1 Introduction

With the development of fractional calculus and the requirement for field applications of physics, mathematics, and chemical engineering, fractional differential equations have attracted great interest in recent years (see, for example, [1–8]).

Based on the nonlinear alternative of a Leray–Schauder model, we have considered and cited previous studies on the existence of solutions of fractional differential equations for investigating the existence and uniqueness of fractional functional equations. Zhou et al. have already used Krasnoselskii’s fixed point theory to study the existence and uniqueness of fractional function equations [9, 10]. Some researchers have studied fractional order impulsive differential equations and discussed the existence solution of nonlinear functional differential equations with multiple delays [11, 12]. Many authors have investigated the existence solutions of fractional function equations with impulse [13–18]. However, there are few studies on the existence of fractional order impulsive functional differential equations multi-delays. The recent development of the theory of fractional differential equations has already affected the present research.

In this paper, we should study the solutions of impulsive fractional order functional differential equations with multiple delays as follows:

$$ \left \{ \textstyle\begin{array}{l} D^{\alpha}x(t)=f(t,x_{t})+\sum_{i=1}^{p}x(t-r_{i}), \quad t\in J=[0,b], t\neq t_{k},\\ \Delta x|_{t=t_{k}}=I_{k}(x(t_{k}^{-})),\quad k=1,2,\ldots,m,\\ x(t)=\phi(t),\quad t\in[-\tau,0], \end{array}\displaystyle \right . $$
(1.1)

where \(p\in\{1,2,\ldots\}\), \(\tau=\max_{1\leq i\leq p}\{r_{i}\}\), \(f: J\times\Omega\rightarrow X\) is a given function, where Ω is a phase space defined in preliminaries. \(0=t_{0}< t_{1}<\cdots t_{m}<t_{m+1}=b\), \(I_{k}\in C(X,X)\) (\(k=1,2,\ldots,m\)) are bounded functions. \(\Delta x|_{t=t_{k}}=x(t_{k}^{+})-x(t_{k}^{-})\), \(x(t_{k}^{+})\) and \(x(t_{k}^{-})\) represent the left and right limits of \(x(t)\) at \(t=t_{k}\), respectively. We assume that the histories \(x_{t}: [-\tau,0]\rightarrow X\), \(x_{t}(s)=x(t+s)\), \(s\in[-\tau,0]\), belong to an abstract phase space Ω.

In this paper, we prove the existence and uniqueness of the solution. The idea of the paper is as follows. In the second part, we give preliminary facts and definition. In the third part, we prove the existence of solutions. In the fourth part, some examples are given to illustrate our main results.

2 Preliminaries

In this section, we give some basic definitions, notations, and results which are used throughout this paper.

Let \(C(J,X)\) be the Banach space of continuous functions x from J into X with the norm \(\|x\|_{\infty}=\sup\{x(t): t\in J\}\), and we introduce the spaces:

$$ \Omega=\bigl\{ \psi: [-\tau,0]\rightarrow X\text{ such that }\psi(t)\text{ is measurable and bounded}\bigr\} , $$

and define \(\|\psi\|_{\Omega}\) by

$$ \|\psi\|_{\Omega}=\sup_{s\in[-\tau,0]}\big|\psi(s)\big|,\quad \forall\psi\in \Omega. $$

We consider the space

$$\begin{aligned} \mathit{PC}={}&\bigl\{ x: [0, b]\rightarrow X\text{ such that }x_{k}\in C\bigl((t_{k},t_{k+1}],X\bigr) , \text{there exist }x\bigl(t_{k}^{+} \bigr)\text{ and }x\bigl(t_{k}^{-}\bigr) \\ &\text{with }x(t_{k})=x\bigl(t_{k}^{-}\bigr), x_{0}=\phi\in\Omega, k=0,1,\ldots,m\bigr\} \end{aligned} $$

to be a Banach space with the norm

$$ \|x\|_{\mathit{PC}}=\max \bigl\{ \|x_{k}\|_{J_{k}}, k=0, 1, \ldots, m \bigr\} , $$

where \(x_{k}\in J_{k}=(t_{k},t_{k+1}]\), \(k=0, 1, \ldots, m\).

Set

$$ \Omega_{b}= \bigl\{ x:[-\tau,b]\rightarrow X \setminus x\in \mathit{PC}(J,X) \cap \Omega \bigr\} , $$

and let \(\|\cdot\|_{b}\) be a seminorm in \(\Omega_{b}\) defined by

$$ \|x\|_{b}=\|\phi\|_{\Omega}+\sup\bigl\{ \big|x(s)\big|: s\in[0,b]\bigr\} , \quad x\in\Omega_{b}. $$

Definition 2.1

The Riemann–Liouville fractional integral operator of order \(\alpha>0\) of a function \(f\in C_{\mu}\), \(\mu\geq1\), is defined as

$$ I^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}f(s)\,ds,\quad t\geq0. $$

Definition 2.2

The fractional derivative of \(f(t)\) in the Caputo sense is defined as

$$ D^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)} \int_{0}^{t}(t-s)^{n-\alpha -1}f^{(n)}(s)\,ds, $$

where \(n-1<\alpha<n\), \(t>0\), \(f\in C_{-1}^{n}\).

Definition 2.3

The function \(x\in\Omega_{b}\) as follows:

$$ x(t)=\left \{ \textstyle\begin{array}{l} \phi(0)+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}f(s,x_{s})\,ds +\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q}\int _{0}^{t-r_{i}}(t-s-r_{i})^{\alpha-1}x(s)\,ds\\ \quad{}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q}\int _{r_{i}}^{0}(t-s-r_{i})^{\alpha-1}\phi(s)\,ds,\quad t\in[0, t_{1}];\\ x(t_{1}^{-})+I_{1}(x(t_{1}^{-}))+\frac{1}{\Gamma(\alpha)}\int _{t_{1}}^{t}(t-s)^{\alpha-1}f(s,x_{s})\,ds\\ \quad{}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q}\int _{t_{1}-r_{i}}^{t-r_{i}}(t-s-r_{i})^{\alpha-1}x(s)\,ds,\quad t\in [t_{1},t_{2}];\\ \vdots\\ x(t_{m}^{-})+I_{m}(x(t_{m}^{-}))+\frac{1}{\Gamma(\alpha)}\int _{t_{m}}^{t}(t-s)^{\alpha-1}f(s,x_{s})\,ds\\ \quad{}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q}\int _{t_{m}-r_{i}}^{t-r_{i}}(t-s-r_{i})^{\alpha-1}x(s)\,ds,\quad t\in[t_{m}, b];\\ \phi(t),\quad t\in[-\tau,0], \end{array}\displaystyle \right . $$
(2.1)

will be called a solution of system (1.1).

For proving the existence of solution of system (1.1), we need to provide the following lemmas.

Lemma 2.1

For \(\sigma\in(0,1]\) and \(0< a\leq b\), we have \(|a^{\sigma}-b^{\sigma}|\leq(b-a)^{\sigma}\).

Lemma 2.2

(Hölder’s inequality)

Assume that \(p,q\geq1\),and \(\frac{1}{p}+\frac{1}{q}=1\). If \(l\in L^{p}(J)\), \(m\in L^{q}(J)\), then for \(1\leq p\leq\infty\), \(lm \in L^{1}(J)\) and \(\|lm\|_{L^{1}(J)}\leq\|l\|_{L^{p}(J)}\|m\|_{L^{q}(J)}\).

For measurable functions \(m: J\rightarrow R\), define the norm

$$\|m\|_{L^{p}(J)}=\left \{ \textstyle\begin{array}{l} (\int_{J}|m(t)|^{p}\,dt )^{\frac{1}{p}}, \quad 1\leq p< \infty, \\ \inf_{\mu(\bar{J})} \{\sup_{t\in J-\bar{J}}|m(t)| \},\quad p=\infty, \end{array}\displaystyle \right . $$

where \(\mu(\bar{J})\) is the Lebesgue measure on J̄. Let \(L^{p}(J, R)\) be the Banach space of all Lebesgue measurable functions \(m: J\rightarrow R\) with \(\|m\|_{L^{p}(J)}<\infty\).

Lemma 2.3

If X is a Banach space, \(U\subset X\) is convex with \(0\in U\), and \(F: U\rightarrow U\) is a completely continuous operator, then either

  1. (i)

    the set \(E= \{ x \in U: x=\lambda F(x), 0<\lambda<1 \}\) is unbounded, or

  2. (ii)

    F has a fixed point.

3 Main result and proofs

In this section, we give the main results on the existence of solutions of system (1.1).

Firstly, to establish our results, we add the following conditions:

\((H_{1})\) :

The map \(f: J\times\Omega\rightarrow X\) is said to be an \(L^{1}\)-Caratheodory map if

  1. (i)

    \(t\rightarrow f(t,u)\) is measurable for each \(u\in\Omega\),

  2. (ii)

    \(u\rightarrow f(t,u)\) is continuous for almost all \(t\in J\),

  3. (iii)

    for each \(r>0\), there exists \(h_{r}\in L^{1}(J, R^{+})\) such that \(\|f(t,u)\|\leq h_{r}(t)\) for all \(\|u\|_{\Omega}\leq r\) and for almost \(t\in J\).

\((H_{2})\) :

\(\|f(t,u)\|\leq p(t)\Psi(\|u\|_{\Omega})\) for almost all \(t\in J\) and all \(u\in\Omega\), where \(p\in L^{1}(J,R_{+})\) and \(\Psi: R_{+}\rightarrow(0,\infty)\) is continuous and increasing with

$$ \int_{t_{k-1}}^{t_{k}}m(s)\,ds\leq \int_{\bar{N}_{k-1}}^{\infty}\frac {ds}{\Psi(s)+s}, $$

where

$$\begin{gathered} \bar{N}_{0}=\upsilon(0)=\|\phi\|_{\Omega}+\frac{qb^{\alpha}}{\Gamma (\alpha+1)}\| \phi\|_{\Omega},\qquad N_{k-1}=\sup_{x\in[-K_{k-2},K_{k-2}]}\big|I_{k-1}(x)\big|+M_{k-2}, \\\bar{N}_{k-1}=N_{k-1}+\frac{q\tau^{\alpha}K_{k-2}}{\Gamma(\alpha+1)}, \qquad M_{k-2}= \Gamma_{k-1}^{-1} \biggl( \int_{t_{k-2}}^{t_{k-1}}m(s)\,ds \biggr), \quad k=2,\ldots, m+2,\end{gathered} $$

and

$$\begin{gathered} K_{0}=\max\bigl(M_{0},\|\phi\|_{\Omega}\bigr), \qquad K_{k}=\max(K_{k-1},K_{k}),\quad k=1,\ldots,m+1, \\m(t)=\max \biggl\{ \frac{b^{\alpha}}{\Gamma(\alpha+1)}p(t),\frac {qb^{\alpha}}{\Gamma(\alpha+1)} \biggr\} , \\\Gamma_{k-1}(z)= \int_{\overline{N}_{k-1}}^{z}\frac{ds}{\Psi(s)+s},\quad z\geq \bar{N}_{k-1}, k \in\{1,2,\ldots,m+2\}.\end{gathered} $$
\((H_{3})\) :

The functions \(I_{k}: X\rightarrow X\) are continuous, there exists a constant \(d_{k}\) such that \(\|I_{k}(x)\|\leq d_{k}\), \(k=1,2,\ldots,m\), for all \(x\in X\).

\((H_{4})\) :

There exist a constant \(\beta\in(0,\alpha)\) and a real-valued function \(l(t)\in L^{\frac{1}{\beta}}(J)\) such that \(\|f(t,\phi)-f(t,\psi)\|\leq l(t)\|\phi-\psi\|_{\Omega}\) for almost every \(t\in[0,b]\) and all \(\phi, \psi\in\Omega\).

\((H_{5})\) :

There exist positive constants \(c_{k}\), \(k=1, 2, \ldots, m\), such that \(\|I_{k}(x(t_{k}^{-}))-I_{k}(y(t_{k}^{-}))\|\leq c_{k}\|x-y\|\) for each \(x,y\in X\).

Next, we give an existence result based on a nonlinear alternative of Leray–Schauder applied to a completely continuous operator.

Theorem 3.1

Assume that conditions \((H_{1})\), \((H_{2})\), and \((H_{3})\) are satisfied, then system (1.1) has at least one solution.

Proof

The proof is carried out in the following steps.

Step 1: Consider the following problem:

$$ \left \{ \textstyle\begin{array}{l} D^{\alpha}x(t)=f(t,x_{t})+\sum_{i=1}^{p}x(t-r_{i}), \quad t\in J=[0,t_{1}], \\ x(t)=\phi(t), \quad t\in[-\tau,0]. \end{array}\displaystyle \right . $$
(3.1)

Define the operator \(F: \Omega_{t_{1}}\rightarrow\Omega_{t_{1}}\) by

$$ F(x) (t)=\left \{ \textstyle\begin{array}{l} \phi(0)+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}f(s,x_{s})\,ds +\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q}\int _{0}^{t-r_{i}}(t-s-r_{i})^{\alpha-1}x(s)\,ds\\ \quad{}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q}\int _{r_{i}}^{0}(t-s-r_{i})^{\alpha-1}\phi(s)\,ds, \quad t\in[0, t_{1}],\\ \phi(t),\quad t\in[-\tau, 0]. \end{array}\displaystyle \right . $$
(3.2)

Claim 1: F is continuous.

Let \(\{x_{n}\}\) be a sequence such that \(x_{n}\rightarrow x\) in \(\Omega_{t_{1}}\). Then, for \(t\in[0,t_{1}]\), we have

$$ \big\| f(s,x_{ns},\cdot)-f(s,x_{s})\big\| \leq\epsilon, \quad n\rightarrow \infty, $$

because f is continuous, for all \(t\in[0, t_{1}]\), we have

$$\begin{gathered} \big|F(x_{n}) (t)-F(x) (t)\big| \\ \quad\leq \frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}\big|f(s,x_{ns})- f(s,x_{s})\big|\,ds\\ \qquad{}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q} \int _{0}^{t-r_{i}}(t-s-r_{i})^{\alpha-1}\big|x_{n}(s)-x(s)\big|\,ds.\end{gathered} $$

Since f is \(L^{1}\)-Caratheodory, we obtain F is continuous.

Claim 2: F maps bounded sets into bounded sets in \(\Omega_{t_{1}}\).

For any \(r>0\), there exists \(l>0\) such that, for each \(x\in B_{r}= \{x\in\Omega_{t_{1}}, \|x\|\leq r \}\), we have \(\|F(x)\| \leq l\).

For each \(t\in[0,t_{1}]\), we get

$$\begin{aligned} \big|F(x) (t)\big| \leq{}& \big|\phi(0)\big|+\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha -1}\big|f(s,x_{s})\big|\,ds \\ &+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q} \int _{0}^{t-r_{i}}(t-s-r_{i})^{\alpha-1}\big|x(s)\big|\,ds\\ &+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q} \int _{r_{i}}^{0}(t-s-r_{i})^{\alpha-1}\big| \phi(s)\big|\,ds \\ \leq{}&\|\phi\|_{\Omega}+\frac{qb^{\alpha}}{\Gamma(\alpha+1)}\|\phi\| _{\Omega}+ \frac{qb^{\alpha}r}{\Gamma(\alpha+1)} +\frac{b^{\alpha}}{\Gamma(\alpha+1)}\big\| h_{r}(t)\big\| _{L^{1}}:=l.\end{aligned} $$

Claim 3: F maps bounded sets into equicontinuous sets in \(\Omega_{t_{1}}\).

Let \(B_{r}\) be a bounded set of \(\Omega_{t_{1}}\) as in Step 2. Then, for each \(s_{1},s_{2} \in[0,t_{1}]\), \(s_{1}< s_{2}\), we obtain

$$\begin{gathered} \big\| F(x) (s_{2})-F(x) (s_{1})\big\| \\ \quad\leq\frac{1}{\Gamma(\alpha)}\big\| h_{r}(t)\big\| _{L^{1}} \biggl( \int_{s_{1}}^{s_{2}}(s_{2}-s)^{\alpha-1}\,ds+ \int_{0}^{s_{1}} \bigl[(s_{2}-s)^{\alpha-1}-(s_{1}-s)^{\alpha-1} \bigr]\,ds \biggr) \\ \qquad{}+\frac{r}{\Gamma(\alpha)}\sum_{i=1}^{q} \biggl( \int_{s_{1}-r_{i}}^{s_{2}-r_{i}}(s_{2}-s-r_{i})^{\alpha-1}\,ds\\ \qquad{}+ \int _{0}^{s_{1}-r_{i}} \bigl[(s_{2}-s-r_{i})^{\alpha -1}-(s_{1}-s-r_{i})^{\alpha-1} \bigr]\,ds \biggr) \\ \quad\leq \biggl[\frac{3\|h_{r}(t)\|_{L^{1}}}{\Gamma(\alpha+1)}+\frac {3qr}{\Gamma(\alpha+1)} \biggr](s_{2}-s_{1})^{\alpha}.\end{gathered} $$

If \(s_{2}\rightarrow s_{1}\), then \(\|F(x)(s_{2})-F(x)(s_{1})\| \rightarrow0\). According to a consequence of Steps 1–3, and together with the Arzela–Ascoli theorem, we can deduce that F is continuous and completely continuous.

Claim 4: (A priori bounds).

Let x be a possible solution of the equation \(x=\lambda P(x)\) with \(\lambda\in(0,1)\). Then, for each \(t\in[0, t_{1}]\), we have

$$\begin{aligned} \big|x(t)\big|\leq{}& \phi(0)+\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}f(s,x_{s})\,ds +\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q} \int _{0}^{t-r_{i}}(t-s-r_{i})^{\alpha-1}x(s)\,ds \\ &+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q} \int _{r_{i}}^{0}(t-s-r_{i})^{\alpha-1} \phi(s)\,ds.\end{aligned} $$

By \((H_{3})\), we have

$$ \big|x(t)\big|\leq \|\phi\|_{\Omega}+\frac{qb^{\alpha}}{\Gamma(\alpha+1)}\|\phi\|_{\Omega }+ \frac{b^{\alpha}}{\Gamma(\alpha+1)} \int_{0}^{t}p(s)\Psi\bigl(\|x_{s}\| _{\Omega}\bigr)\,ds +\frac{qb^{\alpha}}{\Gamma(\alpha+1)} \int_{0}^{t}\big|x(s)\big|\,ds. $$

Let us define \(\omega(t)\) as

$$ \omega(t)=\sup\bigl\{ \big|x(s)\big|: -\tau\leq s\leq t\bigr\} , \quad0\leq t\leq t_{1}. $$

Then we have

$$ \omega(t)\leq \|\phi\|_{\Omega}+\frac{qb^{\alpha}}{\Gamma(\alpha+1)}\|\phi \|_{\Omega }+ \int_{0}^{t}m(s) \bigl(\Psi\bigl(\omega(s)\bigr)+ \omega(s)\bigr)\,ds. $$

Let us take the right-hand side of the above inequality as \(\upsilon(t)\). Then we have

$$ c=\upsilon(0)=\|\phi\|_{\Omega}+\frac{qb^{\alpha}}{\Gamma(\alpha+1)}\| \phi \|_{\Omega}, \qquad\omega(t)\leq\upsilon(t),\quad t\in[0,t_{1}] $$

and

$$ \upsilon{'}(t)\leq m(t) \bigl(\Psi\bigl(\omega(t)\bigr)+\omega(t) \bigr), \quad t\in[0,t_{1}]. $$

Using the nondecreasing character of Ψ, we get

$$ \upsilon{'}(t)\leq m(t) \bigl(\Psi\bigl(\upsilon(t)\bigr)+ \upsilon(t)\bigr),\quad t\in[0,t_{1}]. $$

Then, for each \(t\in[0,t_{1}]\), we have

$$ \Gamma_{1}\bigl(\upsilon(t)\bigr)= \int_{\upsilon(0)}^{\upsilon(t)}\frac{ds}{\Psi (s)+s}\leq \int_{0}^{t_{1}}m(s)\,ds< \int_{\upsilon(0)}^{\infty}\frac {ds}{\Psi(s)+s}. $$

This implies that \(\upsilon(t)<\infty\). So, there is a constant \(K_{0}\) such that \(\upsilon(t)\leq\Gamma_{1}^{-1} (\int_{0}^{t_{1}}m(s)\,ds )=M_{0}\), \(t\in[0,t_{1}]\). So, \(\|x_{t}\|_{\Omega}\leq\omega(t)<\upsilon(t)<M_{0}\), \(t\in[0,t_{1}]\), then\(\|x\|_{\infty}\leq\max\{\|\phi\|_{\Omega}, M_{0}\}=K_{0}\).

Step 2: Consider the following problem:

$$ \left \{ \textstyle\begin{array}{l} D^{\alpha}x(t)=f(t,x_{t})+\sum_{i=1}^{p}x(t-r_{i}), \quad t\in[t_{1},t_{2}], \\ x(t_{1}^{+})=x(t_{1}^{-})+I_{1}(x(t_{1}^{-})),\\ x(t)=x_{m-1}(t),\quad t\in[-\tau,t_{1}]. \end{array}\displaystyle \right . $$
(3.3)

Define the operator \(F_{1}: \Omega_{t_{2}}\rightarrow\Omega_{t_{2}}\) by

$$ F_{1}(x) (t)=\left \{ \textstyle\begin{array}{l} x(t_{1}^{-})+I_{1}(x(t_{1}^{-}))+\frac{1}{\Gamma(\alpha)}\int _{t_{1}}^{t}(t-s)^{\alpha-1}f(s,x_{s})\,ds\\ \quad{}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q}\int _{t_{1}-r_{i}}^{t-r_{i}}(t-s-r_{i})^{\alpha-1}x(s)\,ds, \quad t\in[t_{1}, t_{2}],\\ x(t_{1}^{+})=x(t_{1}^{-})+I_{1}(x(t_{1}^{-})). \end{array}\displaystyle \right . $$
(3.4)

As in Step 1, we can show that \(F_{1}\) is continuous and completely continuous if x is a possible solution of the equation \(x=\lambda F_{1}x\) for some \(\lambda\in(0,1)\).

Note that

$$ \big|x\bigl(t_{1}^{+}\bigr)\big|\leq\sup_{r\in[-K_{0},K_{0}]}\big|I_{1}(r)\big|+K_{0}:=N_{1} $$

and

$$\begin{aligned} \big|x(t)\big|\leq{}& N_{1}+\frac{1}{\Gamma(\alpha)}\sum _{i=1}^{q} \int _{t_{1}-r_{i}}^{t_{1}}(t-s-r_{i})^{\alpha-1}\big|x(s)\big|\,ds+ \frac{1}{\Gamma (\alpha)} \int_{t_{1}}^{t}(t-s)^{\alpha-1}\big|f(s,x_{s})\big|\,ds \\ &+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q} \int _{t_{1}}^{t-r_{i}}(t-s-r_{i})^{\alpha-1}\big|x(s)\big|\,ds \\ \leq{}& N_{1}+\frac{qK_{0}}{\Gamma(\alpha+1)} \bigl\vert (t-t_{1})^{\alpha }-(t-t_{1}+r_{i})^{\alpha} \bigr\vert +\frac{1}{\Gamma(\alpha)} \int_{t_{1}}^{t}(t-s)^{\alpha-1}\big|f(s,x_{s})\big|\,ds \\ & +\frac{q}{\Gamma(\alpha+1)}|t-t_{1}-r_{i}|^{\alpha} \int_{t_{1}}^{t}\big|x(s)\big|\,ds.\end{aligned} $$

By \((H_{2})\), we have

$$ \big|x(t)\big|\leq N_{1}+\frac{q\tau^{\alpha}K_{0}}{\Gamma(\alpha+1)}+\frac{b^{\alpha }}{\Gamma(\alpha+1)} \int_{t_{1}}^{t}p(s)\Psi\bigl(\|x_{s} \|_{\Omega}\bigr)\,ds +\frac{qb^{\alpha}}{\Gamma(\alpha+1)} \int_{t_{1}}^{t}\big|x(s)\big|\,ds. $$

Let us define \(\omega(t)\) as

$$ \omega(t)=\sup\bigl\{ \big|x(s)\big|: -\tau\leq s\leq t\bigr\} ,\quad0\leq t\leq t_{2}. $$

Then we have

$$ \omega(t)\leq N_{1}+ \int_{t_{1}}^{t}m(s) \bigl(\Psi\bigl(\omega(s)\bigr)+ \omega(s)\bigr)\,ds. $$

Let us take the right-hand side of the above inequality as \(\upsilon(t)\). Then we have

$$ c=N_{1}+\frac{q\tau^{\alpha}K_{0}}{\Gamma(\alpha+1)},\qquad \omega(t)\leq\upsilon(t), \quad t \in[t_{1},t_{2}] $$

and

$$ \upsilon{'}(t)\leq m(t) \bigl(\Psi\bigl(\omega(t)\bigr)+\omega(t) \bigr),\quad t\in[t_{1},t_{2}]. $$

Using the nondecreasing character of Ψ, we get

$$ \upsilon{'}(t)\leq m(t) \bigl(\Psi\bigl(\upsilon(t)\bigr)+ \upsilon(t)\bigr),\quad t\in[t_{1},t_{2}]. $$

Then, for each \(t\in[t_{1},t_{2}]\), we have

$$ \Gamma_{2}\bigl(\upsilon(t)\bigr)= \int_{\upsilon(t_{1})}^{\upsilon(t)}\frac {ds}{\Psi(s)+s}\leq \int_{t_{1}}^{t_{2}}m(s)\,ds< \int_{\upsilon (t_{1})}^{\infty}\frac{ds}{\Psi(s)+s}. $$

This implies that \(\upsilon(t)<\infty\). So, there is a constant \(K_{1}\) such that \(\upsilon(t)\leq\Gamma_{2}^{-1} (\int_{t_{1}}^{t_{2}}m(s)\,ds )=M_{1}\), \(t\in[t_{1},t_{2}]\). So, \(\|x_{t}\|_{\Omega}\leq\omega(t)<\upsilon(t)<M_{1}\), \(t\in[t_{1},t_{2}]\), then \(\|x\|_{\infty}\leq\max\{K_{0},M_{1}\}=K_{1}\).

Step 3: We continue this process and take into account that \(x_{m}:=x|_{[t_{m},b]}\) is a solution to the problem

$$ \left \{ \textstyle\begin{array}{l} D^{\alpha}x(t)=f(t,x_{t})+\sum_{i=1}^{p}x(t-r_{i}), \quad t\in[t_{m},b], \\ x(t_{m}^{+})-x(t_{m}^{-})=I_{m}(x_{m-1}(t_{m}^{-})),\\ x(t)=x_{m-1}(t),\quad t\in[-\tau,t_{m}]. \end{array}\displaystyle \right . $$
(3.5)

The solution of system (1.1) is then defined by

$$ x(t)=\left \{ \textstyle\begin{array}{l} x_{0}(t),\quad t\in[-\tau, t_{1}],\\ x_{1}(t),\quad t\in(t_{1}, t_{2}],\\ \vdots\\ x_{m}(t),\quad t\in(t_{m},b]. \end{array}\displaystyle \right . $$

 □

Next, we should use the Banach contraction principle to prove that F has a fixed point.

Theorem 3.2

Assume that conditions \((H_{4})\) and \((H_{5})\) hold, then system (1.1) has a unique solution on J, provided that the following inequality holds:

$$ \frac{1}{\Gamma(\alpha)} \biggl[ \biggl(\frac{1-\beta}{\alpha-\beta} \biggr)^{1-\beta}hb^{\alpha-\beta}+ \frac{b^{\alpha}}{\alpha} \biggr]+\sum_{k=1}^{m} c_{k}< 1,\quad h= \biggl( \int_{0}^{b}l(s)^{\frac{1}{\beta}}\,ds \biggr)^{\beta}. $$

Proof

Let F be the function defined by (3.1), then \(F: \Omega_{b}\rightarrow\Omega_{b}\) is well defined according to Theorem 3.1.

For every \(x,y\in\Omega_{b}\) and \(t\in[-\tau,0]\),

$$ \big\| F\bigl(x(t)\bigr)-F\bigl(y(t)\bigr)\big\| =\big\| \phi(t)-\psi(t)\big\| =0. $$

For each \(t\in[0,t_{1}]\), from conditions \((H_{4})\) and \((H_{5})\), we have

$$\begin{aligned}& \big\| F\bigl(x(t)\bigr)-F\bigl(y(t)\bigr)\big\| \\& \quad\leq\frac{1}{\Gamma(\alpha)} \Biggl\Vert \int_{0}^{t}(t-s)^{\alpha -1} \bigl(f(s,x_{s})-f(s,y_{s})\bigr)\,ds \\& \quad\quad{}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q} \int _{0}^{t-r_{i}}(t-s-r_{i})^{\alpha-1} \big\| x(s)-y(s)\big\| \,ds \Biggr\Vert \\& \quad\leq\frac{l}{\Gamma(\alpha)} \Biggl\Vert \int_{0}^{t}(t-s)^{\alpha-1}\,ds\| x_{s}-y_{s}\|_{\Omega} \\& \qquad{}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q} \int _{0}^{t-r_{i}}(t-s-r_{i})^{\alpha-1}\,ds \big\| x(s)-y(s)\big\| \Biggr\Vert \\& \quad\leq \biggl[\frac{1}{\Gamma(\alpha)} \biggl( \int_{0}^{t} \bigl((t-s)^{\alpha -1} \bigr)^{\frac{1}{1-\beta}}\,ds \biggr)^{1-\beta} \biggl( \int _{0}^{b}l(s)^{\frac{1}{\beta}}\,ds \biggr)^{\beta} +\frac{b^{\alpha}}{\Gamma(\alpha+1)} \biggr]\big\| x(s)-y(s)\big\| \\& \quad\leq\frac{1}{\Gamma(\alpha)} \biggl[ \biggl(\frac{1-\beta}{\alpha-\beta } \biggr)^{1-\beta}ht_{1}^{\alpha-\beta}+ \frac{t_{1}^{\alpha}}{\alpha } \biggr]\big\| x(s)-y(s)\big\| . \end{aligned}$$

Similarly, for each \(t\in[t_{k}, t_{k+1}]\), \(k=1,2,\ldots, m\), we have

$$\begin{gathered} \big\| F\bigl(x(t)\bigr)-F\bigl(y(t)\bigr)\big\| \\ \quad\leq\frac{1}{\Gamma(\alpha)} \biggl\Vert \int_{0}^{t}(t-s)^{\alpha -1} \bigl(f(s,x_{s})-f(s,y_{s})\bigr)\,ds \biggr\Vert \\ \qquad{}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q} \int _{0}^{t-r_{i}}(t-s-r_{i})^{\alpha-1} \big\| x(s)-y(s)\big\| \,ds \\ \qquad{}+\sum_{k=1}^{m} \big\| I_{k} \bigl(x\bigl(t_{k}^{-}\bigr)\bigr)-I_{k}\bigl(y \bigl(t_{k}^{-}\bigr)\bigr)\big\| \\ \quad\leq\frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-s)^{\alpha-1}l(s)\,ds\| x_{s}-y_{s}\|_{\Omega} \\ \qquad{}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{q} \int _{0}^{t-r_{i}}(t-s-r_{i})^{\alpha-1}\,ds \big\| x(s)-y(s)\big\| +\sum_{k=1}^{m}c_{k} \|x-y\| \\ \quad\leq \Biggl[\frac{1}{\Gamma(\alpha)} \biggl( \int_{0}^{t} \bigl((t-s)^{\alpha-1} \bigr)^{\frac{1}{1-\beta}}\,ds \biggr)^{1-\beta} \biggl( \int_{0}^{b}l(s)^{\frac{1}{\beta}}\,ds \biggr)^{\beta} \\ \qquad{}+\frac{b^{\alpha}}{\Gamma(\alpha+1)}+\sum_{k=1}^{m}c_{k} \Biggr]\big\| x(s)-y(s)\big\| \\ \quad\leq\frac{1}{\Gamma(\alpha)} \Biggl[ \biggl(\frac{1-\beta}{\alpha-\beta } \biggr)^{1-\beta}hb^{\alpha-\beta}+ \frac{b^{\alpha}}{\alpha}+\sum_{k=1}^{m}c_{k} \Biggr]\big\| x(s)-y(s)\big\| .\end{gathered} $$

Therefore, F is a contraction operator. Hence, F has a unique fixed point by the Banach contraction principle, that is, system (1.1) has a unique solution. □

\((H'_{4})\) :

There exists a positive constant l such that \(\|f(t,\phi)-f(t,\psi)\|\leq l\|\phi-\psi\|_{\Omega}\) for almost every \(t\in[0,b]\) and all \(\phi, \psi\in\Omega\).

Corollary 3.1

Assume that conditions \((H'_{4})\) and \((H_{5})\) hold, then system (1.1) has a unique solution on J provided that the following inequality \(\frac{(l+1)b^{\alpha}}{\Gamma(\alpha+1)}+\sum_{k=1}^{m} c_{k}<1\) holds.

4 Examples

In this section we provide some examples to illustrate the usefulness of our main results.

Example 4.1

We consider the first fractional impulsive system as follows:

$$ \left\{ \textstyle\begin{array}{l} D^{\alpha}x(t)=\frac{1}{1+e^{t}} (\int_{-1}^{0}x(t+\theta)\,d\theta )^{2}+x(t-1), \quad t\in J=[0,1], t\neq t_{1},\\ \Delta x|_{t=t_{1}}=I_{1}(x(t_{1}^{-})),\\ x(t)=\phi(t),\quad t\in[-1,0], \end{array}\displaystyle \right . $$
(4.1)

where \(f(t,\psi)=\frac{1}{1+e^{t}} (\int_{-1}^{0}\psi(\theta)\,d\theta )^{2}\), \(\alpha=\frac{1}{2}\), \(t_{1}\in(0,1)\). Set \(p(t)=\frac{1}{1+e^{t}}\), \(\Psi(x)=x^{2}+\frac{1}{4}\), then \(f(t,y)\leq\frac{1}{1+e^{t}}\Psi(\|y\|_{\Omega})\), \(y\in\Omega\), \(t\in[0,1]\), \(\int_{0}^{1}m(t)\,dt=\frac{1}{\sqrt{\pi}}\leq\int_{0}^{\infty}\frac {1}{u^{2}+u+\frac{1}{4}}\,du=2\). So, all the assumptions in Theorem 3.1 are satisfied, our results can be applied to problem (4.1).

Example 4.2

We consider the second fractional impulsive system as follows:

$$ \left\{ \textstyle\begin{array}{l} D^{\alpha}x(t)=\frac{1}{10+e^{t}} (\int_{-1}^{0}x(t+\theta)\,d\theta )^{2}+x(t-1), \quad t\in J=[0,1], t\neq t_{1},\\ \Delta x|_{t=t_{1}}=I_{1}(x(t_{1}^{-})),\\ x(t)=\phi(t),\quad t\in[-1,0], \end{array}\displaystyle \right . $$
(4.2)

where \(f(t,\psi)=\frac{1}{10+e^{t}} (\int_{-1}^{0}\psi(\theta)\,d\theta )^{2}\), \(\alpha=\frac{1}{2}\), \(I_{1}(x(t_{1}^{-}))=\frac {1}{3}\), \(t_{1}\in(0,1)\), then \(\|f(t,x)-f(t,y)\|\leq\frac{1}{11}\|x-y\|\), \(\forall x,y\in\Omega _{b}\), \((\frac{1}{11}+1 )\frac{1}{\Gamma(\frac{1}{2}+1)}+0=\frac {24}{11\sqrt{\pi}}<1\). So, all the assumptions in Corollary 3.1 are satisfied, our results can be applied to problem (4.2).

5 Conclusions

In this paper, we use the nonlinear alternative of Leray–Schauder and the Banach fixed point theorem to prove the existence and uniqueness of solution for the fractional order impulsive functional differential equations with multiple delays.