On the periodic boundary value problems for fractional nonautonomous differential equations with non-instantaneous impulses

Abstract

In this paper, we investigate periodic boundary value problems for Caputo type fractional semilinear nonautonomous differential equations with non-instantaneous impulses. By using semigroup theory combined with the measure of noncompactness and some fixed point theorems, the existence of PC-mild solutions for the equations is established. At the end, an example is presented to illustrate the application of our main results.

1 Introduction

Fractional differential equations have gained considerable significance during the past decades. Compared with integer order differential equations, fractional differential equations have memory in time and genetic properties, which are more suitable for describing many problems in anomalous diffusion, viscous fluid mechanics, porous media mechanics, electrical engineering and bioengineering, etc. In [1–6], the authors are committed to fractional differential equations with instantaneous impulsive effects, which can describe sudden changes at certain times such as earthquake, the closing of the switch in the circuit, and so on. Meanwhile, fractional differential equations with non-instantaneous impulses have currently been proven to be useful mathematical models to explain many phenomena occurring in biology, dynamics, control model, etc. For instance, the release and absorption of drugs in the bloodstream is a continuous and gradual process. As recent developments on fractional differential equations with non-instantaneous impulses, we mention the papers [7–16] and the references cited therein.

Cauchy problems for the abstract integer differential equations with non-instantaneous impulses were initially investigated by E. Hernandez and D. O’Regan [7], Pierri et al. [8] as follows:

$$ \textstyle\begin{cases} u'(t)=\operatorname{Au}(t)+f(t,u(t)),& t\in (s_{i},t_{i+1}], i=0,1,2,\ldots,m, \\ u(t)=g_{i}(t,u(t)),& t\in (t_{i},s_{i}], i=1,2,\ldots,m, \\ u(0)=u_{0}\in E, \end{cases} $$

(1.1)

where $A:D(A)\subset E\rightarrow E$ is the generator of a $C_{0}$-semigroup $\{S(t),t\geq 0\}$ on a Banach space E, the prefixed numbers $s_{i}$, $t_{i}$ satisfy $0=s_{0}< t_{1}\leq s_{1}< t_{2}\leq s_{2}<\cdots<t_{m}\leq s_{m}<t_{m+1}=T$, $f:[0,T]\times E\rightarrow E$ and $g_{i}:(t_{i},s_{i}]\times E\rightarrow E$, $i=1,2,\ldots,m$, are continuous functions, the existence of PC-mild solutions has been proved by a fixed point theorem.

Wang and Li [9] studied periodic boundary value problems for differential equations with non-instantaneous impulses via the fixed point theorem:

$$ \textstyle\begin{cases} u'(t)=f(t,u(t)),& t\in (s_{i},t_{i+1}], i=0,1,2,\ldots,m, \\ u(t)=g_{i}(t,u(t)),& t\in (t_{i},s_{i}], i=1,2,\ldots,m, \\ u(0)=u(T). \end{cases} $$

(1.2)

In [10–13], the authors studied the existence of solutions for non-instantaneous impulsive differential equations. Chen et al. [14] studied non-autonomous parabolic evolution equations with non-instantaneous impulses and obtained the existence results of mild solutions. Yu and Wang [15] investigated periodic boundary value problems for integer differential equations with non-instantaneous memory impulses; the existence of PC-mild solutions was established based on the theory of semigroup.

Inspired by these contributions, we consider the following periodic boundary value problems for fractional semilinear nonautonomous differential equations with non-instantaneous impulses:

$$ \textstyle\begin{cases} ^{c}D_{t}^{\beta }x(t)=A(t)x(t)+f (t,x(t),\int _{0}^{t}g(t,s,x(s))\,ds ),& t\in (s_{i},t_{i+1}], i=0,1,2,\ldots ,m, \\ x(t)=h_{i}+U_{\beta }(t,t_{i})\int _{t_{i}}^{t}g_{i}(s,x(s))\,ds,& t \in (t_{i},s_{i}], i=1,2,\ldots ,m, \\ x(0)=x(T), \end{cases} $$

(1.3)

where $\ ^{c}D_{t}^{\beta }$ is the Caputo’s fractional derivative of order β, $\beta \in (0,1]$, $J=[0,T]$, $A(t)$ is a closed linear operator with domain $D(A)$ defined on a Banach space E, f, g, and $U_{\beta }$ are to be specified later, the prefixed numbers $s_{i}$ and $t_{i}$ $(i=1,2,\ldots,m)$ satisfy $0=s_{0}< t_{1}\leq s_{1}< t_{2}\leq \cdots < t_{m}\leq s_{m}< t_{m+1}=T$, $g_{i}:(t_{i},s_{i}]\times E\rightarrow E$, $i=1,2,\ldots ,m$, are continuous and nonlinear functions, $h_{i}\in E$, $i=1,2,\ldots ,m$.

The rest of this paper is organized as follows. In Sect. 2, some basic definitions and auxiliary lemmas that will be needed in the remaining sections are collected. The existence of PC-mild solutions is shown in Sect. 3 based on the theory of resolvent operators, measure of noncompactness and various fixed point theorems. An example is presented to illustrate the main theorems in Sect. 4. Finally, Sect. 5 contains the summary of our results.

2 Auxiliary results

Let $(E,\|\cdot \|)$ be a Banach space, $J=[0,T]$ and $0< T<+\infty$. $C(J,E)$ is the collection of all continuous functions from J into E equipped with the norm $\|x\|_{C}=\max \{\|x(t)\|, t\in J\}$. Let $\operatorname{PC}(J,E)=\{x|x: J\rightarrow E:x\in C((t_{k},t_{k+1}],E),\text{ and there exist } x(t_{k}^{-})\text{ and }x(t_{k}^{+})\text{ with }x(t_{k})=x(t_{k}^{-}), k=1,\ldots ,m\}$ endowed with the PC-norm $\|x\|_{\operatorname{PC}}=\sup \{\|x(t)\|,t\in J\}$.

Definition 2.1

([17, 18])

The Caputo fractional derivative of order β of a function $f:(0,\infty )\rightarrow \mathbb{R}$ is defined as

$$ ^{c}D_{t}^{\beta }f(t)=\frac{1}{\Gamma (n-\beta )} \int _{0}^{t}(t-s)^{n- \beta -1}f^{(n)}(s)\,ds,$$

where $\ n-1<\beta <n$, $n\in N$, $\Gamma (\cdot )$ denotes the gamma function. The Laplace transform of the Caputo fractional derivative of order β is given as

$$ \mathcal{L} \bigl(^{c}D_{t}^{\beta }f(t) \bigr) (s)=s^{\beta }( \mathcal{L}f) (s)-\sum_{j=1}^{n-1}s^{\beta -j-1}x^{(j)}(0), \quad n-1< \beta \leq n,$$

where $(\mathcal{L}f)(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt$ is the Laplace transform of the function $f(t)$.

Definition 2.2

([19, 20])

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

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

In this case, $U_{\beta }(t,s)$ is called the β-resolvent family generated by $A(t)$, denote $M=\max_{0\leq s< t\leq T}\|U_{\beta }(t,s)\|$.

Lemma 2.1

([20, 21])

$U_{\beta }(t,s)$ satisfies the following properties:

• (i) $U_{\beta }(s,s)=I$, $U_{\beta }(t,s)=U_{\beta }(t,r)U_{\beta }(r,s)$ for $0\leq s\leq r\leq t\leq a$;

• (ii) $(t,s)\rightarrow U_{\beta }(t,s)$ is strongly continuous for $0\leq s\leq t\leq a$;

• (iii) If $U_{\beta }(t,s)$ is compact for $t,s>0$, then $U_{\beta }(t,s)$ is continuous in the uniform operator topology.

Definition 2.3

A function $x\in \operatorname{PC}(J,E)$ is said to be a PC-mild solution of problem (1.3) if $x(t)$ satisfies the integral equation

$$ x(t)= \textstyle\begin{cases} U_{\beta }(t,0)[U_{\beta }(T,s_{m})h_{m}+U_{\beta }(T,t_{m})\int _{t_{m}}^{s_{m}}g_{m}(s,x(s))\,ds \\ \qquad {}+\int _{s_{m}}^{T}U_{\beta }(T,s)f (s,x(s),\int _{0}^{s}g(s, \sigma ,x(\sigma ))\,d\sigma )\,ds] \\ \qquad {}+\int _{0}^{t}U_{\beta }(t,s)f (s,x(s),\int _{0}^{s}g(s,\sigma ,x( \sigma ))\,d\sigma )\,ds,\quad t\in [0,t_{1}], \\ h_{i}+U_{\beta }(t,t_{i})\int _{t_{i}}^{t}g_{i}(s,x(s))\,ds,\quad t\in (t_{i},s_{i}], i=1,2,\ldots ,m, \\ U_{\beta }(t,s_{i})h_{i}+U_{\beta }(t,t_{i})\int _{t_{i}}^{s_{i}}g_{i}(s,x(s))\,ds \\ \qquad {}+\int _{s_{i}}^{t}U_{\beta }(t,s)f (s,x(s),\int _{0}^{s}g(s, \sigma ,x(\sigma ))\,d\sigma )\,ds,\\ \quad t\in (s_{i},t_{i+1}], i=1,2, \ldots ,m. \end{cases} $$

(2.1)

Lemma 2.2

([22])

Let $B\subset C(J,E)$ be equicontinuous and bounded, then $\overline{Co}B\subset C(J,E)$ is also equicontinuous and bounded.

Lemma 2.3

([22])

Let E be a Banach space and $D\subset E$ be bounded, then there exists a countable set $D_{0}\subset D$ such that $\alpha (D)\leq 2\alpha (D_{0})$, where α denotes the measure of noncompactness.

Lemma 2.4

([23])

Let $B\subset C(J,E)$ be equicontinuous and bounded, then $\alpha (B(t))$ is continuous on J and

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

3 Main results

First, we demonstrate the existence of PC-mild solutions for problem (1.3) based on the measure of noncompactness and fixed point theorem.

Theorem 3.1

If the following assumptions $(H_{1})$–$(H_{3})$ are satisfied.

$(H_{1})$:

The function $g:D\times E\rightarrow E$ is continuous, $D=\{(t,s)|0\leq s\leq t\leq T\}$, there exists $h(t,\cdot )\in L^{1}(J,\mathbb{R}_{+})$ with $h_{0}=\max_{t\in [0,T]}\int _{0}^{t}h(t,s)\,ds$ for $(t,s)\in D$, $x\in E$ such that

$$ \bigl\Vert g(t,s,x) \bigr\Vert \leq h(t,s) \Vert x \Vert .$$

$(H_{2})$:

The function $f:J\times T_{R}\times T_{R}\rightarrow E$ is bounded and continuous for every $R>0$ such that

$$ \lim_{R\rightarrow \infty }\sup \frac{M(R)}{R}< \frac{1}{\Delta },$$

where $M(R)=\max \{M_{1}(R),M_{2}(R)\}$, $M_{1}(R)=\sup \{\|f(t,x_{1},x_{2})\|:(t,x_{1},x_{2})\in J\times T_{R} \times T_{R}\}$, $M_{2}(R)=\sup \{\|g_{i}(t,x)\|,(t,x)\in J\times T_{R},i=1,2,\ldots ,m \}$, $T_{R}=\{x\in E:\|x\|\leq R\}$, $\Delta =\max \{M^{2}a_{0}(T-t_{m})+Mt_{1}a_{0},Ma_{0}(t_{i+1}-t_{i}),i=1,2, \ldots ,m\}$, $a_{0}=\max \{1,h_{0}\}$.

$(H_{3})$:

For all $R>0$, there exist nonnegative Lebesgue integrable functions $L'_{g},L'_{g_{i}},L'_{1},L'_{2}\in L^{1}(J,\mathbb{R}_{+})$ $(i=1,2,\ldots ,m)$ for all countable and equicontinuous sets $D,D_{i}\subset T_{R}$ $(i=1,2)$ such that

$$\begin{aligned}& \alpha \bigl(g(t,s,D)\bigr)\leq L'_{g}(t)\alpha (D),\\& \alpha \bigl(g_{i}(t,D)\bigr)\leq L'_{g_{i}}(t) \alpha (D), \end{aligned}$$

and

$$ \alpha \bigl(f(t,D_{1},D_{2})\bigr)\leq L'_{1}(t)\alpha (D_{1})+L'_{2}(t) \alpha (D_{2}).$$

Then problem (1.3) has at least one PC-mild solution on $\operatorname{PC}(J,E)$ provided that the resolvent operator $U_{\beta }(t,s)$ is compact for $t,s>0$ and $\rho =\max \{2M^{2}\int _{t_{m}}^{s_{m}}L'_{g_{m}}(s)\,ds+2M^{2}\int _{s_{m}}^{T}(L'_{1}(s)+L'_{2}(s) \int _{0}^{T}L'_{g}(\sigma )\,d\sigma )\,ds +2M\int _{0}^{t_{1}}(L'_{1}(s)+L'_{2}(s) \int _{0}^{T}L'_{g}(\sigma )\,d\sigma )\,ds, 2M\int _{t_{i}}^{s_{i}}L'_{g_{i}}(s)\,ds+2M\times \int _{s_{i}}^{t_{i+1}} (L'_{1}(s)+L'_{2}(s) \int _{0}^{T}L'_{g}( \sigma )\,d\sigma )\,ds,i=1,2,\ldots ,m\}<1$.

Proof

Consider an operator $\mathcal{F}:\operatorname{PC}(J,E)\rightarrow \operatorname{PC}(J,E)$ defined by

$$ (\mathcal{F}x) (t)= \textstyle\begin{cases} U_{\beta }(t,0) [U_{\beta }(T,s_{m})h_{m}+U_{\beta }(T,t_{m})\int _{t_{m}}^{s_{m}}g_{m}(s,x(s))\,ds \\ \qquad {}+\int _{s_{m}}^{T}U_{\beta }(T,s)f (s,x(s),\int _{0}^{s}g(s, \sigma ,x(\sigma ))\,d\sigma )\,ds] \\ \qquad {}+\int _{0}^{t}U_{\beta }(t,s)f (s,x(s),\int _{0}^{s}g(s,\sigma ,x( \sigma ))\,d\sigma )\,ds, \quad t\in [0,t_{1}], \\ h_{i}+U_{\beta }(t,t_{i})\int _{t_{i}}^{t}g_{i}(s,x(s))\,ds,\quad t\in (t_{i},s_{i}], i=1,2,\ldots ,m, \\ U_{\beta }(t,s_{i})h_{i}+U_{\beta }(t,t_{i})\int _{t_{i}}^{s_{i}}g_{i}(s,x(s))\,ds \\ \qquad {}+\int _{s_{i}}^{t}U_{\beta }(t,s)f (s,x(s),\int _{0}^{s}g(s, \sigma ,x(\sigma ))\,d\sigma )\,ds,\\ \quad t\in (s_{i},t_{i+1}], i=1,2, \ldots ,m. \end{cases} $$

It is easy to see that the operator $\mathcal{F}$ is well defined in $\operatorname{PC}(J,E)$.

According to condition $(H_{2})$, there exist $0< r<\frac{1}{\Delta }$ and $R_{0}>0$ for every $R\ge a_{0}R_{0}$ such that

$$ M(R)< rR.$$

Let $\eta =\max \{R_{0}, \frac{M^{2}\|h_{m}\|}{1-M^{2}ra_{0}(T-t_{m})-Mra_{0}t_{1}}, \frac{\|h_{i}\|}{1-Mra_{0}(s_{i}-t_{i})}, \frac{M\|h_{i}\|}{1-Mra_{0}(t_{i+1}-t_{i})},i=1,2,\ldots ,m\}$. For all $x\in B_{\eta }=\{x\in \operatorname{PC}(J,E):\|x\|_{\operatorname{PC}}\leq \eta \}$, $t\in (s_{i},t_{i+1}]$, $i=0,1,\ldots ,m$, then

$$ \Vert x \Vert _{\operatorname{PC}}\leq \eta \leq a_{0}\eta ,$$

which yields

$$ \biggl\Vert \int _{0}^{t}g(t,s,x)\,ds \biggr\Vert \leq \int _{0}^{t}h(t,s) \Vert x \Vert _{\operatorname{PC}}\,ds \leq h_{0}\eta \leq a_{0}\eta .$$

First of all, we show that $\mathcal{F}x\in B_{\eta }$.

For $t\in [0,t_{1}]$,

$$ \begin{aligned} \bigl\Vert (\mathcal{F}x) (t) \bigr\Vert \leq {}& M^{2} \Vert h_{m} \Vert +M^{2} \int _{t_{m}}^{s_{m}} \bigl\Vert g_{m} \bigl(s,x(s)\bigr) \bigr\Vert \,ds\\ &{}+M^{2} \int _{s_{m}}^{T} \biggl\Vert f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x(\sigma )\bigr)\,d\sigma \biggr) \biggr\Vert \,ds \\ &{}+M \int _{0}^{t} \biggl\Vert f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x( \sigma ) \bigr)\,d\sigma \biggr) \biggr\Vert \,ds \\ \leq{} & M^{2} \Vert h_{m} \Vert +M^{2}ra_{0}\eta (s_{m}-t_{m})+M^{2}ra_{0} \eta (T-s_{m})+Mra_{0} \eta t_{1} \\ \leq{} & M^{2} \Vert h_{m} \Vert + \bigl(M^{2}ra_{0}(T-t_{m})+Mra_{0}t_{1} \bigr)\eta \leq \eta . \end{aligned} $$

For $t\in (t_{i},s_{i}]$, $i=1,2,\ldots,m$,

$$ \bigl\Vert (\mathcal{F}x) (t) \bigr\Vert \leq \Vert h_{i} \Vert +M \int _{t_{i}}^{t} \bigl\Vert g_{i} \bigl(s,x(s)\bigr) \bigr\Vert \,ds\leq \Vert h_{i} \Vert +Mra_{0}\eta (s_{i}-t_{i})\leq \eta .$$

For $t\in (s_{i},t_{i+1}]$, $i=1,2,\ldots,m$,

$$\begin{aligned} \bigl\Vert (\mathcal{F}x) (t) \bigr\Vert \leq{} & M \Vert h_{i} \Vert +M \int _{t_{i}}^{s_{i}} \bigl\Vert g_{i} \bigl(s,x(s)\bigr) \bigr\Vert \,ds\\ &{}+M \int _{s_{i}}^{t} \biggl\Vert f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr) \biggr\Vert \,ds \\ \leq{} & M \Vert h_{i} \Vert +Mra_{0}\eta (s_{i}-t_{i})+Mra_{0}\eta (t_{i+1}-s_{i}) \\ \leq {}& M \Vert h_{i} \Vert +Mra_{0}\eta (t_{i+1}-t_{i})\leq \eta . \end{aligned}$$

So $\mathcal{F} :B_{\eta }\rightarrow B_{\eta }$.

Furthermore, we prove that $\mathcal{F} :B_{\eta }\rightarrow B_{\eta }$ is continuous. Let $\{x_{n}\}_{0}^{\infty }$ with $x_{n}\rightarrow x$ in $B_{\eta }$.

For each $t\in [0,t_{1}]$, we obtain

$$\begin{aligned}& \bigl\Vert (\mathcal{F}x_{n}) (t)-(\mathcal{F}x) (t) \bigr\Vert \\& \quad \leq M^{2} \int _{t_{m}}^{s_{m}} \bigl\Vert g_{m} \bigl(s,x_{n}(s)\bigr)-g_{m}\bigl(s,x(s)\bigr) \bigr\Vert \,ds \\& \qquad {}+M^{2} \int _{s_{m}}^{T} \biggl\Vert f \biggl(s,x_{n}(s), \int _{0}^{s}g\bigl(s, \sigma ,x_{n}(\sigma )\bigr)\,d\sigma \biggr) -f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr) \biggr\Vert \,ds \\& \qquad {}+M \int _{0}^{t_{1}} \biggl\Vert f \biggl(s,x_{n}(s), \int _{0}^{s}g\bigl(s, \sigma ,x_{n}(\sigma )\bigr)\,d\sigma \biggr) -f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr) \biggr\Vert \,ds \\& \quad \leq M^{2}(s_{m}-t_{m})\sup _{s\in J} \bigl\Vert g_{m}\bigl(s,x_{n}(s) \bigr)-g_{m}\bigl(s,x(s)\bigr) \bigr\Vert \\& \qquad {}+M^{2}(T-s_{m})\sup _{s\in J} \biggl\Vert f \biggl(s,x_{n}(s), \int _{0}^{s}g\bigl(s,\sigma ,x_{n}(\sigma )\bigr)\,d\sigma \biggr)\\& \qquad {}-f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x(\sigma )\bigr)\,d\sigma \biggr) \biggr\Vert \\& \qquad {}+Mt_{1}\sup _{s\in J} \biggl\Vert f \biggl(s,x_{n}(s), \int _{0}^{s}g\bigl(s, \sigma ,x_{n}(\sigma )\bigr)\,d\sigma \biggr)-f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr) \biggr\Vert . \end{aligned}$$

For each $t\in (t_{i},s_{i}]$, $i=1,2,\ldots ,m$, we obtain

$$ \begin{aligned} \bigl\Vert (\mathcal{F}x_{n}) (t)-( \mathcal{F}x) (t) \bigr\Vert \leq{} & M \int _{t_{i}}^{s_{i}} \bigl\Vert g_{i} \bigl(s,x_{n}(s)\bigr)-g_{i}\bigl(s,x(s)\bigr) \bigr\Vert \,ds \\ \leq {}& M(s_{i}-t_{i})\sup _{s\in J} \bigl\Vert g_{i}\bigl(s,x_{n}(s)\bigr)-g_{i} \bigl(s,x(s)\bigr) \bigr\Vert . \end{aligned} $$

For each $t\in (s_{i},t_{i+1}]$, $i=1,2,\ldots ,m$, we obtain

$$\begin{aligned}& \bigl\Vert (\mathcal{F}x_{n}) (t)-(\mathcal{F}x) (t) \bigr\Vert \\& \quad \leq M \int _{t_{i}}^{s_{i}} \bigl\Vert g_{i} \bigl(s,x_{n}(s)\bigr)-g_{i}\bigl(s,x(s)\bigr) \bigr\Vert \,ds \\& \qquad {}+M \int _{s_{i}}^{t_{i+1}} \biggl\Vert f \biggl(s,x_{n}(s), \int _{0}^{s}g\bigl(s, \sigma ,x_{n}(\sigma )\bigr)\,d\sigma \biggr)-f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr) \biggr\Vert \,ds \\& \quad \leq M(s_{i}-t_{i})\sup _{s\in J} \bigl\Vert g_{i}\bigl(s,x_{n}(s)\bigr)-g_{i} \bigl(s,x(s)\bigr) \bigr\Vert \\& \qquad {}+ M(t_{i+1}-s_{i})\sup _{s\in J} \biggl\Vert f \biggl(s,x_{n}(s), \int _{0}^{s}g\bigl(s,\sigma ,x_{n}(\sigma )\bigr)\,d\sigma \biggr)\\& \qquad {}-f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x(\sigma )\bigr)\,d\sigma \biggr) \biggr\Vert . \end{aligned}$$

Using the fact that the functions $f:J\times E\times E\rightarrow E$, $g:D\times E\rightarrow E$ and $g_{i}:(t_{i},s_{i}]\times E\rightarrow E$ $(i=1,2,\ldots,m)$ are continuous, we have

$$ \lim_{n\rightarrow \infty }\sup_{s\in J} \biggl\Vert f \biggl(s,x_{n}(s), \int _{0}^{s}g\bigl(s,\sigma ,x_{n}(\sigma )\bigr)\,d\sigma \biggr)-f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x(\sigma )\bigr)\,d\sigma \biggr) \biggr\Vert =0$$

and

$$ \lim_{n\rightarrow \infty }\sup_{t\in J} \bigl\Vert g_{i}\bigl(s,x_{n}(s)\bigr)-g_{i} \bigl(s,x(s)\bigr) \bigr\Vert =0\ (i=1,2,\ldots,m).$$

From the above, we deduce that $\|\mathcal{F}x_{n}-\mathcal{F}x\|_{\operatorname{PC}}\rightarrow 0$ as $n\rightarrow \infty$. This shows that $\mathcal{F} :B_{\eta }\rightarrow B_{\eta }$ is continuous.

Now we prove that $\mathcal{F}(B_{\eta })$ is equicontinuous.

For the interval $[0,t_{1}]$, $0\leq e_{1}< e_{2}\leq t_{1}$, $x\in B_{\eta }$, we get

$$\begin{aligned}& \bigl\Vert (\mathcal{F}x) (e_{2})-(\mathcal{F}x) (e_{1}) \bigr\Vert \\& \quad \leq \bigl\Vert U_{\beta }(e_{2},0)-U_{\beta }(e_{1},0) \bigr\Vert \biggl(M \Vert h_{m} \Vert +M \biggl\Vert \int _{t_{m}}^{s_{m}}g_{m}\bigl(s,x(s) \bigr)\,ds \biggr\Vert \\& \qquad {} +M \biggl\Vert \int _{s_{m}}^{T}f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr)\,ds \biggr\Vert \biggr) \\& \qquad {}+ \biggl\Vert \int _{e_{1}}^{e_{2}}U_{\beta }(e_{2},s)f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr)\,ds \biggr\Vert \\& \qquad {}+ \biggl\Vert \int _{0}^{e_{1}}\bigl(U_{\beta }(e_{2},s)-U_{\beta }(e_{1},s) \bigr)f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x(\sigma )\bigr)\,d\sigma \biggr)\,ds \biggr\Vert \\& \quad \leq \bigl\Vert U_{\beta }(e_{2},0)-U_{\beta }(e_{1},0) \bigr\Vert M \biggl( \Vert h_{m} \Vert + \biggl\Vert \int _{t_{m}}^{s_{m}}g_{m}\bigl(s,x(s) \bigr)\,ds \biggr\Vert \\& \qquad {}+ \biggl\Vert \int _{s_{m}}^{T}f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr)\,ds \biggr\Vert \biggr) \\& \qquad {}+Mra_{0}\eta (e_{2}-e_{1})+\sup _{s\in [0,t_{1}]} \bigl\Vert U_{\beta }(e_{2},s)-U_{\beta }(e_{1},s) \bigr\Vert ra_{0}\eta t_{1}. \end{aligned}$$

For the interval $(t_{i},s_{i}]$, $i=1,2,\ldots ,m$, $t_{i}< e_{1}<e_{2}\leq s_{i}$, $x\in B_{\eta }$, we get

$$ \begin{aligned} \bigl\Vert (\mathcal{F}x) (e_{2})-( \mathcal{F}x) (e_{1}) \bigr\Vert \leq {}& \biggl\Vert U_{\beta }(e_{2},t_{i}) \int _{e_{1}}^{e_{2}}g_{i}\bigl(s,x(s) \bigr)\,ds \biggr\Vert \\ &{}+ \biggl\Vert \bigl(U_{\beta }(e_{2},t_{i})-U_{\beta }(e_{1},t_{i}) \bigr) \int _{t_{i}}^{e_{1}}g_{i}\bigl(s,x(s) \bigr)\,ds \biggr\Vert \\ \leq {}&Mra_{0}\eta (e_{2}-e_{1})+ \bigl\Vert U_{\beta }(e_{2},t_{i})-U_{\beta }(e_{1},t_{i}) \bigr\Vert ra_{0}\eta (s_{i}-t_{i}). \end{aligned} $$

For interval $(s_{i},t_{i+1}]$, $i=1,2,\ldots ,m$, $s_{i}< e_{1}<e_{2}\leq t_{i+1}$, $x\in B_{\eta }$, we get

$$\begin{aligned}& \bigl\Vert (\mathcal{F}x) (e_{2})-( \mathcal{F}x) (e_{1}) \bigr\Vert \\& \quad \leq \bigl\Vert U_{\beta }(e_{2},s_{i})-U_{\beta }(e_{1},s_{i}) \bigr\Vert \Vert h_{i} \Vert + \bigl\Vert U_{\beta }(e_{2},t_{i})-U_{\beta }(e_{1},t_{i}) \bigr\Vert \biggl\Vert \int _{t_{i}}^{s_{i}}g_{i}\bigl(s,x(s) \bigr)\,ds \biggr\Vert \\& \qquad {}+ \biggl\Vert \int _{e_{1}}^{e_{2}}U_{\beta }(e_{2},s)f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr)\,ds \biggr\Vert \\& \qquad {}+ \biggl\Vert \int _{s_{i}}^{e_{1}}\bigl(U_{\beta }(e_{2},s)-U_{\beta }(e_{1},s) \bigr)f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x(\sigma )\bigr)\,d\sigma \biggr)\,ds \biggr\Vert \\& \quad \leq \bigl\Vert U_{\beta }(e_{2},s_{i})-U_{\beta }(e_{1},s_{i}) \bigr\Vert \Vert h_{i} \Vert + \bigl\Vert U_{\beta }(e_{2},t_{i})-U_{\beta }(e_{1},t_{i}) \bigr\Vert \biggl\Vert \int _{t_{i}}^{s_{i}}g_{i}\bigl(s,x(s) \bigr)\,ds \biggr\Vert \\& \qquad {}+Mra_{0}\eta (e_{2}-e_{1})+\sup _{s\in (s_{i},t_{i+1}]} \bigl\Vert U_{\beta }(e_{2},s)-U_{\beta }(e_{1},s) \bigr\Vert ra_{0}\eta (t_{i+1}-s_{i}). \end{aligned}$$

We deduce that $\|(\mathcal{F}x)(e_{2})-(\mathcal{F}x)(e_{1})\|\rightarrow 0$ independently of $x\in B_{\eta }$ as $e_{2}\rightarrow e_{1}$, since the compactness of $U_{\beta }(t,s)$ $(t,s>0)$ implies the continuity in the uniform operator topology. This shows that $\mathcal{F}(B_{\eta })$ is equicontinuous. In view of Lemma 2.2, $\overline{Co}\mathcal{F}(B_{\eta })\subset B_{\eta }$ is equicontinuous and bounded.

It remains to prove that $F :\overline{Co}\mathcal{F}(B_{\eta })\rightarrow \overline{Co} \mathcal{F}(B_{\eta })$ is a condensing operator. For any $D\subset \overline{Co}\mathcal{F}(B_{\eta })$, by Lemma 2.3, there exists a countable set $D_{0}=\{x_{n}\}\subset D$ such that

$$ \alpha \bigl(\mathcal{F}(D)\bigr)\leq 2\alpha \bigl(\mathcal{F}(D_{0}) \bigr).$$

Using the fact that $\overline{Co}\mathcal{F}(B_{\eta })$ is equicontinuous, $D_{0}\subset \overline{Co}\mathcal{F}(B_{\eta })$ is equicontinuous. By $(H_{3})$, for $s\in (s_{i},t_{i+1}]$, $i=0,1,\ldots,m$, then

$$ \begin{aligned} \alpha \biggl(f \biggl(s,D_{0}(s), \int _{0}^{s}g\bigl(s,\sigma ,D_{0}( \sigma )\bigr)\,d\sigma \biggr) \biggr)\leq {}&L'_{1}(s)\alpha \bigl(D_{0}(s) \bigr)+L'_{2}(s) \int _{0}^{s}L'_{g}( \sigma )\alpha \bigl(D_{0}(\sigma )\bigr)\,d\sigma \\ \leq {}& \biggl(L'_{1}(s)+L'_{2}(s) \int _{0}^{T}L'_{g}( \sigma )\,d\sigma \biggr)\alpha (D). \end{aligned} $$

For each $t\in [0,t_{1}]$,

$$\begin{aligned}& \alpha \bigl(\mathcal{F}(D_{0}) (t) \bigr)\\& \quad \leq M^{2}\alpha \biggl( \int _{t_{m}}^{s_{m}}g_{m} \bigl(s,D_{0}(s)\bigr)\,ds \biggr)+M^{2}\alpha \biggl( \int _{s_{m}}^{T}f \biggl(s,D_{0}(s), \int _{0}^{s}g\bigl(s, \sigma ,D_{0}(\sigma )\bigr)\,d\sigma \biggr)\,ds \biggr) \\& \qquad {}+M\alpha \biggl( \int _{0}^{t}f \biggl(s,D_{0}(s), \int _{0}^{s}g\bigl(s, \sigma ,D_{0}(\sigma )\bigr)\,d\sigma \biggr)\,ds \biggr) \\& \quad \leq M^{2} \int _{t_{m}}^{s_{m}}L'_{g_{m}}(s) \alpha \bigl(D_{0}(s)\bigr)\,ds+M^{2} \int _{s_{m}}^{T}\alpha \biggl(f \biggl(s,D_{0}(s), \int _{0}^{s}g\bigl(s, \sigma ,D_{0}(\sigma )\bigr)\,d\sigma \biggr) \biggr)\,ds \\& \qquad {}+M \int _{0}^{t}\alpha \biggl(f \biggl(s,D_{0}(s), \int _{0}^{s}g\bigl(s, \sigma ,D_{0}(\sigma )\bigr)\,d\sigma \biggr) \biggr)\,ds \\& \quad \leq \biggl(M^{2} \int _{t_{m}}^{s_{m}}L'_{g_{m}}(s)\,ds+M^{2} \int _{s_{m}}^{T} \biggl(L'_{1}(s)+L'_{2}(s) \int _{0}^{T}L'_{g}( \sigma )\,d\sigma \biggr)\,ds \\& \qquad {}+M \int _{0}^{t_{1}} \biggl(L'_{1}(s)+L'_{2}(s) \int _{0}^{T}L'_{g}( \sigma )\,d\sigma \biggr)\,ds \biggr)\alpha (D). \end{aligned}$$

For each $t\in (t_{i},s_{i}]$, $i=1,\ldots,m$,

$$ \alpha \bigl(\mathcal{F}(D_{0}) (t)\bigr)\leq M\alpha \biggl( \int _{t_{i}}^{t}g_{i} \bigl(s,D_{0}(s)\bigr)\,ds \biggr)\leq M \int _{t_{i}}^{s_{i}}L'_{g_{i}}(s)\,ds \alpha (D).$$

For each $t\in (s_{i},t_{i+1}]$, $i=1,\ldots,m$,

$$ \begin{aligned} \alpha \bigl(\mathcal{F}(D_{0}) (t) \bigr)\leq{} & M\alpha \biggl( \int _{t_{i}}^{s_{i}}g_{i} \bigl(s,D_{0}(s)\bigr)\,ds \biggr)\\ &{}+M\alpha \biggl( \int _{s_{i}}^{t}f \biggl(s,D_{0}(s), \int _{0}^{s}g\bigl(s, \sigma ,D_{0}(\sigma )\bigr)\,d\sigma \biggr)\,ds \biggr) \\ \leq{} & \biggl(M \int _{t_{i}}^{s_{i}}L'_{g_{i}}(s)\,ds+M \int _{s_{i}}^{t_{i+1}} \biggl(L'_{1}(s)+L'_{2}(s) \int _{0}^{T}L'_{g}( \sigma )\,d\sigma \biggr)\,ds \biggr)\alpha (D). \end{aligned} $$

By Lemma 2.4,

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

Hence

$$ \alpha \bigl(\mathcal{F}(D)\bigr)\leq \rho \alpha (D)< \alpha (D).$$

These arguments enable us to infer that $\mathcal{F}:\overline{Co}\mathcal{F}(B_{\eta })\rightarrow \overline{Co}\mathcal{F}(B_{\eta })$ is a condensing operator and by the fixed point theorem of Sadovskii, there exists one fixed point $x^{\star }\in \overline{Co}\mathcal{F}(B_{\eta })\subset \operatorname{PC}(J,E)$ for $\mathcal{F}$. In conclusion, problem (1.3) has at least one PC-mild solution. This completes the proof. □

Now we establish the existence results of PC-mild solutions for problem (1.3) via Krasnoselskii’s fixed point theorem.

Theorem 3.2

Assume that $(G_{1})$–$(G_{4})$ hold and the resolvent operator $U_{\beta }(t,s)$ is compact for $t,s>0$.

$(G_{1})$:

The function $f:J\times E\times E\rightarrow E$ is continuous, there exist nonnegative Lebesgue integrable functions $a,L_{1},L_{2}\in L^{1}(J,\mathbb{R}_{+})$ for $t\in (s_{i},t_{i+1}]$ $(i=0,1,\ldots ,m)$ and $x_{1},x_{2}\in E$ such that

$$ \bigl\Vert f(t,x_{1},x_{2}) \bigr\Vert \leq a(t)+L_{1}(t) \Vert x_{1} \Vert +L_{2}(t) \Vert x_{2} \Vert .$$

$(G_{2})$:

The function $g:D\times E\rightarrow E$ is continuous, $D=\{(t,s)|0\leq s\leq t\leq T\}$, there exist nonnegative Lebesgue integrable functions $b,L_{3}\in L^{1}(J,\mathbb{R}_{+})$ for $(t,s)\in D$, $x\in E$ such that

$$ \bigl\Vert g(t,s,x) \bigr\Vert \leq b(t)+L_{3}(t) \Vert x \Vert .$$

$(G_{3})$:

There exists a function $\omega _{i}(t)$ with $\varpi _{i}=\sup_{t\in [t_{i},s_{i}]}\omega _{i}(t)<+ \infty$ for $t\in (t_{i},s_{i}]$ $(i=1,2,\ldots ,m)$ and $x\in E$ such that

$$ \bigl\Vert g_{i}(t,x) \bigr\Vert \leq \omega _{i}(t).$$

$(G_{4})$:

There exist nonnegative constants $L_{g_{i}}>0$ for $t\in (t_{i},s_{i}]$ $(i=1,2,\ldots ,m)$ and $x,x'\in E$ such that

$$ \bigl\Vert g_{i}(t,x)-g_{i}\bigl(t,x' \bigr) \bigr\Vert \leq L_{g_{i}} \bigl\Vert x-x' \bigr\Vert .$$

Then problem (1.3) has at least one PC-mild solution on $\operatorname{PC}(J,E)$ provided that $\vartheta =\max \{M^{2}\int _{s_{m}}^{T}b_{1}(s)\,ds+M\int _{0}^{t_{1}}b_{1}(s)\,ds, M\int _{s_{i}}^{t_{i+1}}b_{1}(s)\,ds, M^{2}L_{g_{m}}(s_{m}-t_{m}), ML_{g_{i}}(s_{i}-t_{i}), i=1,\ldots ,m\}<1$, where $b_{1}(s)=L_{1}(s)+L_{2}(s)\int _{0}^{T}L_{3}(\sigma )\,d\sigma$.

Proof

We decompose $\mathcal{F}$ as $\mathcal{F}=\mathcal{G}+\mathcal{H}$, where

$$ (\mathcal{G}x) (t)= \textstyle\begin{cases} U_{\beta }(t,0) [U_{\beta }(T,s_{m})h_{m}+U_{\beta }(T,t_{m})\int _{t_{m}}^{s_{m}}g_{m}(s,x(s))\,ds ],\quad t\in [0,t_{1}], \\ h_{i}+U_{\beta }(t,t_{i})\int _{t_{i}}^{t}g_{i}(s,x(s))\,ds,\quad t\in (t_{i},s_{i}], i=1,2,\ldots ,m, \\ U_{\beta }(t,s_{i})h_{i}+U_{\beta }(t,t_{i})\int _{t_{i}}^{s_{i}}g_{i}(s,x(s))\,ds, \quad t\in (s_{i},t_{i+1}], i=1,2,\ldots ,m, \end{cases} $$

and

$$ (\mathcal{H}x) (t)= \textstyle\begin{cases} U_{\beta }(t,0)\int _{s_{m}}^{T}U_{\beta }(T,s)f (s,x(s),\int _{0}^{s}g(s, \sigma ,x(\sigma ))\,d\sigma )\,ds \\ \quad {}+\int _{0}^{t}U_{\beta }(t,s)f (s,x(s),\int _{0}^{s}g(s,\sigma ,x( \sigma ))\,d\sigma )\,ds, \quad t\in [0,t_{1}], \\ 0,\quad t\in (t_{i},s_{i}], i=1,2,\ldots ,m, \\ \int _{s_{i}}^{t}U_{\beta }(t,s)f (s,x(s),\int _{0}^{s}g(s, \sigma ,x(\sigma ))\,d\sigma )\,ds,\quad t\in (s_{i},t_{i+1}], i=1,2, \ldots ,m. \end{cases} $$

Let us fix $R^{\star }>0$ such that

$$ \begin{aligned} R^{\star }\geq {}&\max \biggl\{ \frac{M^{2} \Vert h_{m} \Vert +M^{2}\varpi _{m}(s_{m}-t_{m})+M^{2}\int _{s_{m}}^{T}a_{1}(s)\,ds+M\int _{0}^{t_{1}}a_{1}(s)\,ds}{1-\vartheta }, \\ &{} \Vert h_{i} \Vert +M\varpi _{i}(s_{i}-t_{i}), \frac{M \Vert h_{i} \Vert +M\varpi _{i}(s_{i}-t_{i})+M\int _{s_{i}}^{t_{i+1}}a_{1}(s)\,ds}{1-\vartheta },i=1,2, \ldots ,m\biggr\} ,\end{aligned} $$

where $a_{1}(s)=a(s)+L_{2}(s)\int _{0}^{T}b(\sigma )\,d\sigma$.

We consider the set $B_{R^{\star }}=\{x\in \operatorname{PC}(J,E):\|x\|_{\operatorname{PC}}\leq R^{\star }\}$ for any $x\in B_{R^{\star }}$. From conditions $(G_{1})$ and $(G_{2})$, for all $s\in (s_{i},t_{i+1}]$, $i=0,1,\ldots ,m$, one can find that

$$ \begin{aligned} \biggl\Vert f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x(\sigma )\bigr)\,d\sigma \biggr) \biggr\Vert \leq {}& a(s)+L_{1}(s)R^{\star }+L_{2}(s) \int _{0}^{s}\bigl(b( \sigma )+L_{3}(\sigma )R^{\star }\bigr)\,d\sigma \\ \leq {}& a(s)+L_{2}(s) \int _{0}^{T}b(\sigma )\,d\sigma \\ &{}+ \biggl(L_{1}(s)+L_{2}(s) \int _{0}^{T}L_{3}(\sigma )\,d\sigma \biggr)R^{\star } \\ ={}&a_{1}(s)+b_{1}(s)R^{\star }. \end{aligned} $$

Obviously, $a_{1}(s)$ and $b_{1}(s)$ are nonnegative Lebesgue integrable functions.

According to condition $(G_{3})$ and the above inequities, for any $t\in [0,t_{1}]$, we obtain

$$ \begin{aligned} \bigl\Vert (\mathcal{F}x) (t) \bigr\Vert \leq {}& \bigl\Vert U_{\beta }(t,0) \bigr\Vert \biggl( \bigl\Vert U_{\beta }(T,s_{m})h_{m} \bigr\Vert + \bigl\Vert U_{\beta }(T,t_{m}) \bigr\Vert \biggl\Vert \int _{t_{m}}^{s_{m}}g_{m}\bigl(s,x(s) \bigr)\,ds \biggr\Vert \\ &{}+ \biggl\Vert \int _{s_{m}}^{T}U_{\beta }(T,s)f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr)\,ds \biggr\Vert \biggr) \\ &{}+ \biggl\Vert \int _{0}^{t}U_{\beta }(t,s)f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr)\,ds \biggr\Vert \\ \leq {}& M^{2} \Vert h_{m} \Vert +M^{2}\varpi _{m}(s_{m}-t_{m})+M^{2} \int _{s_{m}}^{T}\bigl(a_{1}(s)+b_{1}(s)R^{ \star } \bigr)\,ds \\ &{}+M \int _{0}^{t}\bigl(a_{1}(s)+b_{1}(s)R^{\star } \bigr)\,ds \\ \leq{} & M^{2} \Vert h_{m} \Vert +M^{2}\varpi _{m}(s_{m}-t_{m})+M^{2} \int _{s_{m}}^{T}a_{1}(s)\,ds+M \int _{0}^{t_{1}}a_{1}(s)\,ds+\vartheta R^{\star } \\ \leq{} & R^{\star }. \end{aligned} $$

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

$$ \begin{aligned} \bigl\Vert (\mathcal{F}x) (t) \bigr\Vert \leq {}& \Vert h_{i} \Vert + \biggl\Vert U_{\beta }(t,t_{i}) \int _{t_{i}}^{t}g_{i}\bigl(s,x(s) \bigr)\,ds \biggr\Vert \\ \leq {}& \Vert h_{i} \Vert +M \int _{t_{i}}^{t} \bigl\Vert g_{i} \bigl(s,x(s)\bigr) \bigr\Vert \,ds \\ \leq{} & \Vert h_{i} \Vert +M\varpi _{i}(s_{i}-t_{i}) \leq R^{\star }. \end{aligned} $$

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

$$ \begin{aligned} \bigl\Vert (\mathcal{F}x) (t) \bigr\Vert \leq {}& \bigl\Vert U_{\beta }(t,s_{i})h_{i} \bigr\Vert + \biggl\Vert U_{\beta }(t,t_{i}) \int _{t_{i}}^{s_{i}}g_{i}\bigl(s,x(s) \bigr)\,ds \biggr\Vert \\ &{}+ \biggl\Vert \int _{s_{i}}^{t}U_{\beta }(t,s)f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr)\,ds \biggr\Vert \\ \leq{} & M \Vert h_{i} \Vert +M\varpi _{i}(s_{i}-t_{i})+M \int _{s_{i}}^{t}\bigl(a_{1}(s)+b_{1}(s)R^{ \star } \bigr)\,ds \\ \leq {}& M \Vert h_{i} \Vert +M\varpi _{i}(s_{i}-t_{i})+M \int _{s_{i}}^{t_{i+1}}a_{1}(s)\,ds+ \vartheta R^{\star }\leq R^{\star }. \end{aligned} $$

From the above inequities, we conclude $\mathcal{F}x=\mathcal{G}x+\mathcal{H}x\in B_{R^{\star }}$.

Next we prove that the operator $\mathcal{G}$ is a contraction on $B_{R^{\star }}$. By $(G_{4})$, for x, $x'\in B_{R^{\star }}$, for any $t\in [0,t_{1}]$, we get

$$ \begin{aligned} \bigl\Vert (\mathcal{G}x) (t)-\bigl( \mathcal{G}x'\bigr) (t) \bigr\Vert \leq{} & M^{2} \int _{t_{m}}^{s_{m}} \bigl\Vert g_{m} \bigl(s,x(s)\bigr)-g_{m}\bigl(s,x'(s)\bigr) \bigr\Vert \,ds \\ \leq{} & M^{2}L_{g_{m}} \bigl\Vert x-x' \bigr\Vert _{\operatorname{PC}}(s_{m}-t_{m}). \end{aligned} $$

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

$$ \begin{aligned} \bigl\Vert (\mathcal{G}x) (t)-\bigl( \mathcal{G}x'\bigr) (t) \bigr\Vert \leq {}& M \int _{t_{i}}^{t} \bigl\Vert g_{i} \bigl(s,x(s)\bigr)-g_{i}\bigl(s,x'(s)\bigr) \bigr\Vert \,ds \\ \leq{} & ML_{g_{i}} \bigl\Vert x-x' \bigr\Vert _{\operatorname{PC}}(s_{i}-t_{i}). \end{aligned} $$

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

$$ \begin{aligned} \bigl\Vert (\mathcal{G}x) (t)-\bigl( \mathcal{G}x'\bigr) (t) \bigr\Vert \leq{} & M \int _{t_{i}}^{s_{i}} \bigl\Vert g_{i} \bigl(s,x(s)\bigr)-g_{i}\bigl(s,x'(s)\bigr) \bigr\Vert \,ds \\ \leq {}& ML_{g_{i}} \bigl\Vert x-x' \bigr\Vert _{\operatorname{PC}}(s_{i}-t_{i}). \end{aligned} $$

From the above inequities with $\vartheta <1$, we have $\|\mathcal{G}x-\mathcal{G}x'\|_{\operatorname{PC}}<\|x-x'\|_{\operatorname{PC}}$. This implies that $\mathcal{G}$ is a contraction.

To prove that $\mathcal{H}$ is completely continuous on $B_{R^{\star }}$, first we claim that $\mathcal{H}$ is continuous applying the arguments employed in the proof of Theorem 3.1. Moreover, $\mathcal{H}$ is uniformly bounded on $B_{R^{\star }}$ since $\|\mathcal{H}x\|_{\operatorname{PC}}\leq R^{\star }$. Next we show that $\mathcal{H}(B_{R^{\star }})$ is equicontinuous. To do this, for $x\in B_{R^{\star }}$, $e_{1},e_{2}\in [0,t_{1}]$ with $e_{1}< e_{2}$, we have

$$\begin{aligned}& \bigl\Vert (\mathcal{H}x) (e_{2})-(\mathcal{H}x) (e_{1}) \bigr\Vert \\& \quad \leq \bigl\Vert U_{\beta }(e_{2},0)-U_{\beta }(e_{1},0) \bigr\Vert M \int _{s_{m}}^{T} \biggl\Vert f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr) \biggr\Vert \,ds \\& \qquad {}+ \biggl\Vert \int _{e_{1}}^{e_{2}}U_{\beta }(e_{2},s)f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr)\,ds \biggr\Vert \\& \qquad {}+ \biggl\Vert \int _{0}^{e_{1}}\bigl(U_{\beta }(e_{2},s)-U_{\beta }(e_{1},s) \bigr)f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x(\sigma )\bigr)\,d\sigma \biggr)\,ds \biggr\Vert \\& \quad \leq \bigl\Vert U_{\beta }(e_{2},0)-U_{\beta }(e_{1},0) \bigr\Vert M \int _{s_{m}}^{T}\bigl(a_{1}(s)+b_{1}(s){R^{ \star }} \bigr)\,ds \\& \qquad {}+M \int _{e_{1}}^{e_{2}}\bigl(a_{1}(s)+b_{1}(s){R^{\star }} \bigr)\,ds \\& \qquad {}+\sup_{s\in [0,t_{1}]} \bigl\Vert U_{\beta }(e_{2},s)-U_{\beta }(e_{1},s) \bigr\Vert \int _{0}^{e_{1}}\bigl(a_{1}(s)+b_{1}(s){R^{ \star }} \bigr)\,ds. \end{aligned}$$

For $e_{1},e_{2}\in (t_{i},s_{i}]$ with $e_{1}< e_{2}$, $i=1,2,\ldots ,m$, we have

$$ \bigl\Vert (\mathcal{H}x) (e_{2})-(\mathcal{H}x) (e_{1}) \bigr\Vert =0.$$

For $e_{1},e_{2}\in (s_{i},t_{i+1}]$ with $e_{1}< e_{2}$, $i=1,2,\ldots ,m$, we have

$$\begin{aligned}& \bigl\Vert (\mathcal{H}x) (e_{2})-(\mathcal{H}x) (e_{1}) \bigr\Vert \\& \quad \leq \int _{e_{1}}^{e_{2}} \bigl\Vert U_{\beta }(e_{2},s) \bigr\Vert \biggl\Vert f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x( \sigma ) \bigr)\,d\sigma \biggr) \biggr\Vert \,ds \\& \qquad {}+ \int _{s_{i}}^{e_{1}} \bigl\Vert U_{\beta }(e_{2},s)-U_{\beta }(e_{1},s) \bigr\Vert \biggl\Vert f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x(\sigma )\bigr)\,d\sigma \biggr) \biggr\Vert \,ds \\& \quad \leq M \int _{e_{1}}^{e_{2}}\bigl(a_{1}(s)+b_{1}(s){R^{\star }} \bigr)\,ds \\& \qquad {}+\sup_{s\in (s_{i},t_{i+1}]} \bigl\Vert U_{\beta }(e_{2},s)-U_{\beta }(e_{1},s) \bigr\Vert \int _{s_{i}}^{e_{1}}\bigl(a_{1}(s)+b_{1}(s){R^{\star }} \bigr)\,ds. \end{aligned}$$

By Lemma 2.1, the compactness of the resolvent operator $U_{\beta }(t,s)$ implies the continuity in the uniform operator topology and together with $a_{1}(s)$, $b_{1}(s)\in L^{1}(J,\mathbb{R}_{+})$, we infer that $\|(\mathcal{H}x)(e_{2})-(\mathcal{H}x)(e_{1})\|\rightarrow 0$ as $e_{2}\rightarrow e_{1}$. Consequently, $\mathcal{H}(B_{R^{\star }})$ is equicontinuous.

Third, we prove that $\mathcal{H}(B_{R^{\star }})$ is precompact.

For $t\in [0,t_{1}]$, $0<\epsilon <t$, $x\in B_{R^{\star }}$, define

$$ \begin{aligned} (\mathcal{H}_{\epsilon }x) (t)={}&U_{\beta }(t,0) \int _{s_{m}}^{T}U_{\beta }(T,s)f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x(\sigma )\bigr)\,d\sigma \biggr)\,ds \\ &{}+ \int _{0}^{t-\epsilon }U_{\beta }(t,s)f \biggl(s,x(s), \int _{0}^{s}g\bigl(s, \sigma ,x(\sigma ) \bigr)\,d\sigma \biggr) \,ds. \end{aligned} $$

Hence

$$ \begin{aligned} \bigl\Vert (\mathcal{H}x) (t)-( \mathcal{H}_{\epsilon }x) (t) \bigr\Vert \leq {}& \biggl\Vert \int _{t- \epsilon }^{t}U_{\beta }(t,s) f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x( \sigma ) \bigr)\,d\sigma \biggr)\,ds \biggr\Vert \\ \leq {}&M \int _{t-\epsilon }^{t}\bigl(a_{1}(s)+b_{1}(s){R^{\star }} \bigr)\,ds. \end{aligned} $$

For $t\in (t_{i},s_{i}]$, $0<\epsilon <t$, $x\in B_{R^{\star }}$, $i=1,2,\ldots ,m$, define $(\mathcal{H}_{\epsilon }x)(t)=0$.

Obviously, $\|(\mathcal{H}x)(t)-(\mathcal{H}_{\epsilon }x)(t)\|=0$.

For $t\in (s_{i},t_{i+1}]$, $0<\epsilon <t$, $x\in B_{R^{\star }}$, $i=1,2,\ldots ,m$, define

$$ (\mathcal{H}_{\epsilon }x) (t)= \int _{s_{i}}^{t-\epsilon }U_{\beta }(t,s) f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x(\sigma )\bigr)\,d\sigma \biggr)\,ds.$$

Thus

$$ \begin{aligned} \bigl\Vert (\mathcal{H}x) (t)-( \mathcal{H}_{\epsilon }x) (t) \bigr\Vert \leq {}& \biggl\Vert \int _{t- \epsilon }^{t}U_{\beta }(t,s) f \biggl(s,x(s), \int _{0}^{s}g\bigl(s,\sigma ,x( \sigma ) \bigr)\,d\sigma \biggr)\,ds \biggr\Vert \\ \leq {}&M \int _{t-\epsilon }^{t}\bigl(a_{1}(s)+b_{1}(s){R^{\star }} \bigr)\,ds. \end{aligned} $$

Since $U_{\beta }(t,s)$ is a compact resolvent operator, then the set $Y_{\epsilon }(t)=\{(\mathcal{H}_{\epsilon }x)(t):x\in B_{R^{\star }}\}$ is relatively compact in E for every $0<\epsilon <t$. Thus $Y(t)=\{(\mathcal{H}x)(t):x\in B_{R^{\star }}\}$ is totally bounded. Hence, $Y(t)$ is relatively compact in E, and so, with the help of the Arzelá–Ascoli theorem, $\mathcal{H}$ is completely continuous on $B_{R^{\star }}$. Therefore, by Krasnoselskii’s fixed point theorem, there exists a fixed point for $\mathcal{F}=\mathcal{G}+\mathcal{H}$, which corresponds to a PC-mild solution of problem (1.3) on $\operatorname{PC}(J,E)$. This completes the proof. □

4 An application

In order to show the application of the main results, we consider the following problem:

$$ \textstyle\begin{cases} {}^{c}D_{t}^{\beta }x(z,t)=t\frac{\partial ^{2}}{\partial z^{2}}x(z,t)+ \frac{t}{4M^{2}(1+t^{2})} x(z,t)+\int _{0}^{t} \frac{e^{t-s} \vert x(z,s) \vert }{8M^{2}e^{4}}\,ds, \\ \quad t\in [0,1)\cup (2,3], z\in (0,1), \\ \frac{\partial }{\partial z}x(0,t)=\frac{\partial }{\partial z}x(1,t)=0, \quad t\in [0,1)\cup (2,3], \\ x(z,t)=y_{1}z+U_{\beta }(t,1)\int _{1}^{t}\frac{ \vert x(z,s) \vert }{8M^{2}(1+t)}\,ds, \quad t\in (1,2], z\in (0,1), \\ x(0,t)=x(3,t), \quad t\in (0,1) , \end{cases} $$

(4.1)

where $E=L^{2}[0,3]$, $0=t_{0}=s_{0}$, $t_{1}=1$, $s_{1}=2$, $\ ^{c}D_{t}^{\beta }$ is the Caputo’s fractional derivative of order β, $0<\beta <1$. The operator $A:D(A)\subset E\rightarrow E$ is defined as $A(t)(z)=t\frac{\partial ^{2}x}{\partial z^{2}}$, where $D(A)=\{x\in E:x''\in E, \ \ x(0)=x(1)=0\}$. It is well known that the operator $A(t)$ generates a β-resolvent family $U_{\beta }(t,s)$ and $\max_{0\leq s< t\leq T}\|U_{\beta }(t,s)\|\leq M$, $(M>1)$.

By setting

$$\begin{aligned}& x(t) (z)=x(z, t), \quad h_{1}z=y_{1}z, \qquad g_{1}\bigl(t,x(t)\bigr) (z)= \frac{ \vert x(z,s) \vert }{8M^{2}(1+t)},\\& f \biggl(t,x(t), \int _{0}^{t}g\bigl(t,s,x(s)\bigr)\,ds \biggr) (z)= \frac{t}{4M^{2}(1+t^{2})} x(z,t)+ \int _{0}^{t} \frac{e^{t-s} \vert x(z,s) \vert }{8M^{2}e^{4}}\,ds, \end{aligned}$$

problem (4.1) can be rewritten as the following abstract form:

$$ \textstyle\begin{cases} ^{c}D_{t}^{\beta }x(t)=A(t)x(t)+f (t,x(t),\int _{0}^{t}g(t,s,x(s))\,ds ),& t\in [0,1)\cup (2,3], \\ x(t)=h_{1}+U_{\beta }(t,1)\int _{1}^{t}g_{1}(s,x(s))\,ds,& t\in (1,2], \\ x(0)=x(3). \end{cases} $$

(4.2)

The function $f:J\times T_{R}\times T_{R}\rightarrow E$ is bounded and continuous, for every $R>0$, such that

$$ \lim_{R\rightarrow \infty }\sup \frac{M(R)}{R}< \frac{1}{Ma_{0}(M+1)}, $$

(4.3)

where $M(R)=\max \{M_{1}(R),M_{2}(R)\}$, $M_{1}(R)=\sup \{\|f(t,x_{1},x_{2})\|:(t,x_{1},x_{2})\in J\times T_{R} \times T_{R}\}$, $M_{2}(R)=\sup \{\|g_{1}(t,x)\|,(t,x)\in J\times T_{R},\}$, $T_{R}=\{x\in E:\|x\|\leq R\}$, $a_{0}=\max \{1,h_{0}\}$.

Let

$$\begin{aligned}& \bigl\Vert g(t,s,x) \bigr\Vert \leq \frac{e^{t-s}}{8M^{2}e^{4}} \Vert x \Vert ,\\& \alpha \bigl(g(t,s,D)\bigr)\leq e^{t}\alpha (D),\\& \alpha \bigl(g_{1}(t,s,D)\bigr)\leq \frac{1}{8M^{2}(1+t)}\alpha (D),\\& \alpha \bigl(f(t,D_{1},D_{2})\bigr)\leq \frac{t}{4M^{2}(1+t^{2})}\alpha (D_{1})+e^{t} \alpha (D_{2}). \end{aligned}$$

Then

$$ \begin{aligned} &2M^{2} \int _{1}^{2}\frac{1}{8M^{2}(1+s)}\,ds+2M^{2} \int _{2}^{3} \biggl( \frac{s}{4M^{2}(1+s^{2})}+e^{s} \int _{0}^{3} \frac{e^{s-\sigma }}{8M^{2}e^{4}}\,d\sigma \biggr)\,ds \\ &\qquad {}+2M \int _{0}^{1} \biggl(\frac{s}{4M^{2}(1+s^{2})}+e^{s} \int _{0}^{3} \frac{e^{s-\sigma }}{8M^{2}e^{4}}\,d\sigma \biggr)\,ds \\ &\quad < \frac{1}{4}+\frac{1}{4}+\frac{1}{4}+ \frac{1}{4}=1, \\ &2M \int _{1}^{2}\frac{1}{8M^{2}(1+s)}\,ds+2M \int _{2}^{3} \biggl( \frac{s}{4M^{2}(1+s^{2})}+e^{s} \int _{0}^{3} \frac{e^{s-\sigma }}{8M^{2}e^{4}}\,d\sigma \biggr)\,ds \\ &\quad < \frac{1}{4M}+\frac{1}{4M}+\frac{1}{4M}< 1.\end{aligned} $$

We have

$$ \begin{aligned} \rho ={}&\max \biggl\{ 2M^{2} \int _{1}^{2}\frac{1}{8M^{2}(1+s)}\,ds+2M^{2} \int _{2}^{3} \biggl(\frac{s}{4M^{2}(1+s^{2})}+e^{s} \int _{0}^{3} \frac{e^{s-\sigma }}{8M^{2}e^{4}}\,d\sigma \biggr)\,ds \\ &{}+2M \int _{0}^{1} \biggl(\frac{s}{4M^{2}(1+s^{2})}+e^{s} \int _{0}^{3} \frac{e^{s-\sigma }}{8M^{2}e^{4}}\,d\sigma \biggr)\,ds, \\ &{}2M \int _{1}^{2}\frac{1}{8M^{2}(1+s)}\,ds+2M \int _{2}^{3} \biggl( \frac{s}{4M^{2}(1+s^{2})}+e^{s} \int _{0}^{3} \frac{e^{s-\sigma }}{8M^{2}e^{4}}\,d\sigma \biggr)\,ds\biggr\} < 1.\end{aligned} $$

Therefore, problem (4.2) satisfies the conditions of Theorem 3.1, then problem (4.2) has a PC-mild solution, which means that problem (4.1) has a mild solution.

5 Conclusion

In this paper, we demonstrate sufficient conditions on the existence of PC-mild solutions for periodic boundary value problems for fractional semilinear nonautonomous differential equations with non-instantaneous impulses. For the proofs of the main theorems, we use the measure of noncompactness together with Sadovskii’s fixed point theorem and Krasnoselskii’s fixed point theorem. Finally, an example is given to illustrate the application of our main results.