert +\vert \psi_{2}(s)\vert +\vert \psi_{3}(s)\vert ]\,ds+ \sum_{k=1}^{m}L_{k}+\Gamma(\alpha-1)\int_{0}^{\eta} \vert \psi(s)\vert \,ds<1\), then BVP (1.1) has a unique solution on J.

Proof

Let \(M=\sup_{t\in J}\vert f(t,0,0,0)\vert + \sup_{t\in J}\vert g(t,0)\vert \) and \(B_{r}=\{u\in \mathit {PC}(J,\mathbb{R}):\Vert u\Vert _{\mathit {PC}^{1}}\leq r\}\), where \(r\geq\frac{1}{1-\rho} [(2+\Gamma(\alpha-1)\eta)M+\sum_{k=1}^{m} \vert I_{k}(0)\vert ]\). Define an operator \(T: B_{r}\rightarrow \mathit {PC}(J, \mathbb{R})\) as (3.2). It is obvious that T is jointly continuous and maps bounded subsets of \(J \times\mathbb{R}\) to bounded subsets of \(\mathbb{R}\). We will prove Theorem 3.1 through the following two steps.

Step 1. We show that \(T(B_{r})\subset B_{r}\). In fact, noting that \(u(0)=u'(0)=D^{\alpha-1}u(0)=0\), we have, for \(u\in B_{r}\), \(t\in J=[0,1]\),

$$\begin{aligned}& \bigl\vert (Tu) (t)\bigr\vert \\& \quad \leq \frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds + \frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\& \quad\quad{} +\frac{t^{\alpha-1}}{\Gamma(\alpha)} \biggl[ \int_{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds +\sum _{t\leq t_{i}}\bigl\vert I_{i} \bigl(u(t_{i})\bigr)\bigr\vert \biggr] \\& \quad \leq \frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \bigl[\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f(s,0,0,0)\bigr\vert +\bigl\vert f(s,0,0,0)\bigr\vert \bigr]\,ds \\& \quad\quad{} +\frac{t^{\alpha-1}}{\Gamma(\alpha)} \int_{0}^{1}(1-s)^{\alpha -2} \bigl[\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f(s,0,0,0)\bigr\vert +\bigl\vert f(s,0,0,0)\bigr\vert \bigr]\,ds \\& \quad\quad{}+\frac{t^{\alpha-1}}{\Gamma(\alpha)}\sum_{t\leq t_{i}} \bigl[\bigl\vert I_{i}\bigl(u(t_{i})\bigr)-I_{i}\bigl(u(0)\bigr) \bigr\vert +\bigl\vert I_{i}\bigl(u(0)\bigr)\bigr\vert \bigr] \\& \quad\quad{}+\frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta} \bigl[\bigl\vert g\bigl(s,u(s) \bigr)-g(s,0)\bigr\vert +\bigl\vert g(s,0)\bigr\vert \bigr]\,ds \\& \quad \leq \frac{r\int_{0}^{1}[\vert \psi_{1}(s)\vert +\vert \psi_{2}(s)\vert +\vert \psi_{3}(s)\vert ]\,ds+M}{\Gamma(\alpha)} +\frac{r\int_{0}^{1}[\vert \psi_{1}(s)\vert +\vert \psi _{2}(s)\vert +\vert \psi_{3}(s)\vert ]\,ds+M}{\Gamma (\alpha)} \\& \quad\quad{} +\frac{1}{\Gamma(\alpha)}\sum_{k=1}^{m} \bigl[L_{k}r+\bigl\vert I_{k}(0)\bigr\vert \bigr] + \frac{r}{\alpha-1} \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds+ \frac{\eta M}{\alpha-1} \\& \quad = \Biggl[\frac{2\int_{0}^{1}[\vert \psi_{1}(s)\vert +\vert \psi_{2}(s)\vert +\vert \psi_{3}(s)\vert ]\,ds}{\Gamma(\alpha)} +\frac{1}{\Gamma(\alpha)}\sum_{k=1}^{m}L_{k}+ \frac{1}{\alpha-1} \int _{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]r \\& \quad\quad{} + \biggl[\frac{2}{\Gamma(\alpha)}+\frac{\eta}{\alpha-1} \biggr]M+\frac{1}{\Gamma(\alpha)}\sum _{k=1}^{m}\bigl\vert I_{k}(0) \bigr\vert \\& \quad \leq \Biggl[2 \int_{0}^{1}\bigl[\bigl\vert \psi_{1}(s) \bigr\vert +\bigl\vert \psi_{2}(s)\bigr\vert +\bigl\vert \psi_{3}(s)\bigr\vert \bigr]\,ds +\sum_{k=1}^{m}L_{k}+ \Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi (s)\bigr\vert \,ds \Biggr]r \\& \quad\quad{} + \bigl[2+\Gamma(\alpha-1)\eta \bigr]M+\sum_{k=1}^{m} \bigl\vert I_{k}(0)\bigr\vert \\& \quad \leq \rho r+(1-\rho)r=r, \\& \bigl\vert (Tu)'(t)\bigr\vert \\& \quad \leq \frac{1}{\Gamma(\alpha-1)} \int_{0}^{t}(t-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds \\& \quad\quad{}+\frac{t^{\alpha-2}}{\Gamma(\alpha-1)} \int _{0}^{1}(1-s)^{\alpha-2} \bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds \\& \quad\quad{}+\frac{t^{\alpha-2}}{\Gamma(\alpha-1)}\sum_{t\leq t_{i}}\bigl\vert I_{i}\bigl(u(t_{i})\bigr)\bigr\vert +t^{\alpha-2} \int_{0}^{\eta }\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\& \quad \leq \int_{0}^{t}\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha -1)} \bigl[\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f(s,0,0,0)\bigr\vert +\bigl\vert f(s,0,0,0)\bigr\vert \bigr]\,ds \\& \quad\quad{}+ \int_{0}^{1}\frac{[t(1-s)]^{\alpha-2}}{\Gamma(\alpha -1)} \bigl[\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f(s,0,0,0)\bigr\vert +\bigl\vert f(s,0,0,0)\bigr\vert \bigr]\,ds \\& \quad\quad{} +\frac{t^{\alpha-2}}{\Gamma(\alpha-1)}\sum_{t\leq t_{i}} \bigl[\bigl\vert I_{i}\bigl(u(t_{i})\bigr)-I_{i}\bigl(u(0)\bigr) \bigr\vert +\bigl\vert I_{i}\bigl(u(0)\bigr)\bigr\vert \bigr] \\& \quad\quad{} +t^{\alpha-2} \int_{0}^{\eta} \bigl[\bigl\vert g\bigl(s,u(s) \bigr)-g(s,0)\bigr\vert +\bigl\vert g(s,0)\bigr\vert \bigr]\,ds \\& \quad \leq \frac{r\int_{0}^{1}[\vert \psi_{1}(s)\vert +\vert \psi_{2}(s)\vert +\vert \psi_{3}(s)\vert ]\,ds+M}{\Gamma(\alpha-1)} +\frac{r\int_{0}^{1}[\vert \psi_{1}(s)\vert +\vert \psi _{2}(s)\vert +\vert \psi_{3}(s)\vert ]\,ds+M}{\Gamma (\alpha-1)} \\& \quad\quad{} +\frac{1}{\Gamma(\alpha-1)}\sum_{k=1}^{m} \bigl[L_{k}r+\bigl\vert I_{k}(0)\bigr\vert \bigr] +r \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds+ \eta M \\& \quad = \Biggl[\frac{2\int_{0}^{1}[\vert \psi_{1}(s)\vert +\vert \psi_{2}(s)\vert +\vert \psi_{3}(s)\vert ]\,ds}{\Gamma(\alpha-1)} +\frac{1}{\Gamma(\alpha-1)}\sum _{k=1}^{m}L_{k}+ \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]r \\& \quad\quad{} + \biggl[\frac{2}{\Gamma(\alpha-1)}+\eta \biggr]M+\frac{1}{\Gamma (\alpha-1)}\sum _{k=1}^{m}\bigl\vert I_{k}(0)\bigr\vert \\& \quad \leq \Biggl[2 \int_{0}^{1}\bigl[\bigl\vert \psi_{1}(s) \bigr\vert +\bigl\vert \psi_{2}(s)\bigr\vert +\bigl\vert \psi_{3}(s)\bigr\vert \bigr]\,ds +\sum_{k=1}^{m}L_{k}+ \Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi (s)\bigr\vert \,ds \Biggr]r \\& \quad\quad{}+ \bigl[2+\Gamma(\alpha-1)\eta \bigr]M+\sum_{k=1}^{m} \bigl\vert I_{k}(0)\bigr\vert \\& \quad \leq \rho r+(1-\rho)r=r \end{aligned}$$

and

$$\begin{aligned}& \bigl\vert D^{\alpha-1}(Tu) (t)\bigr\vert \\& \quad \leq \int_{0}^{t}\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s) \bigr)\bigr\vert \,ds + \int_{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha -1}u(s)\bigr)\bigr\vert \,ds \\& \quad\quad{} +\sum_{t\leq t_{i}}\bigl\vert I_{i} \bigl(u(t_{i})\bigr)\bigr\vert +\Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\& \quad \leq \int_{0}^{t} \bigl[\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f(s,0,0,0)\bigr\vert +\bigl\vert f(s,0,0,0)\bigr\vert \bigr]\,ds \\& \quad\quad{}+ \int_{0}^{1}(1-s)^{\alpha-2} \bigl[\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f(s,0,0,0)\bigr\vert +\bigl\vert f(s,0,0,0)\bigr\vert \bigr]\,ds \\& \quad\quad{} +\Gamma(\alpha-1) \int_{0}^{\eta} \bigl[\bigl\vert g\bigl(s,u(s) \bigr)-g(s,0)\bigr\vert +\bigl\vert g(s,0)\bigr\vert \bigr]\,ds \\& \quad\quad{} +\sum_{t\leq t_{i}} \bigl[\bigl\vert I_{i} \bigl(u(t_{i})\bigr)-I_{i}\bigl(u(0)\bigr)\bigr\vert + \bigl\vert I_{i}\bigl(u(0)\bigr)\bigr\vert \bigr] \\& \quad \leq r \int_{0}^{1}\bigl[\bigl\vert \psi_{1}(s) \bigr\vert +\bigl\vert \psi_{2}(s)\bigr\vert +\bigl\vert \psi_{3}(s)\bigr\vert \bigr]\,ds+M +r \int_{0}^{1}\bigl[\bigl\vert \psi_{1}(s) \bigr\vert +\bigl\vert \psi _{2}(s)\bigr\vert +\bigl\vert \psi_{3}(s)\bigr\vert \bigr]\,ds+M \\& \quad\quad{}+\sum_{k=1}^{m} \bigl[L_{k}r+ \bigl\vert I_{k}(0)\bigr\vert \bigr] +\Gamma(\alpha-1) \biggl[r \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds+ \eta M \biggr] \\& \quad = \Biggl[2 \int_{0}^{1}\bigl[\bigl\vert \psi_{1}(s) \bigr\vert +\bigl\vert \psi_{2}(s)\bigr\vert +\bigl\vert \psi_{3}(s)\bigr\vert \bigr]\,ds +\sum_{k=1}^{m}L_{k}+ \Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi (s)\bigr\vert \,ds \Biggr]r \\& \quad\quad{} + \bigl[2+\Gamma(\alpha-1)\eta \bigr]M+\sum_{k=1}^{m} \bigl\vert I_{k}(0)\bigr\vert \\& \quad \leq \rho r+(1-\rho)r=r, \end{aligned}$$

which imply that \(\Vert Tu\Vert _{\mathit {PC}^{1}}\leq r\), that is, \(T(B_{r})\subset B_{r}\).

Step 2. We show that T is a contraction mapping. Indeed, for all \(u, v\in B_{r}\), for each \(t\in J=[0,1]\), we obtain

$$\begin{aligned}& \bigl\vert (Tu) (t)-(Tv) (t)\bigr\vert \\& \quad \leq \frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f\bigl(s,v(s),v'(s),D^{\alpha-1}v(s) \bigr)\bigr\vert \,ds \\& \quad\quad{}+\frac{t^{\alpha-1}}{\Gamma(\alpha)} \int_{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f\bigl(s,v(s),v'(s),D^{\alpha-1}v(s) \bigr)\bigr\vert \,ds \\& \quad\quad{}+\frac{t^{\alpha-1}}{\Gamma(\alpha)}\sum_{t\leq t_{i}}\bigl\vert I_{i}\bigl(u(t_{i})\bigr)-I_{i} \bigl(v(t_{i})\bigr)\bigr\vert +\frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)-g \bigl(s,v(s)\bigr)\bigr\vert \,ds \\& \quad \leq \Biggl[\frac{2\int_{0}^{1}[\vert \psi_{1}(s)\vert +\vert \psi_{2}(s)\vert +\vert \psi_{3}(s)\vert ]\,ds}{\Gamma(\alpha)} +\frac{1}{\Gamma(\alpha)}\sum _{k=1}^{m}L_{k} \\& \quad\quad{}+\frac{1}{\alpha-1} \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]\Vert u-v\Vert _{\mathit {PC}^{1}} \\& \quad \leq \Biggl[2 \int_{0}^{1}\bigl[\bigl\vert \psi_{1}(s) \bigr\vert +\bigl\vert \psi_{2}(s)\bigr\vert +\bigl\vert \psi_{3}(s)\bigr\vert \bigr]\,ds +\sum_{k=1}^{m}L_{k} +\Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]\Vert u-v\Vert _{\mathit {PC}^{1}} \\& \quad = \rho \Vert u-v\Vert _{\mathit {PC}^{1}}, \\& \bigl\vert (Tu)'(t)-(Tv)'(t)\bigr\vert \\& \quad \leq \int_{0}^{t}\frac {(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f\bigl(s,v(s),v'(s),D^{\alpha-1}v(s) \bigr)\bigr\vert \,ds \\& \quad\quad{} +\frac{t^{\alpha-2}}{\Gamma(\alpha-1)} \int_{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f\bigl(s,v(s),v'(s),D^{\alpha-1}v(s) \bigr)\bigr\vert \,ds \\& \quad\quad{} +\frac{t^{\alpha-2}}{\Gamma(\alpha-1)}\sum_{t\leq t_{i}}\bigl\vert I_{i}\bigl(u(t_{i})\bigr)-I_{i} \bigl(v(t_{i})\bigr)\bigr\vert +t^{\alpha-2} \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)-g \bigl(s,v(s)\bigr)\bigr\vert \,ds \\& \quad \leq \Biggl[\frac{2\int_{0}^{1}[\vert \psi_{1}(s)\vert +\vert \psi_{2}(s)\vert +\vert \psi_{3}(s)\vert ]\,ds}{\Gamma(\alpha-1)} +\frac{1}{\Gamma(\alpha-1)}\sum _{k=1}^{m}L_{k} + \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]\Vert u-v\Vert _{\mathit {PC}^{1}} \\& \quad \leq \Biggl[2 \int_{0}^{1}\bigl[\bigl\vert \psi_{1}(s) \bigr\vert +\bigl\vert \psi_{2}(s)\bigr\vert +\bigl\vert \psi_{3}(s)\bigr\vert \bigr]\,ds +\sum_{k=1}^{m}L_{k} +\Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]\Vert u-v\Vert _{\mathit {PC}^{1}} \\& \quad = \rho \Vert u-v\Vert _{\mathit {PC}^{1}}, \end{aligned}$$

and

$$\begin{aligned}& \bigl\vert D^{\alpha-1}(Tu) (t)-D^{\alpha-1}(Tv) (t)\bigr\vert \\& \quad \leq \int_{0}^{t}\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s) \bigr) -f\bigl(s,v(s),v'(s),D^{\alpha-1}v(s)\bigr)\bigr\vert \,ds \\& \quad\quad{} + \int_{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f\bigl(s,v(s),v'(s),D^{\alpha-1}v(s) \bigr)\bigr\vert \,ds \\& \quad\quad{} +\sum_{t\leq t_{i}}\bigl\vert I_{i} \bigl(u(t_{i})\bigr)-I_{i}\bigl(v(t_{i})\bigr) \bigr\vert +\Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)-g \bigl(s,v(s)\bigr)\bigr\vert \,ds \\& \quad \leq \Biggl[2 \int_{0}^{1}\bigl[\bigl\vert \psi_{1}(s) \bigr\vert +\bigl\vert \psi_{2}(s)\bigr\vert +\bigl\vert \psi_{3}(s)\bigr\vert \bigr]\,ds +\sum_{k=1}^{m}L_{k}+ \Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi (s)\bigr\vert \,ds \Biggr]\Vert u-v\Vert _{\mathit {PC}^{1}} \\& \quad = \rho \Vert u-v\Vert _{\mathit {PC}^{1}}, \end{aligned}$$

which indicates \(\Vert Tu-Tv\Vert _{\mathit {PC}^{1}}\leq\rho \Vert u-v\Vert _{\mathit {PC}^{1}}\), where \(\rho=2\int_{0}^{1}[\vert \psi_{1}(s)\vert +\vert \psi _{2}(s)\vert +\vert \psi_{3}(s)\vert ]\,ds +\sum_{k=1}^{m}L_{k}+\Gamma(\alpha-1)\int_{0}^{\eta} \vert \psi (s)\vert \,ds<1\). Therefore T is a contraction mapping on \(\mathit {PC}^{1}(J,\mathbb{R})\). According to the contraction mapping principle, we conclude that T has a unique fixed point \(u(t)\in \mathit {PC}^{1}(J,\mathbb{R})\), which is the unique solution of BVP (1.1). The proof is complete. □

Now we give a simple and easily verifiable result as follows.

Corollary 3.1

Assume that the conditions \((B_{2})\), \((B_{3})\), and \((B'_{1})\) hold.

\((B'_{1})\) :

\(f\in C(J\times\mathbb{R}^{3}, \mathbb{R})\), for all \((t,u,v,w), (t, \bar{u}, \bar{v}, \bar{w})\in J\times\mathbb{R}^{3}\), there exist some constants \(N_{i}>0\) (\(i=1,2,3\)) such that

$$\begin{aligned} \bigl\vert f(t,u,v,w)-f(t,\bar{u},\bar{v},\bar{w})\bigr\vert \leq N_{1}\vert u-\bar{u}\vert +N_{2}\vert v-\bar{v}\vert +N_{3}\vert w-\bar{w}\vert . \end{aligned}$$

If \(\rho=\frac{\alpha(N_{1}+N_{2}+N_{3})}{\alpha-1}+\sum_{k=1}^{m}L_{k} +\Gamma(\alpha-1)\int_{0}^{\eta} \vert \psi(s)\vert \,ds<1\), then BVP (1.1) has a unique solution on J.

Proof

Let \(M=\sup_{t\in J}\vert f(t,0,0,0)\vert + \sup_{t\in J}\vert g(t,0)\vert \) and \(B_{r}=\{u\in \mathit {PC}(J,\mathbb{R}):\Vert u\Vert _{\mathit {PC}^{1}}\leq r\}\), where \(r\geq \frac{1}{1-\rho} [ (\frac{\alpha}{\alpha-1}+ \Gamma(\alpha-1)\eta )M+\sum_{k=1}^{m} \vert I_{k}(0)\vert ]\). Define an operator \(T:B_{r}\rightarrow \mathit {PC}(J,\mathbb{R})\) as (3.2). It is obvious that T is jointly continuous and maps bounded subsets of \(J \times\mathbb{R}\) to bounded subsets of \(\mathbb{R}\). Similarly, we will prove Corollary 3.1 through the following two steps.

Step 1. We show that \(T(B_{r})\subset B_{r}\). In fact, noting that \(u(0)=u'(0)=D^{\alpha-1}u(0)=0\), we have, for \(u\in B_{r}\), \(t\in J=[0,1]\),

$$\begin{aligned}& \bigl\vert (Tu) (t)\bigr\vert \\& \quad \leq \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds + \frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\& \quad\quad{} +\frac{t^{\alpha-1}}{\Gamma(\alpha)} \biggl[ \int _{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha -1}u(s)\bigr)\bigr\vert \,ds +\sum _{t\leq t_{i}}\bigl\vert I_{i} \bigl(u(t_{i})\bigr)\bigr\vert \biggr] \\& \quad \leq \frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1} \bigl[\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f(s,0,0,0)\bigr\vert +\bigl\vert f(s,0,0,0)\bigr\vert \bigr]\,ds \\& \quad\quad{} +\frac{t^{\alpha-1}}{\Gamma(\alpha)} \int_{0}^{1}(1-s)^{\alpha -2} \bigl[\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f(s,0,0,0)\bigr\vert +\bigl\vert f(s,0,0,0)\bigr\vert \bigr]\,ds \\& \quad\quad{} +\frac{t^{\alpha-1}}{\Gamma(\alpha)}\sum_{t\leq t_{i}} \bigl[\bigl\vert I_{i}\bigl(u(t_{i})\bigr)-I_{i}\bigl(u(0)\bigr) \bigr\vert +\bigl\vert I_{i}\bigl(u(0)\bigr)\bigr\vert \bigr] \\& \quad\quad{} +\frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta} \bigl[\bigl\vert g\bigl(s,u(s) \bigr)-g(s,0)\bigr\vert +\bigl\vert g(s,0)\bigr\vert \bigr]\,ds \\& \quad \leq \frac{2[(N_{1}+N_{2}+N_{3})r+M]}{\Gamma(\alpha+1)} +\frac{1}{\Gamma(\alpha)}\sum_{k=1}^{m} \bigl[L_{k}r +\bigl\vert I_{k}(0)\bigr\vert \bigr] + \frac{r}{\alpha-1} \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds+ \frac{\eta M}{\alpha-1} \\& \quad = \Biggl[\frac{2(N_{1}+N_{2}+N_{3})}{\Gamma(\alpha+1)} +\frac{1}{\Gamma(\alpha)}\sum_{k=1}^{m}L_{k} +\frac{1}{\alpha-1} \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]r \\& \quad\quad{} + \biggl[\frac{2}{\Gamma(\alpha+1)}+\frac{\eta}{\alpha -1} \biggr]M +\frac{1}{\Gamma(\alpha)}\sum _{k=1}^{m}\bigl\vert I_{k}(0) \bigr\vert \\& \quad \leq \Biggl[\frac{\alpha(N_{1}+N_{2}+N_{3})}{\alpha-1} +\sum_{k=1}^{m}L_{k} +\Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi (s)\bigr\vert \,ds \Biggr]r \\& \quad\quad{} + \biggl[\frac{\alpha}{\alpha-1}+\Gamma(\alpha-1)\eta \biggr]M +\sum _{k=1}^{m}\bigl\vert I_{k}(0)\bigr\vert \\& \quad \leq \rho r+(1-\rho)r=r, \\& \bigl\vert (Tu)'(t)\bigr\vert \\& \quad \leq \int_{0}^{t}\frac {(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds +t^{\alpha-2} \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\& \quad\quad{}+\frac{t^{\alpha-2}}{\Gamma(\alpha-1)} \biggl[ \int _{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha -1}u(s)\bigr)\bigr\vert \,ds +\sum _{t\leq t_{i}}\bigl\vert I_{i} \bigl(u(t_{i})\bigr)\bigr\vert \biggr] \\& \quad \leq \int_{0}^{t}\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha -1)} \bigl[\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f(s,0,0,0)\bigr\vert +\bigl\vert f(s,0,0,0)\bigr\vert \bigr]\,ds \\& \quad\quad{} + \int_{0}^{1}\frac{[t(1-s)]^{\alpha-2}}{\Gamma(\alpha -1)} \bigl[\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f(s,0,0,0)\bigr\vert +\bigl\vert f(s,0,0,0)\bigr\vert \bigr]\,ds \\& \quad\quad{}+\frac{t^{\alpha-2}}{\Gamma(\alpha-1)}\sum_{t\leq t_{i}} \bigl[\bigl\vert I_{i}\bigl(u(t_{i})\bigr)-I_{i}\bigl(u(0)\bigr) \bigr\vert +\bigl\vert I_{i}\bigl(u(0)\bigr)\bigr\vert \bigr] \\& \quad\quad{}+t^{\alpha-2} \int_{0}^{\eta} \bigl[\bigl\vert g\bigl(s,u(s) \bigr)-g(s,0)\bigr\vert +\bigl\vert g(s,0)\bigr\vert \bigr]\,ds \\& \quad \leq \frac{2[(N_{1}+N_{2}+N_{3})r+M]}{\Gamma(\alpha)} +\frac{1}{\Gamma(\alpha-1)}\sum_{k=1}^{m} \bigl[L_{k}r+\bigl\vert I_{k}(0)\bigr\vert \bigr] +r \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds+ \eta M \\& \quad = \Biggl[\frac{2(N_{1}+N_{2}+N_{3})}{\Gamma(\alpha)} +\frac{1}{\Gamma(\alpha-1)}\sum _{k=1}^{m}L_{k} + \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]r \\& \quad\quad{}+ \biggl[\frac{2}{\Gamma(\alpha)}+\eta \biggr]M +\frac{1}{\Gamma(\alpha-1)}\sum _{k=1}^{m}\bigl\vert I_{k}(0)\bigr\vert \\& \quad \leq \Biggl[\frac{\alpha(N_{1}+N_{2}+N_{3})}{\alpha-1} +\Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds+ \sum_{k=1}^{m}L_{k} \Biggr]r \\& \quad\quad{}+ \biggl[\frac{\alpha}{\alpha-1}+\Gamma(\alpha-1)\eta \biggr]M +\sum _{k=1}^{m}\bigl\vert I_{k}(0)\bigr\vert \\& \quad \leq \rho r+(1-\rho)r=r, \end{aligned}$$

and

$$\begin{aligned}& \bigl\vert D^{\alpha-1}(Tu) (t)\bigr\vert \\& \quad \leq \int _{0}^{t}\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s) \bigr)\bigr\vert \,ds + \int_{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha -1}u(s)\bigr)\bigr\vert \,ds \\& \quad\quad{}+\sum_{t\leq t_{i}}\bigl\vert I_{i} \bigl(u(t_{i})\bigr)\bigr\vert +\Gamma(\alpha-1) \int _{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\& \quad \leq \int_{0}^{t} \bigl[\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha -1}u(s)\bigr)-f(s,0,0,0)\bigr\vert +\bigl\vert f(s,0,0,0)\bigr\vert \bigr]\,ds \\& \quad\quad{} + \int_{0}^{1}(1-s)^{\alpha-2} \bigl[\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)-f(s,0,0,0)\bigr\vert +\bigl\vert f(s,0,0,0)\bigr\vert \bigr]\,ds \\& \quad\quad{}+\sum_{t\leq t_{i}} \bigl[\bigl\vert I_{i} \bigl(u(t_{i})\bigr)-I_{i}\bigl(u(0)\bigr)\bigr\vert + \bigl\vert I_{i}\bigl(u(0)\bigr)\bigr\vert \bigr] \\& \quad\quad{} +\Gamma(\alpha-1) \int_{0}^{\eta} \bigl[\bigl\vert g\bigl(s,u(s) \bigr)-g(s,0)\bigr\vert +\bigl\vert g(s,0)\bigr\vert \bigr]\,ds \\& \quad \leq r\sum_{i=0}^{3}N_{i}+M+ \frac{r\sum_{i=0}^{3}N_{i}+M}{\alpha-1} +\sum_{k=1}^{m} \bigl[L_{k}r+\bigl\vert I_{k}(0)\bigr\vert \bigr] \\ & \quad\quad{} +\Gamma( \alpha-1) \biggl[r \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds+ \eta M \biggr] \\ & \quad = \Biggl[\frac{\alpha(N_{1}+N_{2}+N_{3})}{\alpha-1} +\sum_{k=1}^{m}L_{k}+ \Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi (s)\bigr\vert \,ds \Biggr]r \\ & \quad\quad{} + \biggl[\frac{\alpha}{\alpha-1}+\Gamma(\alpha-1)\eta \biggr]M +\sum _{k=1}^{m}\bigl\vert I_{k}(0)\bigr\vert \\ & \quad \leq \rho r+(1-\rho)r=r, \end{aligned}$$

which imply that \(\Vert Tu\Vert _{\mathit {PC}^{1}}\leq r\), that is, \(T(B_{r})\subset B_{r}\).

Step 2. We show that T is a contraction mapping. Indeed, for all \(u, v\in B_{r}\), for each \(t\in J=[0,1]\), we obtain

$$\begin{aligned}& \bigl\vert (Tu) (t)-(Tv) (t)\bigr\vert \\ & \quad \leq \frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f\bigl(s,v(s),v'(s),D^{\alpha-1}v(s) \bigr)\bigr\vert \,ds \\ & \quad\quad{} +\frac{t^{\alpha-1}}{\Gamma(\alpha)} \int_{0}^{1}(1-s)^{\alpha -2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f\bigl(s,v(s),v'(s),D^{\alpha-1}v(s) \bigr)\bigr\vert \,ds \\ & \quad\quad{} +\frac{t^{\alpha-1}}{\Gamma(\alpha)}\sum_{t\leq t_{i}}\bigl\vert I_{i}\bigl(u(t_{i})\bigr)-I_{i} \bigl(v(t_{i})\bigr)\bigr\vert +\frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)-g \bigl(s,v(s)\bigr)\bigr\vert \,ds \\ & \quad \leq \Biggl[\frac{N_{1}+N_{2}+N_{3}}{\Gamma(\alpha+1)}+\frac {N_{1}+N_{2}+N_{3}}{\Gamma(\alpha+1)} +\frac{1}{\Gamma(\alpha)}\sum _{k=1}^{m}L_{k} +\frac{1}{\alpha-1} \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]\Vert u-v\Vert _{\mathit {PC}^{1}} \\ & \quad \leq \Biggl[\frac{\alpha(N_{1}+N_{2}+N_{3})}{\alpha-1} +\sum_{k=1}^{m}L_{k} +\Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]\Vert u-v\Vert _{\mathit {PC}^{1}} =\rho \Vert u-v\Vert _{\mathit {PC}^{1}}, \\ & \bigl\vert (Tu)'(t)-(Tv)'(t)\bigr\vert \\ & \quad \leq \int_{0}^{t}\frac {(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f\bigl(s,v(s),v'(s),D^{\alpha-1}v(s) \bigr)\bigr\vert \,ds \\ & \quad\quad{} +\frac{t^{\alpha-2}}{\Gamma(\alpha-1)} \int _{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f\bigl(s,v(s),v'(s),D^{\alpha-1}v(s) \bigr)\bigr\vert \,ds \\ & \quad\quad{} +\frac{t^{\alpha-2}}{\Gamma(\alpha-1)}\sum_{t\leq t_{i}}\bigl\vert I_{i}\bigl(u(t_{i})\bigr)-I_{i} \bigl(v(t_{i})\bigr)\bigr\vert +t^{\alpha-2} \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)-g \bigl(s,v(s)\bigr)\bigr\vert \,ds \\ & \quad \leq \Biggl[\frac{2(N_{1}+N_{2}+N_{3})}{\Gamma(\alpha)} +\frac{1}{\Gamma(\alpha-1)}\sum _{k=1}^{m}L_{k} + \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]\Vert u-v\Vert _{\mathit {PC}^{1}} \\ & \quad \leq \Biggl[\frac{\alpha(N_{1}+N_{2}+N_{3})}{\alpha-1} +\sum_{k=1}^{m}L_{k}+ \Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]\Vert u-v\Vert _{\mathit {PC}^{1}}=\rho \Vert u-v\Vert _{\mathit {PC}^{1}}, \end{aligned}$$

and

$$\begin{aligned}& \bigl\vert D^{\alpha-1}(Tu) (t)-D^{\alpha-1}(Tv) (t)\bigr\vert \\& \quad \leq \int_{0}^{t}\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha-1}u(s) \bigr) -f\bigl(s,v(s),v'(s),D^{\alpha-1}v(s)\bigr)\bigr\vert \,ds \\& \quad\quad{} + \int_{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr) -f\bigl(s,v(s),v'(s),D^{\alpha-1}v(s) \bigr)\bigr\vert \,ds \\& \quad\quad{}+\sum_{t\leq t_{i}}\bigl\vert I_{i} \bigl(u(t_{i})\bigr)-I_{i}\bigl(v(t_{i})\bigr) \bigr\vert +\Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)-g \bigl(s,v(s)\bigr)\bigr\vert \,ds \\& \quad \leq \Biggl[N_{1}+N_{2}+N_{3}+\frac{N_{1}+N_{2}+N_{3}}{\alpha-1} +\sum_{k=1}^{m}L_{k} +\Gamma( \alpha-1) \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]\Vert u-v\Vert _{\mathit {PC}^{1}} \\& \quad = \Biggl[\frac{\alpha(N_{1}+N_{2}+N_{3})}{\alpha-1} +\sum_{k=1}^{m}L_{k}+ \Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr]\Vert u-v\Vert _{\mathit {PC}^{1}}=\rho \Vert u-v\Vert _{\mathit {PC}^{1}}, \end{aligned}$$

which indicates \(\Vert Tu-Tv\Vert _{\mathit {PC}^{1}}\leq\rho \Vert u-v\Vert _{\mathit {PC}^{1}}\), where \(\rho=\frac{\alpha(N_{1}+N_{2}+N_{3})}{\alpha-1} +\sum_{k=1}^{m}L_{k}+ \Gamma(\alpha-1)\int_{0}^{\eta} \vert \psi(s)\vert \,ds<1\). Therefore T is a contraction mapping on \(\mathit {PC}^{1}(J,\mathbb{R})\). According to the contraction mapping principle, we conclude that T has a unique fixed point \(u(t)\in \mathit {PC}^{1}(J,\mathbb{R})\), which is the unique solution of BVP (1.1). The proof is complete. □

For some fixed \(r>0\), considering BVP (1.1) on the cylinder \(\mathcal{R}=[0,1]\times B(0,r)\), we obtain the following theorem.

Theorem 3.2

Assume that conditions \((B_{4})\)-\((B_{6})\) hold. Then BVP (1.1) has at least one solution in J, provided that \(\varrho=\sum_{k=1}^{m}L_{k} +\Gamma(\alpha-1) \int_{0}^{\eta} \vert \psi(s)\vert \,ds<1\).

\((B_{4})\) :

\(f\in C( J\times\mathbb{R}^{3}, \mathbb{R})\), for all \((t,u,u',D^{\alpha-1}u)\in J\times\mathbb{R}^{3}\), there exist \(p\in(0,1)\), \(h\in L_{1/p}([0,1], \mathbb{R^{+}})\) such that \(\vert f(t,u,u',D^{\alpha-1}u)\vert \leq h(t)\), where \(L_{1/p}([0,1], \mathbb{R^{+}})\) denotes space \(1/p\)-Lebesgue measurable functions from \([0,1]\) to \(\mathbb{R^{+}}\) with the norm \(\Vert v\Vert _{1/p}= (\int_{0}^{1}\vert v(s)\vert ^{\frac{1}{p}}\,ds )^{p}\), for \(v\in L_{1/p}([0,1], \mathbb{R^{+}})\).

\((B_{5})\) :

\(I_{k}\in C(\mathbb{R},\mathbb{R})\), for all \(u\in \mathbb{R}\), there exist some constants \(L_{k}>0\) such that \(\vert I_{k}(u)\vert < L_{k}\vert u\vert \), \(k=1,2,\ldots,m\).

\((B_{6})\) :

\(g\in C(J,\mathbb{R})\), for all \((t,u)\in(J, \mathbb{R})\), there exists \(\psi\in L[0,1]\) such that \(\vert g(t,u)\vert \leq \vert \psi(t)\vert \vert u\vert \).

Proof

Let \(B_{\lambda}\) be a closed bounded convex subset of \(\mathit {PC}^{1}([0, 1],\mathbb{R})\) defined by \(B_{\lambda}=\{u:\Vert u\Vert \leq\lambda\}\), \(\lambda\geq \frac{A}{1-\varrho}\), \(A= [1+ (\frac{1-p}{\alpha-p-1} )^{1-p} ]\Vert h\Vert _{1/p}\).

Define the operator \(T: B_{\lambda}\rightarrow \mathit {PC}^{1}([0, 1],\mathbb {R})\) as (3.2). For \(u\in\partial B_{\lambda}\), we have

$$\begin{aligned}& \bigl\vert (Tu) (t)\bigr\vert \\& \quad \leq \int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds + \frac{t^{\alpha-1}}{\alpha-1} \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\& \quad\quad{}+\frac{t^{\alpha-1}}{\Gamma(\alpha)} \biggl[ \int _{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha -1}u(s)\bigr)\bigr\vert \,ds +\sum _{t\leq t_{i}}\bigl\vert I_{i} \bigl(u(t_{i})\bigr)\bigr\vert \biggr] \\& \quad \leq \int_{0}^{1}\frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)}h(s)\,ds + \int_{0}^{1}\frac{(1-s)^{\alpha-2}}{\Gamma(\alpha)}h(s)\,ds \\& \quad\quad{} +\frac{1}{\Gamma(\alpha)}\sum_{t\leq t_{i}}L_{i}\bigl\vert u(t_{i})\bigr\vert +\frac{1}{\alpha-1} \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \bigl\vert u(s)\bigr\vert \,ds \\& \quad \leq \frac{1}{\Gamma(\alpha)} \biggl( \int_{0}^{1}(1-s)^{\frac {\alpha-1}{1-p}}\,ds \biggr)^{1-p} \biggl( \int_{0}^{1}\bigl(h(s)\bigr)^{\frac {1}{p}}\,ds \biggr)^{p} \\& \quad\quad{} +\frac{1}{\Gamma(\alpha)} \biggl( \int_{0}^{1}(1-s)^{\frac{\alpha-2}{1-p}}\,ds \biggr)^{1-p} \biggl( \int_{0}^{1}\bigl(h(s)\bigr)^{\frac{1}{p}}\,ds \biggr)^{p} \\& \quad\quad{} +\frac{1}{\Gamma(\alpha)}\sum_{k=1}^{m}L_{k} \Vert u\Vert _{\mathit {PC}^{1}} +\frac{\Vert u\Vert _{\mathit {PC}^{1}}}{\alpha-1} \int_{0}^{\eta }\bigl\vert \psi(s)\bigr\vert \,ds \\& \quad \leq \biggl[ \biggl(\frac{1-p}{\alpha-p} \biggr)^{1-p}+ \biggl( \frac{1-p}{\alpha-p-1} \biggr)^{1-p} \biggr]\frac{\Vert h\Vert _{1/p}}{\Gamma(\alpha)} +\lambda \Biggl[\frac{1}{\Gamma(\alpha)}\sum_{k=1}^{m}L_{k} +\frac{1}{\alpha-1} \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr] \\& \quad \leq \biggl[1+ \biggl(\frac{1-p}{\alpha-p-1} \biggr)^{1-p} \biggr]\Vert h\Vert _{1/p} +\lambda \Biggl[\sum_{k=1}^{m}L_{k} +\Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr] =A+\lambda\varrho\leq\lambda, \\& \bigl\vert (Tu)'(t)\bigr\vert \\& \quad \leq \int_{0}^{t}\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha-1}u(s)\bigr)\bigr\vert \,ds +t^{\alpha-2} \int_{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\& \quad\quad{}+\frac{t^{\alpha-2}}{\Gamma(\alpha-1)} \biggl[ \int_{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha -1}u(s)\bigr)\bigr\vert \,ds +\sum _{t\leq t_{i}}\bigl\vert I_{i} \bigl(u(t_{i})\bigr)\bigr\vert \biggr] \\& \quad \leq \int_{0}^{1}\frac{(1-s)^{\alpha-2}}{\Gamma(\alpha-1)}h(s)\,ds + \int_{0}^{1}\frac{(1-s)^{\alpha-2}}{\Gamma(\alpha-1)}h(s)\,ds \\& \quad\quad{}+\frac{1}{\Gamma(\alpha-1)}\sum_{t\leq t_{i}}L_{i}\bigl\vert u(t_{i})\bigr\vert + \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \bigl\vert u(s)\bigr\vert \,ds \\& \quad \leq \frac{\alpha-1}{\Gamma(\alpha)} \biggl( \int _{0}^{1}(1-s)^{\frac{\alpha-2}{1-p}}\,ds \biggr)^{1-p} \biggl( \int _{0}^{1}\bigl(h(s)\bigr)^{\frac{1}{p}}\,ds \biggr)^{p} \\& \quad\quad{}+\frac{\alpha-1}{\Gamma(\alpha)} \biggl( \int_{0}^{1}(1-s)^{\frac{\alpha-2}{1-p}}\,ds \biggr)^{1-p} \biggl( \int_{0}^{1}\bigl(h(s)\bigr)^{\frac{1}{p}}\,ds \biggr)^{p} \\& \quad\quad{} +\frac{\alpha-1}{\Gamma(\alpha)}\sum_{k=1}^{m}L_{k} \Vert u\Vert _{\mathit {PC}^{1}} +\Vert u\Vert _{\mathit {PC}^{1}} \int_{0}^{\eta}\bigl\vert \psi (s)\bigr\vert \,ds \\& \quad \leq 2(\alpha-1) \biggl(\frac{1-p}{\alpha-p-1} \biggr)^{1-p} \frac {\Vert h\Vert _{1/p}}{\Gamma(\alpha)} +\lambda \Biggl[\frac{\alpha-1}{\Gamma(\alpha)}\sum _{k=1}^{m}L_{k} + \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr] \\& \quad \leq \biggl[1+ \biggl(\frac{1-p}{\alpha-p-1} \biggr)^{1-p} \biggr]\Vert h\Vert _{1/p} +\lambda \Biggl[\sum_{k=1}^{m}L_{k} +\Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr] =A+\lambda\varrho\leq\lambda, \end{aligned}$$

and

$$\begin{aligned}& \bigl\vert D^{\alpha-1}(Tu) (t)\bigr\vert \\& \quad \leq \int_{0}^{t}\bigl\vert f\bigl(s,u(s),u'(s),D^{\alpha -1}u(s) \bigr)\bigr\vert \,ds + \int_{0}^{1}(1-s)^{\alpha-2}\bigl\vert f \bigl(s,u(s),u'(s),D^{\alpha -1}u(s)\bigr)\bigr\vert \,ds \\& \quad\quad{}+ \sum_{t\leq t_{i}}\bigl\vert I_{i} \bigl(u(t_{i})\bigr)\bigr\vert +\Gamma(\alpha-1) \int _{0}^{\eta}\bigl\vert g\bigl(s,u(s)\bigr)\bigr\vert \,ds \\& \quad \leq \int_{0}^{1}h(s)\,ds + \int_{0}^{1}(1-s)^{\alpha-2}h(s)\,ds+\sum _{t\leq t_{i}}L_{i}\bigl\vert u(t_{i}) \bigr\vert +\Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \bigl\vert u(s)\bigr\vert \,ds \\& \quad \leq \biggl( \int_{0}^{1}\bigl(h(s)\bigr)^{\frac{1}{p}}\,ds \biggr)^{p} + \biggl( \int_{0}^{1}(1-s)^{\frac{\alpha-2}{1-p}}\,ds \biggr)^{1-p} \biggl( \int_{0}^{1}\bigl(h(s)\bigr)^{\frac{1}{p}}\,ds \biggr)^{p} +\sum_{k=1}^{m}L_{k} \Vert u\Vert _{\mathit {PC}^{1}} \\& \quad\quad{} +\Gamma(\alpha-1)\Vert u\Vert _{\mathit {PC}^{1}} \int _{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \\& \quad = \biggl[1+ \biggl(\frac{1-p}{\alpha-p-1} \biggr)^{1-p} \biggr]\Vert h \Vert _{1/p} +\lambda \Biggl[\sum_{k=1}^{m}L_{k} +\Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi(s)\bigr\vert \,ds \Biggr] =A+\lambda\varrho\leq\lambda. \end{aligned}$$

Therefore, \(T(B_{\lambda})\subset B_{\lambda}\). By Lemma 3.2, we see that T: \(B_{\lambda}\rightarrow B_{\lambda}\) is completely continuous. Thus BVP (1.1) has at least one solution by Lemma 2.3. The proof is complete. □

4 Illustrative example

As an application of the main results, we consider the following impulsive fractional differential equation with integral boundary conditions:

$$ \textstyle\begin{cases} {}_{t_{k}}D_{t}^{\frac{5}{2}}u(t)= f(t,u,u',D^{\frac{3}{2}}u), \quad t\neq t_{k}, \\ \Delta D^{\frac{3}{2}}u(t_{k})= I_{k}(u(t_{k}), \quad k=1,\ldots,m,\\ u(0)=u'(0)=0, \quad\quad u'(1)=\int_{0}^{\eta}g(s,u(s))\,ds, \end{cases} $$
(4.1)

here \(t\in J=[0,1]\), \(t_{k}=1-\frac{1}{2^{k}}\) (\(k=1,2,\ldots,m\)), \(\alpha=\frac{5}{2}\), \(\eta=\frac{1}{2}\).

Case 1

Let

$$\begin{aligned}& \begin{aligned} f\bigl(t,u,u',D^{\frac{3}{2}}u\bigr)&=\frac{t^{4}u(t)}{10+h_{1}(u(t),u'(t),D^{\frac {3}{2}}u(t))} + \frac{e^{-t}D^{\frac{3}{2}}u(t)}{20+h_{3}(u(t),u'(t),D^{\frac{3}{2}}u(t))} \\ &\quad{} +\frac{u'(t)}{(1+t)^{3}(15+h_{2}(u(t),u'(t),D^{\frac{3}{2}}u(t))}, \end{aligned} \\& I_{k}\bigl(u(t_{k})\bigr)=\frac{\vert u(t_{k})\vert }{5^{k}+\vert u(t_{k})\vert }+1, \quad\quad g(t,u)= \frac{t^{2}u^{2}(t)}{1+u^{2}(t)}, \end{aligned}$$

where \(h_{i}(u,u',D^{\frac{3}{2}}u)\geq 0\) (\(i=1,2,3\)). By simple computation, we have \(\psi_{1}(s)=\frac{s^{4}}{10}\), \(\psi_{2}(s)=\frac{1}{15(1+s)^{3}}\), \(\psi_{3}(s)=\frac{1}{20e^{t}}\), \(L_{k}=\frac{1}{5^{k}}\), \(\psi(s)=2s^{2}\),

$$\begin{aligned} \rho &=2 \int_{0}^{1}\bigl[\bigl\vert \psi_{1}(s) \bigr\vert +\bigl\vert \psi _{2}(s)\bigr\vert +\bigl\vert \psi_{3}(s)\bigr\vert \bigr]\,ds +\sum_{k=1}^{m}L_{k}+ \Gamma(\alpha-1) \int_{0}^{\eta}\bigl\vert \psi (s)\bigr\vert \,ds \\ &=\frac{1}{25}+\frac{1}{20}+\frac{1-e^{-1}}{10}+\frac {1}{4} \biggl(1-\frac{1}{5^{m+1}}\biggr)+\frac{\Gamma(\frac{3}{2})}{12} \\ &< \frac{264+25\sqrt{\pi}}{600}< 1. \end{aligned}$$

Thus, all the assumptions of Theorem 3.1 are satisfied. Hence BVP (4.1) has a unique solution on \(J=[0,1]\).

Case 2

Take

$$\begin{aligned} f\bigl(t,u,u',D^{\frac{3}{2}}u\bigr)=\frac{t^{2}(u+u+D^{\frac {3}{2}}u)^{2}}{1+(u+u+D^{\frac{3}{2}}u)^{2}}. \end{aligned}$$

\(I_{k}(u(t_{k}))\) and \(g(t,u)\) are the same as Case 1. It is clear that \(\vert f(t,u,u',D^{\frac{3}{2}}u)\vert \leq t^{2}\triangleq h(t)\) and \(\varrho=\sum_{k=1}^{m}L_{k}+\Gamma(\alpha-1)\int_{0}^{\eta} \vert \psi(s)\vert \,ds<\frac{6+\sqrt{\pi}}{24}<1\). Thus, BVP (4.1) has at least one solution in \(J=[0,1]\) by Theorem 3.2.

5 Conclusions

Compared with previous papers involving impulsive fractional order differential equations, the impulse of our boundary value problem (1.1) is related to the fractional order derivative, namely, \(\Delta D^{\alpha-1}u(t_{k})= I_{k}(u(t_{k}))\). It is difficult and challenging to find the Green function of (1.1). Our results are new and interesting. Our methods can be used to study the existence of positive solutions for the high order or multiple-point boundary value problems of nonlinear fractional differential equation with the impulses involving the fractional order derivative. However, there exist some difficulties and complexities to address the structure of the Green function for these boundary value problems.