1 Introduction

In the last decades, many researchers have been attracted to studying the fractional differential equations, and a lot of good results have been obtained, see [1–17] and the references therein. In [1–9], authors studied the fractional differential equations with instantaneous impulses, which have been applied to describe abrupt changes such as the shocks and natural disasters. For more details on this subject, see [1–9]. The differential equations with instantaneous impulse cannot explain some dynamics problems of evolution process. For instance, the drug delivery in the bloodstream is a gradual and continuous process. However, the models with non-instantaneous impulses can explain these problems.

The differential equations with non-instantaneous impulses of the following form were initially investigated by the authors in [18, 19]:

$$\textstyle\begin{cases} x'(t)=Ax(t)+f(t,x(t)),\quad t\in(s_{k},t_{k+1}], k=0,1,\ldots,N,\\ x(t)=g_{k}(t,x(t)), \quad t\in(t_{k},s_{k}], k=1,2,\ldots,N,\\ x(0)=x_{0}\in E, \end{cases} $$

where \(A:D(A)\subset E\rightarrow E\) is the generator of a \(C_{0}\)-semigroup \(\{S(t)_{t\geq0}\}\) on a Banach space E. In [18, 19], the existence results have been established by using the fixed point theorems.

In [20], authors investigated the following periodic boundary value problem of integer nonlinear evolution equations with non-instantaneous impulses:

$$\textstyle\begin{cases} x'(t)=Ax(t)+f(t,x(t)),\quad t\in(s_{k},t_{k+1}], k=0,1,\ldots,N,\\ x(t)=S(t-t_{k})g_{k}(t,x(t)),\quad t\in(t_{k},s_{k}], k=1,2,\ldots,N,\\ x(0)=x(T), \end{cases} $$

where the semigroup \(\{S(t)_{t\geq0}\}\) is compact and the linear operator A is independent of t. The existence results were obtained by using the fixed point theorems. Now, many researchers are studying the fractional differential equations with non-instantaneous impulses, and a lot of good results have been obtained [21–27]. In [21, 23], the authors studied the stability of the fractional differential equations with non-instantaneous impulses. The existence results of the fractional differential equations with non-instantaneous impulses are discussed by authors in [22, 24–26].

Inspired by the above said work, we consider the following periodic boundary value problem for nonlinear fractional evolution equations with non-instantaneous impulses:

$$ \textstyle\begin{cases} {}^{c}D_{t}^{\beta}u(t)=A(t)u(t)+f(t,u(t))+\int _{0}^{t}q(t-s)h(s,u(s))\,ds,\\ \quad t\in(s_{i},t_{i+1}], i=0,1,\ldots,m,\\ u(t)=U_{\beta}(t,t_{i})g_{i}(t,u(t)),\quad t\in(t_{i},s_{i}],i=1,2,\ldots,m,\\ u(0)=u(T), \end{cases} $$
(1.1)

where \({}^{c}D_{t}^{\beta}\) is the Caputo’s fractional derivative of order \(\beta\in(0,1]\), \(A(t)\) is dependent on t and a closed and linear unbounded operator with domain \(D(A)\) defined on a Banach space E, the fixed points \(s_{i}\) and \(t_{i}\) satisfying \(0=s_{0}< t_{1}\leq s_{1}< t_{2}\leq\cdot\cdot\cdot<t_{m}\leq s_{m}< t_{m+1}=T\) are pre-fixed numbers. f, \(U_{\beta}\), h, and \(g_{i}\) (\(i=1,2,\ldots,m\)) are to be specified later, \(q:[0,T]\rightarrow X\) is continuous.

In [18–20], differential equations are all integer order, the linear operator A is independent of t, and the semigroup is compact. In [24–26], the linear operator A is independent of t. In this paper, we consider the existence of mild solutions for the fractional differential equations (1.1) under the conditions of the compact and noncompact semigroup; meanwhile, the linear operator \(A(t)\) is dependent on t. Therefore, the results presented in this paper improve and generalize the main results in [18–20, 24–26] by using a different method.

2 Preliminaries

Let \(J=[0,T]\), \(C(J,E)=\{u:J\rightarrow E\mbox{ is continuous}\}\), \(\operatorname{PC}(J,E)=\{u:J\rightarrow E:u\in C((s_{i},t_{i+1}],E), [4]\mbox{ and there exist } u(t_{i}^{-})\mbox{ and }u(t_{i}^{+})\mbox{ with }u(t_{i}^{-})=u(t_{i}), i=1,2,\ldots,m\}\) with the PC-norm \(\|u\|_{\mathrm{PC}}=\sup\{\|u(t)\|:t\in J\}\).

Lemma 2.1

([28])

Let E be a Banach space, \(D\subset E\) be a bounded closed and convex set. Assume that \(Q:D\rightarrow D\) is a strict set contraction mapping. Then Q has at least one fixed point in D.

Definition 2.1

([29, 30])

The Riemann–Liouville fractional integral of f of order \(\nu>0\) is defined by

$$J_{t}^{\nu}f(t)=\frac{1}{\Gamma(\nu)} \int_{0}^{t}(t-s)^{\nu-1}f(s)\,ds,\qquad f(t) \in L^{1}(J;E). $$

The integral operators \(\{J_{t}^{\nu}\}_{\nu\geq0}\) have the following semigroup property:

$$J_{t}^{\nu}J_{t}^{\mu}=J_{t}^{\nu+\mu}, \qquad \nu,\mu\geq0. $$

Definition 2.2

([29, 30])

The Caputo fractional derivative of order \(n-1<\nu<n\) with the lower limits zero for a function \(f\in C^{n}[0,+\infty)\) can be written as

$${}^{c}D_{t}^{\nu}f(t)=\frac{1}{\Gamma(n-\nu)} \int_{0}^{t}(t-s)^{n-\nu -1}f^{(n)}(s) \,ds,\qquad t>0, n\in\Bbb{N}. $$

Lemma 2.2

([31])

Let E be a Banach space and \(B\subset C[J,E]\) be equicontinuous and bounded, then \(\mathop{\overline{\mathrm{Co}}}B \subset C[J,E]\) is also equicontinuous and bounded (where Co̅B denotes the closed convex hull of B).

Definition 2.3

([32])

Let S be a bounded set of E, \(\alpha(S)=\inf\{\delta>0:S\mbox{ can be expressed as } [4]\mbox{the union of a finite number of sets such that the diameter of each set does not exceed }\delta, [4]\mbox{ i.e., }S=\bigcup_{i=1}^{m}S_{i}\mbox{ with }\operatorname{diam}(S_{i})\leq\delta, i=1,2,\ldots,m\}\), \(\alpha(S)\) is called the Kuratowski measure of noncompactness of set S. Obviously, \(0<\alpha(S)<\infty\).

Lemma 2.3

([33])

Let E be a Banach space, and let \(D \subset E\) be bounded, then there exists a countable set \(D_{0}\subset D\) such that \(\alpha(D)\leq 2\alpha(D_{0})\).

Lemma 2.4

([31, 34])

Let E be a Banach space, and let \(B\subset C[J,E]\) be equicontinuous and bounded, then \(\alpha (B(t))\) is continuous on J, and

$$\alpha \biggl( \int_{J}B(s)\,ds \biggr)\leq \int_{J}\alpha\bigl(B(s)\bigr)\,ds, \qquad \alpha(B)=\max _{t\in J} \alpha\bigl(B(t)\bigr). $$

Definition 2.4

([5, 35])

Let \(A(t)\) be a closed and linear operator with domain \(D(A)\) defined on a Banach space E, \(\beta>0\) be a constant. Let \(\rho[A(t)]\) be the resolvent set of \(A(t)\), we call \(A(t)\) the generator of a β-resolvent operator family if there exist \(\omega\geq0\) and a strongly continuous function \(U_{\beta}:\Bbb{R}_{+}^{2}\rightarrow B(E)\) such that \(\{\lambda^{\beta}: \operatorname{Re} \lambda>\omega\}\subset\rho(A)\) and

$$\bigl(\lambda^{\beta}I-A(s)\bigr)^{-1}u= \int_{0}^{\infty}e^{-\lambda(t-s)}U_{\beta}(t,s)u\,dt, \quad \operatorname{Re}(\lambda)>\omega, u\in E. $$

In this case, \(U_{\beta}(t,s)\) is called the β-resolvent family generated by \(A(t)\).

Remark 2.1

By [5, 36], the β-resolvent operator family \(U_{\beta}(t,s)\) satisfies the following properties:

  1. (1)

    \(U_{\beta}(s,s)=I\), \(U_{\beta}(t,s)=U_{\beta}(t,r)U_{\beta}(r,s)\) for \(0\leq s\leq r\leq t\leq a\).

  2. (2)

    \((t,s)\rightarrow U_{\beta}(t,s)\) is strongly continuous for \(0\leq s\leq t\leq a\).

Definition 2.5

A function \(u\in \mathrm{PC} (J,E)\) is said to be a mild solution of problem (1.1) if u satisfies the following equations:

$$\begin{aligned}& \begin{aligned} u(t)&=U_{\beta}(t,0) \biggl[U_{\beta}(T,t_{m})g_{m} \bigl(s_{m},u(s_{m})\bigr)\\ &\quad {}+ \int _{s_{m}}^{T}U_{\beta}(t,s) \biggl(f \bigl(s,u(s)\bigr)+ \int_{0}^{s}q(s-\tau)h\bigl(\tau,u(\tau )\bigr)\,d \tau \biggr)\,ds \biggr] \\ &\quad{}+ \int_{0}^{t}U_{\beta}(t,s) \biggl(f \bigl(s,u(s)\bigr)+ \int_{0}^{s}q(s-\tau)h\bigl(\tau ,u(\tau)\bigr)\,d \tau \biggr)\,ds,\quad t\in[0,t_{1}], \end{aligned} \\& u(t)=U_{\beta}(t,t_{i})g_{i}\bigl(t,u(t)\bigr),\quad t \in(t_{i},s_{i}], i=1,2,\ldots,m, \\& u(t)=U_{\beta}(t,t_{i})g_{i}\bigl(s_{i},u(s_{i}) \bigr)+ \int_{s_{i}}^{t}U_{\beta}(t,s) \biggl(f \bigl(s,u(s)\bigr)+ \int_{0}^{s}q(s-\tau)h\bigl(\tau,u(\tau)\bigr)\,d \tau \biggr)\,ds, \\& \quad t\in(s_{i},t_{i+1}], i=1,\ldots,m. \end{aligned}$$

3 \(U_{\beta}(t,s)\) is noncompact

We define an operator \(F:\operatorname{PC}(J,E)\rightarrow \operatorname{PC}(J,E)\) by

$$ (Fu) (t)= \textstyle\begin{cases} U_{\beta}(t,0)[U_{\beta}(T,t_{m})g_{m}(s_{m},u(s_{m}))\\ \quad{}+\int_{s_{m}}^{T}U_{\beta}(T,s) (f(s,u(s))+\int_{0}^{s}q(s-\tau )h(\tau,u(\tau))\,d\tau )\,ds]\\ \quad{}+\int_{0}^{t}U_{\beta}(t,s) (f(s,u(s))+\int_{0}^{s}q(s-\tau)h(\tau ,u(\tau))\,d\tau )\,ds,\quad t\in[0,t_{1}],\\ U_{\beta}(t,t_{i})g_{i}(t,u(t)),\quad t\in(t_{i},s_{i}],i=1,2,\ldots,m,\\ U_{\beta}(t,t_{i})g_{i}(s_{i},u(s_{i}))+\int_{s_{i}}^{t}U_{\beta}(t,s) (f(s,u(s))\\ \quad {}+\int_{0}^{s}q(s-\tau)h(\tau,u(\tau))\,d\tau )\,ds, \quad t\in(s_{i},t_{i+1}], i=1,\ldots,m. \end{cases} $$
(3.1)

Theorem 3.1

Assume that the following conditions \((H_{1})\) and \((H_{2})\) hold.

\((H_{1})\) :

For any \(R>0\), functions f, h and \(g_{i}\ (i=1,2,\ldots,m):J\times E\rightarrow E\) are bounded and continuous on \(J\times T_{R}\), and

$$ \limsup_{R\rightarrow\infty}\frac{M(R)}{R}< \frac{1}{\Delta} , $$
(3.2)

where

$$\begin{aligned} \Delta&=\max \biggl\{ M^{2} \biggl(1+(T-s_{m})+ \int_{s_{m}}^{T} \int _{0}^{s}q(s-\tau)\,d\tau \,ds \biggr)+M \biggl(t_{1}+ \int_{0}^{t} \int_{0}^{s}q(s-\tau)\,d\tau \,ds \biggr), \\ &\quad M, M \biggl(1+(t_{i+1}-s_{i})+ \int_{s_{i}}^{t} \int _{0}^{s}q(s-\tau)\,d\tau \,ds \biggr), i=1,2, \ldots,m \biggr\} , \end{aligned} $$

\(M(R)=\sup\{\|f(t,u)\|, \|h(t,u)\|,\|g_{i}(t,u)\|(i=1,2,\ldots,m):(t,u)\in J\times T_{R}\}\), \(T_{R}=\{u\in E:\|u\|\leq R\}\), the resolvent operator \(U_{\beta}(t,s)\) is noncompact for \(t,s>0\), \(M=\max_{0\leq s< t\leq T}\| U_{\beta}(t,s)\|<+\infty\).

\((H_{2})\) :

For all \(R>0\), there exist nonnegative Lebesgue integrable functions \(L_{f}, L_{h}, L_{g_{i}} \in L^{1}(J,\Bbb{R}^{+})\) (\(i=1,2,\ldots,m\)) such that, for all equicontinuous and countable sets \(D\subset T_{R}=\{u\in E:\|u\|\leq R\}\),

$$ \alpha(f(t,D)\leq L_{f}(t)\alpha(D),\qquad \alpha(h(t,D)\leq L_{h}(t)\alpha(D) $$
(3.3)

and

$$ \alpha(g_{i}(t,D)\leq L_{g_{i}}(t)\alpha(D) $$
(3.4)

with

$$ \begin{aligned}[b] \varrho&= \max \biggl\{ M^{2}L_{g_{m}}(t)+M^{2} \int _{s_{m}}^{T}L_{f}(s)\,ds+M \int_{0}^{t}L_{f}(s)\,ds\\ &\quad {}+ \bigl(M^{2}+M\bigr) \int_{0}^{t} \int_{0}^{s}q(s-\tau )L_{h}(\tau)\,d\tau \,ds, \\ &\quad ML_{g_{i}}(t),L_{g_{i}}(t)+ \int_{s_{i}}^{t}L_{f}(s)\,ds+ \int_{s_{i}}^{t} \int _{0}^{s}q(s-\tau)L_{h}(\tau)\,d\tau \,ds,\\ &\quad i=1,2,3,\ldots,m \biggr\} \\ &< 1. \end{aligned} $$
(3.5)

Then problem (1.1) has at least one mild solution \(u\in \mathrm{PC} (J,E)\).

Proof

By (3.2), there exist \(0< r<\Delta^{-1}\) and \(R_{0}>0\) such that, for any \(R\geq R_{0}\), we have

$$M(R)< rR. $$

Let \(R^{*}\geq R_{0}\). For any \(u\in B_{R^{*}}=\{u\in \operatorname{PC}(J,E) : \|u\| _{\mathrm{PC}}\leq R^{*}\}\), we prove that \(Fu\in B_{R^{*}}\).

Step 1. By (3.2), for any \(t\in[0,t_{1}]\), we have

$$\begin{aligned} \bigl\Vert (Fu) (t) \bigr\Vert &\leq \bigl\Vert U_{\beta}(t,0) \bigr\Vert \bigl\Vert U_{\beta}(T,t_{m})g_{m} \bigl(s_{m},u(s_{m})\bigr) \bigr\Vert \\ &\quad{}+ \bigl\Vert U_{\beta}(t,0) \bigr\Vert \biggl\Vert \int_{s_{m}}^{T}U_{\beta}(T,s) \biggl(f \bigl(s,u(s)\bigr)+ \int_{0}^{s}q(s-\tau)h\bigl(\tau,u(\tau)\bigr)\,d \tau \biggr)\,ds \biggr\Vert \\ &\quad{}+ \biggl\Vert \int_{0}^{t}U_{\beta}(t,s) \biggl(f \bigl(s,u(s)\bigr)+ \int_{0}^{s}q(s-\tau )h\bigl(\tau,u(\tau)\bigr)\,d \tau \biggr)\,ds \biggr\Vert \\ &\leq M^{2}rR^{*} \biggl(1+(T-s_{m})+ \int_{s_{m}}^{T} \int_{0}^{s}q(s-\tau)\,d\tau \,ds \biggr) \\ &\quad{}+MrR^{*} \biggl(t_{1}+ \int_{0}^{t} \int_{0}^{s}q(s-\tau)\,d\tau \,ds \biggr) \\ &\leq R^{*}. \end{aligned} $$

For any \(t\in(t_{i},s_{i}]\), \(i=1,2,\ldots,m\), we have

$$\bigl\Vert (Fu) (t) \bigr\Vert \leq \bigl\Vert U_{\beta}(t,t_{i})g_{i} \bigl(t,u(t)\bigr) \bigr\Vert \leq MrR^{*}\leq R^{*}. $$

For any \(t\in(s_{i},t_{i+1}]\), \(i=1,2,\ldots,m\), we have

$$\begin{aligned} \bigl\Vert (Fu) (t) \bigr\Vert &\leq \bigl\Vert U_{\beta}(t,t_{i})g_{i}\bigl(s_{i},u(s_{i}) \bigr) \bigr\Vert \\ &\quad{} + \biggl\Vert \int_{s_{i}}^{t}U_{\beta}(t,s) \biggl(f \bigl(s,u(s)\bigr)+ \int_{0}^{s}q(s-\tau)h\bigl(\tau,u(\tau)\bigr)\,d \tau \biggr)\,ds \biggr\Vert \\ &\leq MrR^{*} \biggl(1+(t_{i+1}-s_{i})+ \int_{s_{i}}^{t} \int_{0}^{s}q(s-\tau)\,d\tau \,ds \biggr) \\ &\leq R^{*}. \end{aligned} $$

Step 2. We prove that \(F:B_{R^{*}}\rightarrow B_{R^{*}}\) is continuous. Let \(\{u_{n}\} _{0}^{\infty}\subset \operatorname{PC}(J,E)\) and \(u_{n}\rightarrow u\), \(u\in \operatorname{PC}(J,E)\). By the continuity of the functions f, \(g_{i}\) (\(i=1,2,\ldots,m\)), h, we have

$$\begin{gathered} \lim_{n\rightarrow\infty}\sup_{t\in J} \bigl\Vert f\bigl(t,u_{n}(t)\bigr)-f \bigl(t,u(t)\bigr) \bigr\Vert =0, \\ \lim_{n\rightarrow\infty}\sup_{t\in J} \bigl\Vert g_{i}\bigl(t,u_{n}(t)\bigr)-g_{i}\bigl(t,u(t) \bigr) \bigr\Vert =0, \\ \lim_{n\rightarrow\infty}\sup_{t\in J} \bigl\Vert h \bigl(t,u_{n}(t)\bigr)-h\bigl(t,u(t)\bigr) \bigr\Vert =0. \end{gathered} $$

For any \(t\in[0,t_{1}]\), we have

$$\begin{aligned} &\bigl\Vert (Fu_{n}) (t)-(Fu) (t) \bigr\Vert \\ &\quad \leq M^{2}\sup_{t\in J} \bigl\Vert g_{m}\bigl(t,u_{n}(t) \bigr)-g_{m}\bigl(t,u(t)\bigr) \bigr\Vert \\ &\qquad {}+ M^{2} \int_{s_{m}}^{T} \bigl\Vert f\bigl(s,u_{n}(s) \bigr)-f\bigl(s,u(s)\bigr) \bigr\Vert ds \\ &\qquad{}+M^{2} \int_{s_{m}}^{T} \int_{0}^{s} \bigl\Vert h\bigl(s,u_{n}(s) \bigr)-h\bigl(s,u(s)\bigr) \bigr\Vert q(s-\tau)\,d\tau \,ds \\ &\qquad{}+M t_{1} \sup_{t\in J}\bigl\Vert f\bigl(t,u_{n}(t)\bigr)-f \bigl(t,u(t)\bigr) \bigr\Vert \\ &\qquad{}+M \int_{0}^{T} \int_{0}^{s} \bigl\Vert h\bigl(s,u_{n}(s) \bigr)-h\bigl(s,u(s)\bigr) \bigr\Vert q(s-\tau )\,d\tau \,ds. \end{aligned} $$

For any \(t\in(t_{i},s_{i}]\), \(i=1,2,\ldots,m\), we have

$$\bigl\Vert (Fu_{n}) (t)-(Fu) (t) \bigr\Vert \leq M \sup_{t\in J}\bigl\Vert g_{i}\bigl(t,u_{n}(t)\bigr)-g_{i}\bigl(t,u(t) \bigr) \bigr\Vert . $$

For any \(t\in(s_{i},t_{i+1}]\), \(i=1,2,\ldots,m\), we have

$$\begin{aligned} \bigl\Vert (Fu_{n}) (t)-(Fu) (t) \bigr\Vert &\leq M \sup_{t\in J}\bigl\Vert g_{i}\bigl(t,u_{n}(t) \bigr)-g_{i}\bigl(t,u(t)\bigr) \bigr\Vert \\ &\quad {}+ M \int_{s_{i}}^{T} \bigl\Vert f\bigl(s,u_{n}(s) \bigr)-f\bigl(s,u(s)\bigr) \bigr\Vert ds \\ &\quad{}+M \int_{s_{i}}^{T} \int_{0}^{s} \bigl\Vert h\bigl(s,u_{n}(s) \bigr)-h\bigl(s,u(s)\bigr) \bigr\Vert q(s-\tau)\,d\tau \,ds. \end{aligned} $$

By the above inequalities, we know that

$$\Vert Fu_{n}-Fu \Vert _{\mathrm{PC}}\rightarrow0 \quad ( n \rightarrow\infty). $$

Therefore, the operator \(F:B_{R^{*}}\rightarrow B_{R^{*}}\) is continuous.

Step 3. We will prove that \(F(B_{R^{*}})\) is equicontinuous. Case 1. For interval \([0,t_{1}]\), \(0\leq e_{1}< e_{2}\leq t_{1}\), \(u\in B_{R^{*}}\), we have

$$ \begin{aligned}[b] &\bigl\Vert (Fu) (e_{2})-(Fu) (e_{1}) \bigr\Vert \\ &\quad \leq \bigl\Vert U_{\beta}(e_{2},0)-U_{\beta}(e_{1},0) \bigr\Vert \bigl\Vert U_{\beta}(T,t_{m})g_{m}\bigl(s_{m},u(s_{m}) \bigr) \bigr\Vert \\ &\qquad{}+ \bigl\Vert U_{\beta}(e_{2},0)-U_{\beta}(e_{1},0) \bigr\Vert \\ &\qquad {}\times \biggl\Vert \int _{s_{m}}^{T}U_{\beta}(T,s) \biggl(f \bigl(s,u(s)\bigr)+ \int_{0}^{s}q(s-\tau)h\bigl(\tau,u(\tau )\bigr)\,d \tau \biggr)\,ds \biggr\Vert \\ &\qquad{}+\sup_{s\in[0,t_{1}]} \bigl\Vert U_{\beta}(e_{2},s)-U_{\beta}(e_{1},s) \bigr\Vert \\ &\qquad {}\times \biggl\Vert \int_{0}^{e_{1}} \biggl(f\bigl(s,u(s)\bigr)+ \int_{0}^{s}q(s-\tau)h\bigl(\tau,u(\tau )\bigr)\,d \tau \biggr)\,ds \biggr\Vert \\ &\qquad{}+ \biggl\Vert \int_{e_{1}}^{e_{2}}U_{\beta}(e_{2},s) \biggl(f\bigl(s,u(s)\bigr)+ \int _{0}^{s}q(s-\tau)h\bigl(\tau,u(\tau)\bigr)\,d \tau \biggr)\,ds \biggr\Vert . \end{aligned} $$
(3.6)

Case 2. For interval \((t_{i},s_{i}]\), \(i=1,2,\ldots,m\), \(0\leq e_{1}< e_{2}\leq t_{1}\), \(u\in B_{R^{*}}\), we have

$$ \begin{aligned}[b] &\bigl\Vert (Fu) (e_{2})-(Fu) (e_{1}) \bigr\Vert \\ &\quad \leq \bigl\Vert U_{\beta}(e_{2},t_{i})g_{i}\bigl(e_{2},u(e_{2}) \bigr)-U_{\beta}(e_{1},t_{i})g_{i} \bigl(e_{1},u(e_{1})\bigr) \bigr\Vert \\ &\quad \leq \bigl\Vert U_{\beta}(e_{2},e_{1})U_{\beta}(e_{1},t_{i})g_{i} \bigl(e_{2},u(e_{2})\bigr)-U_{\beta}(e_{1},t_{i})g_{i}\bigl(e_{1},u(e_{1}) \bigr) \bigr\Vert \\ &\quad \leq M \bigl\Vert U_{\beta}(e_{2},e_{1})g_{i} \bigl(e_{2},u(e_{2})\bigr)-g_{i} \bigl(e_{1},u(e_{1})\bigr) \bigr\Vert . \end{aligned} $$
(3.7)

Case 3. For interval \((s_{i},t_{i+1}]\), \(i=1,2,\ldots,m\), \(0\leq e_{1}< e_{2}\leq t_{1}\), \(u\in B_{R^{*}}\), we have

$$ \begin{aligned}[b] &\bigl\Vert (Fu) (e_{2})-(Fu) (e_{1}) \bigr\Vert \\ &\quad \leq \bigl\Vert U_{\beta}(e_{2},t_{i})-U_{\beta}(e_{1},t_{i}) \bigr\Vert \bigl\Vert g_{i}\bigl(s_{i},u(s_{i}) \bigr) \bigr\Vert \\ &\qquad{}+\sup_{s\in(s_{i},t_{i+1}]} \bigl\Vert U_{\beta}(e_{2},s)-U_{\beta}(e_{1},s) \bigr\Vert \\ &\qquad {}\times \int_{s_{i}}^{e_{1}} \biggl(f\bigl(s,u(s)\bigr)+ \int_{0}^{s}q(s-\tau)h\bigl(\tau ,u(\tau)\bigr)\,d \tau \biggr)\,ds \\ &\qquad{}+ \biggl\Vert \int_{e_{1}}^{e_{2}}U_{\beta}(e_{2},s) \biggl(f\bigl(s,u(s)\bigr)+ \int _{0}^{s}q(s-\tau)h\bigl(\tau,u(\tau)\bigr)\,d \tau \biggr)\,ds \biggr\Vert . \end{aligned} $$
(3.8)

By (1)–(2) of Remark 2.1 and the above inequalities (3.6)–(3.8), we derive that \(\|(Fu)(e_{2})-(Fu)(e_{1})\|\rightarrow0\) as \(e_{2}\rightarrow e_{1}\). Therefore \(F(B_{R^{*}})\) is equicontinuous. From Lemma 2.2, we know that \(\mathop{\overline{\mathrm{Co}}}F(B_{R^{*}})\subset B_{R^{*}}\) is bounded and equicontinuous.

Step 4. We will prove that \(F:\mathop{\overline{\mathrm{Co}}}F(B_{R^{*}})\rightarrow\mathop{\overline{\mathrm{Co}}}F(B_{R^{*}})\) is a condensing operator. For any \(D\subset\mathop{\overline{\mathrm{Co}}}F(B_{R^{*}})\), by Lemma 2.3, there exists a countable set \(D_{0}=\{u_{n}\}\subset D\) such that

$$ \alpha\bigl(F(D)\bigr)\leq2\alpha\bigl(F(D_{0})\bigr). $$
(3.9)

From the equicontinuity of \(\mathop{\overline{\mathrm{Co}}}F(B_{R^{*}})\), we have that \(D_{0}\subset\mathop{\overline{\mathrm{Co}}}F(B_{R^{*}})\) is equicontinuous. Case 1. For any \(t\in[0,t_{1}]\), by Lemma 2.4, we get

$$ \begin{aligned}[b] \alpha\bigl(F(D_{0}) (t)\bigr)&\leq M^{2}\alpha\bigl(g_{m}\bigl(s_{m},D_{0}(s_{m}) \bigr)\bigr) \\ &\quad{}+M^{2}\alpha \biggl( \int_{s_{m}}^{T} \biggl(f\bigl(s,D_{0}(s) \bigr)+ \int_{0}^{s}q(s-\tau )h\bigl(\tau,D_{0}( \tau)\bigr)\,d\tau \biggr)\,ds \biggr) \\ &\quad{}+M\alpha \biggl( \int_{0}^{t} \biggl(f\bigl(s,D_{0}(s) \bigr)+ \int_{0}^{s}q(s-\tau)h\bigl(\tau ,D_{0}( \tau)\bigr)\,d\tau \biggr)\,ds \biggr) \\ &\leq M^{2}L_{g_{m}}(t)\alpha(D) \\ &\quad{}+M^{2} \biggl( \int_{s_{m}}^{T}L_{f}(s)\alpha \bigl(D_{0}(s)\bigr)+ \int_{0}^{s}q(s-\tau )L_{h}(\tau)\alpha \bigl(D_{0}(\tau)\bigr)\,d\tau \,ds \biggr) \\ &\quad{}+M \biggl( \int_{0}^{t}L_{f}(s)\alpha \bigl(D_{0}(s)\bigr)+ \int_{0}^{s}q(s-\tau )L_{h}(\tau)\alpha \bigl(D_{0}(\tau)\bigr)\,d\tau \,ds \biggr) \\ &\leq \biggl(M^{2}L_{g_{m}}(t)+M^{2} \int_{s_{m}}^{T}L_{f}(s)\,ds+M \int _{0}^{t}L_{f}(s)\,ds \\ &+\bigl(M^{2}+M\bigr) \int_{0}^{t} \int_{0}^{s}q(s-\tau)L_{h}(\tau)\,d\tau \,ds \biggr)\alpha(D). \end{aligned} $$
(3.10)

Case 2. For any \(t\in(t_{i},s_{i}]\), \(i=1,2,\ldots,m\), we get

$$ \alpha\bigl(F(D_{0}) (t)\bigr)\leq ML_{g_{i}}(t) \alpha(D). $$
(3.11)

Case 3. For any \(t\in(s_{i},t_{i+1}]\), \(i=1,2,\ldots,m\), we get

$$ \begin{aligned}[b] \alpha\bigl(F(D_{0}) (t)\bigr) &\leq ML_{g_{i}}(t)\alpha(D)\\ &\quad {}+M \biggl( \int_{s_{i}}^{t}L_{f}(s)\alpha \bigl(D_{0}(s)\bigr)+ \int_{0}^{s}q(s-\tau)L_{h}(\tau)\alpha \bigl(D_{0}(\tau)\bigr)\,d\tau \,ds \biggr) \\ &\leq M \biggl(L_{g_{i}}(t)+ \int_{s_{i}}^{t}L_{f}(s)\,ds+ \int_{s_{i}}^{t} \int _{0}^{s}q(s-\tau)L_{h}(\tau)\,d\tau \,ds \biggr)\alpha(D). \end{aligned} $$
(3.12)

By Lemma 2.4, we have

$$\alpha\bigl(F(D_{0})\bigr)=\max_{t\in J} \alpha \bigl(F(D_{0}) (t)\bigr). $$

Therefore

$$\alpha\bigl(F(D)\bigr)\leq\varrho\alpha(D). $$

By (3.5), we get that \(F:\mathop{\overline{\mathrm{Co}}}F(B_{R^{*}})\rightarrow \mathop{\overline{\mathrm{Co}}}F(B_{R^{*}})\) is a strict set contraction mapping. Thus, by Lemma 2.1, F has at least one fixed point \(u^{*}\in \mathop{\overline{\mathrm{Co}}}F(B_{R^{*}})\subset \operatorname{PC}(J,E)\), which means that problem (1.1) has at least one mild solution. □

Remark 3.1

In Theorem 3.1, we assume that the resolvent operator \(U_{\beta}(t,s)\) is noncompact for \(t>0\). In the following Theorem 4.1, the resolvent operator \(U_{\beta}(t,s)\) is compact for \(t>0\).

4 \(U_{\beta}(t,s)\) is compact

Theorem 4.1

Assume that the following conditions \((H_{3})\)–\((H_{5})\) hold.

\((H_{3})\) :

The resolvent operator \(U_{\beta}(t,s)\) is compact for \(t,s>0\) and \(M= [4]\max_{0\leq s< t\leq T}\|U_{\beta}(t,s)\|<+\infty\).

\((H_{4})\) :

The functions \(f, h:J\times E\rightarrow E\) are continuous, and there exist nonnegative Lebesgue integrable functions \(a,b, L_{f},L_{h}\in L^{1}(J,\Bbb{R}^{+})\) such that, for all \(u \in E \), \(t\in J\),

$$\begin{aligned} &\bigl\Vert f(t,u) \bigr\Vert \leq a(t)+L_{f}(t) \Vert u \Vert ^{\gamma},\quad 0< \gamma\leq1, \\ &\bigl\Vert h(t,u) \bigr\Vert \leq b(t)+L_{h}(t) \Vert u \Vert ^{\mu},\quad0< \mu \leq1. \end{aligned} $$
\((H_{5})\) :

The functions \(g_{i}, i=1,2,\ldots,m:J\times E\rightarrow E\) are continuous, there exist nonnegative functions \(c_{i}\), \(L_{g_{i}}\), \(i=1,2,\ldots,m\), such that

$$\bigl\Vert g_{i}(t,u) \bigr\Vert \leq c_{i}(t)+L_{g_{i}}(t) \Vert u \Vert ^{\nu_{i}},\quad 0< \nu_{i}\leq1, i=1,2,\ldots,m, $$

and

$$\overline{C}=\sup_{t\in[t_{i},s_{i}],i=1,2,\ldots,m}c_{i}(t)< \infty, \qquad \overline{L}=\sup_{t\in[t_{i},s_{i}],i=1,2,\ldots,m}L_{g_{i}}(t)< \infty. $$

Then problem (1.1) has a mild solution on \(\operatorname{PC}(J,E)\).

Proof

The proof includes the following several steps.

Step 1. \(F:\operatorname{PC}(J,E)\rightarrow \operatorname{PC}(J,E)\) is continuous. The proof is similar to Step 1 of Theorem 3.1.

Step 2. For all \(R>0\), we will prove that \(F(B_{R})\) is equicontinuous. Case 1. For interval \([0,t_{1}]\), \(0\leq e_{1}< e_{2}\leq t_{1}\), \(u\in B_{R}\), we get

$$\begin{aligned} & \bigl\Vert (Fu) (e_{2})-(Fu) (e_{1}) \bigr\Vert \\ &\quad \leq \bigl\Vert U_{\beta}(e_{2},0)-U_{\beta}(e_{1},0) \bigr\Vert \bigl\Vert U_{\beta}(T,t_{m})g_{m}\bigl(s_{m},u(s_{m}) \bigr) \bigr\Vert \\ &\qquad{}+ \bigl\Vert U_{\beta}(e_{2},0)-U_{\beta}(e_{1},0) \bigr\Vert \\ &\qquad {}\times \biggl\Vert \int _{s_{m}}^{T}U_{\beta}(T,s) \biggl(f \bigl(s,u(s)\bigr)+ \int_{0}^{s}q(s-\tau)h\bigl(\tau,u(\tau )\bigr)\,d \tau \biggr)\,ds \biggr\Vert \\ &\qquad{}+\sup_{s\in[0,t_{1}]} \bigl\Vert U_{\beta}(e_{2},s)-U_{\beta}(e_{1},s) \bigr\Vert \\ &\qquad {}\times \biggl\Vert \int_{0}^{e_{1}} \biggl(f\bigl(s,u(s)\bigr)+ \int_{0}^{s}q(s-\tau)h\bigl(\tau,u(\tau )\bigr)\,d \tau \biggr)\,ds \biggr\Vert \\ &\qquad{}+ \biggl\Vert \int_{e_{1}}^{e_{2}}U_{\beta}(e_{2},s) \biggl(f\bigl(s,u(s)\bigr)+ \int _{0}^{s}q(s-\tau)h\bigl(\tau,u(\tau)\bigr)\,d \tau \biggr)\,ds \biggr\Vert \\ &\quad \leq M\bigl(\overline{C}+\overline{L}R^{\nu_{m}}\bigr) \bigl\Vert U_{\beta}(e_{2},0)-U_{\beta}(e_{1},0) \bigr\Vert \\ &\qquad{}+M \bigl\Vert U_{\beta}(e_{2},0)-U_{\beta}(e_{1},0) \bigr\Vert \\ &\qquad {}\times \int_{s_{m}}^{T} \biggl(a(s)+L_{f}(s)R^{\gamma}+ \int_{0}^{s}q(s-\tau) \bigl(b(\tau)+L_{h}( \tau)R^{\mu}\bigr)\,d\tau \biggr)\,ds \\ &\qquad{}+\sup_{s\in[0,t_{1}]} \bigl\Vert U_{\beta}(e_{2},s)-U_{\beta}(e_{1},s) \bigr\Vert \\ &\qquad {}\times \int_{0}^{e_{1}} \biggl(a(s)+L_{f}(s)R^{\gamma}+ \int_{0}^{s}q(s-\tau) \bigl(b(\tau )+L_{h}(\tau)R^{\mu}\bigr)\,d\tau \biggr)\,ds \\ &\qquad{}+M \int_{e_{1}}^{e_{2}} \biggl(a(s)+L_{f}(s)R^{\gamma}+ \int_{0}^{s}q(s-\tau ) \bigl(b(\tau)+L_{h}( \tau)R^{\mu}\bigr)\,d\tau \biggr)\,ds. \end{aligned}$$
(4.1)

Case 2. For interval \((t_{i},s_{i}]\), \(i=1,2,\ldots,m\), \(0\leq e_{1}< e_{2}\leq t_{1}\), \(u\in B_{R}\), we have

$$ \begin{aligned}[b] \bigl\Vert (Fu) (e_{2})-(Fu) (e_{1}) \bigr\Vert &\leq \bigl\Vert U_{\beta}(e_{2},t_{i})g_{i}\bigl(e_{2},u(e_{2}) \bigr)-U_{\beta}(e_{1},t_{i})g_{i} \bigl(e_{1},u(e_{1})\bigr) \bigr\Vert \\ &\leq M \bigl\Vert U_{\beta}(e_{2},e_{1})g_{i} \bigl(e_{2},u(e_{2})\bigr)-g_{i} \bigl(e_{1},u(e_{1})\bigr) \bigr\Vert . \end{aligned} $$
(4.2)

Case 3. For interval \((s_{i},t_{i+1}]\), \(i=1,2,\ldots,m\), \(0\leq e_{1}< e_{2}\leq t_{1}\), \(u\in B_{R}\), we have

$$\begin{aligned} &\bigl\Vert (Fu) (e_{2})-(Fu) (e_{1}) \bigr\Vert \\ &\quad \leq \bigl\Vert U_{\beta}(e_{2},t_{i})-U_{\beta}(e_{1},t_{i}) \bigr\Vert \bigl\Vert g_{i}\bigl(s_{i},u(s_{i}) \bigr) \bigr\Vert +\sup_{s\in(s_{i},t_{i+1}]} \bigl\Vert U_{\beta}(e_{2},s)-U_{\beta}(e_{1},s) \bigr\Vert \\ &\qquad {}\times \biggl\Vert \int_{s_{i}}^{e_{1}} \biggl(f\bigl(s,u(s)\bigr)+ \int_{0}^{s}q(s-\tau )h\bigl(\tau,u(\tau)\bigr)\,d \tau \biggr)\,ds \biggr\Vert \\ &\qquad{}+ \biggl\Vert \int_{e_{1}}^{e_{2}}U_{\beta}(e_{2},s) \biggl(f\bigl(s,u(s)\bigr)+ \int _{0}^{s}q(s-\tau)h\bigl(\tau,u(\tau)\bigr)\,d \tau \biggr)\,ds \biggr\Vert \\ &\quad \leq\bigl(\overline{C}+\overline{L}R^{\nu_{i}}\bigr) \bigl\Vert U_{\beta}(e_{2},t_{i})-U_{\beta}(e_{1},t_{i}) \bigr\Vert +\sup_{s\in(s_{i},t_{i+1}]} \bigl\Vert U_{\beta}(e_{2},s)-U_{\beta}(e_{1},s) \bigr\Vert \\ &\qquad {}\times \int_{s_{i}}^{e_{1}} \biggl(a(s)+L_{f}(s)R^{\gamma}+ \int _{0}^{s}q(s-\tau) \bigl(b(\tau)+L_{h}( \tau)R^{\mu}\bigr)\,d\tau \biggr)\,ds \\ &\qquad{}+M \int_{e_{1}}^{e_{2}} \biggl(a(s)+L_{f}(s)R^{\gamma}+ \int_{0}^{s}q(s-\tau ) \bigl(b(\tau)+L_{h}( \tau)R^{\mu}\bigr)\,d\tau \biggr)\,ds. \end{aligned}$$
(4.3)

By Remark 2.1 and the above inequalities (4.1)–(4.3), we derive that \(\|(Fu)(e_{2})-(Fu)(e_{1})\|\rightarrow0\) as \(e_{2}\rightarrow e_{1}\). Therefore, \(F(B_{R})\) is equicontinuous.

Step 3. We will prove that \(F(B_{R})\) is precompact. For all fixed t (\(0< t\leq T\)) and \(0<\epsilon<t\), let \(u\in B_{R}\) and define

$$(F_{\epsilon}u) (t)= \textstyle\begin{cases} U_{\beta}(t,0)[U_{\beta}(T,t_{m})g_{m}(s_{m},u(s_{m}))\\ \quad{}+\int_{s_{m}}^{T}U_{\beta}(T,s) (f(s,u(s))+\int_{0}^{s}q(s-\tau )h(\tau,u(\tau))\,d\tau )\,ds]\\ \quad{}+\int_{0}^{t-\epsilon}U_{\beta}(t,s) (f(s,u(s))+\int _{0}^{s}q(s-\tau)h(\tau,u(\tau))\,d\tau )\,ds,\quad t\in[0,t_{1}],\\ U_{\beta}(t,t_{i})g_{i}(t,u(t)),\quad t\in(t_{i},s_{i}],i=1,2,\ldots,m,\\ U_{\beta}(t,t_{i})g_{i}(s_{i},u(s_{i}))+\int_{s_{i}}^{t-\epsilon}U_{\beta}(t,s) (f(s,u(s))+\int_{0}^{s}q(s-\tau)h(\tau,u(\tau))\,d\tau )\,ds,\\ \quad t\in(s_{i},t_{i+1}], i=1,\ldots,m. \end{cases} $$

By the compactness of \(U_{\beta}(t,s)\) for \(t,s>0\), the set \(Y_{\epsilon}(t)=\{(F_{\epsilon}u)(t): u\in B_{R}\}\) is relatively compact in E for any ϵ (\(0<\epsilon<t\)). For any \(u\in B_{R}\), from the following inequalities

$$\bigl\Vert (Fu) (t)-(F_{\epsilon}u) (t) \bigr\Vert \leq \textstyle\begin{cases} M\int_{t-\epsilon}^{t} (a(s)+L_{f}(s)R^{\gamma}+\int_{0}^{s}q(s-\tau)(b(\tau )+L_{h}(\tau)R^{\mu})\,d\tau )\,ds,\\ \quad t\in[0,t_{1}],\\ 0,\quad t\in(t_{i},s_{i}],i=1,2,\ldots,m,\\ M\int_{t-\epsilon}^{t} (a(s)+L_{f}(s)R^{\gamma}+\int_{0}^{s}q(s-\tau)(b(\tau )+L_{h}(\tau)R^{\mu})\,d\tau )\,ds,\\ \quad t\in(s_{i},t_{i+1}], i=1,\ldots,m, \end{cases} $$

we get \(Y(t)=\{(Fu)(t): u\in B_{R}\}\) is totally bounded. Thus, \(Y(t)\) is relatively compact in E. By using the Arzelá–Ascoli theorem, we have that \(F:\operatorname{PC}(J,E)\rightarrow \operatorname{PC}(J,E)\) is completely continuous.

For \(0<\lambda<1\), let \(u=\lambda(Fu)\), we have

$$u(t)= \textstyle\begin{cases} \lambda U_{\beta}(t,0)[U_{\beta}(T,t_{m})g_{m}(s_{m},u(s_{m}))\\ \quad{}+\lambda\int_{s_{m}}^{T}U_{\beta}(T,s) (f(s,u(s))+\int _{0}^{s}q(s-\tau)h(\tau,u(\tau))\,d\tau )\,ds]\\ \quad{}+\lambda\int_{0}^{t}U_{\beta}(t,s) (f(s,u(s))+\int_{0}^{s}q(s-\tau )h(\tau,u(\tau))\,d\tau )\,ds,\quad t\in[0,t_{1}],\\ \lambda U_{\beta}(t,t_{i})g_{i}(t,u(t)),\quad t\in(t_{i},s_{i}],i=1,2,\ldots,m,\\ \lambda U_{\beta}(t,t_{i})g_{i}(s_{i},u(s_{i}))+\lambda\int_{s_{i}}^{t}U_{\beta}(t,s) (f(s,u(s))+\int_{0}^{s}q(s-\tau)h(\tau,u(\tau))\,d\tau )\,ds,\\ \quad t\in(s_{i},t_{i+1}], i=1,\ldots,m, \end{cases} $$

and then

$$\bigl\Vert u(t) \bigr\Vert \leq \textstyle\begin{cases} M^{2}(\overline{C}+\overline{L}R^{\nu_{m}})+M^{2}\int_{s_{m}}^{T} (a(s)+L_{f}(s)R^{\gamma}+\int_{0}^{s}q(s-\tau)(b(\tau)+L_{h}(\tau)R^{\mu})\,d\tau )\,ds\\ \quad{}+M\int_{0}^{t} (a(s)+L_{f}(s)R^{\gamma}+\int_{0}^{s}q(s-\tau)(b(\tau )+L_{h}(\tau)R^{\mu})\,d\tau )\,ds,\quad t\in[0,t_{1}],\\ M(\overline{C}+\overline{L}R^{\nu_{i}}),\quad t\in (t_{i},s_{i}],i=1,2,\ldots,m,\\ M(\overline{C}+\overline{L}R^{\nu_{i}})+M\int_{s_{i}}^{t} (a(s)+L_{f}(s)R^{\gamma}+\int_{0}^{s}q(s-\tau)(b(\tau)+L_{h}(\tau)R^{\mu})\,d\tau )\,ds,\\ \quad t\in(s_{i},t_{i+1}], i=1,\ldots,m. \end{cases} $$

Let

$$\begin{aligned} \rho&=\max \biggl\{ M^{2}\bigl(\overline{C}+ \overline{L}R^{\nu _{m}}\bigr)\\ &\quad {}+M^{2} \int_{s_{m}}^{T} \biggl(a(s)+L_{f}(s)R^{\gamma}+ \int_{0}^{s}q(s-\tau) \bigl(b(\tau )+L_{h}(\tau)R^{\mu}\bigr)\,d\tau \biggr)\,ds \\ &\quad{}+M \int_{0}^{t} \biggl(a(s)+L_{f}(s)R^{\gamma}+ \int_{0}^{s}q(s-\tau) \bigl(b(\tau )+L_{h}(\tau)R^{\mu}\bigr)\,d\tau \biggr)\,ds,\\ &\quad {} M\bigl( \overline{C}+\overline{L}R^{\nu_{i}}\bigr),\\ &\quad {} M\bigl(\overline{C}+ \overline{L}R^{\nu_{i}}\bigr)\\ &\quad {} +M \int_{s_{i}}^{t} \biggl(a(s)+L_{f}(s)R^{\gamma}+ \int_{0}^{s}q(s-\tau ) \bigl(b(\tau)+L_{h}( \tau)R^{\mu}\bigr)\,d\tau \biggr)\,ds,\\ &\quad {} t\in(s_{i},t_{i+1}], i=1,\ldots,m \biggr\} . \end{aligned} $$

Then there exists a constant \(M_{*}>\rho\) such that \(\|u\|_{\mathrm{PC}}\neq M_{*} \). Let \(V=\{u\in \operatorname{PC}(J,E):\|u\|_{\mathrm{PC}}< M_{*}\}\). Obviously there is no \(u\in \partial V\) such that \(u=\lambda(Fu)\) for \(\lambda\in(0,1)\). It thus follows from the nonlinear alternative for single-valued maps that F has a fixed point u in V̅, which is a mild solution of problem (1.1). □

5 An example

We give an example to illustrate our main results in this paper.

$$ \textstyle\begin{cases} {}^{c}D_{t}^{\alpha}u(x,t)=t\frac{\partial^{2}}{\partial x^{2}}u(x,t)+\frac {t}{1+t^{2}}\sin u(x,t)+\int_{0}^{t}e^{t-s}\frac{e^{s}}{ \Vert u(x,s) \Vert }\,ds,\\ \quad t\in[0,1)\cup(2,3],\\ \frac{\partial}{\partial x}u(0,t)=\frac{\partial}{\partial x}u(1,t)=0,\quad t\in[0,1)\cup(2,3],\\ u(x,t)=U_{\beta}(t,1)\frac{1}{3}u(x,t),\quad x\in[0,\pi], t\in(1,2], x\in(0,1),\\ u(0,x)=u(3,x)=0, \quad x\in(0,1) , \end{cases} $$
(5.1)

where \(E=L^{2}[0,3]\), \(J=[0,3]\), \(0=t_{0}=s_{0}\), \(t_{1}=1\), \(s_{1}=2\), \(T=3\), \(0<\alpha\leq1\), the operator \(A:D(A)=\{u\in E:u''\in E, u(0)=u(1)=0\}\subset E\rightarrow E\) defined by \(A(t)(z)=t\frac{\partial^{2}u}{\partial x^{2}}\). Then \(A(t)\) generates a β-resolvent family \(U_{\beta}(t,s)\) on E.

Denote

$$\begin{gathered} u(t)=u(\cdot, t), \qquad f\bigl(t,u(t)\bigr)=\frac{t}{1+t^{2}}\sin u(\cdot,t), \qquad h\bigl(t,u(t)\bigr)=\frac{e^{s}}{ \Vert u(\cdot,t) \Vert }, \\ q(t-s)=e^{t-s}, \qquad g_{1}\bigl(t,u(t)\bigr)=\frac{1}{3}u(\cdot,t). \end{gathered} $$

Therefore, Eq. (5.1) takes the following abstract form:

$$ \textstyle\begin{cases} {}^{c}D_{t}^{\beta}u(t)=A(t)u(t)+f(t,u(t))+\int _{0}^{t}q(t-s)h(s,u(s))\,ds,\\ \quad t\in(s_{i},t_{i+1}], i=0,1,\ldots,m,\\ u(t)=U_{\beta}(t,t_{i})g_{i}(t,u(t)),\quad t\in(t_{i},s_{i}],i=1,2,\ldots,m,\\ u(0)=u(T). \end{cases} $$
(5.2)

If we assume that problem (5.2) satisfies the conditions of Theorems 3.1 and 4.1, then by Theorems 3.1 and 4.1, we know that problem (5.2) has a mild solution, which means that problem (5.1) has a mild solution.

6 Conclusion

This paper investigates the existence of periodic boundary value problems for fractional semilinear integro-differential equations with non-instantaneous impulses by the measure of noncompactness, the theory of the resolvent family, and the fixed point theorem. The main results presented in this paper improve and generalize many results in [18–20, 24–26].